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Post-tensioned Concrete Floors

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Harbour Exchange, London

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P o s t - t e n s i o n e d C o n c r e t e F l o o r s

S a m i K h a n

Director, Bunyan Meyer and Partners

M a r t i n W i l l i a m s

Lecturer, Univers i ty of Oxford, Department of Engineering Science

and Fel low of New Col lege, Oxford

~

, U T T E R W O R T H

E ! N E M A N N

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B u t t e r w o r t h - H e i n e m a n n L t d

L i n a c r e H o u s e , J o r d a n H i ll , O x f o r d O X 2 8 D P

- ~ A m e m b e r of t h e R e e d E l se v ie r p lc g r o u p

O X F O R D L O N D O N B O S T O N

M U N I C H N E W D E L H I S I N G A P O R E

T O K Y O T O R O N T O W E L L I N G T O N

S Y D N E Y

F i r s t p u b l i s h e d 1 9 95

9 B u t t e r w o r t h - H e i n e m a n n L t d 1 99 5

All rights reserved. No part of this publication

may be reproduced in any m aterial form (including

photocopying or storing in any medium by electronic

means and whether or not transiently or incidentally

to some oth er use of this publication) without the

written permission of the copyright hold er except

in accordance with the provisions of the Copyright,

Designs and Paten ts Act 1988 or und er the terms of a

licence issued by the Copyright Licensing Agency Ltd,

90 Tottenham Co urt Road, London, England W1P 9HE.

Applications for the co pyrigh t hol der 's written permission

to reproduce any part of this publication should be addressed

to the publishers

Brit ish Libra ry Cataloguing in Publicat ion D ata

K a h n , S a m i

P o s t - t e n s i o n e d C o n c r e t e F l o o r s

I . T i t l e I I . W i l l i a m s , M a r t i n

6 9 3 . 5 4 2

I S B N 0 7 5 0 6 1 6 8 1 4

Library of Congress Cataloguing in Publicat ion Data

K a h n , S a m i .

P o s t - t e n s i o n e d c o n c r e t e fl o o r s / S a m i K a h n , M a r t i n W i l l i a m s .

p . c m .

I n c l u d e s b i b l i o g r a p h i c a l r e f e r e n c e s a n d i n d e x .

I S B N 0 7 5 0 6 1 6 8 1 4 ( p b k . )

1. F l o o r s , C o n c r e t e . 2 . P o s t - t e n s i o n e d p r e s t r e s s e d c o n c r e t e .

I . W i l l i a m s , M a r t i n . I I . T i t l e.

T H 2 5 2 9 . C 6 K 4 8 1 9 9 5

6 9 0 ' .1 6 - d c 2 0 9 4 - 3 6 8 5 4

C I P

T y p e s e t b y V i s i on T y p e s e t t i n g , M a n c h e s t e r

P r i n t e d a n d b o u n d i n G r e a t B r i t a i n b y

H a r t n o l l s L i m i t e d , B o d m i n , C o r n w a l l

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C O N T E N T S

INTRODUCTION

NOTATIONS

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

1.10

1.11

1.12

1.13

1.14

1.15

THE BASIC PRINCIPLES

Introduction

Prestressing in principle

Stress reversal

Tendons

Prestress losses

Initial and final stresses

Pre-tensioning and post-tensioning

Reinforced and post-tensioned concrete floors

Bonded and unbonded post-tensioning

Stressing stages

Construction tolerances

Fire resistance

Holes through completed floors

Post-tensioning in refurbishment

Some misconceptions about post-tensioned floors

2

2.1

2.2

2.3

2.4

2.5

2.6

MATERIALS AND EQUIPMENT

Formwork

Dense concrete

Lightweight concrete

Post-tensioning tendons

Prestressing hardware

Equipment

3

3.1

3.2

3.3

3.4

SLAB CONFIGURATION

General

Structural elements of a floor

Panel configuration

Span to depth ratio

p a g e ix

xi

1

1

2

5

6

7

8

9

10

14

17

18

18

18

19

20

24

24

26

35

39

47

55

61

61

64

70

74

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v i CONTENTS

4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

6

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7

7.1

7.2

7.3

PLANNING A STRUCTURE

Design objectives and buildability

Restraint from vertical elements

Dispersion of the prestressing force

Column moments

Movements in a concrete floor

Crack prevention

Tendon profile

Access at the live end

Transfer beams

Durability

Fire protection

Minimum and maximum prestress

Additional considerations for structures in seismic zones

Example 4.1

TENDON PROFILES AND EQUIVALENT LOADS

General

Equivalent load

Secondary moments

Concordance

Tendon profile elements

Composite profiles

Tendon deviation in plan

Clash of beam and slab tendons

FLEXURE IN THE SERVICEABILITY STATE

The design process

Options in a design

Computer programs

Partial prestressing

Permissible stresses in concrete

Permissible stresses in strand

Analysis

Simply supported span

Continuous spans

Example 6.1

Example 6.2

Example 6.3

Example 6.4

PRESTRESS LOSSES

General

Friction losses

Anchorage draw-in

79

79

82

85

87

90

91

93

94

96

97

99

102

103

107

108

108

109

112

114

114

122

129

130

132

132

134

136

137

138

141

142

144

147

149

151

153

156

160

160

163

164

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C O N T E N T S v i i

7.4

7.5

7.6

7.7

7.8

7.9

Elastic shortening

Shrinkage of concrete

Creep of concrete

Relaxation of tendons

Tendon elongation

Tendon force from elongation

Example 7.1

8

8.1

8.2

8.3

8.4

8.5

8.6

8.7

ULTIMATE FLEXURAL STRENGTH

Failure mechanisms

Level of prestress

Applied loads

Procedure for calculating the strength

Ultimate stresses

Strain compatibility

Anchorage zone

Example 8.1

Example 8.2

9

9.1

9.2

DEFLECTION AND VIBRATION

Deflections

Vibration

Example 9.1

Example 9.2

Example 9.3

10 SHEAR

10.1 Shear strength of concrete

10.2 Beams and one-way slabs

10.3 Two-way slabs

10.4 Alternatives to conventional shear reinforcement

Example 10.1

Example 10.2

11 SLABS ON GRADE

11.1 The design process

11.2 Factors affecting the design

11.3 Traditional RC floors

11.4 Post-tensioned ground floors

11.5 Elastic analysis

11.6 Construction

Example 11.1

12 DETAILING

12.1 Drawings and symbols

168

170

171

172

173

173

174

177

177

180

181

182

185

189

191

194

197

198

198

206

213

216

219

221

222

224

230

240

244

245

249

250

251

257

259

261

267

269

271

271

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v i i i C O N T E N T S

12 .2 Min imum re in for cement

12 .3 Tendon spac ing and pos i t i on

12.4 Def lec t ion and c ladding

12 .5 Mo vem ent jo in t s

12.6 Detai l ing for seismic resistance

13 S IT E A C T I V IT I ES A N D D E M O L I T I O N

13.1 Storage of m ater ials

13.2 Ins ta l la t ion

13.3 Concre t ing

13.4 Stressing

13 .5 Grou t ing

13.6 Finishing opera t ions

13 .7 Demol i t i on

13.8 Cut t ing holes

R E F E R E N C E S

I N D E X

274

276

277

277

281

284

286

287

288

289

294

295

295

303

306

309

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I N T R O D U C T I O N

This book dea l s wi th the des ign o f concre te bu i ld ing s t ruc tu res inc orpo ra t ing

post - te ns ion ed f loors . Po st - tens io ning is the m ost versat i le form o f pres t ress ing, a

techniq ue w hich enables eng ineers to mak e the m ost effective use of the ma ter ial

p roper t i es o f concre te , an d so to des ign s t ruc tu ra l e l ements w hich a re s t rong ,

s lender and eff ic ient . Design in post - tens ioned concrete i s not d i ff icul t and, i f

done p roper ly , can con t r ibu te s ign i f i can t ly to the economy and the aes the t i c

qual i t ies of a bui ld ing. As a result , pos t - tens io ned f loors have fou nd w idespre ad

use in o ff ice bu i ld ings a nd car pa rk s t ruc tu res , and a re a l so f requen t ly em ployed

in wareh ouses and pub l i c bu i ld ings . H owe ver , in sp it e o f th is , mos t p res t ressed

concre te t ex t s devo te compara t ive ly l i t t l e a t t en t ion to f loors , concen t ra t ing

ins tead on beam e lements . Th i s book therefo re a ims to answer the need fo r a

com prehens ive t re a tm ent o f pos t - t ens ioned f loor des ign .

The f i rs t four chap ters o f the book g ive a de ta i l ed , non-m athem at ica l accou n t

o f the p r inc ip les o f p res t ress ing , the mater i a l s an d e qu ipm ent used , and the

p lann ing o f bu i ld ings inco rpora t ing pos t - t ens ioned f loors. The fo llowing chap ters

ou t l ine the de ta i l ed des ign p rocess , inc lud ing numerous worked examples , and

the book conc ludes wi th chap ters descr ib ing s i t e p rocedures fo r cons t ruc t ion ,

dem ol i t ion and a l t e ra t ion . W hi le the reader i s as sum ed to have a g rasp o f the

basics of reinforced con crete des ign, no p r ior k now ledge of pres t ress ing is

requ i red . The book i s thus su i t ab le fo r use by a rch i t ec t s , con t rac t managers and

qua n t i ty su rveyo rs who m ay wish to ga in an und ers t and ing o f the p rinc ip les

wi tho u t g o ing in to the m athem at ica l aspec t s o f the des ign p rocess , as well as

s t ructural engineers requir ing detai led des ign guidance. I t i s a lso in tended for use

as an educa t iona l t ex t by s tuden t s fo l lowing c iv i l eng ineer ing , a rch i t ec tu re and

bui ld ing courses .

The t i tle of the boo k ref lects the fact that i t s em pha sis i s on the be ha viou r a nd

design of the f loors themselves . T hus , whi le the effect of post - te ns ion ed f loors on

o ther s t ruc tu ra l e l ements such as co lumns and wal l s i s cons idered , de ta i l ed

gu idance on the des ign o f these e l ements is no t g iven ; such in fo rm at ion can be

ob ta ined f rom any one o f the m an y exce l len t re in fo rced concre te des ign t ex t s

a l ready ava i l ab le . Ne i ther does th i s book dea l wi th the p res tress ing o f bu i ld ing

e lements o ther than f loors , such as foundat ions , moment - res i s t ing co lumns o r

ver t i ca l hangers . These e l ements a re compara t ive ly ra re , o r a re no t usua l ly

pres t ressed. I f guida nce on des ign of such eleme nts is requ ired, reference should

be made to special is t l i terature.

In any book on pos t - t ens ion ing compar i sons wi th re in fo rced concre te a re

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x I N T R O D U C T I O N

inevi tab le . Pos t - tens ioning offers numerous advantages , bu t i t would be fool i sh

to suggest that i t i s the best design opt ion in al l cases . Therefore, detai led

guidanc e is g iven on the re la t ive mer i t s of pos t - tens io ned and re inforced con crete

f loors , and the reasons for choo s ing one or the o ther in a par t i cu lar s i tuat ion are

discussed.

Al tho ugh the pos t - tens ioning tec hnique i s now qui te well es tab l i shed , research

and deve lopm ent ac t iv it ies cont inue to of fer possib il it ies for fu ture im prov em ents ,

a r ecen t example be ing t he deve lopme n t o f po lym er p rest r es s ing t endons . M any

of these advanc es requi re con s iderable fur ther researc h before they are rea dy for

pract ica l use . In th i s book such developments are d i scussed br ief ly , bu t the

emphas i s i s p laced f i rmly on the curren t pract ice .

W hi le the ge nera l pr incip les and m etho ds of pres t ress ing are universa l , de ta i led

des ign p rocedures u sua l l y fo l l ow na t i ona l code r ecommenda t i ons , wh ich va ry

from count ry to count ry . For a t ex t to be usefu l as a des ign guide , i t mus t

necessar ily mak e reference to nat ion al codes . In th i s book , ful l des ign proce dure s

com pat ib l e wi th bo th t he Br it ish S t and ard BS 8110 :1985 and t he Amer i can code

ACI 318-1989 a re desc r ibed . How ever , t he design methods a re a lways i n t roduced

in a way which emphas izes the pr incip les on which they are based ra ther than

s imply re i tera t ing the code guidel ines . Whi le every ef for t has been made to

descr ibe as fu l ly as poss ib le the provis ions of BS 8110 and ACI 318, the book

should c er ta in ly not be reg arde d as a rep lacem ent of e i ther code. The re levant

s tan da rd or code of pract ice should a lways be consul ted to check specif ic

requ i remen t s .

As far as poss ib le , the equat ions and data presented are g iven in SI uni t s

fo l lowed by imp er ia l equivalen ts. E xt ract s f rom BS 8110 are presented in m et r ic

form only , s ince i t is un l ikely they wi ll be em ployed in count r ies us ing impe r ia l

uni t s . ACI 318 formulae and data are g iven in both SI and imper ia l un i t s .

The he lp o f ma ny i nd iv idua l s and o rgan i za t i ons i n t he p repara t i on o f th i s book

is gra tefu l ly acknowledged. The f in i shed d iagrams were produced wi th the

generous as s i st ance o f t he Br i ti sh C eme n t Assoc i a t ion , C ro wtho rne . M os t o f t he

pho tog ra phs and some ske tches were con t r i bu t ed by VS L In t e rna t i ona l o f Berne ,

S wi t zer l and . Ex t remely help fu l comm ent s on t he manu sc r ip t were made by P a l

Ch ana o f t he Concre t e R esea rch and In nov a t i on Cen t re , Imper i a l Co ll ege; B . K .

Bard han -Ro y o f J an Babrow sk i and P a r tne r s ; P e t e r M at thew o f S wift S t ruc tu res

L t d ; a n d D a v i d R a m s a y o f D H V B u r r o w - C r o c k e r C o n s u lt in g .

Ext rac t s f rom BS 8110 have been re pro duc ed by permiss ion of the Br i t ish

S t anda rds In s t i tu t i on , 2 P a r k S t ree t , Lo ndo n , W lA 2BS. Ex t rac t s f rom ACI 318

have been rep rod uced by pe rmi s s ion o f t he Am er i can Concre t e In s t i t u te , Box

19150, 22400 W est Seven Mi le R oad , D et ro i t , M ichigan 48219, USA . Copies o f

t he codes may be pu rchased f rom these o rgan i za t ions a t t he add res ses g iven . The

authors are a l so gra tefu l to the Br i t i sh Cement Associa t ion and the Concrete

Socie ty for thei r permiss ion to include ex t ract s f rom thei r pub l ica t ions ; to B r idon

Wi re fo r supp ly ing da t a on t he i r p roduc t s ; and t o Bunya n , M eyer & P a r tne r s fo r

thei r suppor t .

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N O T A T I O N S

The fol lowing symbols a re common to a l l chapters . Fur ther symbols , used

local ly, are given in the relevant chapters.

h

c

Ap

A~

A s

asv

b

br

bv

C~

D

d

d~

dx

d~

dp

dr

ep

Ec

Er

Es

fob

f t c i

f ' c

Li

Lu

gb

ge

ft

Area of concre te sec tion

Area o f tend on s teel

Area of bonded s tee l

Area of compress ion s teel

Area of l inks per uni t length of m em ber

Width of concre te in compress ion

W idth of concre te on tens ion face, r ib w idth

width of sect ion effect ive in shear , r ib width for T- , I - or L-sect ions

Creep coefficient

Overa l l depth of sec t ion

Depth of tens ion s tee l f rom compress ion face

D epth f rom ext reme com press ion f ibre to cent re of com press ion

Depth of rec tangular s t ress block

Depth of neut ra l axis

Depth o f t endon cen t ro id

De pth o f bonded rod r e in forcement in t ens ion

Eccent r ic i ty of tendon

M odu lus of e last ic i ty of concre te a t 28 days

M od ulus of elas tic ity of concre te a t s t ress ing

Modulus of e last ic i ty of steel

Equiva lent s t ress , assuming a rec tangular s t ress block

Init ia l concrete cyl inder strength

28-day cyl inder s t rength

Ini t ia l concre te cube s t rength

28-day cube s t rength

St ress in tendon in the ul t imate s ta te

Ini t ia l s t ress in tendon, af ter immediate losses

Stress in tendon af ter a l l losses

Ul t ima te s t reng th o f tendon pe r un i t a r ea

Tendon yie ld s t ress

St ress in bonded re inforcement in ul t imate s ta te

Tensi le s t reng th of concre te

St rength of rod re inforcement

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x i i N O T A T I O N S

f y ' r

k

K

L

L t

M

M e r

Mo

M r

M u

P i

Pf

Pj

Po

Px

Pay

Pcu

Sv

V

vo

Voo

Vor

Vp

W

w

We

w e

Z

Zt

Zb

#

'~m

Strength of s teel used in l inks

M om en t o f i ne r ti a o f concre te s ec ti on

S u b g r a d e m o d u l u s

Wobble fr ict ion coefficient , per uni t length

S pan l eng th

Tendon l eng th be tween anchorages

M o m e n t

C r a c ki n g m o m e n t

M om en t due to se lf weight of concre te sect ion

M o m e n t w i th f a c t or e d l o a d

Ul t imate f l exural s t rength

In i t i a l t endon force , af ter immediate losses

Final tendon force, after al l losses

Jack ing fo rce in t end on

T e n d o n f o rc e a t a n c h o r a g e

Tendon fo rce a t d i s t ance x

Average s t ress in concre te due to pres t ress

S t ress in concre te compress ion b lock

Spacing of l inks

Ul t ima t e shea r fo rce

Ul t imate shear res i s tance

Ul t im ate sh ear res i s tance of sect ion uncra cke d in fl exure

Ul t imate shear res i s tance of sect ion cracked in f l exure

Ver t ica l shear due to pres t ress

Ul t im ate sh ear res i s tance of a re inforced , non -pres t ressed , sec t ion

Concen t ra t ed l oad o r t o t a l d i s t r i bu t ed l oad

Load in tens i ty

Concre t e dens i ty

Equivalen t load in tens i ty

S ec ti on modu lus

Sect ion modulus for top f ibre

S ec t i on modu lus fo r bo t t om f i b re

Ra t io

d c / d .

Coeff ic ien t of f r ic t ion between te nd on and sheath

Def lect ion or d i sp lacement

Par t ia l safe ty fac tor

Poisson ' s ra t io

Sign conv ent ions

Posi t ive s igns apply to:

y -ax is go ing upw ards

L o a d a c t i n g d o w n w a r d s o n c o n c r e t e m e m b e r

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S uppor t r e a c t ions a c t ing upwa r ds on me mbe r

S a g g i n g m o m e n t

Compress ive s t r e ss

NOTATIONS

xi i i

D e f i n i t i o n s a n d c o n v e r s i o n s

Rod re inforcement" bonded non-pres t re ssed s tee l

Assum ed re la t ion sh ip be tween cy l inder and cube s t r ength" f~ , = 0 .8f t,

The fo l lowing approximate convers ions a re used"

1 m etre (m) = 39.37 inches ( in)

1 k i logra m (kg) = 2 .2 pou nds ( lb )

1 k i lon ew ton (kN) = 220 .0 pou nds ( lb )

1 new ton pe r sq .m m (N/ m m 2) = 145.0 pou nds pe r sq . in (ps i)

1 k i lon ew ton pe r sq .m m (kN /m m 2) = 145.0 k i lop oun ds (k ip) pe r sq .in (ksi )

1 k i lon ew ton pe r sq .m (kN /m 2) = 20 .89 pou nds pe r sq .f t (psf )

1 k i logra m per cu .m (kg/m 3) = 0 .0625 pou nds pe r cu .f t (pcf )

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1 T H E B A S I C P R I N C I P L E S

1 . 1 I n t r o d u c t i o n

P os t - t e ns ion ing ha s be e n in u se i n f l oo r c ons t r uc t ion f o r s e ve r a l de c a de s now,

espec ia l ly in the Uni ted S ta te s , Aus t ra l ia , the Fa r Eas t and , to some ex ten t , in

Euro pe . I t s econ om ic an d technica l advantag es a re be ing inc reas ing ly apprec ia ted ,

a nd the p r op o r t io n o f c onc r e t e fl oo r s being pos t - t e ns ione d is g rowing .

In th is cha pte r the bas ic pr inc ip les of p res t re ss ing a re expla ined , a nd the

var iou s m e th od s o f pres t re ss ing a re br ie f ly d iscussed . This i s fo l lowed by

c ompa r i sons be twe e n the a l t e r na t ive f o r ms o f u s ing c onc r e t e a s a s t r uc tu r a l

m e d ium . I t a n swe r s t he q ue s t ions f r e q ue n t ly a ske d by those in t e r e s te d e no ugh in

the sub je c t t o w i sh to kn ow m or e a bo u t i t bu t w i th no ne e d f or a de t a i le d de s ign

ins igh t .

P os t - t e ns ion ing i s a t e c hn iq ue o f p r e - loa d ing the c onc r e t e i n a ma nne r wh ic h

e l imina tes , o r r educes , the tens i le s t r e sses tha t a re induced by the dead and l ive

loads ; the pr inc ip le i s fur the r d iscussed la te r in th is chapte r . F igure 1 .1 i s a

d i a g r a m m a t i c r e p r e se n ta t ion o f t he p r oce s s . H igh s t r e ng th s t e el r ope s , ca l le d

strands, a r e a r r a nge d to pa s s t h r oug h the c onc r e t e fl oo r . W he n the c onc r e te ha s

ha r de n e d , e a c h se t o f s t r a nds is g r ippe d in t he j a ws o f a hyd r a u l i c j a c k a nd

s t r et c he d to a p r e - de t e r mine d f o rc e . T he n the s t r a nd i s l oc ke d in a pu r p ose - m a de

device , ca l led an anchorage, which has been cas t in the concre te ; th i s induces a

compress ive s t r e ss in the concre te . The s t r and i s the rea f te r he ld pe rmanent ly by

the a nc ho r a ge .

T he non - j a c k ing e nd o f t he s t r a nd m a y be bon de d in c onc r et e , o r it m a y be

f i t ted wi th a pre - locked anchorage which has a l so been cas t in the concre te . The

anc ho rage a t the j ack ing end i s ca l led a l ive ancho rage whereas the one a t the

non - j a c k ing e n d i s t e r me d a dead anc horage. T o a l low the s t r a n d to s t r e t c h in the

ha r de n e d c on c r e t e unde r t he l oa d a pp l i e d by the j a c k , bo nd be twe e n the s t r a nd

a nd c on c r e t e is p r e ve n te d b y a tube th r ou gh wh ic h the s t r a nd pa s se s . T he tube ,

t e r me d a duct o r sheathing, m ay be a m e ta l o r p la s t ic p ipe , o r i t m ay con s is t o f a

p las t ic ex t rus ion m ou lde d d i rec t ly on the rope . I f ex t rud ed , the s t r a nd i s in j ec ted

wi th a rus t - inh ib i t ing grease . Af te r s t r e ss ing , the she a th ing , i f no t o f the ex t ru ded

k ind , i s g r ou te d w i th c e me n t mor t a r u s ing a me c ha n ic a l pump .

T he t e r ms tendon a n d cable, a r e the ge ne r a l a nd in t e r c ha nge a b le na m e s f o r t he

h igh s t r e ng th s te el l e ng ths u se d in pos t - t e ns ion ing - - - e q u iva l e n t t o reinforcement

in r e inforced concre te . A tend on m ay con s is t o f ind iv id ua l wires , so l id rods or

r ope s . I t m a y c on ta in one o r mor e r ope s o r w i r es house d in a c om m on she a th ing .

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2 POST-TENSIONED CONCRETE FLOORS

Excep t in g rou nd s labs , tendo ns do no t run in s t ra igh t l ines. They a re norm al ly

drap ed be tween su ppor t s wi th a sha l low sag , jus t as a rope ha ngs w hen l igh tly

s t re tched be tw een two sup por t s . T he geom et r i c shape o f the t endon in e l eva t ion

is calle d its

profile;

i t is usua lly , bu t n o t necessar i ly , parabo l i c . A t any po in t a long

i ts l eng th , the ver ti ca l d i s t ance be tween the cen t ro id o f the concre te sec t ion and

the cen tre of the tend on is cal led i t s eccentrici ty; by convent ion i t i s said to be

posi t ive when the tendon is below the sect ion centroid .

The requ i rements fo r concre te , rod re in fo rcement and fo rmwork fo r pos t -

t ens ion ing a re s imi la r to those fo r re in fo rced concre te , excep t fo r minor

d if fe rences. Ea r ly s t reng th o f concre te is an adv an tag e in pos t - t ens ion ing ; the

qua n t i ty o f rod re in fo rcement i s m uch smal le r ; and the shu t t e r ing needs a ho le in

the ver ti ca l edge shu t t e r a t each jack ing end o f the t endon , and the anch orage s

need to be a t t ach ed to the edge shu t te rs . T he d i ffe rences in the mater i a l s , t ho ugh

m inor , a re d i scussed in de ta il i n Cha p ter 2 . Pos t - t ens ion ing a l so needs s t res s ing

tendo ns to be m ade o f h igh t ens il e s t reng th ; these and the assoc ia ted h ard w are

are br ief ly d iscussed in th is chapter and in detai l in Chapter 2 .

The bas ic fo rm of a pos t - t ens ioned f loor is s imi la r to tha t o f a re in fo rced

concre te f loor . S labs can be so l id , r ibbed o r waf f l e ; beams can be downs tand ,

upstand or s t r ips wi thin the s lab th ickness .

1.2 Pres tressing in pr in c iple

In re in fo rced concre te co ns t ruc t ion , the l ack o f s treng th o f concre te in t ens ion i s

compensa ted fo r by p rov id ing bonded s t ee l re in fo rcement near the t ens ion faces

of the con crete sect ion. The s teel , being s t ron g in tens ion, b ears the tens i le forces

and the concre te t akes the com press ive fo rces . U nd er n o- load cond i t ion the s t ee l

i s uns t ressed ; as a re in fo rced concre te member i s loaded i t defo rms , induc ing

com press ive a nd tens ile st resses . The s t resses in concrete an d s teel, therefore , vary

wi th the load .

In p res tress ing , a perm ane n t ex te rna l ax ia l fo rce , o f p rede te rm ined m agn i tude ,

i s app l i ed to the concre te member , which induces a compress ive s t res s in the

conc rete sect ion. W hen the service load is appl ied, the gene rated tens i le s t ress has

to overcome the compress ive prestress before the concre te i s d r iven in to any

tension. T he tens i le s t reng th o f conc rete i s, therefore, effectively enh anc ed. Th e

pres t ress ing force does not s ignif icant ly change wi th the load wi thin the

serviceabi l i ty l imi t . The pr inciple i s i l lus t rated in Figure 1 .2(a) .

Con s ider a s imple beam, requ i red to car ry a dow nw ard ac t ing impo sed load ; a t

th is s tage, assu m e th at the sel f-weight of the be am is negl ig ible . An axial

pres t ress ing force is appl ied at the cen troid of the sect ion, which induce s a

uniform com press ive s t ress across the sect ion, Fig ure 1 .2(a) . At the top o f the

beam , the f lexura l com press ion i s add ed to the prestress and the concre te on the

com press io n face is subjected to the sum of the pres t ress an d the f lexural s t ress ,

i .e . t he concre te has a h igher compress ive s t res s than i t would have wi thou t the

pres t ress . At the bo t to m of the beam , the f l exura l tens ion i s in oppos i t ion to the

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Reinforcement

Strand

Split pocket

f o r m e r , ~

Anchorage

casting

Wedges

Figure 1.1 P o s t - t e n s i o n i n g o f a f l o o r

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4 POST-TENSIONED CONCRETE FLOORS

(a) Axial prestress

(D

t ._

X

< ~ .

" O

-J Z

+2 .0 +4 .0 +6 .0

H+s-

+ 2 .0 - 4 . 0 - 2 . 0

+ 2 . 0 - 3 . 0 - 1 . 0 + 4.0 + 3.0

I

+2 .0 +3 .0 +5 .0 - 4 .0 +1 .0

Eccentr ici ty

D/4

.v.,,

(b) Ecce ntr ic prestress ~ -=

< ~, ~. 9 ~, .J z

Figure

1.2 The pr inc ip le of prestress ing

Stresses (N/mm 2)

compress ion from the pres t ress and, therefore, the s t ress in the concrete i s lower

than the t ens ion i t would have under f l exure a lone .

If the extern al force is appl ied eccen tr ical ly , as show n in Figu re 1 .2(b) , then the

com press ive s t res s induced a t the bo t tom of the sec t ion i s h igher fo r the same

axial force. I f the ec centr ici ty is suff iciently large, the top of the be am develop s a

s l ight tens ion. When the imposed load is appl ied to such a beam, i t s top f ibre i s

sub jec ted to the d i ffe rence be tween the f lexura l compress ion and the t ens ion f rom

pres t ress . Thus the f l exura l compress ion mus t overcome the p res t ress - induced

tens ion befo re the concre te goes in to com press ion . The b o t tom of the beam

remains in compress ion .

An eccentr ical ly appl ied force, therefore, increases the capaci ty of the sect ion

for f l exura l t ens ion a t the bo t tom and fo r compress ion a t top . I t i s much more

ef f i c i en t than an ax ia l p res t ress . The apparen t enhancement in s t res s capac i ty o f

con crete al lows a smal ler conc rete sect ion to be used than is poss ible in reinforced

concrete . Pres t ress i s general ly appl ied eccentr ical ly , except in very special

c i rcumstances .

The advan tage can perhaps be bes t i l lus t ra t ed by the numer ica l va lues shown

in Figure 1 .2 . The app l i ed load p ro duces a compress ive s t res s o f + 4 .0 N /m m 2 a t

the top and a tens il e s tress o f - 4 . 0 N/m m 2 a t the bo t tom . W i th an ax ia l p res t ress

of 2 .0 N/m m 2 the com bined ne t st res ses wou ld be + 6 .0 N /m m 2 and - 2 .0 N/m m 2

at top and bo t tom respec t ive ly , F igure 1 .2 (a) .

I f the sam e p res t ress ing fo rce is app l i ed eccen t r i ca lly then a m om ent i s induced ,

whose m agn i tude is the p rodu ct o f the p res t ressing fo rce and i ts eccen tr i c ity .

As s u mi n g an eccen t ri c it y o f o n e -qu a r t e r o f t h e mem b er d ep t h , t h e mo m en t

p ro d u ces f lex u ra l st re ss es o f - 3 .0 N / m m 2 t en s io n a t t h e to p an d + 3 .0 N / m m 2

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THE BAS IC PRINCIPLES 5

co m p res s i o n a t t h e b o t t o m . T h es e co m b i n e w i t h t h e ax i a l co m p res s i o n o f

+ 2 . 0 N / m m 2 t o p r o d u c e p re s tr e ss s tr es se s o f - 1 . 0 N / m m 2 a t t o p a n d

+ 5 .0 N / m m 2 a t b o t t o m . T h e f i n al s tr e ss e s, d u e t o p re s t r e s s an d ap p l i ed l o ad , a r e

n o w + 3 . 0 N / m m 2 c o m p re s si on a t to p a n d + l . 0 N / m m 2 c o m p re s si on a t

b o t t o m , F i g u re 1 . 2 (b ) .

W i t h a D/4eccen t r ic i ty (where D is the dep th ) the s t ress due to p res t ress a lon e

h as in c rea s ed f ro m + 2 .0 N / m m 2 t o + 5 .0 N / m m 2 a t b o t t o m , t h e r a t i o o f m ax i -

m um to avera ge s t ress be ing 2 .5. Th is ra t io is dep end en t on the sha pe o f the sec-

t ion and the eccen t r ic i ty . Ra t ios in the range o f 2 .5 to 4 .0 a re co m m on ly ach ieved .

N o t e t h a t w i t h eccen t r ic p re s t r e ss b o t h f in a l st re s se s ( t o p + 3.0 N / m m 2 an d

b o t t o m + 1 .0 N / m m 2) a r e l es s t h an t h ey wo u l d h av e b een wi t h o u t p re s tr e s s

( + 4 . 0 an d - 4 . 0 N / m m 2 r e s p ec t iv e l y ). I n f ac t, t h e b o t t o m f ib re is st il l i n

com press ion and , fo r a f inal t ens ion o f - 2 .0 N/ m m 2 as in the ax ia lly p res t ressed

case , the p res t ress ing fo rce can be reduced by 60%.

In f loo rs , the l eve l o f com press ion due to p res t ress i s usua l ly in the ran ge o f 1 .0

to 5 .0 N /m m 2 (150 to 700 ps i) , the av erage be ing a r ou nd 3 .0 N /m m 2 (450 ps i) .

The low er l eve ls o f s t ress a re genera l ly used in g rou nd s labs and the h igher in

p o s t - t en s i o n ed b eam s . T h i s r an g e i s qu i t e l ow co m p are d wi t h t h a t i n, s ay, b ri d g es

wh e re t h e av e rag e s t r es s m ay b e m u ch h i g h e r , b ecau s e o f t h e lo n g e r s p an s an d t h e

h igher loads .

1 .3 S t r e s s r e v e r s a l

In con t inuo us s t ruc tu r es , i f the se l f-weigh t o f the f loo r is smal l co m pa red wi th the

app l ied loads o r i f there i s a wide va r ia t ion in span leng ths , the n i t is poss ib le fo r

t h e l o ad - i n d u ced s t r e s s e s i n a s p an t o b e r ev e r s ed u n d e r a c e r t a i n l o ad

co m b i n a t i o n ; t h e s t r es s a t t h e b o t t o m i n t h e m i d d l e o f t h e s p an m ay b e

co m p res s i v e an d a t t h e t o p i t m ay b e t en s i l e . I n s u ch a m em b er i n r e i n fo rced

concre te , su f f icien t t ens ion s tee l wo u ld be p ro v ide d on each face to cope w i th the

rev e rs a l . R e i n fo rcem en t a t e ach f ace wo u l d b e d e s i g n ed i n d ep en d en t l y o f t h e

rev e r s e m o m en t b ecau s e t h e t wo co n d i t i o n s can n o t ex i s t s i m u l t an eo u s l y .

C o n s i d e r w h a t h ap p en s i f t h e ap p l i ed l o ad i s r ev e r s ed in a p re s tr e s s ed m em b er .

T ak i n g t h e ex am p l e i n F i g u re 1 . 2 (b ) , a s s u m e t h a t t h e r ev e r s e l o ad p ro d u ces a

t e n si o n o f - 4 . 0 N / m m 2 a t t he t o p a n d a c o m p r e s s i o n o f + 4 . 0 N / m m 2 a t th e

bo t tom. The f ina l s t resses in th i s case wou ld be as g iven in Tab le 1 .1 .

In the abse nce o f any p res t ress , the f l exura l s tresses fo r the reverse load (ac t ing

u p w a r d s ) w o u l d b e - 4 . 0 N / m m 2 a t t op a n d + 4 . 0 N / m m 2 a t b o t t o m ; w i th

pres t ress they a re - 5 .0 and + 9.0 N /m m 2 respec t ive ly . C lear ly , th is p res t ressed

m em b er i s wo r s e o f f w i t h t h e r ev e rs e l o ad i n g .

The p rob lem is cause d by the h igh eccen t r ic i ty , wh ich induces a t ens ion on the

top face . Th is t ens ion ge t s add ed to the f lexura l t ens ion o f the reverse load . W i th a

lower eccen t r ic i ty the s t resses under the reversed load a re a l so lower , though the

s t resses und er the no rm al loa d w ould increase . In th is exam ple the reverse load i s

equ a l in m ag n i t u d e t o t h e n o rm a l l o ad a n d s o t h e be s t r e s u lt s wo u l d b e o b t a i n ed

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6 POST- TENSIONED CONC RETE FLOO RS

Table 1.1

Effect o f load reversal

Prestress + Bendin9 = Final

Normal l oad

Top

Btm

Reversed load

Top

Btm

- 1.0 + 4.0 = + 3.0N /m m 2

+5 .0 - 4 .0 = +1 .0

- 1.0 - 4.0 = - 5.0N /m m z

+ 5.0 + 4.0 = + 9.0

w i t h a n a x i a l ly a p p l ie d p r e s t r es s . I f t h e p r e s tr e s s p r o d u c e d a u n i f o r m c o m p r e s s i o n

o f + 2 .0 N / r a m 2 o v e r th e w h o l e s ec t i o n t h en t h e f i na l s t re s s e s w o u l d b e - 2 . 0 an d

+ 6 . 0 N / r a m 2 i n e a c h c a se , a s in F i g u r e 1 .2 (a ). T h i s i s a n i m p r o v e m e n t o n t h e

eccen t r i c a l l y ap p l i ed p r e s t r e s s b u t n o t an e f f ic i en t u se o f m a t e r i a l s .

T h i s c l ea r l y i l l u s tr a t e s t h a t p r e s t r e s s i n g i s n o t s o e ff ec t iv e w h en r ev e r s a l o f l o ad

i s i n v o lv e d . F o r t u n a t e l y , l o a d r e v e rs a l s e l d o m o c c u r s i n b u i l d i n g fl o o rs a n d w h e n

i t d o es , t h e d ead l o ad i s u s u a l l y s u f f ic i en t e i t h e r t o k eep t h e n e t l o a d s ti ll a c t i n g

d o w n w a r d s , o r t o g r e a t l y r e d u c e t h e e f fe ct of t h e r e v er s e lo a d i n g . I n c o n t i n u o u s

s p a n s , o f s h o r t l e n g t h c a r r y i n g h e a v y l iv e l o a d s , s u c h a s i n w a r e h o u s e s , t h e d e a d

l o a d m a y n o t b e l a r g e e n o u g h t o a v o i d s tr e ss r e v e rs a l a n d , t h e r ef o r e, r e v e r s a l m a y

b e a c r it i c a l co n d i t i o n . I n s u ch ca s e s , a r ed u ced eccen t r i c i t y o f p r e s t r e s s p r o v i d e s

t h e b e t t e r s o l u t i o n .

1 . 4 T e n d o n s

I n p r e s t r e s s ed s t r u c t u r e s , t h e ex t e r n a l p r e s t r e s s i n g f o r ce i s g en e r a l l y ap p l i ed b y

s t r e t ch i n g s t e e l r o d s , w i re s o r r o p es

( s t r a n d )

ag a i n s t t h e co n c r e t e s ec t i o n , w h i ch

g o es i n t o co mp r es s i o n . T h e h i g h s t r en g t h s t e e l r o d s , w i r e s , o r s t r an d s a r e

co l l ec t i ve ly ca l l ed t e n d o n s o r cab le s . I n p o s t - t en s i o n ed f l o o r s , h o w ev e r , u s e o f

s t r a n d i s n o w a l m o s t u n i v e r s a l .

T h e t e r m s t r a n d c a n b e r a t h e r c o n f u s i n g - - i t a p p l i e s t o t h e ro p e c o n s i s t i n g o f a

n u m b e r o f i n d i v i d u a l w i re s w o u n d t o g e t h e r , it d o e s n o t m e a n t h e in d i v i d u a l w i r e

co mp r i s i n g a r o p e . A t y p i ca l s t r an d co n s i s t s o f 7 w i r e s w o u n d i n t o a r o p e ; t h e

c o m m o n l y u s e d s iz es ar e n o m i n a l 13 m m a n d 15 m m (0 .5 a n d 0 . 6 i n ) i n d i a m e t e r .

T h e a c t u a l s i z e s a n d s t r a n d c h a r a c t e r i s t i c s a r e g i v e n i n C h a p t e r 2 .

I n p o s t - t en s i o n i n g , b ecau s e t h e p r e s t r e s s i s ap p l i ed a f t e r t h e co n c r e t e h a s

g a i n e d s u f f i c i e n t s t r e n g t h , b o n d c a n n o t b e a l l o w e d t o d e v e l o p b e t w e e n t h e

c o n c r e t e a n d t h e t e n d o n s b e f o re s t r e ss i n g , a n d t h e re f o r e, t h e t e n d o n s a r e h o u s e d

i n a b o n d - b r e a k i n g d u c t o r s h e a t h i n g . M o r e t h a n o n e w i r e o r s t r a n d m a y b e

h o u s ed i n o n e co m m o n d u c t ; t h e g r o u p o f o n e o r m o r e i s a l so ca l led a

t e n d o n

o r

cable .

T h e s t r a n d i s s i m i l a r t o r o d r e i n f o r c e m e n t w i t h r e g a r d t o i t s m o d u l u s o f

e l a s t i c i t y an d co e f fi c ien t o f t h e r m a l ex p an s i o n b u t i t is ab o u t f o u r t i mes s t r o n g e r

t h a n r e i n f o r c e m e n t s t ee l. S t r a n d m a y h a v e a n u l t i m a t e s t r e n g t h o f 1 8 60 N / m m 2

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THE BAS IC PRINCIPLES 7

(270 ks i ) co m pa red wi th 460 N /m m 2 (67 ks i) fo r rod re inforcem ent . A t se rv ice

l o a d s t h e s t r a n d m a y h a v e a s tr es s o f l l 0 0 N / m m 2 ( 16 0 k si ) w h il e r o d

r e in fo r c e me n t m a y c a r r y o n ly a b ou t 250 N /m m 2 (36 ks i) .

I n a dd i t i on to t he t e ndons , som e bon de d r od r e in f o r c e me n t is a l so p r ov ide d in

pos t - t e ns ione d f loo r sma r ound the a nc ho r a ge s , i n t he s l a b a s t i e s , a s s e c onda r y

re inforcem ent , o r i f needed to ach ieve the requi red u l t im a te s t r ength .

The w e ight o f s tee l (s t r ands an d rod re inforcement) r equi red in a pos t - tens io ned

f loor is typ ica l ly on ly ha l f , o r le ss , o f tha t needed in a r e inforced concre te f loor .

Tho se used to w ork ing w i th re inforced concre te a re su rpr i sed o n the i r f i rs t v i s it to

a po s t - tens ione d f loor s i te because the re i s so l it t le s teel . Ins tea d of heavy ~

re inforceme nt ba r s space d a t 150 m m cen t res , they see a couple of 15 m m s t rand s

every met re or so .

S t r a nd be ing so muc h s t r onge r t ha n r od r e in f o r c eme n t , t he q ue s t ion a r i se s a s

to w hy i t i s no t used in re inforced concre te . A t a wo rk in g s t r e ss of 1100 N /m m 2 ,

the ave rage tensi le s t r a in in the s t r and wo uld be ap pro xim ate ly 0 .0056. I f used as

non - p r e s t r e s se d b ond e d r e in f o rc e me n t , t he c r a c ks i n c onc r e t e r e su l ti ng f r om th i s

s t r a in ma y a pp r oa c h 1 .0 m m in w id th , wh ic h wo u ld be una c c e p ta b le . T o l imi t t he

c r a c ks to a c c e p ta b le w id th s t he s t r a nd w ou ld ha ve to be u se d a t a muc h r e duc e d

and ine f f ic ien t s t r e ss leve l . In pres t re ss ing , the tens ion in the tendon i s used to

induc e c om pr e s s ion in t he c onc r e t e a nd , t he r e f o re , a dva n ta ge c a n be t a ke n o f i ts

re la t ive ly h igh s t r ength .

1 . 5 P r e s t r e s s l o s s e s

In pres t re ss ing , the compress ive s t r e ss in the concre te due to the pres t re ss i s

ma in t a ine d by the t e ns ion in t he t e ndons , so t ha t a ny c ha nge in t he c onc r e t e

s t ra in i s r e flec ted in the te nd on forces and v ice ve rsa . I f the ten do ns a re r e leased or

a l lowed to s l ip then the i r tens i le force i s los t o r r educed , and the concre te

com press ion u nderg oes a cor res pon din g loss . A s imi la r e ffec t occurs i f the length

o f t he p r e s tr e s se d m e m be r r e duce s , fo r e xa mple due to sh r inka ge o r c r e ep o f

c onc r e t e ; the t e nd on l e ng th , being e q ua l t o t ha t o f t he c onc r e te m e m be r , a l so

r e duc es a nd som e t e ns ion in the t e nd on ( a nd c om pr e s s ion in the c onc r e t e) i s l o s t.

Du r ing the s t re s s ing o f a t e ndon , f ri c ti on be twe e n the t e ndo n a n d the duc t

causes a loss in the tend on force so tha t i t reduces aw ay f rom the j ack in g end . The

me c ha n ic a l a c t ion o f s e c u r ing a t e ndon in t he a nc ho r a g e a l lows the t e ndon to

dra w in, typical ly b y 6 -8 m m ( 88 o 3 in) , which resul ts in a fu r the r loss in the

ten do n force. Also , a s each success ive tend on in a mem ber i s s t r e ssed , the length

o f t he conc r e t e me m be r r e duc e s by a sma l l a m ou n t , w i th the c onse q ue nc e tha t t he

p r e v ious ly s t r e s se d t e ndons lo se some t e ns ion . I n a dd i t i on to t he se imme d ia t e

lo sse s, pos t - t e ns ion e d c onc r e t e un de r goe s a g r a d ua l r e duc t ion in l eng th be c a use

of shr ink age an d c reep , w hich in theory cont in ues for i ts li fe , tho ug h a s ign i fican t

pr op or t io n of i t occurs w i th in the f ir s t f ew weeks . S tee l tend on s a l so un derg o

long - t e r m c r ee p e lon ga t ion ( ca ll ed relaxation) bu t o f a sma l l e r ma gn i tude . T he se

ac t ions cause a long- te rm loss in the tendon force .

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8 POST-TENSIONEDCONCRETE FLOORS

T h e l o s s e s can b e s een t o o ccu r i n t h r ee d i s t i n c t s t ag es .

9 T h e p r e s t r e s s i n g f o r ce i s a t i t s max i mu m w h en a t en d o n i s b e i n g s t r e s s ed an d

b e fo r e it is a n c h o r e d . F r i c t i o n r e d u c e s t h e f o rc e a w a y f r o m t h e j a c k i n g e n d . T h e

p r ev i o u s l y s t re s s ed t en d o n s , o f co u r s e , u n d e r g o a l o s s d u e t o e l a s t i c s h o r t en i n g

o f co n c r e t e a s e ach s u cces s i v e t en d o n i s s t r e ss ed .

9 W h e n t h e t e n d o n is a n c h o r e d t h e n t h e d r a w - i n c a u s e s a n i m m e d i a t e l o s s i n

p r e s t r e s s .

9 S h r i n k ag e an d c r eep o f co n c r e t e , an d r e l ax a t i o n o f s t e el , c au s e a f u r t h e r l o s s b u t

t h i s i s l o n g t e r m .

T h e m a g n i t u d e o f l os s es d e p e n d s o n m a n y f a c t o rs , s u ch a s t h e c o n c r e t e

p r o p e r t i e s , t h e ag e a t w h i ch t h e p r e s t r e s s i s ap p l i ed , t h e av e r ag e p r e s t r e s s v a l u e ,

t h e len g t h o f t h e t en d o n s an d t h e t en d o n p r o fi l es .

1 . 6 I n it i a l a n d f i n a l s t r e s s e s

T h e l o s s i n p r e s t r e s s a r i s i n g f r o m f r i c t i o n , a n c h o r a g e d r a w - i n a n d i m m e d i a t e

e l a st ic s h o r t e n i n g o f t h e m e m b e r i s t e r m e d

in i t ia l loss .

T h e t o t a l l o n g - t e r m l o ss is

ca l l ed f i na l l o s s ; i t i n c l u d es t h e i n i t i a l l o s s an d t h e l o s s e s d u e t o s h r i n k ag e an d

c r e e p o f c o n c r e t e , a n d r e l a x a t i o n o f s t r a n d .

A s e c c e n t r i c p r e s t r e s s i s a p p l i e d t o a m e m b e r , t h e m e m b e r t e n d s t o d e f l e c t

u p w ar d s s o t h a t i t s se l f -w e i g h t i s b o r n e b y t h e p r e s t r e s s . I n fac t , p r e s t r e s s a l m o s t

n e v e r a c t s a l o n e ; i t is a c c o m p a n i e d b y t h e s e lf -w e i g ht o f t h e m e m b e r p r o d u c i n g

s t r es s e s o p p o s i t e t o t h o s e o f t h e p r e s t r e s s .

I n t h e e x a m p l e s h o w n i n F i g u r e 1 .2 , n o a l l o w a n c e w a s m a d e f o r lo s se s . A s s u m e

a 5 % l o ss d u e to a n c h o r a g e d r a w - i n a n d a f u r t h e r 1 5 % t h e r e af t e r. A l so , a s s u m e

t h a t t h e s e l f - w e i g h t o f t h e f l o o r i n d u ces s t r es s e s o f _ 1 .6 N / m m 2 an d t h a t t h e

i m p o s e d l o a d p r o d u c e s a fu r t h e r _ 2 .4 N / m m 2. T h e s t re s se s a t v a r io u s s t a ge s a r e

s h o w n i n T ab l e 1 .2

N o t e t h a t t h e b o t t o m o f t h e s e c ti o n is i n c o m p r e s s i o n a t j a c k i n g a n d i n i ti a l

s t ag es ; a s l o ad i s ap p l i ed t h e co m p r es s i o n r ed u ces . I f t h e ap p l i ed l o ad i s f u r t h e r

i n c r e a s e d t h e n t h e b o t t o m o f t h e m e m b e r m a y g o i n t o t e n s io n . O f t e n , th e h i g h e s t

co m p r es s i v e s t r e s s ex i s ts a t t h e b o t t o m o f t h e s ec t i o n a t t h e j a ck i n g o r i n i t i a l

Table 1.2 Initial and fina l stresses

Stage Top/b tm Prestress Sel f -wt Imposed N et s tress

Jack ing To p - 1.00 + 1.60 + 0.00 = + 0.60

Btm + 5.00 - 1.60 + 0.00 = + 3.40

Ini tia l Top -0 .9 5 + 1 .60 +0.0 0 = +0.65

Btm +4.75 - 1 .60 +0.0 0 = + 3 .15

Final Top -0 .8 1 + 1.60 + 2.40 = + 3.19

Btm + 4.04 - 1.60 - 2.40 = + 0.04

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THE BASIC PRINCIPLES 9

s tage , and th is occurs a t an ea r ly s tage in the l i fe of the concre te wh en i t is no t

ma tu r e . I n suc h c a se s pos t - t e ns ione d f loo r s c a n be c ons ide r e d to ha ve be e n

loa d - t e s t e d du r ing c ons t r uc t ion .

The s i tua t ion i s r eve rsed a t the top of the sec t ion ; i f the se l f -we ight is sma l l then

the top f ib re m a y be in t e ns ion in i t ia l ly , a nd wou ld go in to c om pr e s s ion a s fu r the r

load i s imp osed . There fore , the in i t ia l and f ina l st r esses represen t tw o ex t rem es in

the se rv iceabi l ity load in g of a pos t - ten s ion ed f loor . S tre sses a re com pu ted for

both s tages in des ign and a re genera l ly r equi red to be wi th in presc r ibed l imi ts .

Fo r the in i t ia l stage the load cons is t s o f on ly the se lf -we ight o f the m em ber ; no

othe r imposed load i s cons ide red . At the f ina l s tage , the va r ious loads a re

c ombine d , i n a c c o r da nc e w i th t he na t iona l s t a nda r d , t o a r r ive a t t he mos t

a dve r se c ond i t i on .

1.7 Pre - tension ing and post- tensioning

The tendons can be s t r e ssed e i the r be fore cas t ing the concre te or a f te r the

c onc r e t e ha s be en c a s t a nd ha s ga ine d som e s t re ng th . I n pre-tensioning the wires

o r s t r a nds a r e s t re s se d a ga in s t e x t e rna l a nc ho r po in t s ( o r som e t ime s a ga in s t t he

m ou ld ) a n d c on c r e t e is t he n c a s t in d i re c t c on ta c t w i th t he t e ndon s , t hus a l l owing

bo nd to de ve lop . W he n the c onc r e te ha s ga ine d su f fi ci en t s tr e ng th , t he t e ndo ns

a r e r e l e a se d f r om the t e mpor a r y e x t e r na l a nc ho r a ge s , t he r e by t r a ns f e r r ing the

force to the concre te , induc ing a compress ive s t r e ss in i t . The tens ion in the

t e ndons a nd the c o r r e spond ing c ompr e s s ion in t he c onc r e t e i s t he n so l e ly

d e p e n d e n t o n b o n d b e t w e e n c o n c r e t e a n d t e n d o n , a n d n o o t h e r m e c h a n i c a l

device is used.

P r e - t e ns ione d t e ndons u sua l ly r un in s t r a igh t l i ne s . I n o r de r t o de v ia t e f r om

this , ex te rna l d e f lec t ing devices a re needed . W ith them , a prof i le cons is t ing of a

se ries of s t r a igh t l ines can be ob ta ined . These devices s low dow n the m anu fac tur in g

p r oc e s s a nd the y a dd to t he c os t s . T he r e q u i r e me n t f o r e x t e r na l t e mpor a r y

a nc ho r s a nd the p r ob le m s in p ro f i li ng the t e ndons m a ke p r e - t e ns ion ing d i ff ic u lt

f o r a pp l i c a t ion in situ. The process i s a lmost exc lus ive ly conf ined to precas t ing ,

and i s no t d iscussed any fur the r .

In post- tensioning, c onc r e t e is no t a l l owe d to c om e in c on ta c t w i th the t e ndo ns .

T he t e nd ons a r e p l ac e d in duc ts , o r she a th s , wh ic h p r e ve n t bo nd , a nd c onc r e t e is

cas t so tha t the duc t i tse l f i s bo nd ed bu t the tend on ins ide remains f ree to m ove .

W hen the concre te has ga ined suf f ic ien t s t r ength the tend on s a re s t re ssed d i rec t ly

a ga in s t t he c onc r e t e a nd the y a re me c ha n ic a l ly l oc ke d in anchorages cas t a t each

e nd . A f te r th i s s ta ge , t e n s ion in t he t e ndons , a nd he nc e the i nduc e d c om pr e s s ion

in the c onc r e t e , i s ma in t a ine d by the a nc ho r a ge s .

In bonded pos t - t e ns ion ing the duc t i s g r ou te d a f t e r t he t e ndons ha ve be e n

s t ressed , so tha t the s t re ssed tendon s becom e bond ed . In unbonded pos t - t e ns ion ing ,

a s i t s na me imp l i e s , t he t e ndons a r e ne ve r bonde d . A de t a i l e d c ompa r i son

be twe e n bonde d a nd unbonde d sys t e ms i s g ive n in S e c t ion 1 .9 .

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10 POST-TENSIONED CONCRETE FLOORS

1 .8 R e i n f o r c e d a n d p o s t - t e n s i o n e d c o n c r e t e f lo o r s

Reinforced concre te t echno logy i s wide ly ava i l ab le and i s wel l unders tood .

Pos t - t ens ion ing i s an advance on re in fo rced concre te t echno logy and i t i s o f t en

discussed in the context of reinforced concrete .

1 .8 . 1 G ene ra l

Pos t - t ens ion ing o ffers some very useful techn ica l and e conom ic adv an tag es over

re in fo rced co ncre te , par t i cu la r ly fo r long spans , wh ere con t ro l o f def l ec t ion is

des i rab le , o r i f cons t ruc t ion dep th mu s t be min imised . I t is , howe ver , no t the bes t

so lu t ion in al l c i rcums tances and the var ious a l t e rna t ive fo rms o f con s t ruc t ion

shou ld be carefu l ly cons idered fo r each s t ruc tu re befo re making the cho ice .

Fo r pos t - t ens ion ing , i t is im po r tan t to con s ider ava i lab i l ity o f the hard w are

and the technical expert i se required. Except ing very special des ign object ives ,

pos t - t ens ion ing is un l ike ly to be econom ica l fo r shor t spans . Of ten a com bina t ion

of pos t - t ens ion ing and ano ther fo rm of cons t ruc t ion o ffers a good so lu t ion . Fo r

exam ple, in a f loor consis t ing of rec tang ular bays , i f the shor t sp an is smal l

enough , the bes t so lu t ion may be to span the s l ab in the long d i rec t ion in

pos t - t ens ioned concre te and use re in fo rced concre te bea m s t rips in the shor t - sp an

direct ion.

Ec ono m y of cons t ruc t ion var ies from one s it e to the nex t , depend ing on

accessib il ity and ava i l ab i li ty o f m ater i a l an d l ab our , and o f course on the des ign

load ing and cons t ra in t s tha t may be imposed by o ther d i sc ip l ines , such as a

res t r ic t ion on the d epth of the s t ructure . I t is , therefore, not p oss ible to g ive a

g en e ra l co s t co mp ar i s o n b e t w een t h e t w o fo rms o f co n s t ru c t i o n . T h e au t h o r s

have fou nd tha t fo r span l eng ths o f ab ou t 9 m (30 ft ) the cos t o f a f loor car ry ing a

super imposed load o f 5 kN /m 2 (100 psf ) is s imi la r in re info rced and pos t - t ens ioned

concre tes . Pos t - t ens ioned concre te i s more economica l than re in fo rced concre te

above th is span l eng th . In cases o f res t r ic t ed f loor dep th o r h igh loads , the span

length for equa l cos ts ma y be as low as 7 .5 m (25 ft ). Fu rth er savings resul t f rom

the l igh te r weigh t and l esser cons t ruc t ion dep th o f the pos t - t ens ioned f loor.

For re in fo rced concre te , on ly the u l t imate s t reng th ca lcu la t ions a re normal ly

carr ied out and deflect ion in the serviceabi l i ty s tate i s deemed to be sat is f ied by

conf in ing the span- to -dep th ra t io wi th in l imi t s p rescr ibed in the na t iona l

s t anda rds . On ly in ra re cases is i t necessary to ca lcu la te def lec tions. C rack con t ro l

i s usua l ly governed by .deemed- to -sa t i s fy ru les fo r bar spac ing .

In pos t - t ens ioned concre te des ign , se rv iceab i l i ty ca lcu la t ions a re car r i ed ou t

for the in i t ia l and f inal loading condi t ions , for def lect ion and cracking, and the

u l t imate s t reng th is checked a f t e r thi s. S t ruc tu ra l des ign o f p res t ressed concre te ,

therefore, requires more effor t .

The sha l low dep th o f a pos t - t ens ioned f loor is a par t i cu la r ad van tage in

mul t i s to rey bu i ld ings ; in some cases i t has been poss ib le to add an ex t ra f loor

where there was a res t r i c t ion on bu i ld ing he igh t . Even where there i s no such

res t r ic t ion, the red uced bui ld ing v olum e gen erates savings in the cos t of services

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THE BA SIC PRINCIPLES 11

a nd c l a dd ing , a nd in subse q ue n t r unn ing a nd ma in t e na nc e c os t s . T he r e duc t ion

in the w e ight o f the bu i ld in g genera tes fur the r sav ings in the cos t o f fou nd a t ion s ;

the we igh t o f c onc r e t e i n f loo r s i n a mu l t i s to r e y bu i ld ing m a y be a s mu c h a s ha l f o f

the t o t a l w e igh t o f t he bu i ld ing . T he c os t pe r un i t a r e a o f a pos t - t e ns ion e d f lo o r ,

c ons ide r e d in i so l a t ion , m a y be h ighe r t ha n tha t o f a r e in f o r ce d f loo r o f t he s a me

span and ca r ry ing the same load , bu t i t i s qu i te poss ib le for the bu i ld ing wi th

pos t - tens io ned f loors to prov e the more co s t ef fect ive wh en the o th e r sav ings a re

t a ke n in to a c c oun t .

1 .8 .2 Des ign and serv icea b i l i t y

There a re nu m ero us d i f fe rences be tween the beh avio ur of f loors un der se rv ice

loa ds i n t he two f o r ms o f c ons t r uc t ion :

9 As d iscussed ea r l ie r, po s t - tens ione d concre te i s a t a s l igh t d isadv anta ge in

f loors where s t r e sses due to appl ied load can be reve rsed . This would be the

c a se fo r sho r t c on t inuo us spa ns sub j e ct t o he a vy a pp l i e d l i ve l oa ds c om pa r e d

wi th the se l f -we ight o f the f loor , such as in warehouses .

9 P r e s t re s se d c onc r e t e und e r goe s m or e s ho r t e n ing o f l e ng th c om pa r e d w i th

re inforced concre te , because of the in i tia l ax ia l com press io n . In a r e inforced

concre te member , the c reep s imply a f fec ts i t s de f lec t ion , bu t in pres t re ssed

concre te i t a lso a f fec ts the length of the m em ber .

9 Re in f o rc e d c onc r e t e f l oo r s on a ve r a ge t e nd to ha ve a spa n - to - de p th r a t i o

be twee n 20 and 25 whereas p os t - ten s ion ed f loors a re usua l ly in the range o f 30

to 40 . It i s f eas ib le for the ra t io to ap pro ach 60 for ligh t ly loaded long spans .

Pos t - tens ion ing i s r a re ly used in spans under 6 m (20 f t ) because the sha l low

dep ths do no t pro vid e suff icient eccentr ic i ty for the eff icient use of prestress ing.

9 T he u pw a r d f o rc e e xer t e d by a c u r ve d t e nd on a c ts a ga in s t t he a pp l i e d loa ds .

The de f lec t ion of the f loor i s, the re fore , lower because i t cor res po nd s to the ne t

d i ff er enc e be twe e n the a pp l i e d do w nw a r d loa d a n d the up w a r d f o rc e f r om the

t e ndons .

T he d r a pe o f t he t e ndon s a n d the p r e st r e s sing f o rc e c a n be t a il o r e d to c o n t r o l

de f lec t ion w here so des i red . I t is poss ib le to des ign a p os t - ten s ion ed f loor w hich

wi ll have n o de f lec t ion und er a g iven load , a l th ou gh th is is un l ike ly to r e su l t in

economica l use of p res t re ss ing .

9 I n pos t - t e ns ione d c ons t r uc t ion , t he c onc r e t e s e c t ion unde r wor k ing loa d i s

e i the r i n c omp r e s s ion o r i t ha s a sma l l a m ou n t o f t e ns ion on one f ac e. I n e i the r

case i t i s un l ike ly to h ave an y c racks , a nd i f any do deve lo p they wi l l no t

pe ne t r a t e de e p ly in to t he s e c t ion . B y c ompa r i son , i n r e in f o r c e d c onc r e t e

c ons t r u c t ion the c onc r e te m us t c r a c k be f o r e the r e in f o r c e me n t c a n be s tr e s se d

to the design level .

T he w ho le o f t he pos t - t e ns ion e d c onc r e t e s e c tion , be ing unc r a c ke d , is

effec tive in f lexure , so tha t a po st- te ns ion ed f lo or wil l hav e less def lec t ion th an a

re inforced concre te f loor of the same de pth a nd subjec t to the same load .

9 P os t - t e n s ion ing ke e ps t he c onc r e t e in c om pr e s s ion , wh ic h c on t r o l s sh r inka ge

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POST-TENSIONEDCONCRETE FLOORS

crack ing an d reduces the poss ib i li ty o f open ing up o f cons t ruc t ion jo in t s .

W hen t ens il e st res ses do deve lop in a pos t - t ens ioned m em ber , the i r mag n i tude

i s much smal le r than in an equ iva len t re in fo rced concre te member . A

pos t - t ens ioned f loor , therefo re , has be t t e r water t igh tness than a re in fo rced

concre te f loor. Th i s is par t i cu la r ly im po r tan t in car parks w here de- i c ing sa l ts

o f t en cause cor ros ion o f re in fo rcement in a re in fo rced concre te f loor . The

hai r l ine c racks over suppor t s in a re in fo rced concre te con t inuous f loor may

al low w ater to pene t ra te and f reeze, caus ing spa l ling o f concre te .

9 The u ncrac ked concre te o f a pos t - t ens ioned f loor p rov ides a be t t e r p ro tec t ion

aga ins t co r ros ion o f s t ee l than tha t g iven by a c racked re in fo rced concre te

sec t ion . In unbonded pos t - t ens ion ing the g rease packed p las t i c ex t rus ion

prov ides exce l l en t p ro tec t ion to the s t rand .

9 A

pos t - t ens ioned f loor, be ing l igh te r than a re in fo rced c oncre te f loor o f s imi la r

span and car ry ing the same app l i ed load , imposes smal l e r loads on the

co l u mn s an d fo u n d a t i o n s .

The c o lumn s pa r t i cu la r ly benef it f rom the pos t - t ens ion ing o f the f loor ,

because the t endon curva tu re an d the h igher c reep o f the f loor com bine to

r ed u ce th e co l u m n mo m en t s , a s s h o w n i n Ch ap t e r 5 . Th e s ize o f a co l u mn an d

i ts re in fo rcement a re usua l ly governe d by the bend ing m om en t . A reduc t ion in

moment can, therefore, resul t in s ignif icant savings .

S t if f co lum ns and wal ls m ay a t t rac t s ign if i can t m agni tud es o f l a t e ra l fo rces .

These shou ld be checked .

9 D r ap e d t en d o n s d i rec tl y ca r ry s ome o f t h e s h ea r fo r ce - -n u m er i ca l l y equ a l t o

the ver t ica l com pon en t o f the t endon fo rce nea r the suppo r t . The concre te

sect ion, therefore, carr ies a smal ler shea r force an d so dr op panels a re less l ikely

to be needed in post - ten s ione d con st ruc t ion. O f course, th is effect can b e offset

by the fact that post-tension ed floors are shallowe r than reinforced conc rete floors.

The p resence o f an ax ia l compress ive s t res s on the concre te sec t ion enh ances

i t s punch ing s t reng th .

9 At h igh t em pera tu res , say above 150~ s t rand loses i ts s t reng th fas t e r than rod

re in fo rcement . Th i s i s com pen sa ted fo r by spec i fy ing a deeper concre te cover to

tendons than to rod re in fo rcement in re in fo rced concre te .

9 In re in fo rced concre te , micro -cra cks m us t de ve lop befo re the re in fo rceme nt

can func t ion at i ts required level of s t ress . Po st - ten s ion ed conc rete , as s tated

earl ier , is expec ted to rem ain crack-free in service. In case of an iso lated

over load ing caus ing c racks in a pos t - t ens ioned f loor, the c racks a re expec ted to

close once the overloading is e l iminated.

1.8.3 Materials and equipment

Before d iscuss ing the s it e opera t ions , cons ider the mater i a l s an d eq u ipm ent used

in re in fo rced and pos t - t ens ioned concre te .

9 Bo th fo rms o f cons t ruc t ion use s imi la r g rades o f concre te , bu t e ar ly s t reng th is

a def in i te adva n tage in pos t - t ens ion ing . C om pac t ion , f in i sh ing and cur ing a re

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THE BASIC PRINCIPLES

13

ident ical . Post - tens ioned const ruct ion requires less concrete; th is can lead to

s ignif icant savings , as the placing of conc rete bec om es m ore ex pensive as the

bui ld ing height increases .

9 Fo rm w ork is a lso s imi la r , excep t tha t pos t - t ens ion ing requ i res ho les to be

dr i ll ed in the ver ti ca l edge board s fo r the tendon s to p ro jec t th rou gh , and the

an ch o rag e b l o ck s n eed t o b e a t t a ch ed t o t h e fo rmw o rk .

9 Po s t - t en s i o n i n g r equ i r e s s p ec i a l ma t e r i a l s - - an ch o rag es , d u c t s an d s t r an d ,

which have to be p rocu red and s to red on s it e. Spec ial equ ipm ent is al so needed ,

such as s t res s ing jacks a nd g rou t pu m ps , which need to be s to red and m oved

from one pos i t ion to ano ther .

1.8.4 Site activiti es

The s i te act iv i t ies for reinforced and post - tens ioned concretes are now brief ly

co mp ared .

9 A majo r adv an tag e in favour o f re in fo rced concre te is tha t know ledge o f i ts

cons t ruc t ion is read i ly ava i lab le . Pos t - t ens ion ing norm al ly needs the se rv ices

of t rained opera t ives for ins tal la t ion, w hich involves an ad di t ion al t rade on s i te .

9 Access to l ive anch ora ges at f loor edges is need ed for s tress ing tend ons , a nd

g ro u t i n g d u c t s fo r b o n d ed t en d o n s . Us u a l l y a p l a tfo rm ab o u t a m e t r e in w i d th

is sufficient . This w ould not be a prob lem wh ere scaffolding is erected for o th er

t rades . Where no scaffolding is provided, the access p lat form is a special

r equ i r emen t an d p u rp o s e -m ad e s t ag i n g may h av e t o b e e r ec ted a t e ach f lo o r

leve l fo r the du ra t ion o f the s tres sing and g rou t ing opera t ions , o r over - runs fo r

f ly ing forms may be ut i l ized for the purpose.

9 Po st - ten s ion ed conc rete f loors require m uch less w ork for assem bly of s teel.

The rea son i s tha t the q uan t i ty o f s tee l in t endon s a nd rod re in fo rcement fo r a

pos t - t ens ioned f loor is abou t ha l f tha t o f the re in fo rcement in an equ iva len t

re in fo rced concre te f loor . Ap ar t f rom the smal l e r qua n t i ty o f s tee l to be s to red ,

hand led and assembled , there i s a fu r ther sav ing in t ime because the same

tendons usua l ly run over a se ri es o f spans , un l ike re in fo rced concre te w here

shor t l eng ths o f bars , o r cages , a re hand-a ssem bled in each span and over each

s u p p o r t .

9 Post - tens ioning al lows large f loor areas to be cas t in one operat ion. This also

means fewer cons t ruc t ion jo in t s .

9 Add i t iona l s i t ework is cer ta in ly invo lved in the opera t ions o f l ay ing and

s t ress ing o f t endon s , and subsequen t g rou t ing a nd m aking good o f the

anch orag e pocke t s . H owe ver , th i s wo rk is norm al ly car r i ed ou t wi th in a per iod

of three o r four day s for single s tage s t ress ing o r wi thin a for tn ig ht for two s tage

s t ress ing (see Sect ion 1 .10 for d iscuss ion of s tress ing s tages) . Besides , these

o p e ra t i o n s d o n o t i mp ed e co n s t ru c ti o n o f co l u mn s an d t h e fl o o r ab o v e an d s o

they can e f fec tive ly be p rog ram m ed ou t o f the c r i ti ca l pa th .

9 Pres t ress i s appl ied wi thin three or four days o f cas t ing con crete and at th is

s tage the f loor becom es capa ble of sup por t ing i ts own w eight . Soffi t shut ters

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14 POST-TENSIONED CONCRETE FLOORS

c a n , t he r e f o r e , be r e m ove d e a r l ie r f o r r e - u se e ls e w he r e . S o m e o f t he ve r t i c al

p r o ps a r e r e t a i ne d f o r c on s t r uc t i on l oa ds . F e w e r s e ts o f so ff it shu t t e r s w ou l d be

ne e de d f o r a pos t - t e n s i one d f l oo r .

9 S om e o f t he spe ci a l e q u i p m e n t , pa r t i c u l a r l y f o r u se i n bo nd e d pos t - t e n s i on i ng ,

m a y n e e d c r a n e t i m e f o r h a n d l i n g , t r a n s p o r t i n g a n d i n u s e .

1 .9 B o n d e d a n d u n b o n d e d p o s t - te n s i o n i n g

P o s t - t e n s i o n e d f l o o r s m a y u s e b o n d e d o r u n b o n d e d t e n d o n s , o r b o t h t y p e s o f

t e ndons m a y be u se d i n a f l oo r . Th i s s e c t i on i n t r oduc e s t he t w o t ype s a nd t he n

c om pa r e s t he m f o r t he i r d i f f e r e nc e s a nd m e r i t s .

Ov e r a l l f igu r e s a re no t r e a d i l y a va i l a b l e f o r the t o t a l a r e a o f f l oo r s c ons t r uc t e d

u s i n g u n b o n d e d a n d b o n d e d s y s t em s b u t , j u d g i n g f r o m t h e sa le o f h a r d w a r e , i n

t h e U K a n d i n t h e U S A b o t h s y s t e m s ar e w i d e ly us e d. U n b o n d e d p o s t - t e n s i o n i n g

a p p e a r s t o b e m o r e i n f a v o u r i n t h e U K a n d t h e U S A . I n A u s t r a l ia t h e t r e n d i s

m u c h m o r e t o w a r d s b o n d e d s y s t e m s i n f l o o r c o n s t r u c t i o n .

1 .9 . 1 G e n e r a l

I n p o s t - t e n s io n i n g , b o n d b e t w e e n c o n c r e t e a n d t e n d o n s is p r e v e n t e d u n t il a f te r

t h e c o n c r e t e h a s a c q u i r e d a d e q u a t e s t r e n g t h a n d t h e t e n d o n s h a v e b e e n s t re s s ed .

To a c h i e ve t h is , t h e t e ndo ns a r e ho use d i n sheathin9 o r duc t s a nd c onc r e t e i s c a s t

a r o u n d t h e m . T h e d u c t s t h e m s e lv e s a re b o n d e d t o t he c o n c re t e b u t t h e y p r e c lu d e

c o n t a c t b e t w e e n c o n c r e t e a n d t h e t e n d o n s , w h i c h r e m a i n f r e e t o m o v e . T h e

t e ndons a r e s t r e s se d a t t he su i t a b l e t i m e a nd a r e t he n l oc ke d i n pos i t i on i n

anchorage

assembl ie s .

I n bond ed post-tensionin9 t h e duc t s a r e g r ou t e d a f t e r s t r e s s i ng a nd so bond i s

e s t a b l i she d be t w e e n t he t e ndons a nd t he c onc r e t e s e c t i on , w i t h t he duc t s a s

i n t e r ve n i ng c om pone n t s . As i t s na m e i m p l i e s , i n unb ond ed post- tensioning t he

duc t , u sua l l y r e f e r r e d t o a s a she a t h i ng i n t h i s c a se , i s ne ve r g r ou t e d a nd t he

t e ndon i s he l d i n t e n s i on so l e l y by t he e nd a nc ho r a ge s .

D u c t s f o r u se in bon de d p os t - t e n s i one d f l oo r s m a y be c i r c u l a r i n s e ct i on , 50 to

75 m m (2 t o 3 i n ) i n d i a m e t e r , o r ob l o ng , 75 x 20 m m (3 x 3 / 4 i n ), a nd m a de o f

c o r r u g a t e d g a l v a n i z e d i r o n o r p l a s t i c . T h e c o r r u g a t i o n s i m p a r t t o t h e d u c t a

longi tud ina l f l ex ib i l i t y , so tha t i t can be eas i ly ben t to a des i red prof i l e , whi l e

r e t a i n i ng r i g i d i ty t o ke e p t he s e c t ion c i r c u l a r. W i t h a c o r r u ga t e d duc t i t i s pos s i b l e

for the t endon to be curved to a rad ius a s sma l l a s 2 .5 rn (8 f t ) whereas wi th an

unc o r r uga t e d m e t a l duc t i t i s d i f f i c u l t t o ha ve a ny p r o f i l e o t he r t ha n a v i r t ua l

s t ra igh t l i ne .

The a c t ua l s i ze o f t he duc t va ri e s , de p e nd i ng o n t he m a nu f a c t u r e r a nd t he

nu m be r o f s t r a nds t o be hous e d i n it . C i r c u l a r duc t s , f o r u se i n no r m a l f l oo r s , a r e

in m ost sys t ems ava il ab le in severa l d i amete rs ; t he mos t com m on s izes accom m od a te

2 t o 4 s t r a nds , 5 t o 7 s t r a nds a nd 8 t o 12 s t r a nds . The ob l ong o r f iat duc t i s

n o r m a l l y d e s ig n e d t o a c c o m m o d a t e u p t o a m a x i m u m o f f o u r o r f iv e s t r a n d s .

Af t e r t he t e ndon s ha ve be e n s t r e sse d a nd t he exc es s l e ng t h o f s t r a n d t r i m m e d ,

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THE BASIC PRINCIPLES 15

t he duc t s a r e g r ou te d w i th a ne a t c e me n t pa s t e o r a s a nd - c e me n t mor t a r

c on ta in ing su i t a b le a dm ix tu r e s , wh ic h bonds the s t r a nds t o t he duc t . S a nd i s no t

inc lude d in the g r o u t f o r the sma l l d i a me te r duc t s no r ma l ly u se d in pos t - t e ns ione d

f loors . The d uc t i t se lf be ing a l ready b on ded , the ten do n b ecom es e f fec tively

bon de d to t he c onc r e t e s e c t ion . B o nd i s c a pa b le o f c a r r y ing the t o t a l p r e s t r e ss ing

force , should the anchorage become ine f fec t ive for some reason .

F o r un bo nd e d pos t - t e ns io n ing u se , a s pa r t o f i ts m a nuf a c tu r ing p r oc e s s, t he

s t r a nd ha s a t h in po lyp r o py le ne s le eve e x t r ude d on to i t a nd the spa ce be twe e n the

s t r a nd a nd the s l e e ve i s g r e a se pa c ke d . T he se me a su r e s p r e ve n t bond w i th t he

c onc r e te a n d inh ib i t c o r r o s ion . T h i s s t r a nd is ca s t d i re c t ly i n t he c onc r e te w i thou t

a n in t e r ve n ing du c t a nd i t r e ma ins ung r ou te d th r o ug ho u t t he li fe o f t he s t r uc tu r e .

A s in the b ond e d sys t e m, the t e nd ons a r e s tr e s se d w he n the c onc r e t e ha s ga ine d

suf f ic ien t s t r ength bu t the pres t re ss i s ma in ta ined so le ly th rough the mechanica l

a nc ho r a ge s a t e a c h e nd .

I n bo th sy s t e ms , bonde d a nd unbonde d , t he a nc ho r a ge s a r e s e t i n poc ke t s a t

the f loor edge , and s t r e ss ing i s ca r r ied ou t us ing hydraul ic equipment . Af te r

s t r e ss ing , the s t r an ds a re cu t c lose to the anch orag e face us ing a h igh-spee d d isc

c u t t e r . T he a nc ho r a ge a s se mbl i e s a r e t he n sp r a ye d w i th a c o r r o s ion inh ib i t a n t

a nd c ov e r e d w i th g r e a se -pa c ke d p l a st i c c a ps. La s t ly t he poc ke t s a r e m a de g ood

wi th c eme n t s a n d m or t a r f o r p r o t e c t ion a ga in s t c o r r o s ion a n d f ir e. T he c o r r o s ion

inh ib i t a n t , t he g r e a se c a p a nd the mor t a r p r ov ide a de q ua te p r o t e c t ion to t he

a nc ho r a ge a s se mbly in a pos t - t e ns ione d f loo r .

B onde d t e ndons r e q u i r e me ta l o r p l a s t i c duc t s a nd the a dd i t i ona l s i t e

ope r a t ion o f g r ou t ing , bu t s e ver a l s t r a nds a r e a nc ho r e d in a c o m m on a s se mb ly ,

the r e by sa v ing on the n um be r o f i nd iv idua l a nc ho r a ge s . T he a dd i t i on a l c os t o f

the duc t a n d g r ou t ing m a y be o ff se t by the s a v ing in t he c os t o f a nc ho r a g e s ; t h i s

de pe nds on the r e la t ive c os ts a nd the t e ndon l e ng ths . N o tw i th s t a nd ing the cos t ,

bonde d t e ndons a r e c ons ide r e d to be mor e s e c u r e a ga in s t a c c ide n ta l da ma ge .

They a re , the re fore , ve ry use fu l in brea king a f loor in to smal le r a reas for conf in ing

a c c ide n ta l da ma ge . B onde d t e ndons a r e o f t e n u se d in be a m s t r i p s whe r e a s

unbonde d t e ndons ma y be u se d in s l a bs suppor t e d on the se be a m s t r i p s .

1 .9 .2 Des ign and serv ice ab i l i t y

9 B onde d t e ndons u sua l ly c ons i s t o f s e ve r a l s t r a nds p l a c e d in a c ommon duc t .

The duc t i s la rge r in d iamete r than a s ing le s leeved s t r and and when p laced in

pos i t i on i t g ive s a sma l l e r e c c e n t r i c i t y t ha n a n unbonde d t e ndon . T he r e f o r e ,

where maximum eccent r ic i ty i s des i red , bonded tendons a re le ss e f f ic ien t .

9 A g iven c onc r e t e s e c t ion w i th bon de d t e ndon s w ou ld ha ve a h ighe r l oc a l

u l t im a t e s t r e n g t h t h a n t h e sa m e s e c ti o n u si n g u n b o n d e d t e n d o n s . A p p r o a c h i n g

u l t ima te l oa d , t he s t r a in , a nd he nc e the te ns il e st re s s, in a bon de d t e nd on mu s t

inc re a se w i th the s t r a in in t he a d j a c e n t c onc re t e . A n un bo nd e d t e ndo n c a n s l ip

re la tive to the concre te , so tha t the inc rease in i ts tens ile s t re ss i s m uch smal le r .

A n unbonde d t e ndon ma y c a r r y a l a r ge r f o r c e i n t he s e r v i c e a b i l i t y s t a t e

because of lower f r ic t ion .

9 Gr ou t ing o f duc t s i n t he pa s t ha s no t a lwa ys be e n c a r r ie d ou t w i th t he c a r e it

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16 POST-TENSIONEDCONCRETE FLOORS

de ser ves a nd the r e ha ve be e n c a ses o f t e nd on c o r r o s io n r e su l t ing f r om t r a ppe d

a i r a nd e xce ss wa ter . F r e e z ing o f wa te r t r a p pe d in a duc t m a y c a use bu r s t i ng o f

the duc t an d sp a l l ing of the su r ro un din g con cre te ; th i s i s a poss ib i l i ty wh ere the

f loor is expos ed to icy con di t ion s , such as the u ppe r decks of a ca r pa rk .

T he se p r ob le ms ha ve be e n d i s c ove r ed du r ing de mo l i t i on o f pos t - t e n s ione d

bu i ld ings o r whe n a r e f u r b i shme n t r e q u i r e d som e t e ndons to be c u t . A w a r e ne s s

o f t he po te n t i a l p r ob le m ha s c a use d the i n s t al l er s t o pa y m uc h m or e a t t e n t ion

to g r ou t ing tha n the y d id i n e a r li e r da ys . H o we ve r , t he f a ct r e ma ins t ha t onc e a

duc t ha s be e n g r ou te d the r e is no e a sy wa y o f c he c k ing i ts a de q ua c y .

9 T he u nb ond e d t e ndo ns r e m a in to t a l l y de pe nde n t o n the i n t e g r it y o f t he i r

a nc ho r a g e s ; i f a n a n c ho r a g e su ff er s da m a ge , a c c ide n ta l o r due to c o r r o s ion ,

the n a l l p r e s tr e s s in t he pa r t i c u l a r t e n don is l o s t. A g r ou te d t e n don is bond e d to

the c onc re t e s e c t ion a nd i f i ts a nc ho r a g e ge t s da m a ge d the n the bo nd shou ld be

a b le t o r e t a in t he p r e s t r e s s ing f o r c e be yond the bond l e ng th .

9 I f a n u nbo nd e d s t r a nd is da m a ge d in a c on t inu ous f l oo r t he n the who le s e ri es o f

assoc ia ted spans suf fe r f rom the loss . For a bonded tendon the loss i s conf ined

to the pa r t i c u l a r spa n whe r e t he da m a ge ha s oc c u r r e d . A su ff ic ie n t q ua n t i t y o f

rod re inforcement should be provided in unb ond ed f loors to conta in such dam age .

A s imi la r s i tua t io n m ay a r i se if t en do ns a re de l ibe ra te ly cu t , fo r ins tance , to

ma ke a ho le . T he a d j a c e n t c on t inuous spa ns wou ld in t h i s c a se r e q u i r e

p r o p p i n g if t h e t e n d o n s w e re n o t b o n d e d .

9 In case of da m age to a s t r a nd in a com ple ted f loor , fo r shor t lengths i t i s

p o s s i b l e t o w i t h d r a w t h e u n b o n d e d s t r a n d f r o m i t s e x t r u d e d s h e a t h i n g a n d

i n s er t a n o t h e r , p o s si b ly a c o m p a c t s t r a n d i n s te a d o f t h e n o r m a l - - t h e c o m p a c t

s t ran d h as a s l igh t ly sm al le r d iam ete r . R eplacem ent , o f course , is no t poss ib le i f

t he t e ndon i s bonde d .

9 I n c a se o f f ir e, the bo nde d t e ndo n ha s t he p r o t e c t ion o f t he a dd i t i on a l c on c r e t e

c ove r , be c a use the duc t is m uc h l a r ge r t ha n the s t r a nd a r e a a nd s t re s s ing t e nds

to pu l l t he t e nd on in to t he m a ss o f t he c onc r e t e , a wa y f r om the su r fa c e . T he

duc t i t se lf ac ts a s a he a t s ink to a smal l ex ten t .

A n un bo nd e d t e ndo n ha s l es s c onc r e t e c ove r bu t t he she a th ing ha s a l imi t e d

insu la t ing va lue .

9 In the case of ove r loadin g , the d is t r ibu t ion of c racks i s s imi la r in r e inforced an d

bonde d pos t - t e ns ione d f loo r s . W i th unbonde d t e ndons , t he c r a c ks a r e w ide r

a nd f u r the r a pa r t , un l e s s bonde d r od r e in f o r c e me n t ha s be e n p r ov ide d to

d is t r ibu te the c racks .

1 .9 .3 Mate r i a l, equ ipm ent and si te ac t iv i t ies

9 B o nde d t e ndo ns ne e d e x t r a ha r dw a r e in the f o r m o f me ta l o r p l a s t ic duc t s w i th

the c onse q ue n t e x t r a ha nd l ing a nd s i t e s to r a ge r e q u i r e me n t .

9 I n a bond e d sys t e m, duc t s a r e u sua l ly a s se mble d in pos i t i on f ir st a nd the n the

s t r a nds a r e t h r e a de d th r ough . S ubse q ue n t ly , t he duc t s ne e d g r ou t ing . I n a n

un bo nd e d sys t em the t e ndons , c ons i s t i ng o f i nd iv idua l s t r a nds , a re a s se mble d

d i r e c t ly on the f o r mwor k .

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THE BASIC PRINCIPLES 17

9 Bo nded t endo ns be ing m ul t i s t rand , the i r s tres sing jacks a re too heavy to be

hand led manual ly , so c rane t ime i s requ i red dur ing the s t res s ing opera t ion .

Th e g ro u t i n g equ i p men t , r equ i r ed fo r t h e b o n d ed s y s t em, may a l s o n eed

cranage to move i t f rom one pos i t ion to the nex t .

9 In the bo nde d sys tem a ll s t rands con ta ined in a duc t a re s t res sed in one

opera t ion , ins t ead o f one a t a time in the case o f unb ond ed t endons . Therefo re ,

fewer opera t ions a re needed to s t res s bonded t endons .

1 .10 S t ress ing s tages

Con cre te , tho ugh wea k in t ens ion , i s capab le o f wi ths tand ing a t ensi le s t res s o f the

ord er of 10% of i ts com press ive s t reng th . In reinforc ed conc rete the tens i le

s t rength cannot be ut i l ized, because the high s t rains on the tens ion face cause the

concre te to c rack . On ce c racked , the concre te i s as sum ed incapa b le o f car ry ing

any t ens ion . In po s t - t ens ion ing , the s t ra ins on the t ens ion face o f concre te a re low

and the tens i le s t ren gth of conc rete i s u t i lized in calculat ions for the serviceabi l i ty

state.

I t is es sen ti a l to avo id m icrocracks in concre te which is to be pos t - t ens ioned ;

they tend to dev elop du ring the set t ing of conc rete an d w i thin the f irs t few days of

cas t ing thereaf ter , before i t has gained suff icient s t rength to wi ths tand the

tempera tu re and sh r inkage s t ra ins . Pres t ress , by induc ing compress ion in

concre te , reduces the p ro bab i l i ty o f sh r inkage c racks , and the norm al p rac t i ce in

pos t - t ens ion ing i s to a pp ly some of the p res t ress a t as ear ly an age as p rac t i cab le .

Too ear ly a s t ress ing, before the concrete has at ta ined suff icient s t rength , can

cause c racks th rou gh overs t ress ing . The ac tua l t iming o f the s tres s ing op era t ion

is o f ten a compro mise be tween the two conf l ic t ing cons idera t ion s - -des i rab i l i ty o f

s t ress ing ear ly and the need to reduce the poss ibil i ty o f dam age from early s t ress ing.

In general , f loors are s t ressed wi thin three or four days of cas t ing. At a cube

st reng th o f 25 to 30 N /m m 2 (fr 3000 to 3500 ps i) it is poss ible to app ly full

p res t ress . Whe re ful l p res t ress cann o t be app l i ed in one opera t ion , a p ropo r t ion ,

say 50% , is app l i ed w i th in th ree days . At the f ir s t opp or tu n i ty thereaf te r , usua l ly

within 14 day s of cas t ing concre te , the pres t ress i s increase d to i t s fu ll des ign value.

In the f i rs t s tage o f the tw o-s tage s t ress ing opera t ion , e i ther a ll t endons m ay be

s t ressed to hal f the des ign force, or ha l f of the ten don s m ay b e fully st ressed. The

la t t e r requ i res the concre te to be s t rong en ough fo r the bear ing s t res ses o f the

ancho rages , in which case a ll ancho rages can be s tres sed , and the on ly advan tage

of two -s tage s t ress ing wou ld be tha t o f a s l ight redu ct ion in pres t ress ing losses .

M os t po s t - t ens ion ing ancho rages a re des igned fo r s t res s ing to a 100% force a t a

concre te cube s t reng th o f 25 to 30 N /m m 2 ( fr = 3000 to 3500 ps i) , a l thoug h some

manufac tu rers supp ly a phys ica l ly l a rger anchorage range which can be s t res sed

a t a concre te s t reng th o f abo u t 15 N /m m 2 ( f r 1750 ps i) .

Two-s tage s t ress ing is used wi th some reluctance because i t involves two vis i t s

to each f loor , the p la t fo rm fo r access to anchorages mus t be main ta ined fo r a

l o n g e r p e r i o d an d t h e ten d o n s t o b e b o n d ed r ema i n u n g ro u t ed fo r a lo n g e r p e r i o d

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18 POST-TENSIONEDCONCRETE FLOORS

wi th the consequen t da nge r o f wa ter ge t t ing in to the duc t s . Tw o-s tage s t res s ing ,

however , has a des ign advan tage in tha t some shr inkage and c reep wi l l have

a l ready occur red befo re the second s t age and so the t endons wi l l undergo a

smal ler loss in pres t ress from these causes .

Tw o-s tag e s t ress ing is requ ired whe re a relat ively high level of pres t ress i s

needed , fo r example when a par t i cu la r ly s l im concre te sec t ion i s des i red o r the

app l i ed load is m uch h igher in in tens i ty th an the se lf -weigh t o f the m em ber , such

as in t rans fer beams . F loors con ta in ing l igh tweigh t concre te a re a lmos t a lways

s t ressed in two s tages .

I t i s des i rable to s t ress al l tendons ful ly in one operat ion. BS 8110 speci f ies a

m in im um concre te s t reng th o f 25 N /m m 2 a t s t ress ing.

1 .1 1 C o n s t r u c t io n t o l e r a n c e s

The permiss ib le cons t ruc t ion to le rances fo r dev ia t ion in member s i ze and in

pos i t ion ing o f the s tee l a re norm al ly speci fi ed in the na t iona l s t anda rds . These

to le rances a re no t rep roduced here .

The to le rances spec i fi ed fo r pos t - t ens ioned f loors , wi th rega rd to the d ep th o f

the m em ber and placing of s teel are general ly s imi lar to those for reinforced concrete .

Ul t imate s t reng th ca lcu la t ions fo r pos t - t ens ioned and re in fo rced concre te

fol low ident ical procedures for shear and for f lexure. Therefore, a g iven

d imens iona l var i a t ion can be expec ted to have a s imi la r e f fec t on the u l t imate

s t reng th in bo th sys tems .

1 .1 2 F i r e r e s i s t a n c e

Pos t - t en s ioned concre te , be ing l a rge ly f ree o f micro -crack s , p rov ides be t t e r

p ro tec t ion to s t ee l than re in fo rced concre te . F ro m the durab i l i ty po in t o f v iew,

therefo re , the con cre te covers norm al ly speci fi ed fo r re in fo rced concre te a re qu i t e

adequate fo r pos t - t ens ioned concre te .

In case o f f ire, however , a t h igh t em pera tu res pos t - t ens ion ing t endo ns lose a

m uch g re a te r p rop or t io n o f the i r s t reng th than re in fo rcement , and , fo r th is

reason , mo s t o f the na t iona l s t anda rds spec ify an increased cover in pos t - tens ioned

co n c re t e . Th e i n c rea s e i n co v e r d ep en d s o n t h e f l o o r co n f i g u ra t i o n - -b eam o r

s lab , so l id o r r ibbed , s imply-suppor ted o r con t inuous . The requ i red add i t iona l

cover var ies f rom n i l fo r a ha l f -hour res i s tance to a m axim um of 20 m m for a

s imply suppor ted beam des igned to res i s t two-hours f i re .

1 .1 3 H o l e s t h r o u g h c o m p l e t e d f lo o r s

Holes , no t p lanned a t the des ign s t age , may be requ i red th rough a comple ted

floor by a tenant for addi t ional services , for ins tal l ing new l i f t s , s ta i rcases or a i r

cond i t ion ing duc t s .

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THE BASIC PRINCIPLES 19

P o s t - t en s i o n i n g i s o f t en s a i d t o l i m i t t h e f lex i b il it y o f a b u i l d i n g b y r e s t r i c t i n g

t h e p o s s i b il i t y o f m ak i n g h o l e s t h r o u g h a fl o o r a ft e r i ts co m p l e t i o n . F o r s ma l l

se rv i ce ho l es , a pos t - t ens ioned f l oo r , i n f ac t , o f f e r s a g rea t e r f l ex ib i l i t y t han a

r e i n f o r ced co n c r e t e s l ab , w h e t h e r i t i s i n a s t e e l f r ame o r a co n c r e t e f r ame . T h e

t e n d o n s a r e n o r m a l l y s p a c e d m u c h f a r t h e r a p a r t t h a n t h e ro d r e i n f o r c e m e n t i s i n

a r e i n f o r ced co n c r e t e f l o o r , an d l a r g e r h o l e s c an , t h e r e f o r e , b e mad e i n a

p o s t - t en s i o n ed f l o o r w i t h o u t cu t t i n g an y o f t h e s te e l.

C a r e , o f co u r s e , n eed s t o b e ex e rc i s ed t o en s u r e t h a t a t en d o n i s n o t d am ag e d .

C o n c e n t r a t i o n o f t e n d o n s u s u a l ly o c c u r s a l o n g th e b e a m l in e s w h i c h s h o u l d n o t

b e t o o d i ff icu lt t o i d en ti f y o n s it e o r f r o m t h e co n s t r u c t i o n d r aw i n g s . S o m e t i me s ,

t en d o n p o s i t i o n s a r e m ar k ed o n t h e f l o o r so ff it . I n an y ca s e , a t en d o n can b e ea s il y

l o ca t ed o n s i te w i t h a co v e r m e t e r o r a m e t a l d e t ec t o r ; i n f ac t th i s s h o u l d b e d o n e

t o c o n f i r m t h e t e n d o n p o s i t i o n e v e n w h e n i t i s k n o w n .

La r g e h o l e s a r e d i f f i cu l t t o mak e i n an y f l o o r co n s t r u c t i o n , p o s t - t en s i o n ed o r

n o t . I t i s a l m o s t c e r t a i n t h a t a la r g e h o l e w i ll req u i r e a n ew s y s t em o f b eam s t o

s u p p o r t t h e cu t ed g es , an d t h a t t h e b eams w i l l h av e t o b e s u p p o r t ed d i r ec t l y o n

t h e ex i s ti n g co l u mn s . I f t h e p r o p o s ed h o l e r eq u i r e s o n e o f t h e ex i s t in g b eam s t o b e

cu t t h en t h e p r o b l em b eco mes ev en mo r e co mp l ex . T h i s r eq u i r e s a c a r e f u l d e s i g n

ch eck o n t h e ex i s ti n g s t r u c t u r e i r r e s p ec ti v e o f w h e t h e r t h e fl o o r an d t h e b eam s a r e

o f s t r u c t u r a l s t e e l w o r k , r e in f o r ced co n c r e t e o r p o s t - t en s i o n ed co n c r e t e .

P o s t - t en s i o n i n g s p ec i a l i s t t r a i n ed o p e r a t i v e s a r e n eed ed t o cu t an d s ecu r e t h e

t en d o n s . B o n d ed t en d o n s a r e r e la t i v e l y ea s y t o cu t , b ecau s e b o n d p r ev en t s l o s s o f

p r e s t r e s s . U n b o n d e d t e n d o n s m a y n e e d a d j a c e n t c o n t i n u o u s s p a n s t o b e

p r o p p ed b e f o r e t h e t en d o n s a r e d e - t en s i o n ed , cu t an d r e - s t r e s s ed u s i n g n ew

a n c h o r a g e s .

1 .14 Post - tens ion ing in re fu rb ish men t

P o s t - t e n s i o n i n g h a s b een s u ccess f ul l y u s ed i n th e r e f u r b i s h m en t o f ex i s t in g

r e i n fo r ced co n c r e t e b u i l d i n g s , w h e r e t h e fl o o r s n eed ed s t r en g t h en i n g o r u p g r ad i n g .

T h e l o ad cap a c i t y o f an ex i s t in g r e i n fo r ced o r p o s t - t e n s i o n ed f l o o r c an b e

i n c r ea s ed b y t h e ap p l i c a t i o n o f p r e s t r e s s in t h e ma n n e r s h o w n i n F i g u r e 1 .3 (a ) .

P o c k e t s a r e cu t a t t h e en d o f t h e s l ab f o r an ch o r ag e s , a n d s l o p i n g h o l e s a r e d r il l ed ,

t h r o u g h w h i c h t h e s t r a n d i s t h r e a d e d . A t e a c h p o i n t w h e r e a t e n d o n c h a n g e s

d i r e c t io n , t h e s t r a n d i s s e a t e d o n a m a n d r e l t o a v o i d s t re s s c o n c e n t r a t i o n w h i c h

w o u l d o t h e r w i s e o c c u r. T h e m a n d r e l s a l so p r o v i d e a s im p l e m e a n s o f c o n t r o l l in g

t h e t en d o n eccen t r i c i t y ; t h e t en d o n s a r e l o ca t ed b e l o w t h e s l ab an d , t h e r e f o r e ,

t h ey a r e u t i l iz ed i n a v e r y e ff ic i en t m an n e r . T h i s m e t h o d i n c r ea s e s t h e d ep t h o f t h e

s t r u c t u r a l z o n e o n t h e u n d e r s i d e o f t h e fl o o r ; th e t o p l ev el r em a i n s u n ch a n g ed ,

and , t here fo re , t he s t a i r cases and l i f t s a r e no t a f f ec t ed .

A s i m i l a r a r r a n g e m e n t c a n b e us e d fo r s t r e n g t h e n i n g t h e d o w n s t a n d b e a m s i n a

f l o o r . I n t h i s c a s e , t h e man d r e l s a r e l o ca t ed i n h o l e s d r i l l ed t h r o u g h t h e

d o w n s t a n d w e b o f t h e b e a m . T h e t e n d o n s r e m a i n w i t h i n t h e ex i st i ng d e p t h o f t h e

s t r u c t u r e .

F o r a r i b b ed f l o o r , an a r r an g emen t s i m i l a r t o F i g u r e 1 .3 ( a ) c an b e u s ed , b u t

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20 POST-TENSIONED CONCRETE FLOORS

Existing floor Anchorage

/ \

,

Deflector mandrel

New tendons (external)

(a)

New post-tensionedslab ~ Soft-filled gap

Slip membrane around column

Existing floor

(b)

Figure 1.3

St reng then ing o f a f l oor

w i t h t h e t en d o n s l o ca t ed b e t w e en t h e r i b s . I n t hi s c a se t h e r e is n o i n c r ea s e i n t h e

s t r u c t u r a l d e p t h .

T h e m a i n a d v a n t a g e o f u s i ng p o s t - te n s i o n i n g i n t h e a b o v e m a n n e r i s t h a t m o s t

o f t h e w o r k i s c a r r i ed o u t f r o m t h e o u t s i d e , o r f r o m b e l o w ; t h e f l o o r it s e lf c an

c o n t i n u e t o b e u s e d d u r in g t h e s t re n g t h e n i n g o p e r a t i o n s . T h e m e t h o d a l so u s e s a

m i n i m u m o f ma t e r i a l s a n d v e r y li tt l e w e t t r ad es . I t is b a s i ca l l y ap p l i c ab l e t o s i n g le

s p an s w h e r e acce s s i s av a i l ab l e t o t h e s l ab ed g es . I n co n t i n u o u s s p an s , t h e

t e n d o n s m a y b e l o c a t e d in s p ec i a ll y p r e p a r e d g r o o v e s o v e r th e s u p p o r t s b u t t h i s

w o u l d n eed acce s s t o t h e t o p .

I n t h e a b o v e m e t h o d s , t h e t e n d o n s b e i n g e x t e r n a l a n d u n b o n d e d , s o m e

p r o t e c t i o n m u s t b e p r o v i d e d t o p r e v e n t p h y s i c a l o r f ir e d a m a g e , a n d t o c o n t a i n

the s t r and in case o f a f a i l u re .

W h er e an ex i s t i n g sl ab h a s d e t e r i o r a t ed , i t i s p o s s i b l e t o u s e i t a s a p e r m an en t

s h u t t e r i n g an d ca s t a n ew p o s t - t en s i o n ed f l o o r o n t o p , s ee F i g u r e 1 .3 ( b ) . T h e

p r e s t r e s s c an b e s o d e s i g n ed t h a t t h e n ew s l ab d o es n o t i mp o s e an y l o ad o n t h e o l d

o n e i n s e r vi ce , ex cep t a t t h e s u p p o r t s w h e r e t h e o l d s l ab may b e r eq u i r e d t o ca r r y

t h e s h e a r a t c o l u m n s . T h i s a r r a n g e m e n t is p a r t i c u l a r l y s u i t a bl e f o r u p g r a d i n g o l d

r e i n f o r ced co n c r e t e f i a t s lab s . I t s d i s ad v an t a g e , o f co u r s e , i s t h a t o f t h e i n c r ea s e i n

the s t ruc tu ra l t h i ckness above t he s l ab t op l eve l , wh ich a f f ec t s s t a i r cases , l i f t s and

r a m p s .

1 .15 S om e m isconcep t ions ab ou t p os t - tens ioned f loo rs

P r es t r e s s i n g i s s o me t i mes co n ce i v ed a s ap p l i c a t i o n o f a l a r g e co mp r es s i v e f o rce t o

con cre t e i n i ts g reen s t a t e , sub jec t i ng i t t o t he poss ib i l i t ies o f sud de n f a i l u re an d

b u c k l i n g. F o r m w o r k , i t is a s s u m e d , m u s t b e s t r o n g e n o u g h t o s u s t a in t h e i n i ti a l

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THE BASIC PRINCIPLES

21

pres t ress ing fo rce . Anchorages a re a l so cons idered to be a po ten t i a l source o f

da ng er - - t he y m ay shoo t ou t as h igh ve loc ity missi les in case o f fa ilu re o f a t endo n

o r an an ch o rag e .

While each of the above poss ibi l i t ies exis ts in pres t ress ing in general , they do

no t occur in pos t - t ens ioned f loors .

F o r m w o r k

Moulds a re occas iona l ly used to anchor the p res t ress ing t endons in the p recas t

p re tens ion ing p rocess , where the t endon s a re s t ressed befo re the concre te is cas t ,

the fo rce be ing t rans fer red to concre te by re leas ing the anch or when the concre te

has gained suff icient s t reng th . In th is case the mo ulds m ust be des igned to su s tain

the pres t ress ing force, wi th a safety margin .

In an

in s i tu

post - tens ioned f loor , the s t ress ing is carr ied out

a f t e r

t he concre te

has ga ined sufficient s t rength to safely sus tain the pres t ress ing force. The force is

app l i ed d i rec tly to the concre te ; the fo rm wo rk i s ne ve r requ i red to car ry a ny o f the

pres t ress ing fo rce . Apar t f rom the few ho les needed fo r the t end ons , the des ign o f

fo rmwork i s the same as fo r re in fo rced concre te .

Bu ck l in g u n d er p rest ress

Co ncern has , qu i t e incor rec t ly , been expressed abo u t the poss ib i l ity o f buck l ing

of a pos t - t ens ioned f loor und er the p res t ressing fo rce .

In fac t , buck l ing i s imposs ib le to occur when the t endons a re in con tac t wi th

the concrete , as they are in post - tens ione d f loors . The po ss ibi l ity of buck l ing

would ex i s t on ly a t the t ime o f s t res s ing where the p res t ress was app l i ed th rou gh

s t ra igh t bonded t endons , because a t tha t s t age the t endons would , theore t i ca l ly ,

no t be in con tac t wi th the shea th ing . Once bo nded , the t endo ns p rov ide a l a t e ra l

s u p p o r t t o t h e co n c re t e o p p o s i n g an y t en d en cy t o b u ck l e ; t h e a r r an g emen t

becomes sel f-correct ing.

The s i tuat ion in pract ice never ar ises because:

9 Fo r bu ckl in g to occu r , the s lenderness r at io o f the s lab and the level of pres t ress

need to be much h igher than i s commonly used .

9 Lon g s t ra igh t t e ndon s in s lender mem bers a re used on ly in g rou nd s labs .

Ho w ev e r , g ro u n d s l ab s a l mo s t a l w ay s u s e u n b o n d ed t en d o n s , w h i ch a r e i n

con tac t w i th concre te . E ven i f bon ded t endon s a re used , the s t ress l eve l is too

low, o f the o r der o f 1 .5 N /m m 2 (abou t 200 psi ), t o overco me the se lf weigh t o f

the floor.

9 Fo r f loors con ta in ing unb ond ed t endons , the poss ib i li ty o f buck l ing does no t

ex i s t because the t endons a re a lways in con tac t wi th the concre te .

Sudden fa i lure

Sudd en and b r i t tl e fa i lu re o f a concre te m em ber occurs whe n i t is sub jec ted to

h igh comp ress ive s t res ses ap pro ach ing the l imi t o f i ts s t reng th . In b r idges

com pr i s ing p recas t g i rders , the g i rders a re o f ten shaped and des igned to mak e the

m axim um use o f concre te , which i s heav i ly s t ressed imm edia te ly a f t e r , o r dur ing ,

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22 POST-TENSIONED CONCRETE FLOORS

stressing. The poss ibi l i ty of fa i lure occurs a t this ini t ia l s tage , before the gird er is

pu t in se rv ice ; the rea f te r , se rv ice load s u sua l ly

reduce

the s tresses. Sudden fa i lure

can o ccur d ur in g se rv ice if a sub s tan t ia l a rea of the concre te i s los t f rom the

c o m p r e s s i o n z o n e d u e t o d a m a g e .

In po s t - tens ione d f loors , the leve l o f p res t re ss ing i s genera l ly too low for the

concre te to ap pro ach fa ilu re in com press io n . Th e ave rage s t re ss in a s lab is o f the

o r de r o f 2 N /m m 2 (300 p si ) c om pa r e d w i th 8 N /m m 2 (1200 ps i) o r m or e in a

br idge g i rde r .

T e n d o n f a i l u re

I n c ase o f a l oca l da ma g e to a bon de d t e ndo n , t he g r ou t shou ld be a b l e t o t a ke the

f ul l f o rc e in t he t e ndon , a n d the r e by c on ta in a ny t e nde nc y o f t he t e ndo n to m ove

whic h wo u ld c a use a lo s s o f p r e s tr e s s a wa y f r om the da ma ge . T he m e c ha n ic a l

a nc h o r a ge a s se mbly o f a bond e d t e ndo n m a y in th i s se nse be c ons ide r e d

r e du nda n t . I f a bo nde d t e nd on o r i ts a nc ho r a ge f ai ls , i t is un like ly to be n o t i c e d a t

t he t ime . T h is i s e v ide n t f rom the num be r o f de fe c tive t e ndons in r oa d b r idge s

d i sc ove r e d on ly whe n the y a r e r e fu r b i she d , s t r e ng the ne d o r de mol i she d . U sua l ly ,

the f i rs t ind ica t ion of a prob lem is an inc rease in de f lec t ion .

W i th unb on de d t e ndon s , e xpe r ime n ta l s tud i e s (s e e C h a p te r 13) ha ve show n

tha t t he e x t r ude d p l a s ti c she a th ing a bs o r bs m os t o f t he ene r gy r e le a sed whe n a

s t r a n d i s b r o k e n . T h e m a x i m u m m o v e m e n t o f t h e s tr a n d a t a n e n d is of t h e o r d e r

of 150 mm (6 inches) .

T he e ne r gy be ing s to r e d in t he s t r a nd a n d n o t i n t he a nc ho r a ge , i f a n a nc ho r a g e

fa i ls then i t wil l not f ly out .

In bu i ld ings , the anchorage i s covered wi th a concre te p lug in the s t r e ss ing

poc k e t a nd the c l a dd ing o f t he bu i ld ing . T he pos s ib i li t y o f a ny mis s il es e me r g ing

in case of f a i lu re of a tend on or an an cho rage i s so remo te tha t i t can be cons ide red

non -exis ten t . This has been con f i rmed bo th b y fu l l- s ize te s ts on ac tua l s t ruc tures

a n d u n d e r l a b o r a t o r y c o n d i t i o n s .

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THE BASIC PRINCIPLES 23

Figure 1.4 Hang Seng Bank- Hong Kong

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2 M A T E R IA L S A N D

E Q U I P M E N T

Pos t - t ens ioned f loors use a l l t he mater i a l requ i red in a re in fo rced concre te

f l o o r ~ f o r m w o r k , r o d r e i n f o r c e m e n t a n d c o n c r e t e ~ a n d , a d d i t i o n a l l y , t h e y u s e

high tens i le s teel s t rand and the hardware speci f ic to post - tens ioning.

As a mater i a l , rod re in fo rcement in pos t - t ens ioned f loors is exac t ly the sam e as

tha t in reinforced c oncr ete in every respect . The n orm al h igh tens i le s teel , as used

in rod re in fo rcement , has a y ie ld s t res s o f 460 N /m m 2 (66 000 ps i) and a m odu lus

of e l as ti c ity o f 200 kN /m m 2 (29000 ks i) . I t has a Po i s son ' s ra t io o f 0 .3 a nd a

coeff icient of the rm al exp ans ion o f 12.5 x 10 -6 p er ~ (7 x 10 -6 p e r ~ Th e

st reng th o f h igh tens i le s teel i s affected by the r ise in tem per atu re, dro pp ing from

100% a t 300~ to on ly 5% a t 800~

The t echn o logy fo r the p roduc t ion , c om pac t ion and cur ing o f concre te i s wel l

und ers too d and i s no t d i scussed here . Only the p roper t i es o f concre te w hich a re

impor tan t fo r pos t - t ens ion ing a re cons idered .

N orm al dense concre te , o f 2400 kg /m 3 dens i ty (150 pcf ) , is m ore com m on in

pos t - t ens ion ing . L igh tweigh t concre te , however , has cer t a in advan tages in the

r igh t c i rcums tances . Bo th a re dea l t wi th in separa te sec t ions . The p roper t i es o f

the two concre tes a re qu i t e d i ffe ren t and i t is no t a good p ra c t i c e to use the two

s ide by s ide ; there may be p rob lems f rom d i f fe ren t i a l movement and the

difference in thei r mod ul i o f e las t ic ity , shr inka ge an d creep.

2 . 1 F o r m w o r k

In a pos t - tens ioned f loor, the vert ica l edge board s o f the fo rmw ork nee d to have

ho les d ri ll ed th rou gh fo r the t endo ns to pass a t live anchorages . Du r ing s t res s ing ,

the concre te undergoe s a s ligh t reduc t ion in leng th due to the ax ia l com pon en t o f

t h e pr e st re s s . Th o u g h t r ap p i n g o f fo rmw o rk b e t w een an y d o w n s t an d s is n o t a

ser ious p rob lem , the des ign o f the fo rm wo rk shou ld recogn ize the poss ib il i ty .

D urin g s t ress ing, the post - tens io ned s lab li ft s off the form w ork , so that there i s a

re -d i s t r ibu t ion o f i ts weigh t. Th i s m ay im pose h eav ier loads on par t s o f the

fo rm wo rk th an those due to the weigh t o f the wet concre te . In o ther respec t s , the

form w ork for a post - tens ioned f loor is s imi lar to th at for a reinforced concrete f loor .

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MATERIALS AND EQUIPMENT

25

Ea ch l ive an ch ora ge is set in a recess, or anchorage pocket, in the s lab edge. Th e

pocket i s fo rmed us ing p ropr ie t a ry p las t i c fo rmers supp l i ed by the spec ia l i s t

p res t ress ing hardware supp l i e r . The fo rmers a re removed when the fo rmwork i s

s t r ipped . Expanded po lys ty rene b locks have somet imes been used fo r th i s

purpose bu t , in the au thors ' exper i ence they a re o f t en d i f f i cu l t to remove

af te rwards an d p ieces o f po lys ty rene a re le ft in the anc hora ge pocke t . At t em pts to

remov e these by burn ing causes hea t ing o f the anchora ge and the s t rand a t a

cr it ical point and use of chemicals m ay have a det r ime ntal effect on concrete or s teel.

The poc ke t s a re norm al ly t aper ing in shape to a llow remo val o f the fo rme r , and

are larger than the s ize of the anc hor age in elevat ion to a l low sufficient ro om for

the j ack to be coup led . Th e ac tua l s ize depend s on the anch orag e d im ens ions an d

the c learance req u i red fo r the j ack ; bo th o f these vary be tween the v ar ious

pres t ress ing sys tems . The re levan t da ta can be ob ta ined f rom the par t i cu la r

ma nufac tu re r . The poc ke t dep th i s su ff ic ien t to accom m oda te the ancho rage , the

pro jec t ing s t rand (abou t 30 m m, 1 .25 in ) and the g rease cap , and to p rov ide

adequate cover to the assembly .

Anchorage cas t ings a re t emporar i ly a t t ached to the ver t i ca l edge board . A

s ing le s t rand cas t ing m ay be a t t ach ed by long nai ls pass ing th rou gh ho les a t the

corners o f the anch orag e cas t ing , o r by p rop r ie t a ry mean s . The na i ls ge t cas t in

t h e co n cre t e an d can n o t b e r em o v ed w h en t h e ed g e b o a rd is s t ri p p ed . An ch o rag es

fo r mul t i s t rand t endons a re heav ier and a re suppor ted by bo l t s which pass

thro ug h s leeves of the sam e length as the pocke t . Th e s leeve i tsel f gets concreted

and can no t be remov ed bu t the bo l t is rem oved w hen the edge shu t t e r is s t r ipped .

A re in fo rced concre te f loor does no t undergo any s ign i f i can t long i tud ina l

shor ten ing dur ing the f ir s t few days o f cast ing ; sh r inkage does no t o ccur w hi le the

concre te is wet dur ing cur ing . In con t ras t , a pos t - t ens ioned f loor does shor ten in

length during s t ress ing. The s t rain depends on the average pres t ress level . For a

s l ab wi th a n average s t res s o f 2 .0 N /m m 2 (300 ps i) , t he s t ra in m ay be a bou t

0 .0001 , i .e . a t en -met re bay m ay sho r ten by 1 mm . A ten-met re l eng th o f beam

s tressed to a n average o f 6.0 N /m m 2 (900 ps i) wo uld shor ten by 3 m m (~ inch).

These s t ra ins a re no t l a rge , bu t they ma y jus t t rap the fo rm wo rk be tw een two

ver ti ca l faces o f a dow ns tan d . Rem oval o f such a t rapp ed sof fit shu t t e r m ay need

some fo rce, which can dam age the a r ri ses o f the dow ns tand s .

The di ff icul ty can be avoided by a f i l ler s t r ip in the formwork, which can be

rem oved before st res s ing , o r by incorp ora t ing a s t rip o f comp ress ib le m ater i a l . In

r ibbe d an d waffle floors , rem ova l of the form s is eas ier i f the s ides of the r ibs are

given a gen erou s s lope, say no t less than 10 ~

A pos t - t ens ioned f loor lif ts o f f i ts fo rm wo rk when p res t ress i s app l i ed - -us ua l ly

th ree o r four days a f t e r cas t ing . Sof fit fo rms , therefo re , become red und an t a t th is

s t age and a re norma l ly remov ed ; some props a re re ta ined fo r cons t ruc t ion loads .

Ea r l y r emo v a l o f f o rmw o rk a ll ow s a f a s te r t u rn -a ro u n d , an d i t is n o rma l l y

poss ible to use fewer sets tha n w ould be require d in a s im i lar project in t radi t ion al

re in fo rced concre te . Each fo rmwork se t would , therefo re , be used many more

t imes and i t shou ld be des igned accord ing ly .

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POST-TENSIONEDCONCRETE FLOORS

2 .2 D e n s e c o n c r e t e

Concre te for a pos t - tens ioned f loor i s s imi la r to tha t in a r e inforced concre te

s t r uc tu r e . I n r e in f o r c e d c onc r e t e t he impor t a n t p r ope r t i e s a r e i t s s t r e ng th a nd

dur a b i l i t y ; o the r p r ope r t i e s suc h a s r a t e o f de ve lop m e n t o f s t r e ng th , e l a s ti c

m o d u l u s , s h r i n k a g e a n d c re e p a re o f se c o n d a r y i m p o r t a n c e . F o r p o s t - te n s i o n i n g ,

the de s i ra b l e p r ope r t i e s o f c onc r e te a r e h igh e a r ly s t r e ng th , l ow sh r inka ge a nd

creep , and h igh m od ulu s of e la s tic i ty . These prop er t ie s an d the i r s ign if icance in

pos t - t e ns ion ing a r e d i s c us se d be low .

Air en t ra ined concre te may be a so lu t ion for s t ruc tures subjec t to f r eeze - thaw

c ycles . I n e a c h c a se , t he sho r t - t e r m a nd the l ong - t e r m p r ope r t i e s o f the pa r t i c u l a r

c onc r e t e shou ld be de t e r mine d .

I t shou ld be a pp r e c i a t e d tha t be t t e r q ua l i t y c on t r o l ne e ds t o be e xer c is e d f o r

pos t - t e ns ion e d f loo rs . I t i s im po r t a n t t o a vo id the ne e d fo r r e pa ir s a r i s ing ou t o f

de fec ts such as sur face c raz ing , shr inkage c racks or broken a r r i ses , because ,

un l ike re inforced concre te , the tens ile s t r ength of concre te i s usua l ly re l ied u po n

in se rv iceabi l i ty s ta te ca lcu la t ions for C lass 2 s t ruc tures des igned in accordance

wi th BS 8110. A c razed or c racked concre te or a r epa i red pa tch cannot be re l ied

up on to w i th s t a n d a n y t e ns ion e ve n if i ts a ppe a r a nc e is a c c e p tab le . M e a su r e s

shou ld the r e f o re be t a ke n to m in imize the pos s ib i li t y o f t he c onc r e t e c r a c k ing

af ter the ini t ia l se t .

2.2 . 1 Therm al p rop er t ies

In re inforced concre te , the coe f f ic ien t o f the rmal expans ion i s convenien t ly

assu m ed to equa l tha t for s tee l , 12 .5 x 10 -6 pe r ~ (7 x 10 -6 pe r ~ In fac t, i t s

va lue fo r c onc r e t e va r ie s w i th t he t ype o f a gg r e ga t e , t he p r op o r t i on o f c e me n t

pa s t e , the m o i s tu r e c on te n t , t he a ge o f c onc r e t e , a nd e ve n the t e mpe r a tu r e o f

concre te . Because the the r m al pro per t ie s of p la in con cre te d if fe r f rom those o f

stee l, the coeff ic ient of ex pan sio n of re inforced con cre te is a lso affec ted by th e

re inforcement conten t . Wi th so many va r iab les , i t i s d i f f icu l t to g ive a r ea l i s t ic

guidance on the va lue for the d i f fe ren t concre tes .

Nevi l le (1981) g ives the coe f fic ien t o f the rm al e xpan s ion for nea t ce m ent a s

18 .5 • 10 -6 pe r ~ (10 x 10 -6 pe r ~ tha t for cem ent pas te a s va ry ing be tw een

11 x 10 -6 and 20 x 10 -6 pe r ~ (6 x 10 -6 and 11 x 10 -6 pe r ~ dep end ing on

the sand conten t , and the va lues g iven in Table 2 .1 for 1 :6 concre tes made wi th

dif ferent aggregates .

Table 2 .2 shows the BS 8110 va lues for the coef f ic ien ts o f the rm al ex pan s ion of

typ ic a l r oc k g r oups a nd c onc r e t e s ma de f r om the m.

2.2 .2 Concrete s t rength

C on c r e t e u se d in pos t - t e ns ione d f loo r s ge ne ra l ly ha s a 28 - da y c ube s t r e ng th f t ,

i n t he r a nge 25 to 50 N /m m 2 (c y l inde r s t r e ng th f c '= 3000 to 6000 ps i ), t he

m os t c o m m on be ing 40 N /m m 2 (fc ' = 4500 ps i) . H ighe r s t r e ng ths , we ll ove r

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MATERIALS AND EQUIPMENT 27

Table 2.1 Coefficient of thermal expansion ~ rom Nevil le, 1981)

Aggregate

Air -cured W et -cu red Ai r -c ur ed nd we tted

10-6/~

lO-6/~ 10-6/~ lO-6/~ 10-6/~ lO-6/~

G rav el 13.1 7.3 12.2 6.8 11.7 6.5

G ra nit e 9.5 5.3 8.6 4.8 7.7 4.3

Q ua rtz ite 12.8 7.1 12.2 6.8 11.7 6.5

Do lerite 9.5 5.3 8.5 4.7 7.9 4.4

Sa nd ston e 11.7 6.5 10.1 5.6 8.6 4.8

Lim esto ne 7.4 4.1 6.1 3.4 5.9 3.3

Po rtl an d 7.4 4.1 6.1 3.4 6.5 3.6

Blas t-furna ce 10.6 5.9 9.2 5.1 8.8 4.9

Fo am ed slag 12.1 6.7 9.2 5.1 8.5 4.7

Table 2.2 Coefficient o f thermal expansion (from BS 8110)

Typical coefficient o f expansion x 1O- 6 / 0 C

Aggregate type Aggregate Conc rete

Flint, qua rtzite 11 12

Gran ite, basal t 7 10

Limestone 6 8

1 00 N / m m 2 ( fc ' ~ 1 2 0 0 0 p si ) a r e b e i n g p r o d u c ed f o r s p ec i a li z ed u s e an d t h e t r en d

i n p o s t - t en s i o n i n g i s a l s o t o w ar d s h i g h e r g r ad es .

W i t h h i g h e r s t r en g t h co n c r e t e , i t b eco mes p o s s i b l e t o u s e a s h a l l o w er s ec t i o n ,

w h i ch m ay b e m o r e eco n o m i ca l ev en if t h e co s t o f a u n i t v o l u m e o f co n c r e t e i s

h i g h . H o w ev e r , t h e m o d u l u s o f e l a s t ic i t y d o es n o t i n c r ea s e i n d i r ec t p r o p o r t i o n

w i t h t h e s t r en g t h an d , t h e r e f o r e , a s h a l l o w f l o o r mad e f r o m h i g h s t r en g t h

co n c r e t e d e fl ec ts mo r e t h a n a d eep e r o n e co n t a i n i n g a lo w er g r ad e o f co n c r e t e . I n

p o s t - t en s i o n i n g , d e f l ec t i o n can b e co n t r o l l ed t o a l a r g e ex t en t b y ch o o s i n g a

s u i t ab l e co m b i n a t i o n o f p r e s t r e s s i n g fo r ce an d ecc en t r i c it y . H i g h s t r en g t h

co n c r e t e s ma y , t h e re f o r e , p r o v e mo r e eco n o m i ca l i f p o s t - t en s i o n ed t h an r e in f o rced .

T h e s h a l l o w er s ec t i o n w o u l d h av e a l o w er n a t u r a l v i b r a t i o n f r eq u en cy , p o s s i b l y

c r eep i n g i n t o t h e u n d es i r ab l e p e r cep t i b i l i t y r an g e ; t h i s i s d i s cu s s ed i n C h ap t e r 9 .

F o r p o s t - t en s i o n i n g , co n c r e t e i s d e s i g n ed t o g a i n h i g h ea r l y s t r en g t h b eca u s e i t

p e r m i t s s t r e s s i n g a t an ea r l y ag e , t h e r eb y r ed u c i n g t h e p o s s i b i li t y o f s h r i n k a g e

c r ack s . I t a l s o a l l o w s f o r mw o r k t o b e r e l ea s ed ea r l y f o r u s e e l s ew h e r e . I n

p o s t - t en s i o n e d co n s t r u c t i o n , a m i n i m u m s t r en g t h i s s p ec if i ed a t w h i ch t h e

p r e s t r e s s c an b e ap p l i ed , an d cu b es o r cy l i n d e r s a r e t e s t ed b e f o r e s t r e s s i n g ,

u s u a l l y ab o u t t h r ee d ay s a f t e r c a s t i n g .

T h e r a t e a t w h i ch co n c r e t e g a i n s s t r en g t h d e p en d s o n t h e p r o p e r t i e s o f i ts

c o n s t i t u e n ts a n d t h e m a n n e r o f c u r i n g ; th e m o s t i m p o r t a n t c o n s t i t u e n t i s t h e

cemen t . A s a g en e r a l g u i d a n ce f o r t h e r a t e o f d ev e l o p m en t o f s t r en g t h , B S 8 1 1 0

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POST-TENSIONEDCONCRETE FLOORS

Table 2.3 Strength o f concre te and age

Cube s t reng th N /m m 2 a t various ages

7 day s 28 days 2 mo nths 3 mo nths 6 mon ths 1 year

20.0 30 33.0 35.0 36.0 37.0

28.0 40 44.0 45.5 47.5 50.0

36.0 50 54.0 55.5 57.5 60.0

g ive s t he va lue s show n in T a b le 2 .3 . T he se c ond c o lum n r e p r e se n t s t he nom ina l

grad e of concre te . N o va lue i s g iven for an age of th ree days b ecause the v a r ia t ion

in test resul ts is too wide to be re l iable .

T he 28 - da y s t r e ng th o f c onc r e t e is de t e r m ine d by t e s ts on c ube s o r c y l inde r s

wh ic h a r e c r u she d by a pp ly ing a c om pr e s s ive loa d in one d i r e c t ion on ly . I t is the n

assu m ed th a t the s t r ength appl ie s to s t re sses on one ax is a s we ll a s to b iax ia l and

tr iaxia l s t resses.

C on c r e t e , c om m on ly a s sume d to h a ve no t e ns i le s t r e ng th , in fa c t is c a pa b le o f

sus ta in ing a s ign i f ican t leve l o f tens ion . I t a l so has the p rop er ty tha t i f, un der an

a bn or m a l l oa d ing o f sho r t du r a t ion , mic r o - c r a c ks a pp e a r on i t s t e n s ion fa ce the n

on r e m ova l o f t h is l oa d the c r a c ks he a l ove r a pe r iod , p r ov ide d tha t t he c onc r e t e i s

m a in t a in e d in c ompr e s s ion . T h e p r oc e s s is no t f u lly und e r s too d b u t i t p r o ba b ly i s

due to the f ree l ime in concre te .

T r a d i t i ona l ly t e s t s a r e no t c a r r i e d ou t f o r me a su r ing the t e ns i l e s t r e ng th o f

concre te , pe rhaps because i t i s no t u t i l ized in re inforced concre te des ign . I t i s ,

how ever , o f pa r t ic u la r in te res t in pos t - ten s ion in g . The tens i le s t r en gth i s a

f unc t ion o f t he s t r e ng th o f m or t a r be tw e e n a gg r e ga t e p ie c es a nd i t doe s n o t

de pe n d on the s t r e ng th o f a gg r e ga t e t o t he s a m e e x te n t a s the c om pr e s s ive

s t r e ng th o f c onc r e t e doe s . I n n o r m a l we igh t c onc r e te s , t he a gg r e ga t e i s ge ne ra l ly

s t r onge r t ha n the m or t a r , so t ha t t e n s ion c r a c ks de ve lop a t t he a gg r e ga t e - m or t a r

interface .

Tes ts show tha t con cre te i s capab le of w i ths ta nd ing tens i le st r esses of the o rde r

of 10% of i t s com press ive s t r eng th ; the re la t ionsh ip , how ever , i s no t l inea r . The

r a t e o f ga in o f t e ns i le s t r e ng th w i th a ge s lows dow n f a s te r t ha n tha t f o r

c omp r e s s ive s t r e ng th , w i th t he r e su l t t ha t t he r a t i o o f t e ns i le - to - c ompr e s s ive

s t reng th reduces w i th age . The tens ile s t r eng th a lm ost r eaches i ts peak a t an age of

a b o u t o n e m o n t h .

A t t e m p t s t o m e a su r e t e ns il e s tr e ng th d i r e ct ly ha ve p r ove d un r e l ia b l e . A mo r e

sa t i s f a c to r y me thod i s t o a pp ly two l i ne l oa ds t o a c onc r e t e c y l inde r a long two

d ia me t r i c a l ly oppos i t e s ide s ; a na ly s is i nd i c a te s t h a t unde r t h is l oa d in g a s t a t e o f

a lmos t un i f o r m t e ns ion e x i s t s i n t he c y l inde r . A no the r me thod i s t o t e s t a

4 in x 4 in x 12 in (100 x 100 x 300 m m ) p la in concre te beam to bend ing f rac ture ;

the tens i le s t r ength thus de te rmined i s te rmed the

m o d u l u s o f r u p tu r e .

I t var ies

be tw een 1 .75 and 2 .25 t imes the d i rec t tens i le s t r ength and a va lue of 2 .0 can be

assu m ed for th is f ac tor . I t sho uld be no ted tha t the d i rec t tensi le s t r eng th an d the

m odu lus o f r up tu r e a r e no t i n t e r c ha nge a b le ; t he f o r me r a pp l i e s to d i r e c t te n s ion

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MATERIALS AND EQUIPMENT

29

s i tuat ions , such as the pr incip al tens i le s t ress , and the la t ter to f lexure, such as the

c r a c ki n g m o m e n t .

AC I 318 spec if ies a m odulu s o f rup tu re va lue o f 0 .7( fr ~ fo r nor m al weigh t

concre te . BS 8110 does no t g ive any re la t ionsh ip be tw een the mo dulus o f rup tu re

an d the cu be s t reng th . I t a l lows a pr in cipal tens ile s t ress of 0.24(fCu)~ which

includes a par t ia l safety factor , presumably 1 .25.

The t ens i l e s t reng th can be improved by add ing g lass , po lypropy lene o r s t ee l

f ibre . Glass f ibre i s a t tacked by the alkal i in concrete . Special mater ial has been

developed w hich is mo re res i s t an t than nor m al g lass, bu t the s t reng th o f concre te

containing glass f ibre tends to fal l wi th t ime. Short length , f ine polypropylene

fibre i s c laime d to s ignif icantly reduce ear ly plas t ic mo ve m en t of con crete; the

s t reng th o f m atu re conc rete i s not affected. Steel fibre i s m ore sui table for use in

pos t - t ens ioned f loors ; a 2% con ten t appears to increase the t ens i l e s t reng th by

ab o u t 5 0 %, t h o u g h i t may n o t b e eco n o mi ca l .

In a real s t ructur e, the s t resses are almo st a lways biaxial , an d conc rete s t reng th

in one di rec t ion is affected by the s t resses in the othe r d i rect ion. Tasuj i , Slate and

Ni l son (1978) have g iven the ra t io o f b iax ia l to un iax ia l s t reng th sh own in F igure

2.1. The effect of t ransv erse s t ress on tens ile s t reng th is a t presen t ign ored in the

des ign o f pos t - t ens ioned concre te .

2 .2. 3 E las t i c p rop e r t i es

Poisson ' s ra t io may be needed in ca lcu la t ions fo r unusua l ly complex s t ruc tu res

on ly. I t s va lue fo r concre te var i es be tween a ppro x im ate ly 0 .11 an d 0 .21 , h igher

concre te s t reng ths have the lower va lue . A va lue o f 0 .20 is comm only used fo r al l

g rades in pos t - t ens ion ing .

The m od ulus of e las t ic i ty , a lso cal led Young's m odulus, of conc rete (Er i s the

mo s t imp or ta n t e l ast ic p rope r ty requ i red in ca lcu la t ions . I t is a measu re o f the

shor t - t e rm s t ra in p roduced in concre te by an app l i ed s t res s . I t i s requ i red fo r

ca lcu la t ing e las ti c beha v iour o f a s t ruc tu re , such as def l ec tion , and fo r es t ima t ing

losses of pres t ress . C onc rete wi th a low m odu lus of e las t ic ity has a larger s t rain for

a g iven st res s ; a f loor con ta in ing such concre te has a l a rger def l ec tion and m ore o f

i ts p res t ress ing fo rce is los t th roug h e last ic shor ten ing , and o f course th rou gh

shr inkage an d c reep defo rm at ions , o f concre te .

The va lue o f m odulu s o f e l as ti c ity is de te rmine d by co mp ress ion t est s and the

same va lue i s as sum ed to app ly in t ens ion . M odu lus o f e l as ti c ity depends on a

n u m b er o f f ac to r s , s uch a s qu a l it y o f ag g reg a t e an d cemen t , ad mi x t u re s , m an n e r

of cu r ing an d the age o f concre te . Br i t ish and Eu rop ean code gu ide l ines have

t rad i t iona l ly p roposed a cube roo t re l a t ionsh ip be tween Er and the 28-day

concrete s t rength (Nevi l le , 1981), but in BS 8110 a s imple relat ionship was

adop ted fo r genera l use , g iven in Equat ion (2 .1 ) .

E c = K o + 0 .2fcu k N /m m 2

(2.1)

where K o = a con s tan t rep resen t ing aggrega te qua l i ty

f~ = the 28-day cube s t reng th in N/m m 2

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30

POST- TENSIONED CONCRETE FLOORS

~.__.L2 1.2 ~ .~ .-

feu

1 .

I

I

i

0 .8 , , ,

0 .6

0 .4

0.2

- 0 . 2 t o 1 2

- 0 . 2

I i

i i

i i

i i

0 . 4 0 . 6 0 . 8

1.0

a l / f e u

F i g u r e

2.1

Biax ia l s t rength o f concre te

The value of Ko is c losely related to the m odu lus of e las t ic i ty of the ag greg ate . I t

v a ri e s b e tw een 14 an d 2 6 k N / m m 2 fo r t he U K ag g reg a te s an d is t ak en a s

2 0 k N / m m 2 fo r n o rm a l w e i g h t co n c re te .

In or der to assess pres t ress loss , the m odu lus of e las t ic ity value is req uired at an

age other than 28 days . Equat ion (2 .2) g ives the BS 8110 relat ionship .

Eet

" - -

Er + 0 .6 fr162 kN /m m 2

(2.2)

wh ere Ect = m odu lus of e las t ic ity at the des i red age

Y e t =

cu b e s t r en g th a t t h a t ag e in N / m m 2

Based on the s t reng ths show n in Tab le 2 .3 and on the above equa t ion s , m odul i o f

elas t ic i ty are tabulated in Table 2 .4 for d i fferent concrete grades at var ious ages

w i th K o = 2 0 k N/ m m 2. I t mu s t b e r emem b ered t h a t t h e p a r t i cu l a r co n c re te b e i n g

used may no t deve lop the s t reng ths a t the ra t e as sumed in Tab le 2 .3 .

I t shou ld be no ted tha t BS 8110 does no t perm i t an increase in s t reng th be yon d

28 days in sat isfying l imi t s ta te requ irem ents . W her e calcula t ions o f def lect ion or

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MATERIALS AND EQUIPMENT 31

Table 2.4 M odu lus o f e last ic i ty o f normal weigh t concrete (kN /m m 2) (from BS 8110)

Grade 7 day s 28 day s 2 mo nths 3 months 6 mo nths 1 year

30 20.8 26.0 27.6 28.6 29.1 29.6

40 23.0 28.0 29.7 30.3 31.2 32.2

50 25.0 9 30.0 31.4 32.0 32.7 33.6

de f o r m a t ion a r e t o be m a de , s t r e ng th a t h ighe r a ge ma y be u se d . I n c a l c u l a t ing E r

f rom E qu a t io ns (2.1) and (2 .2) , the ac tua l va lue o f Ko for the pa r t ic u la r aggreg a te

shou ld be u se d . F o r un kn ow n a gg r e ga t e s a r a nge o f va lue s va r y ing f rom 14 to

26 kN /m m 2 shou ld be u se d ; t h is is e q u iva l e n t t o a va r i a t i on o f _ 6 kN /m m 2 in t he

28-day va lue of Er for a l l g rades of concre te .

A C I 318 r e c om m e nds a sq ua r e r oo t r e l a t ionsh ip be twe e n E c a nd f ' c .

Ect = the m od ulu s of e la s t ic i ty a t the des i red age

= 57000brr ~ in psi un its

= 4700(f~'t)~ i n N / m m 2 u n i ts

( 2 . 3 )

where fc't = cy l inder s t r eng th a t the age .

I t is gene ra l ly ac cepte d th at the ac tu al value of E c m ay vary by ___2 0 % f ro m t h a t

g ive n by e q ua t ions ( 2.3 ) de pe nd ing on the m odu lus o f e l a st ic i ty o f t he a gg r e ga te .

A c ompa r i son be twe e n E q ua t ions ( 2 .1 ) a nd ( 2 .3 ) , a s suming a r a t i o be twe e n

cyl inder and cube s t r en gths of 0 .8 , is show n in F igu re 2 .2 .

2.2.4 Shrinkage

W ater i s needed in concre te for the chem ica l r eac t ions to take p lace for the in i t ia l

s e t ti ng o f t he m or t a r a nd de ve lop m e n t o f s t re ng th . W i th on ly su ff ic ie n t wa te r f o r

th i s pu r pose , t he c onc r e t e i s t oo d r y f o r p r ope r p l a c ing a nd c ompa c t ion ;

the r ef o r e, t he a c tua l q u a n t i t y o f wa te r i n a mix is m uc h m or e th a n tha t r e q u i r e d

so le ly f o r the c he m ic a l r e a ct ion . T he e xce ss wa te r g r a dua l ly m ig r a t e s t h r o ugh the

pores an d e vap ora te s f rom the sur face . This loss of wa te r in tu rn causes

s h r i n k a g e - - a r e duc t ion in t he vo lum e o f t he ha r de n e d c onc r e te . C onve r se ly , if t he

mo i s tu r e c on te n t o f ha r de ne d c onc r e te is i nc re a se d the n i t e xpa nds . I n c o r p o r a t io n

of som e p las t ic ize r s and poly prop ylen e f ib res i s c la imed to reduce the ea r ly

sh r inka ge o f c onc re t e .

Shr in kag e i s genera l ly expressed as a s t r a in . The to ta l ex ten t o f shr ink age a nd

i ts ra t e de pe n d on the a m ou n t o f wa te r i n i ti a ll y p re se n t i n t he mix , on the a m b ie n t

humid i ty a nd on the s e c t ion de p th . C e r t a in a gg r e ga t e s a r e a l so a f f e c t e d by

humid i ty a nd c onc r e t e s c on ta in ing a gg r e ga t e s w i th h igh sh r inka ge p r ope r t i e s

ha ve a c o r r e spond ing ly h ighe r sh r inka ge .

I f the con cre te sur face is sea led a t an ea r ly age , such as dur in g cu r ing o r by

app l ica t ion of chemica ls to g ive the sur face any des i red p rop er ty or by a sc reed ,

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32 POST-TENSIONED CONCRETE FLOO RS

40

30

20

. . _ . - - - . - . . ' 7 . . - . .' : "

... 9

_ _

L ' ' ' ' '

, o , !

20

I I I I I 1 I 1 I I I I I

30 40

Concrete strength fcu (N/mm2)

50

Figure 2.2 Var ia t ion o f Ec wi th 28-day s t rength

then the ra te of shr ink age i s r educed . An a rb i t r a r y p e r iod o f 30 yea r s is , how ever ,

normal ly a ssumed to e lapse be fore concre te r eaches a s tab le s ta te .

S e a sona l v a r i a t i on o f hum id i ty ha s a c o r r e s pon d ing e ffe ct on c onc r e t e i f i t is

e xpose d to t he e l e me n t s a nd i t shows the s e a sona l e f f e c t supe r impose d on the

no r m a l l ong - t e r m sh r inka ge . I n t he U K c l ima te , the s e a sona l sh r inka ge va r i a t i o n

in pa r ti c u l a r ly e xpose d loc a t ions c a n be as muc h a s 40% o f t he l ong - t e r m

sh r inka ge s t r ain .

F o r no r m a l i nd oo r e xposu r e o f p l a in c onc re t e , a n u l t ima te sh r inka g e s t r a in o f

300 • 10 -6 i s an accep ted f igure for mixes con ta in i ng a bo ut 190 l i tr e s of w a te r

pe r c ub ic me t r e o f c onc r e t e ( 19% by vo lume ) . F o r ou tdo o r e xpo su r e a va lue o f

100 • 10 -6 i s genera l ly taken . Wh ere concre te i s kn ow n to have a d i f fe ren t wa te r

c o n t e n t , s h r in k a g e m a y b e r e g a rd e d a s p r o p o r t i o n a l t o w a t e r c o n t e n t w i th i n t h e

range 150 to 230 l i t r e s /m 3 (15 to 2 3% by v olum e) .

F o r a mor e a c c u r a t e a s se s sme n t , F igu r e 2 .3 shows the B S 8110 r e l a t i onsh ip

be twee n 30-yea r shr inkag e , hum idi ty a nd the e ffec tive th ickness of concre te . Th e

f igu r e r e la t es t o p l a in c onc r e te o f no r m a l w or ka b i l i t y ma d e w i thou t wa te r

r e duc ing a dm ix tu r e s , w i th a wa te r c o n te n t o f 190 l it re s pe r c ub ic me t r e ( 19% by

vo lume ) . I n the U K , r e l at ive hum id i t i e s o f 45% a nd 85% a r e c ons ide r e d su i t a b l e

f o r i ndoo r a nd ou tdoo r e xposu r e r e spe c t ive ly . T he e xa mple , shown do t t e d ,

ind i c at e s t ha t a t 60% hum id i ty a 300 mm th i c kne s s o f p l a in c onc r e te i s e xpe c t e d

to have a to ta l shr in kage s t r a in o f 325 x 10 -6 . In o th e r count r ie s , o r wh ere h igh

sh r inka ge a gg r e ga t e s a r e t o be u se d , sh r inka ge c ha r a c t er i s ti c s shou ld be ob ta ine d

f rom the con cre te supp l ie r if kn ow n a t the des ign s tage .

Fo r no n- re c tan gu la r sec t ions , the e ffec tive th ickness m ay be taken as twice the

ra t io of vo lu m e to exposed sur face a rea and where o n ly one sur face is expo sed for

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~VIATERIALS AND EQUIPMENT 33

->/

150 300 600

Thickness (mm)

20 40 60 80 100

% am bient humidity

Figure

2.3 Ultimate shrinkage of plain c oncrete

m ois tu r e e va po r a t ion , suc h a s i n g r ou nd s l abs la id on wa te r p r o o f me mb r a ne s ,

the e f fec t ive th ickness should be doubled .

Shr in kage s t r a in i s a l so a ffected by the re inforcemen t co nten t o f concre te . This

i s d iscussed in the ch apte r d ea l ing w i th pres t re ss ing losses where i t is o f m ore

imme d ia t e u se .

A C I 318 c ombine s t he sh r inka ge s t r a in s w i th c r e e p a nd i t s p r ov i s ions a r e

d iscussed in the next sec t ion .

2 .2 .5 C reep o f conc re te

Creep

i s the grad ua l chang e in length of concre te un der a sus ta ined load . I t is a

long t e r m ph e no m e no n , l ike sh r inka ge ; i t is q u it e r a p id i f t he c onc r e t e i s l oa de d a t

an ea r ly age an d i t s ra te i s neg l ig ib le a fte r ab ou t 30 yea r s . I t is pa r t ly r ecov erab le ;

i f t he l oa d ing is r e mov e d the n a f te r one ye a r t he r e c ove r y m a y be o f the o r de r o f

the s t r a in c o r r e spon d ing to a s t re s s o f 30% o f t he a c tua l s t r e ss r e duc t ion . F o r t he

U K c ond i t i ons , f o r the c onc r e te s no r ma l ly u se d in fl oo r s, it c a n be a s sum e d th a t

40% , 60% a nd 80% o f t he fi na l c r ee p de ve lops du r ing the fi rs t m on th , 6 mo n ths

a nd 30 m on th s . F o r h ighe r g r a de s o f c onc r e t e , s ay a bove 80 N /m m 2 ( f' c = 9000

psi) , c reep occurs a t an ear l ier age .

The f ina l c reep i s dependent on the same fac tor s tha t a f fec t shr inkage and ,

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34 POST-TENSIONED CONCRETE FLOORS

ad d i t i o n a l l y , o n t h e av e r ag e s t r e s s . T h e s t r e s s - t o - c r eep r e l a t i o n s h i p i s n ea r l y

l i n ea r f or s tr e s s l eve l s u p t o o n e - t h i r d o f t h e cu b e s t r en g t h . I n t h e U K , r e l a t i v e

h u m i d i t i e s o f 4 5 % a n d 8 5 % a r e c o n s id e r e d s u i t a b l e fo r i n d o o r a n d e x p o s u r e

respec t ive ly .

C r eep i s meas u r ed a s a s t r a i n p e r u n i t s t r e s s an d t h e c r eep d e f o r ma t i o n i n a

memb er i s g i v en b y t h e f o l l o w i n g l i n ea r r e l a t i o n s h i p .

6~ = t o t a l c r eep d e f o r m a t i o n

= e~ .a .L (2.4)

where e r = c reep s t r a in per un i t s t r ess

a = s t ress

L = l e n g t h o f m e m b e r

Al t e rna t i ve ly , t he e f fec t o f c reep c an be def ine d i n t e rm s o f a f ac to r , ca l l ed

creep

coe f f i c i en t ,

w h i ch i s u s ed t o mo d i f y t h e mo d u l u s o f e l a s t ic i t y o f co n c r e t e . U s i n g

t h i s ap p r o ach , t h e c r eep d e f o r ma t i o n i n a memb er i s "

6r = Cr162 (2.5)

where Cr = c reep coef f i c i en t

T h i s a p p r o a ch i s p r e f e r r ed b ecau s e , o n ce t h e v a l u e o f E c i s mo d i f i ed t o E J C ~ ,

ca l cu l a t i o n o f a ll d e f o r m a t i o n s i n c l u d i n g d e f l ec t io n can f ol l o w t h e n o r m a l

m e t h o d s . T h e v a l u e o f C r v a ri e s a s s h o w n i n F i g u r e 2 .4 w h i ch i s b a s ed o n

r e c o m m e n d a t i o n s g i ve n i n B S 81 10 . T h e e x a m p l e s h o w n d o t t e d i n d i c a t e s t h a t a t

6 0 % h u m i d i t y f o r c o n c re t e l o a d e d a t a n a g e o f 2 8 d a y s a n d o f 3 00 m m t h i c k n e ss

the c reep coef f i c i en t i s 2 .0 .

I t i s i m p o r t a n t t o n o t e t h a t t h e d e f o r m a t i o n g i v en b y E q u a t i o n s ( 2.4) an d ( 2 .5 )

t. b

/

f

150 300 600 . 20

Thickness (mm)

F i g u r e 2 . 4

Creep coef f ic ient for p la in concrete

365

I I ] I I I I I

40 60 80 100

% ambient humidi ty

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Table 2.5 Creep coefficient Cc (from ACI 318)

MATERIALS AND EQUIPMENT

35

Load dura t ion

< 3 months 6 months 12 months > 5 years

Coefficient C c

1.0 1.2 1.4 2.0

i s tha t due to c reep on ly and i t does no t inc lude the immedia te e l as t i c

deform at ion . I f to t a l defo rm at ion inc lud ing the imm edia te e l as ti c defo rm at ion is

des i red the n Cc in Eq ua t ion (2.5) should be replace d w i th (1 + Co).

In ACI 318 , fo r re in fo rced concre te members the mul t ip l i e r i s Cr + 50.p') ,

where p ' i s the compress ion re in fo rcement ra t io . Here Cr depends on ly on the

dura t ion o f load ing as shown in Tab le 2 .5 . For p os t - t ens ion ing the sam e va lue o f

the mul t ip l i e r may be t aken as fo r re in fo rced concre te .

2 . 3 L i g h twe i g h t co n c re te

The t e rm l igh tweigh t concre te app l i es to concre te con ta in ing l igh tweigh t

aggrega tes , to ae ra ted co ncre te an d to no- f ines concre te . On ly the fi rs t m en t ione d

concre te is genera lly used in pos t - t ens ioned f loors .

D ense c oncre te , of 2400 kg/m 3 densi ty (150 pcf) , a t present tends to be the f i rst

choice for a post - tens ioned f loor , because th is i s the t radi t ional mater ial , i t i s

read i ly ava i l ab le a nd i t is cheaper . L igh tweigh t concre te , o f 1800 to 2000 kg /m 3

densi ty (110 to 125 pcf) , i s chosen in preference w here i t i s im po rta nt to save on

the self-weight of a f loor , such as on a s ite wi th po or gro un d c ond i t ions , or on v ery

long spans where the se l f -weigh t would o therwise be the dominan t load .

2.3.1 General

A l igh tweigh t re in fo rced concre te f loor has a l a rger def l ec t ion than a normal

weigh t concre te f loor o f the same sec tion , because o f i ts lower m odulus o f

elas t ic i ty . In long spans where deflect ion is cr i t ical , the l ightweight reinforced

concre te f loor m ay , therefo re , have to be deeper. Th i s l imi ta t ion does n o t a pp ly in

pos t - t ens ion ing , because p res t ress can be des igned to cope wi th the o therwise

marginal ly h igh deflect ion.

Us ing l igh tweigh t concre te , a pos t - t ens ioned so l id f loor can be sha l lower ,

and hence l igh te r than a normal weigh t concre te f loor on two accoun t s - - l es ser

dep th a nd lesser concrete d ensi ty . In a waff le f loor the overal l de pth of the

s t ruc tu re i s de te rm ined by the beam s; i f the s lab pane l is m ade equa l in dep th

to the beam , then i t is deeper than i t need be f rom the s t ruc tu ra l po in t o f v iew.

Using l ightweight concrete , the saving in weight can be s ignif icant and,

therefore, l ightweight concrete can be a very eff ic ient mater ial in such an

app l i ca t ion .

Another reason fo r choos ing l igh tweigh t concre te i s env i ronmenta l . Natu ra l

aggrega tes fo r use in dense concre te a re be ing dep le ted in some areas whereas

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36 POST-TENSIONEDCONCRETE FLOORS

l i gh twe igh t c onc r e t e o f t en m a ke s u se o f a gg r e ga t e s ma n uf a c tu r e d f r om w a s t e

produc ts . There fore , i t i s seen as a more envi ronment- f r iendly a l te rna t ive .

L igh twe igh t a gg r e ga t e s a r e mor e e xpe ns ive tha n na tu r a l g r a ve l , a nd l i gh t -

we igh t conc r e t e ne e ds mor t, ce me n t pa s t e f o r t he s a me s t r e ng th tha n the no r m a l

we ight concre te . These fac tor s con t r ibu te to the h ighe r un i t cos t o f l igh tw e ight

concre te .

L igh twe igh t c onc r e t e s ha ve ma ny a l t e r na t ive a gg r e ga t e s , w i th d i f f e r ing

charac te r i s t ic s an d , con sequ ent ly , the re i s a la rge r range of va r i a t ion in the i r

p r ope r t i e s . H owe v e r , t he p r ope r t i e s f o r a ny p a r t i c u l a r t ype o f l i gh twe igh t

aggrega te can be eas i ly and accura te ly e s tab l i shed by te s t s . I t i s , the re fore ,

im po r t a n t t ha t t he va lue s g ive n her e a re t a ke n f o r ge ne r al gu ida nc e on ly a n d tha t

the cha rac te r i s t ic s a re de te rmined by te s t s on the pa r t icu la r concre te to be used .

A l t e r na t ive ly , pub l i she d da t a f o r t he pa r t i c u l a r c onc r e t e ma y be u se d .

Ch arac te r i s t ic s of l igh twe ight concre te a re of ten de f ined in te rms of those of the

de nse c onc r e t e o f t he s a me s t r e ng th . C om pa r e d w i th de nse c onc r e t e, i t ha s a

s lowe r ra t e o f de ve lop m e n t o f s t r e ng th , a h ighe r sh r inka ge s t r a in a nd a h ighe r

c reep coe f fic ien t . These cha rac te r i s t ic s d i ffe r f rom the pro per t ie s desc r ibed ea r l ie r

a s de s ir a b l e fo r pos t - t e ns ion ing . N e ve r the l e s s , me a su r e s c a n be t a ke n to r e duc e

the undes i rab le e lem ents of the i r in f luence and l igh twe ig ht concre te can be , and

is , successfu l ly used in pos t - tens ioned f loors .

T he ba s i c c ha ra c t e r is t i c o f a ll li gh twe igh t a gg r e ga t es , na tu r a l o r m a n- m a de , i s

the i r h igh poros i ty , which t r ans la te s in to h igh pe rmeabi l i ty . L igh twe ight concre te ,

c o m p a r e d w i t h a n o r m a l w e ig h t c o n cr e te o f th e s a m e m o r t a r p a s te c o n t e n t ,

w o u l d b e p r o n e t o a h i g h e r ra t e o f c a r b o n a t i o n a n d w o u l d b e m o r e p e r m e a b l e t o

c he mic a l s , suc h a s c h lo r ide s . T he se a ppa r e n t d r a wba c ks a r e , howe ve r , pa r t l y

ne ga te d by the h ighe r c e me n t pa s t e c on te n t i n s t r uc tu r a l l i gh twe igh t c onc r e t e s

des igned for a g iven s t r ength . A lso , the e la s tic p rop er t ie s o f m or ta r a re of ten

ne a r e r t o t hose o f l i gh twe igh t a gg r e ga t e t ha n na tu r a l a gg r e ga t e ; the m or t a r i n a

l igh tw e ight concre te , the re fore , r em ains in a c lose r con tac t w i th the aggreg a te . I t

is m uc h l es s p r on e to t he de ve lopm e n t o f m ic r o - c r a c ks a t t he a gg r e ga t e - pa s t e

in t e r f a c e tha n no r ma l de nse c onc r e t e . T he no r ma l p r a c t i c e , howe ve r , i s t o

inc rease the concre te cover to s tee l by ab ou t 10 m m (0 .375 in). Car p a rk f loors ,

whe r e s a l t ma y be u se d in w in t e r , shou ld be g ive n some p r o t e c t ion , a s i nde e d

those in de nse c onc r e t e shou ld be . W i th the se p r e c a u t ions , t he r e i s no r e a son

why l i gh twe igh t c onc r e t e shou ld be l e s s du r a b le , pa r t i c u l a r ly i n t he i n t e r na l

e nv i r onm e n t o f a n e nc lo se d bu i ld ing . I n f a c t, l igh twe igh t c onc r e te ha s be e n u se d

in o i l p l a t f o r ms whe r e t he e nv i r onme n t i s muc h mor e s e ve r e t ha n in a no r ma l

bui ld ing .

L igh tw e igh t a gg r e ga t e ha s a l owe r re s i st a nc e to a b r a s ion be c a use o f t he po r o us

na tu re of i ts surface . I t is adv isab le to use som e form of surface f inish in high traf fic

areas.

Lightwe ight concre te has a be t te r f i r e r e s i s tance than concre te cons is t ing of

no r m a l a gg r e ga t e s, be c a use o f i ts l owe r he a t c ond uc t iv i ty , a nd i t s lowe r r a t e o f

s t r ength loss wi th r i se in tempera ture .

Fo r coe f f ic ien t o f the rm al e xpa ns ion , see Sec t ion 2 .2 .1 .

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MATER IALS AND EQUIPMENT

37

2.3 .2 Product ion and hand l ing

Ligh tweigh t aggrega tes , be ing porous , have a l a rger capac i ty to ho ld mois tu re

than norm al ag grega tes ; consequen t ly , the worka b i l i ty o f l igh tweigh t concre te i s

qui te sensi tive to it s w ater co ntent . I f the mo is ture con tent of the aggr ega te is too

low at the t ime i t i s p laced in the mixer , then i t rapidly absorbs water and the

concre te mix is too h arsh . On the o ther ha nd , i f the mois tu re con ten t o f the

aggrega te i s too h igh then the concre te i s too wet and p rone to segrega t ion .

De term ina t ion o f the op t im um quan t i ty o f wa ter to be used in a concre te is m ade

rath er d i ff icul t by the differences in abso rpt io n capaci t ies and rates of ab sor pt io n

of the var ious aggreg a tes used in p roducing l igh tweigh t concre te . I t is im po r tan t

to t est the charac te r i st i cs o f the par t i cu la r aggrega te to be used , and to accu ra te ly

measure i t s mois tu re con ten t when the concre te i s p roduced .

Care i s a lso needed in handl ing l ightweight concrete on s i te . I t needs more

vibrat ion th an norm al concrete , because it s lower densi ty releases a co rrespondingly

smal le r am oun t o f po ten t i a l energy dur ing com pact ion and because the ligh tweigh t

aggrega te t ends to have a rough su r face . On the o ther h and , the spec if ic g rav i ty o f

l igh tweigh t aggrega te is lower than tha t o f m or ta r pas te and , therefo re , a wet mix

or excessive v ib ra t ion causes segrega t ion th roug h the l igh te r ag grega te f loa t ing

u p w ard s .

The w orkab i l i ty o f a d ry , harsh mix is o f t en impro ved by a i r en t ra inm ent , bu t

th is reduces the concre te s t reng th and m us t be com pensa ted fo r by m ore cem ent .

Workab i l i ty and compact ib i l i ty can be improved wi th su i t ab le p las t i c i zers .

L igh tweigh t concre te has a lower hea t conduct iv i ty and , therefo re , i t has a

s teeper tem pera ture g radien t im me diately after cas t ing. The s i tuat ion is aggra vated

by i t s h igh cement con ten t because more hea t has to be d i s s ipa ted fo r a g iven

vo lume of concre te .

L igh tweigh t concre te i s m ore v u lnerab le to c rack ing a t an ea r ly age than dense

concrete . I t s rate of gain of s t reng th in the f i rs t few days af ter cas t ing is s lower an d

i t has a h igher sh r inka ge s t rain . These factors can set up large tens i le s t rains in the

bo dy o f the l ightweight co ncrete be fore i t has g ained sufficient s t rength . T he

prob lem i s a l l ev ia ted by care in hand l ing , compact ion and cur ing , and by

app l i ca t ion o f some pres tress a t an ear ly age. Tw o-s tage s t res s ing is com m only

used fo r p os t - t ens ioned f loors , usua l ly a t 15 and 25 N /m m 2 cube s t reng ths

( f r an d 3 00 0 p si ) f or t h e co m m o n l y emp l o y ed G ra d e 4 0 co n c re te

(f~' = 4500 psi).

2.3.3 Strength

Ligh tweigh t concre te used in pos t - tens ioned f loors genera l ly has the same 28 -day

s t reng th as tha t o f dense concre te , i. e. i n the range 30 to 50 N/m m 2 ( f , = 3500 to

6000 ps i) ; t he m os t c om m on be ing 40 N /m m 2 (4500 ps i ). H igher s t reng ths a re

poss ib le bu t a re no t in common use a t p resen t in pos t - t ens ioned f loors .

The rat io of i ts tens i le to com press ive s t re ngth is of the sam e ord er as tha t for

norm al dense co ncre te . The e last ic p roper t i es o f l igh tweigh t aggrega te a re o f the

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38 POST-TENSIONEDCONCRETE FLOORS

s ame o rd e r a s t h o s e o f cemen t m o r t a r . Th e re fo re , a b e t t e r ag g reg a t e -m o r t a r

con tac t ex is ts in ligh tweigh t than in norm al co ncre te . L igh tweigh t ag grega tes a re

Weaker in s t reng th an d so tens ion c racks t end to go th roug h the aggrega te

p a r t i c l e s r a t h e r t h an a ro u n d t h em as h ap p en s i n n o rma l co n c re t e .

Ae ra t ion im proves the t ens il e s treng th . T he d i scuss ion on the ra t e o f ga in o f

tens i le s t rength for normal weight concrete also appl ies to l ightweight concrete ,

see Section 2.2.2.

Fo r l igh tweigh t concre te , the mo dulus o f rup tu re i s t aken as 1 .8 t imes the

cyl inder sp l i tt ing s t reng th , i f specif ied; if the spl i t t ing s t reng th is no t specif ied th en

the mod ulus o f rup tu re i s t aken as 80% tha t o f the dense concre te o f equa l s t reng th .

The ra te o f ga in o f s t reng th o f l igh tweigh t c oncre te i s simi la r to th a t fo r norm al

concre te fo r the same cons t i tuen t s in mor ta r pas te . In l igh tweigh t concre te ,

Por t l and cement i s somet imes par t i a l ly rep laced by o ther cements , and sand in

f ine aggrega te may be rep laced by o ther mater i a l , such as g round coarse

aggrega te . Such compos i t e cements , and to a l es ser ex ten t the f ine aggrega te ,

a f fec t the ra t e o f deve lopm ent o f s t reng th .

2.3.4 Elast ic prop ert ies

Poisson ' s ra t io fo r l igh tweigh t conc re te i s as sum ed to be 0 .2 , the sam e as fo r dense

concre te .

Fo r m odu lus o f e l as ti c ity o f l igh tweigh t concre te , BS 8110 recom m ends the

relat ionship given in equat ion (2 .6) .

E c = m od ulus of e las t ic i ty for l ightwe ight conc rete

- - E c n • ( w e / 2 4 0 0 ) 2 k N / m m 2 (2.6)

= 0 .63 • Er for wr of 1900 kg/m 3

where Er = m odulus o f e l as ti c ity fo r norm al dense concre te hav ing the same

s t reng th

wc = dens i ty of l ightweigh t co ncrete in kg/m 3

The In s t i tu t ion o f S t ruc tu ra l Eng ineers an d the Conc re te Soc ie ty (1987) g ive a

s imple express ion, E qu at io n (2.7) , re lat ing the m odu lus of e las t ic ity to the

concre te dens i ty and cube s t reng th .

Er = W c 2 x ( L u ) 0 .5 x 1 0 - 6 (2.7)

AC I 318 recom m ends a more d i rec t re la t ionsh ip fo r the modu lus o f e l as ti c ity , as

given in equ at io n (2 .8) , for values o f wr in the ran ge of 1500 to 2500 k g/m 3 (90 to

155 pcf) , which covers concre te of no rm al densi ty .

Er = (wr a '5 • 43(fc') ~ x 10 -6 (N -m m units) (2.8)

- (we)x's x 33(fc') ~ (psi un its)

Fo r the c om m only used dens i ty o f 1900 kg /m 3, the re l a t ionsh ip in equ a t ion (2.8 )

becomes Er = 3560 x (f ,)o .5 , which is equivalent to 0 .75 t imes the value for

dense concrete .

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MATERIALS AN D EQUIPMENT 39

Eq ua tio ns (2.6), (2.7) and (2.8) yield very different v alues for E c. If a va lue is

ava i l ab le f rom tes t s on the par t i cu la r l igh tweigh t concre te to be used then i t

should be used in preference.

2 .3 .5 Sh r inkage and c reep o f l i gh tw e igh t conc re te

Conc re tes con ta in ing l igh tweigh t aggrega tes have a h igher capac i ty fo r m ois tu re

and so they are general ly suscept ib le to larger in i t ia l and to tal shr inkage

mo v emen t t h an t h o s e co n t a i n i n g n o rma l ag g reg a t e . Th e s h r i n k ag e fo r s o me

concre tes ma y be up to 40 % h igher . Creep o f l igh tweigh t concre te a l so t ends to be

h igher than tha t fo r normal concre te bu t no t to the same ex ten t as sh r inkage .

Large va r i a t ion o f sh r inkage and c reep be tween c oncre tes con ta in ing d i f feren t

f ine and c oarse ag grega tes p rec lude the p resen ta t ion o f m ean ingfu l gu ide lines.

Charac te r i s ti cs fo r the par t i cu la r concre te to be used shou ld be ob ta in ed f rom the

supp l i e r , de te rmined by t es t s o r t aken f rom pub l i shed da ta .

The h igher sh r inkage and c reep o f l igh tweigh t concre te , com pare d w i th those

of dense concre te , cause a l a rger long- te rm def lec t ion . T h i s , howeve r , is more o f a

p rob lem in re in fo rced concre te than in pos t - t ens ioned concre te , because in the

lat ter i t i s poss ible to con trol def lect ion by a judicio us c om bin at ion of pres t ress ing

force and eccentr ici ty .

The high creep also causes a greater loss in pres t ress ing force. This has to be

accepted and, therefore, the l ightweight concrete requires a s l ight ly h igher in i t ia l

p res t ressing fo rce than w ould be requ i red fo r a norm al c oncre te me m ber o f the

same sec t ion and car ry ing the same load . However , the smal l e r se l f -weigh t o f

l ightweight concrete would require a smal ler pres t ress ing force. In pract ice the

increase in the requ i red p res t ress ing fo rce due to h igher losses may be ba lanced by

the saving d ue to lesser weight . The di fference in pres t ress ing force, i f any , wou ld

be smal l and, i f consid ered necessa ry , it should b e poss ible to ta i lor the concrete

sec tion to m ake the bes t use o f the t endons .

2 . 4 Po s t - ten s i o n i n g ten d o n s

In i ts ear ly s tages o f deve lopm ent , p res t ress ing was a t t em pted by t ens ion ing mi ld

s tee l rods , w hich ha d a wo rk ing s tres s o f the o rd er o f on ly 140 N /m m 2 (20000

psi ) . The problem in us ing such a low s t rength s teel in a tendon was that a

s ign i f i can t par t o f the p res t ress was los t as sh r inkage and c reep occur red in

concre te an d re laxa t ion in s tee l. Concre te in compress ion m ay su ffe r a long- te rm

s t ra in o f the o rder o f 400 x 10- 6 f rom shr inka ge a nd c reep , and p res t ress ing s tee l

wou ld a l so un dergo a cor respond ing s t ra in . W i th a mod ulus o f e l ast ic i ty o f

200 kN /m m 2, a s t ra in o f 400 x 10 -6 is equ iva len t to a s t ress loss o f 80 N /m m 2 in

the ten don . In th is ins tance, in mi ld s teel wi th an in i tia l s t ress of 140 N /m m 2 the

loss o f 80 N /m m 2 am oun ts to a 57% reduct ion o f s tress . A p resen t -day h igh

tensile tend on m ay h ave an in i tia l s t ress of 130 0 N /m m 2 (190 ks i ) and a loss of

80 N /m m 2 in i t i s only 6% . The highe r the in i tia l s t ress in the tend on, the less

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40

POST-TENSIONED CONCRETE FLOORS

s ignif icant are the shrinkage an d creep losses . Fo r th is reason pra ct ical appl ica t ion

of pres t ress ing w as held up unt i l the pro du ct io n of h igh -s t reng th s teels .

Ear ly p res t ressing t endo ns cons i s ted o f h igh-s t reng th wi res ancho red s ing ly o r

in mul t ip les in sui table anchorage assembl ies . Wires are s t i l l used in some

pos t - t ens ion ing sys tems bu t the i r mos t common use i s in p re tens ioned p recas t

p roduct s . In pos t - t ens ioned f loors , strand--a rope compr i s ing severa l wi res - - i s

almost universal ly used.

As s t a t ed in Chap ter 1 , the t e rm s t rand can be ra ther confus ingmi t app l i es to

the rope cons i s t ing o f a num be r o f ind iv idua l wi res wou nd toge ther , i t does n o t

m ean the ind iv idua l wi re compr i s ing the rope . A t end on m ay cons i s t o f a s ing le

s t rand o r severa l s t rands housed in the same duc t (o r shea th ing) . Monostrand

t endon s , cons i s ting o f a single s trand , a re used main ly in unbon ded con s t ruc t ion

w h ereas multistrand t endons , fo r use in bonded app l i ca t ions , normal ly con ta in

severa l s t rands . Mul t i s t rand t endons can be e i ther c i rcu la r o r f i a t . The th ree

tendons a re shown in F igure 2 .5 .

The genera l p roper t i es o f t endo n s tee l a re s imi la r to those o f rod re in fo rcement .

I t has a P o i s son ' s ra t io o f 0 .3, and a coeff ic ien t o f therm al e xpans io n o f

12.5 x 10 -6 pe r ~ (7 x 10 -6 per ~

2 .4 . 1 S t r a n d

St rand , com m only in use in pos t - t ens ioned f loors , is m ade f rom seven co ld d raw n

high ca rbo n s teel wires . Six of the wires are spu n to geth er in a hel ical form ar ou nd

a s l ight ly larger seventh s t raight centre wire . The s t rand is then given ei ther a

Grease packing Flat duct

Cement grout

,( Strands

Sheath Strand

Monostrand system Flat duct system

(unbonded) (bonded)

Round duct

Cement grout

(~ Strands

Multistrand system

(bonded)

Figure 2.5

Unbonded and bonded tendons

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MATERIALS AND EQUIPMENT

41

Normal Compact

Figure

2.6

Prestress ing s trand

s t res s re l i ev ing t rea tment o r i t i s run th rough a con t ro l l ed t ens ion and low

tempera tu re hea t t rea tment p rocess which g ives i t t he low re laxa t ion p roper ty .

(Relaxat ion is d iscussed in some detai l in Sect ion 2 .4 .4 .)

Fo r a m o re co m p ac t t y p e o f p ro d u c t , t h e s ev en -w i re s t r an d i s d r aw n t h ro u g h a

d ie unde r con t ro l l ed cond i t ions o f t ens ion and t em pera tu re . In the p rocess the

ind iv idua l wi res a re co ld worked and compacted in to the charac te r i s t i c shape o f

the d ie -d rawn s t rand . Com pact (or drawn) strand, co mp ared w i t h n o rma l s t r an d ,

has a h igher s teel area for a g iven overal l d iameter and, therefore, i t has a h igher

fo rce to d iame ter ra t io . F igu re 2 .6 shows the tw o types o f s t rand in sect ion .

Dr aw ing a s t rand th rou gh a d ie increases i ts s t reng th bu t reduces i ts duc t il ity .

The p roduct i s b r i t t l e and l i ab le to fa i lu re wi thou t much warn ing . In o rder to

avoid br i t t le and sudden fai lure the s t rand is annealed to increase i t s duct i l i ty .

For non-bonded app l i ca t ion s t rand i s impregnated wi th a p ro tec t ive f lu id

which pene t ra tes to the cen t re wi re , the s t rand i s then coa ted wi th a co r ros ion

res i s tan t g rease and f ina lly a con t inuo us po lyp ropy len e shea th ing , o f 0 .75-1 .00

mm min imum th ickness , i s ho t ex t ruded to cover the s t rand . The g rease g ives

long- te rm p ro tec t ion and lubr ica t ion fo r ease o f m ove m ent w hen s t res s ing . The

shea th ing has a h igh impact res i s t ance ; i t p ro tec t s the s t rand f rom phys ica l

damage , and i t p reven t s bond wi th the concre te .

In the ear ly years o f deve lopm ent o f the unb ond ed sys tem, s t rand was g reased

b y h an d an d s t r ip s o f w a t e rp ro o f p ap e r w e re w o u n d a ro u n d t h e g rea s ed s t r an d t o

p rev en t b o n d . Th e p ro b l em w i t h t h i s s y s t em w as t h a t t h e p ap e r g o t d amag ed

d u r i n g h an d l i n g , a s s emb l y o f ten d o n s , o r d u r i n g co n c re t in g b y s h a rp ag g reg a t e

or v ib ra to rs ; th i s a llowed the s t rand to deve lop loca l bond wi th concre te and l ed

to diff icul ties wi th s t ress ing. The next a t te m pt c onsis ted o f push ing the gre ased

s t rand th roug h a p las t ic p ipe ; in theory th i s shou ld hav e work ed bu t sub sequen t

exposure o f such s t rand showed tha t wa ter had fo und i t s way in to the p last ic p ipe

and caused ser ious cor ros ion o f the s t rand . W ater cou ld have go t in poss ib ly

whi le the t end on was s to red o n s i te , dur ing c oncre t ing i f the p ipe go t dam aged , o r

whi le the t endon was wai t ing to be s t res sed and the anchorage to be p ro tec ted .

PVC was somet imes used to house the s t rand ; when s t res sed , PVC may re lease

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42

POST-TENSIONEDCONCRETE FLOORS

ch lo r ine which wou ld cause cor ros io n o f the s tee l. F or these reasons , s t rand

housed in p las t i c p ipes o r hand wrapped in paper i s to be avo ided .

St rand con ta in ing m ore th an seven wi res and o f l a rger si ze is a l so ma nufa c tu re d

for special ized use where large concentrated pres t ress ing forces are required. I t s

s ize ranges up to 40 m m and i t has a nom ina l b rea k ing loa d o f the o rde r o f 1000

kN (1 .6 in and 225 k ip approx imate ly ) . In pos t - t ens ioned f loors such fo rce

co n cen t r a t i o n s a r e n o t n eed ed an d t h e sev en -w i re s t ran d i s d o m i n an t t h ro u g h o u t

the wor ld .

For special ized use galvanized, s ta in less s teel and epoxy-coated s teel s t rands

are a l so ava i l ab le . Curren t ly , t es t s a re be ing car r i ed ou t on s t rand made f rom

carb on f ib re ; th is has a 10% h igher s t reng th th an s tee l s t rand and is m uch l igh te r

in weight , but a rel iable an ch ora ge sys tem sui table for com m ercial use i s s ti ll to be

d ev e l o p ed . O t h e r n o n - f e r ro u s s t r an d s a r e a l s o u n d e r d ev e l o p men t .

The idea l s t rand wo uld cons i s t o f s t ab le m ater i a l , i ner t as fa r as possib le aga ins t

a t t ack f rom chemica l s in concre te and the env i ronment , i t would have a h igh

s t reng th , low re lax a t ion , a low m odu lus o f e l as ti c ity , h igh fa t igue s t ren g th , a

coeff icient of ther m al ex pan sion s imi lar to that for concrete , g ood f ire res is tance,

no t sub jec t to b r i t t le fa i lu re , and wo uld be capab le o f be ing re l iab ly and eas ily

anch ored . Q ui t e a l is t, bu t m an y o f the p roper t i es a re a l ready ava i l ab le in s t rand

being produced for special ized use.

2 . 4 . 2 C o r r os i on o f s t r and

Pres t ress ing s t rand is suscep tib le to c o r ros ion f rom the same chem ica l sources as

rod re in fo rcement , such as ox ida t ion and a t t ack by ch lo rides . High-s t reng th s tee l

is a lso a f fec ted by hyd roge n ions , which cause b r i t tl eness. H ydr oge n i s libera ted

by the chemica l reac t ion o f ac ids on i ron and o ther m eta l s which ma y be

con ta ined in the p res t ress ing wi res , and by the a lumin ium con ta ined in

h igh-a lumina cements and in t endon g rou t s , where i t was o f t en added as an

ex p an d i n g ag en t .

The a lka l ine env i ro nm ent in the concre te , due to the p resence o f f ree lime,

p rov ides a very good p ro tec t ion to s tee l. The a lka l in i ty is g radu a l ly los t th ro ugh

carb ona t ion and the n the s tee l becomes suscep t ib le to a t tack . A smal l am ou nt o f

hard ox ida t ion a l so p ro tec t s the s t ee l and improves bond wi th the g rou t .

A ra ther d ang erous phe nom eno n , charac te r i s t i c o f h igh ly st ressed s tee l, is

s t r e s s co r r o s i o n , which causes a sudd en fra cture o f the s teel. St ress corr os io n

c rack s r e su l t fr o m t h e co mb i n ed ac t io n o f co r r o s i o n - - f r o m n i tr a t e s, ch l o r id e s ,

su lf ides and o ther co r ros ive chemica l age n t s - -a nd h igh st res s. Hea t - t rea te d w i res

are more p rone to s t res s co r ros ion than d rawn wi res .

S t rand , as used in pos t -t ens ion ing , cons i st s o f smal l wi res abou t 4 mm in

d iam eter . Co rros ion to a dep th o f, say , 0 .1 mm represen t s a much g rea te r loss o f

area in a 4 m m wi re than i t does in the case o f a 20 m m re in fo rcement bar . F or th is

reason , an d to gua rd aga ins t s tress co r ros ion , i t is very im po r tan t tha t

pres t ress ing s t rand is wel l protected.

Bonded t endons a re sa fe r in th i s respec t , p rov ided tha t the g rou t does no t

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MATERIALS AND EQUIPMENT

43

I Test esults j,

[ BS 8110 "

/ l~f . .~ B D L

t "

/

/ ~

/

/ , ,

0.5% 1.0% Strain

- - C

Figure 2.7

Stress-strain curve for strand. Yr, is the part ial safety factor for strength.

conta in any of fens ive chemica ls and tha t no voids a re le f t in the shea th ing .

Gr ou t ing shou ld be c a r r i e d ou t a s soon a f t e r s t r e s s ing a s pos s ib l e t o a vo id

de le t e r ious ma t t e r f i nd ing i t s wa y in to t he she a th ing .

2. 4 . 3 S t r e s s - s t r a i n r e l a t i o n s h i p

C ur ve O A D in F igu r e 2 .7 shows the t yp i c a l s tr e s s -s t r a in r e l a t i onsh ip f o r s t r a nd .

At low s t re ss level the l ine is s t r a igh t , then a t so m e po in t i t beg ins to curve which

ind ic at e s onse t o f no n - p r op o r t io na l s t ra in . I t t he n c ha nge s t o a ge n t le c u rve ,

a lmost a s t r a igh t l ine , lead ing up to the u l t ima te s t r e ss a t f a i lu re .

The i ns ta n tan eou s m od ulu s o f e la s t ic i ty is clea r ly de f ined by the s lope of l ine

OA , or tan gen t o f angle ct, bu t the p o in t o f onse t o f non - l inea r i ty i s no t c lea r ly

def ined and , the re fore , the s t r e ss - s t r a in cha rac te r i s t ic can no t be g iven in te rm s of

the y ield po in t w hich is used for m an y tens i le ma te r ia l s . The lo ad a t 1% extens ion

a nd the l oa d a t wh ic h 0 .1% non- l ine a r s t r a in oc c u r s ( 0 .1%

proof oad)

a re used by

m a nuf a c tu r e r s t o c on t r o l t he i r p r oc e ss . H igh va lues o f t he se two pa r a m e te r s

ind ica te th a t the s t r e ss - s t r a in re la t ion sh ip i s l inea r in the no rm al r an ge of s t re sses .

T he to t a l s t r a in a t fa il u re m a y be o f t he o r de r o f 6% whe n m e a su r e d in

l a bo r a to r y t e st s on 600 mm (2 f t) l ong sa m ple s sub je c t t o t e ns ion . A C I 318 a nd B S

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4 4

POST-TENSIONED CONCRETE FLOORS

8110 requi re the to ta l s t r a in a t f a i lu re to be no t le ss than 3 .5%. In a r ea l

a pp l i c a t ion , f a il u re p r ob a b ly oc c u r s a t one o r m or e d i s c r e te po in t s i n t he l e ng th o f

a me m be r a n d the a c tua l i nc r ea se i n t e ndon l e ng th de pe nds on the l e ng ths o f l oc a l

y ie ld zones . The 6% s t ra in i s o f no s ign if icance in such a s i tu a t ion ; how ever , i t is

ind ica t ive of the duc t i l i ty of the ma te r ia l .

In BS 8110 the s t r e ss - s t r a in curve i s idea l ized for convenience in te rms of th ree

s t ra igh t l ines , r epresen ted by OA, AB and BC in F igure 2 .7 .

S t r a nd is ma n uf a c tu r e d f r om the s a me q ua l i t y o f s te e l a s h igh - te ns i le w i r e bu t

i t s ap pa ren t m od ulu s o f e la s t ici ty is s ligh t ly lower , because sp i ra l wi res of the

s t ran d ten d to s t r a igh ten s l igh t ly un der a tens ile force . The m od ulu s of e la s t ic i ty

( slope o f t he s t r a igh t pa r t o f t he c u r ve ) f o r s t r a nd va ri e s be twe e n 190 kN /m m 2

a nd 200 kN /m m 2; the m e a n va lue o f 195 kN /m m 2 is no r m a l ly u se d in c a l c u la t i ons .

2 . 4 . 4 R e l a x a t i o n

Stra nd subjec t to a sus ta ined tens ile force und ergo es a gradu a l inc rease in s t r a in .

C onve r se ly , if a s t r a nd is st r es se d a nd a nc h o r e d be twe e n tw o f ixe d po in t s t he n i t

g r a dua l ly l o se s f o r c e . T he phe nome non , t e r me d r e l a x a t i o n , i s s imi la r to tha t o f

c r ee p in conc r e t e , e xce p t t ha t r e l a xa t ion o f s t r a nd i s m uc h sma l l e r in m a gn i tude

tha n c r ee p o f c onc r e te . T he r e l a xa t ion o f s t re s s in a pa r t i c u l a r s t r a n d de pe nd s on

the in i ti a l st re s s, on a m b ie n t t e m pe r a tu r e a nd on the l e ng th o f t ime . Dur in g

m a nuf a c tu r e s t r a nd m a y be g ive n a he a t t r e a tm e n t de s igne d to re duc e r e l a xa t ion ;

these s t r ands a re ca l led

l o w - r e l a x a t i o n s t r a n d s .

Re la xa t ion is de t e r mine d in l a bo r a to r y t e st s ove r a pe r iod o f 1000 hou r s (s ix

we e ks ). T h i s pe r iod is ve ry sho r t c om pa r e d w i th t he n o r m a l l ife e xpe c t a nc y o f a

bu i ld ing o f 30 to 50 ye ar s . E x t r a po la t ing f r om l a bo r a to r y t es t s, t he r e l a xa t ion o f

s t ress a t 70% of the tens ile s t r eng th a t 20~ (68~ m ain t a ine d for a pe r io d of

500 000 h ou rs (57 yea rs ) i s expec ted to be of the ord e r of 1 .8% for low -re lax a t ion

( type 2 ) s t r a nds . F o r no r m a l r e l a xa t ion s t r a nds ( type 1 ) t he lo s s m a y be a s h igh a s

14% .

Re la xa t ion r a p id ly i nc r e a se s w i th a mb ie n t t e mpe r a tu r e , a s i s e v ide n t f r om

T a b le 2 .6 wh ic h shows long - t e r m c ha r a c t e r is t i cs f o r l ow- r e l a xa t ion s t r a n d a t a n

in it i al s t re s s o f 70% o f s t r e ng th . In a mb ie n t t e m pe r a tu r e s a bo ve 80~ ( 176~

low- r e l a xa t ion s t r a nd i s no t r e c om m e nde d , un l e s s wo r k ing loa ds a r e s ign if i ca n t ly

r e duc e d . A t a n a m b ie n t t e m pe r a tu r e o f 30~ ( 85~ low- r e l a xa t ion s t r a nd wo u ld

lose ab ou t 2 .3 % s t re ss and in n o r m a l r e l a x a t i o n s t r a n d t he lo s s wou ld be ove r 20 % .

I n mos t bu i ld ings , t e mpe r a tu r e i s c on t r o l l e d a t a s t a b l e c omf o r t l e ve l a nd

s t r a nd i s no t sub j e c t t o c ond i t i ons wh ic h wou ld c a use h igh - r e l a xa t ion lo s s .

H ow e ve r , in bu i ld ings w i tho u t a i r - c ond i t i on ing , pa r t i c u l a r ly i n ho t c o un t r i e s , i t

i s impor t a n t t o u se t he a pp r op r i a t e l o s s f i gu r e s .

Table 2.6

Lo no-term relaxation and tem perature at an ini t ial s tress of 0 .7fp,

Temperature 20~ 40~ 60~ 80~ 100~

Re laxa t ion 1.8% 3.5% 5.1% 7.5% 10.7%

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MATERIALS AND EQUIPMENT 4 5

20

15

.-. 10

4

*- 2

O

1.0

t -

O 0.8

m

'~

0 . 6 _

X

t~ 0 .4

Q )

r r

0.2

0.1

10

axation strand

Low-relaxat ion strand

1 w eek 6 weeks ~ 1 y e a r ~

J

9 1DAD O O O O 41D D~ ~ a l B

~ aID ~ O ~ g OD Oq

J~

11.4 ye ar s - 5 7 years

100 1000 10,000

100 ,000 500 '000

Test duration (hours)

F | g u re 2 . 8 Relaxation and t ime

T h e r e l a t i o n s h i p b e t w een t i me an d r e l ax a t i o n p l o t t ed o n a l o g - l o g s ca l e i s

g en e r a l l y a s s u m ed t o b e li n ea r . F i g u r e 2 .8 s h o w s t y p i ca l r e l a t i o n s h i p f o r an i n i t i a l

l o a d o f 7 0 % o f b r e a k i n g l o a d a t a n a m b i e n t t e m p e r a t u r e o f 2 0 ~ ( 6 8~

F i g u r e 2 .8 i n d i ca t e s t h a t f o r n o r ma l s t r an d t h e 5 0 0 0 0 0 h o u r ( 5 7 y ea r s )

r e l ax a t i o n can b e 2 .6 ti mes t h e 1 0 00 h o u r v a l u e an d f o r l o w - r e l ax a t i o n s t r an d t h e

r a t i o can b e 1 .6 . F o r p o s t - t en s i o n ed w o r k , B S 8 11 0 r ec o m m en d s v a l u e s o f 2 .0 an d

1.5 respect ively .

2 . 4 .5 S t r a n d s i z e a n d s t r e n g th

T ab l e 2 .7 g i ve s t h e d a t a f o r B r i t is h s t r an d co m m o n l y u s ed i n p o s t - t en s i o n ed f l oo r s .

I t s h o u l d b e n o t e d t h a t f o r t h e s a m e n o m i n a l d i a m e t e r t h e s t r e n g t h a n d o t h e r

p r o p e r t i e s o f s t r an d a r e v e r y d i ff e ren t f o r th e t h r ee t y p es an d g r ad es . I t i s

i m p o r t a n t t h a t s t r a n d i s s p e c i f i e d b y t y p e , d i a m e t e r , s t r e n g t h a n d r e l a x a t i o n

ch a r ac t e r i s t i c s . M i x i n g o f d i ff e ren t s t r an d s i n a me m b er , p a r t i cu l a r l y u s e o f

s t r an d s o f t h e s am e d i am e t e r b u t o f d i ff e r en t t y p es , s h o u l d b e av o i d e d .

Table 2.7 British low-relaxation strand

Type Dia. Area Breaking load 0.1% proof Weight

mm mm 2 N/mm 2 kN load kN kg/m

Standard

Super

Compact

15.2 139 1670 232 197 1.090

12.5 93 1770 164 139 0.730

15.7 150 1770 265 225 1.180

12.9 100 1860 186 158 0.785

15.2 165 1820 300 255 1.295

12.7 112 1860 209 178 0.890

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46 POST-TENSIONEDCONCRETE FLOORS

Table 2.8 P rop er t ie s o f US s t rand

Grade N om ina l d ia . Area Breaking load

in in 2 lb

250 0.5 O. 144 36 000

0.6 0.216 54 000

270 0.5 0.153 41 300

0.6 0.217 58 600

Co m pact 0.5 0.174 47 000

0.6 0.256 67 440

Amer ican s t r and i s ve ry s imi la r in s ize and s t r ength . I t i s ava i lab le in two

gr a de s : g r a de 250 wh ic h ha s a nom ina l s t r e ng th o f 250 000 p si a nd g r a de 270 w i th

a nom ina l s t r e ng th o f 2700 00 ps i . T he da t a a r e g iven in T a b le 2 .8 .

W he r e ve r pos s ib l e , pa r t i c u l a r ly f o r s t r a nd wh ic h doe s no t c o m ply w i th B r i ti sh

o r A me r i c a n s t a nda r ds , s t r a nd c ha r a c t e r i s t i c s shou ld be ob ta ine d f r om the

manufac ture r . This a l so appl ie s to i t s long- te rm re laxa t ion and the e f fec t o f

a m b i e n t t e m p e r a t u r e .

S t reng th o f pres t re ss ing s t r and i s se r ious ly a ffected by a ri se in tem per a tu re

abo ve 150~ i t f al ls a lm ost l inea r ly f rom 100 % a t 150~ to ze ro a t 700~ A t low

te mp e r a tu r e s , t he s t r e ng th inc re a ses bu t duc t i l i ty de c re a se s . T yp ic a l ly , c om pa r e d

wi th t he s t r e ng th a t 20~ s t r a nd s t r e ng th a t - 80~ is 5% h ighe r a nd a t - 160~

it is 12% higher .

2 .4 .6 T ransm iss ion leng th

T he l e ng th o f bon de d s t r a nd ne e de d to de v e lop a g iven fo rc e ( u sua lly t a ke n a s t he

in i t ia l p res t re ss ing force ) in the s t r and i s te rmed the t r a n s m i s s i o n l e n g t h . A s

discussed in Sec t ion 2 .5 .2 , s t r and s t r e ssed f rom one end only i s o f ten bonded to

the c onc r e t e a t t he o the r e nd . E xc e p t a t bonde d de a d e nds , t he t r a nsmis s ion

le ng th ha s no r e l e vanc e in unbo nd e d p os t - t e ns ion ing . I n bon de d a pp l i c a t ions , i t

i nd i c at e s t he d i s ta nc e be y ond wh ic h the s t r a nd c a n be e xpe c te d to ha v e r e t a ine d

its force in case of damage.

N o r m a l s t r a nd , w i th r ou nd w i r es , ha s a tr a nsm is s ion l e ng th o f a bo u t 30 t ime s

the s t r a nd d i a m e te r a t a f o rc e o f 70% o f i ts b r e a k ing loa d whe n c a s t in c onc r e t e a t

a c ube s t r e ng th o f 50 N /m m 2 ( f c '= 6000 ps i) . C om pa c t s t r a nd ha s a sm oo the r

sur face and a la rge r a re a of s teel ; i ts t r ansm iss io n length i s 45 d iamete r s . These

l e ng ths ar e fo r s t r a nd bo nde d in no r m a l de nse c onc r e t e ; f o r l i gh twe igh t c onc r e t e

the le ng ths shou ld be inc r e a se d in p r op o r t i on to t he pe r mis s ib le bon d s tr e ss . B S

8110 s t ipu la te s tha t for l igh twe ight concre te the bond s t r e sses should no t exceed

80% o f t hose c a l c u la t e d f o r no r m a l - we ig h t c onc r e t e ; th i s a m ou n t s t o a n inc r e ase

o f 20% in t r a nsmis s ion l e ng th f o r l igh twe igh t c onc r e te .

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4 7

2 .5 Pres t ress ing hard wa re

The p res t ress ing h ard wa re i s ava i lab le f rom a num ber o f loca l and in te rna t ion a l

m anufac tu rers . Bas ica lly , i t cons is t s o f ancho rages ( live , dead and in te rmed ia te ,

as d i scussed be low) , and shea th ing fo r bonded sys tems . In o rder to p ro tec t the

anchora ge assembly f rom cor ros ion , m anufac tu rers p rov ide an in tegra l p ro tec t ion

sys tem to su i t the i r hard wa re . G rou t ing o f the shea th ing requ i res smal l tubes to

be cas t in the concre te w hich p ro jec t a t the top o f the mem ber and p rov ide a pa th

to the sheathing.

The fol lowing sect ion gives typical detai l s for the hardware, which would be of

in teres t to the des igner or the user . The s izes and descr ip t ions are meant for

genera l gu idance on ly and i t i s recommended tha t ac tua l da ta a re ob ta ined f rom

the suppl iers in the area.

Pro tec t ion to an anch orag e i s requ i red on ly a t the live end , the dead end i s cas t

in the concrete . T he l ive anch ora ge is hou sed in a recess, or pock et , w hich is wide

enoug h fo r the s tres sing jack and deep enough so tha t there w ould be adequ ate

concrete cover to the assembly when the recess i s made good. After s t ress ing, the

s t ran d is cut off c lose to the face of the w edge us ing a d isc cut ter and the who le

assembly i s sp rayed wi th a co r ros ion inh ib i t an t . The assembly i s then covered

wi th a g rease- f i l l ed cap and the recess made good wi th mor ta r con ta in ing a

non-shr ink ing agen t .

I f a t end on is very long , then the losses a t the fa r end m ay be h igh . Three

opt io ns are poss ible: acce pt the high losses an d use ten don s at a lesser effic iency;

p rov ide l ive ancho rages a t bo th ends and s tres s the t endon f rom bo th ends ; o r use

in te rmedia te anchorages , ca l l ed

couplers.

Tw o-e nd s t ress ing, whi le al lowing s t ran d to be used a t a greate r eff ic iency, uses

two l ive anchorages (a l ive anchorage cos t s more than a dead anchorage) and

requ i res more s i t e opera t ions . The cho ice be tween these a l t e rna t ives would be

m ade a f te r cons ider ing the mer i t s and deme r i ts o f each fo r the pa r t i cu la r p ro jec t.

2.5 . 1 L ive anc hora ge

A few decades ago mos t manufac tu rers had the i r own pa ten t dev ices fo r

anch or ing t end ons . T hey inc luded bars wi th th rea ded ends , en la rged ends o f wi re

which passed th ro ugh ho les in th ick p la tes such tha t the bu t to n sa t on the p la te ,

wedges , concre te m ale and female cones g r ipp ing a num ber o f wi res a r rang ed in a

c irc le , and a too thed con ica l wedge in a bar re l . The c one an d b ar re l sys tem is now

the mos t com m only em ployed dev ice fo r pos t - t ens ion ing f loors and i t is ava il ab le

f ro m mo s t man u fac t u re r s .

The anchorage cas t ing i s in d i rec t con tac t wi th the concre te and i s the

component which t ransfers the pres t ress ing force to the concrete . I t i s s l ight ly

con ica l in shape which a l lows s t rands to f l a re ou t fo r anchor ing f rom the i r

com pact fo rm at ion in the duc t . The bar re l i s o f ten an in tegra l par t o f the

anch orage cas t ing in tha t the cas t ing has s lop ing ho les d r i ll ed in i t, and m ach ined

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4 8 POST-TENSIONEDCONCRETE FLOORS

[~~C onc re te fa ce

ar

. , . " p .

'~':.~ .. St rand

Sleeve

I .

i

- - - f , " _ ]

e

Wedges

G r eas e

T L . . . . : I ' -

Joint ing tape

Cap

F i g u r e 2 .9 Typical monostrand anchorage assem bly

sm oo th , i n a pp r op r i a t e pos i t i ons t o t a ke the we dge s d ir e ct ly . S om e sys te ms use

the c a s t i ng to t r a ns f e r t he l oa d to c onc r e t e a nd a s e pa r a t e he a d ing b loc k

m a c h i n e d t o a c c o m m o d a t e t h e w e d g e s .

F igu r e 2 .9 shows a t yp i c a l s e c t ion a t t he e nd o f a pos t - t e ns ione d m e m be r

c o n t a i n i n g a m o n o s t r a n d a n c h o r a g e a s s em b l y , c o m p l e te i n c lu d i n g th e c o r r o s i o n

p r o t e c t ion a nd F igu r e 2 .10 shows a t yp i c a l mu l t i s t r a nd a nc ho r a ge . T he i r

represen ta t ive d imens ions a re g iven in Tables 2 .10 and 2 .11 .

Le t u s b ri ef ly look a t t he pu r pose o f e a c h c om pon e n t . A we dge ( o r c one o r

j aws) ac tu a l ly gr ips the s t r an d; i t i s r e ta ined in a locked p os i t ion by be ing pul led

by the s t r an d in to a conica l ho le , e i the r in a sepa ra te b a r re l o r in a hea der b lock .

T he he a d e r b loc k o f a mu l t i s t r a n d a nc h o r a ge s it s on the c onc r e te , w i th a c on ic a l

guide ( somet imes re fe r red to a s a trumpet) which a l lows the seve ra l s t r ands in a

_ L _1 ~_ B _1

r 1

Duct d iameter

F i g u r e 2 .10 Multistrand anchorage

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M A T E R I A L S A N D E Q U I P M E N T

49

t e ndon t o f l a r e ou t t o a spa c i ng w he r e t he y c a n be c onve n i e n t l y a nc ho r e d . The

guide m ay b e e i the r a cas t ing o r i t m ay be fabr i ca ted f rom s t ee l p l a t e ; i t is cas t i n to

the concre te .

S om e sy s t e m s m a y ha ve se pa r a t e ba r r e l s i n t o w h i c h t he w e dge s fi t. The ba r r e l s

s i t on a c om m on t h i c k be a r i ng b l oc k w h i c h i n t u r n be a r s on t he c onc r e t e . Th i s

a r r a n g e m e n t i s n o w g e t t i n g l e s s c o m m o n . M o s t s y s t e m s h a v e n o w d i s c a r d e d

se pa r a t e ba r r e l s ; i n s t e a d t he y u se a s te e l b l oc k w h i c h ha s c on i c a l ho l e s m a c h i ne d

in to i t t o rece ive the wedges and th i s b lock i s sea ted on the gu ide .

As the s t ra nd i s s t re ssed , the wedge cone (cons i s t ing of two or th ree j aws )

m o ve s ou t o f t he ba r r e l ; a t t h e r e q u i r e d t e n do n f o r c e the c one i s push e d i n t o t he

ba r r e l so t ha t t he s t r a n d i s l i gh tl y g r i ppe d i n t he s e r r a t i ons o f t he c one . The

hyd r a u l i c j a c k i s t he n r e l e ase d ; t h is a c t i on pu l l s the s t r a n d a nd t he c one f u r t he r

i n t o t he ba r r e l , by 6 t o 8 m m ( 0.25 t o 0 .375 i n ) so t ha t t he s t r a n d i s g r i ppe d t i gh t e r

as a re su l t . The 8 m m draw - in , o f course , causes a loss in the pres t re ss ing force.

The s t r e s s i ng ope r a t i on i s show n d i a g r a m m a t i c a l l y i n F i gu r e 2 . 11 .

The m e c ha n i c s o f t r a n s f e r r i ng t he t e nd on f o rc e t o c onc r e t e is w o r t h a cl o se r

l ook a s i t b r i ngs t o f oc us s e ve r a l i m por t a n t po i n t s w h i c h t he de s i gne r shou l d be

a w a r e o f . The r e a r e t h r e e m a i n c o m po ne n t s , o r i n te r fa c e s o f c om po ne n t s , w h ose

in tegr i ty i s e ssen t i a l fo r the re l i ab i l i ty o f the t endon .

F i r s t l y , t he w e dge c one , w h i c h is m a de f r om pa r t i c u l a r g r a de s o f s te e l a nd is

Slide ba rrels and

wedges on strand

Position b arrels and

wedge s on the anchorage,

and place jack

G rip strands in jack,

take up slack, stress

tendon and record

extension

Lock

wedges

~ Barrels

Wedges

[ ; = , 4

i : : = 4 Z

1 : : : : 1

Anchorage

Spiral reinforcement

Hydraulic lac k J

Extension

I . / , 2 ~ ~ _

mea suring ruler

l _ _ - - -

Extension

Figure 2.11

St ress ing o f a tendon

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50 POST-TENSIONEDCONCRETE FLOORS

g i v en a s p ec i a l h ea t t r e a t men t t o h a r d en i t s s e r r a t i o n s . T h e co n e w ed g e i s

l o n g i t u d i n a ll y s p li t i n t o t w o o r t h re e p i e c e s - - d e p e n d i n g o n t h e m a n u f a c t u r e r . I f

n o t s p li t th en , o f co u r s e , it w o u l d n o t g r i p t h e s t r an d . T h i s p e r h a p s i s t h e mo s t

c r it ic a l c o m p o n e n t i n t h e a n c h o r a g e a s s e m b l y a n d m o s t f ai lu re s d u r i n g s t r e s s in g

o ccu r d u e t o t h e s e r r a t i o n s o f t h e co n e n o t g r i p p i n g t h e s t r an d . T h e s e r r a t i o n s

m ay s h ea r i f t h e s t e el is n o t o f g o o d q u a l i t y , t h ey m ay f ai l t o b i t e i n t o t h e s t r a n d i f

t h ey h av e n o t b een h a r d en ed p r o p e r l y , o r t h ey may f a il b ecau s e t h e co n e an g l e is

s u c h t h a t t h e s t r a n d d o e s n o t c o m e i n t o c o n t a c t w i t h th e w h o l e o f t h e t o o t h e d

l e n g t h i n t h e c o ne . T h e i m p o r t a n c e o f o b t a i n i n g c o n e s fr o m a r e li a b le s o u r c e m u s t

b e e m p h a s i z e d .

S eco n d l y , t h e co n e s l i d e s i n t h e co n i ca l h o l e i n t h e b a r r e l ( o r t h e an ch o r ag e

b l o ck ) p r o d u c i n g b u r s t i n g s t r e s s e s i n t h e b a r r e l cy l i n d e r ; i t i s e s s en t i a l t h a t t h e

s l i d i n g s u r f ace s a r e s mo o t h , t h e i r s l o p es ma t ch ex ac t l y an d t h a t t h e b a r r e l c an

w i t h s t a n d t h e b u r s t i n g f o r ce w i t h ad e q u a t e m ar g i n o f s a fe t y . I t i s u n u s u a l f o r a

b a r r e l , o r a c a s t i n g , t o f a il i n b u r s t i n g b u t a l l t o o ea s y t o u s e a co n e f o r t h e w r o n g

s iz e o f s t r a n d ~ a s s h o w n e a r l ie r , s t r a n d s o f t h e sa m e n o m i n a l d i a m e t e r d if fe r f r o m

each o t h e r i n f r ac t i o n s o f a m i l l i me t r e i n d i ame t e r .

T h i r d l y , t he f la n g es o f t h e a n c h o r a g e c a s t i n g s h o u l d b e s t r o n g e n o u g h n o t t o

f ai l a n d y e t l a r ge e n o u g h n o t t o o v e r l o a d t h e c o n c re t e . T h e t w o r e q u i r e m e n t s a r e

r a t h e r co n f l i c t i n g i n t h a t t o r ed u ce t h e b ea r i n g s t r e s s o n co n c r e t e a l a r g e co n t ac t

a r ea i s r eq u i r ed , w h i ch g en e r a t e s h i g h e r s t r e s s e s i n t h e an ch o r ag e ca s t i n g i t s e l f .

T h i s a l s o se t s a l i m i t f o r t h e m i n i m u m s t r en g t h o f co n c r e t e a t w h i ch t en d o n s can

b e s t r e s s e d . B e a r i n g s t r e s s e s a p p r o a c h i n g c o n c r e t e s t r e n g t h a r e c o m m o n a n d

m o s t a n c h o r a g e s a r e d e s i g n e d t o b e s t r e s s e d a t a m i n i m u m c u b e s t r e n g t h o f

2 5 N / m m 2 ( f c ' = 3 0 00 p si ). S o m e m a n u f a c t u r e r s o ff er a la r g e r s iz e d a n c h o r a g e ,

u s u a l l y t o s p ec i a l o r d e r , w h i ch a l l o w s s t r e s s i n g a t a co n c r e t e cu b e s t r en g t h o f

15 N / m m 2 (fc' = 1750 psi ) .

C o m p a c t s t r a n d is c a p a b l e o f a h i g h e r fo r ce f or a g i ve n n o m i n a l d i a m e t e r a n d i t

g en e r a t e s a h i g h e r b ea r i n g s t r e ss in co n c r e t e , an d , o f co u r s e , i n a l l co m p o n e n t s o f

t h e a n c h o r a g e a s s e m b l y . N o t a ll m a n u f a c t u r e r s h a v e t e s t ed a ll t h e i r a n c h o r a g e s

f o r u s e w i t h co mp ac t s t r an d .

T h e r e a r e t h r ee t en d o n co n f i g u r a t i o n s i n u s e i n p o s t - t en s i o n ed f l o o r s : s i n g l e

s t r an d f o r u n b o n d ed u s e , f la t t en d o n s co n s i s t i n g o f f o u r o r fi ve s t r an d s f o r b o n d ed

u s e, an d m u l t i s t r an d c i r cu l a r t en d o n s co n t a i n i n g u p t o 1 2 s t r an d s . T h e mu l t i s t r an d

Table 2.9

Typical 13 mm (0 .5 in) l i ve anchorages (VSL Monost rand pres t ress ing sys tem)

Type L x B x H

millimetres inches

Single str an d 65 130 60 (2.5 5.0 2.3)

Fla t 5 stra nd 260 240 70 (10.2 9.5 2.8)

M ulti 3 str an d 180 120 120 (7.1 4.7 4.7)

M ulti 4 str an d 175 135 135 (6.9 5.3 5.3)

M ulti 7 str an d 210 165 165 (8.3 6.5 6.5)

M ult i 12 str an d 275 215 215 (10.8 8.5 8.5)

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MATERIALS AND EQUIPMENT 51

Table 2.10 T y p i c a l 15 m m (0 .6 in ) l iv e anc hor ages (V SL M o nos t r a nd p r e s tr e s s ing s y s t em )

T y pe L x B x H

mi l l imetres inches

Single stra nd 65 130 60 (2.5 5.0 2.3

Flat 5 stran d 277 240 70 (10.9 9.5 2.8

M ulti 3 str an d 175 135 135 (6.9 5.3 5.3

M ulti 4 stra nd 210 150 150 (8.3 5.9 5.9

M ulti 7 stra nd 230 190 190 (9.1 7.5 7.5

M ulti 12 stran d 320 250 250 (12.6 9.9 9.9

t e ndo ns a r e de s i gne d f o r a m a x i m um o f 3 , 4 , 7 , o r 12 s t r a nds o f 13 m m a nd 15 m m

n o m i n a l d i a m e t e r. A n c h o r a g e s s u i ta b l e fo r l a r g er t e n d o n s a r e m a n u f a c t u r e d b u t

f o r o t he r u se s be c a use t he m a g n i t ud e o f c on c e n t r a t e d f o rc e a va i l a b l e f rom t h e m i s

no t ne e de d i n pos t - t e n s i one d f l oo r s . The a nc ho r a ge s i z e s va r y be t w e e n

manufac ture rs , Tables 2 .8 and 2 .9 show the typ ica l . At the des ign s t age , i t i s

a dv i sa b l e t o c he c k t he a c t ua l d i m e ns i on s o f t he ha r d w a r e a va i l a b l e in t he a re a .

I t is no t ne c e s sa r y t o ha ve se ve l~ s t r a nds i n a 7 - s t r a nd a nc ho r a ge ; i t c a n a c c e p t

a n y n u m b e r u p t o a m a x i m u m o f s ev e n. O f c o u r s e if t h e n u m b e r o f s t r a n d s i s le ss

tha n the ca pac i ty o f the an ch ora ge then i t is be ing us ed a t a reduc ed co s t ef fi ci ency .

2.5.2 Dead anchorage

G e n e r a l r e m a r k s a b o u t b e a r i n g s t r e s s e s a n d a n c h o r a g e d i m e n s i o n s ( S e c t i o n

2 . 5 . 1 ) a r e a l so a pp l i c a b l e t o de a d a nc ho r a ge s w he r e t he s a m e c a s t i ng i s u se d .

A t t he t e ndon e nd w he r e a c c e s s i s no t r e q u i r e d f o r a ny ope r a t i on a f t e r

c onc r e t i ng , t he a nc ho r a ge a s se m b l y i s c a s t i n c onc r e t e . Th i s a s se m b l y c a n be a

s imp le r dev ice beca use a l l t ha t i s r equ i re d of i t i s secure ly to h o ld the end of the

t e nd on a nd t r a n s f e r t he f o r ce t o t he c onc r e t e . I t m us t , ho w e ve r , be r e li a bl e . I f a

wedg e cone s l ips a t the l ive end i t can be rep laced . I f a dead an ch ora ge fa il s then

t he c ho i c e i s e i t he r t o a ba ndon t ha t t e ndon o r t o c u t t he c onc r e t e ou t t o ga i n

a c ce s s t o t he a nc ho r a g e a n d t r y t o re p l a c e t he pa r t t h a t ha s f a il ed . One m a y no t be

a c c e p t a b l e f r om t he de s ign po i n t o f v i ew a nd t he o t he r w ou l d r e q u i r e a n

unsc he du l e d s i t e ope r a t i on .

A p r e - l oc ke d l ive a nc ho r a ge c a n be , a nd o f t e n is , u se d a t t he de a d e nd . Be i ng

i na c c e s s i b l e , i t w ou l d be unw i se t o u se t he no r m a l w e dge c one f o r s e c u r i ng t he

s t r a nd , e ve n i f i t is p r e - l oc ke d ; a h i ghe r g r a de w e dge i s p r e fe r r e d . M os t o f t he

p r e s t r e s s i ng sy s t e m s o f f e r a c he a pe r a nd m or e s e c u r e de a d a nc ho r a ge w he r e a

p u r p o s e - m a d e b a r r e l is s w a g e d o n t o t h e s t r a n d e n d a t th e i r w o r k s . T h e s w a g e d

ba r r e l be a r s o n a s te e l p l a t e w i t h ho l e s d r i ll e d f o r t he s t r a nds t o pa s s t h r ou gh . The

p l a t e t ra n s f e r s t he l oa d t o t he c onc r e t e . A de s ign a d va n t a ge o f suc h a de a d e nd i s

t ha t t he c onc r e t e is p r e s tr e s se d a l m o s t t o t he e nd o f t he f l oo r.

I n a va r i a t i on on t he a bove a r r a nge m e n t , F i gu r e 2 . 12 ( a ) , t h e she a t h i ng i s a l so

c u t sho r t t o a l l ow t he s t r a nds t o sp r e a d ou t , t h e s t r a nds a r e bonde d ove r t he

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52 POST-TENSIONEDCONCRETE FLOORS

. • S w a g e d

arrel

n ~ Bearing plate

. . . .

(a) Swaged barrel and plate

i Crimped wire

j Spacer mesh

Strand

~b) Unravelled strand with crimped wire ends

Figure 2.12

Dead-end anchorages

exposed l eng th. The overa ll l eng th o f t hi s t ype o f dead ancho rage m ay be o f the

ord er of 400 mm (16 in). The con crete in th is ar ra ng em en t is no t p res t ressed r ight

to the end, but i t i s cheaper.

Where i t i s no t essent ia l for concre te to be pres t ressed r ight to the end , the

shea th ing may be cu t sho r t and t he exposed l eng th o f t he s t rand bon ded wi thou t

the swaged barre l an d the s teel p la te , F igure 2 .12(b) . In or der to reduce the bon d

length , the surface area of s t ran d m ay be increased by u nravel l ing the seven

indiv idual wi res forming the s t rand . S om et imes the ends of the unravel led w i res

a re a l so c r imped t o p rov ide a mechan i ca l anchorage . The l eng th o f s t r and

bo nd ed in concre te var ies between d i f feren t suppl iers, bu t usual ly it is o f the o rde r

o f 900 mm (3 f t) . The un rave l led s t r and dead end i s no t r e com me nded fo r

un bo nd ed use , because the grease and the lubr ica nt are d i ff icu lt to remo ve f rom

the impregna t ed s t r and and t he i r p resence impa i r s bond .

Ano the r me thod o f s av ing on t he num ber o f dead a nchorages is t o l oop t he

tendo ns w here the dead ends w ould hav e been , as show n in F igure 2 .13 . The l ive

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MATERIALS AND EQU IPMENT

53

!

I

l

I

I

!

I

I

Figure 2.13 Plan showing looped dead ends

a nc ho r a g e s c a n be a r r a nge d e i t her a long one e dge o f a fl oo r o r a long bo th

edges.

2 .5.3 Coup le rs and i n te rmed ia te ancho rages

In long lengths of s lab , i t i s o f ten convenien t to cas t the concre te in seve ra l

ope ra t ions . E ach len gth m ay con s is t o f seve ra l sec t ions of d i f fe ren t ages and the

o lde r one m a y be a p p r oa c h in g the onse t o f sh r inka ge . I t is de s i ra b l e t o a p p ly

pres t re ss to each sec t ion in success ion . This r educes the poss ib i l i ty of shr ink age

cracks and , because shor te r tendon lengths a re now be ing s t r e ssed , the pres t re ss

losses are smaller .

T he t e r m

coupler

a pp l ie s t o a m e c ha n ic a l de v ic e wh ic h a l lows a ne w t e ndo n to

be c oup le d to t he e nd o f a t e ndo n w h ic h ha s a l r e a dy be e n loc ke d . A n intermediate

anchorage i s a device which anch ors a tend on in the midd le of i ts length a t a

c onve n ie n t c ons t r uc t ion j o in t a n d the s ame t e n don c on t inue s pa s t t he in t e r me d ia t e

anchorage . The la t te r i s convenien t on ly for s ing le - s t r and sys tems and i s used

m o s t l y in u n b o n d e d a p p l ic a t io n s .

A n in t e r me d ia t e a n c ho r a ge , f o r u se w i th unb on de d sys t ems , c ons is t s o f a

s lo t t e d p l at e wh ic h i s d r op pe d ove r t he ba r e s t r a nd . T he s t r a nd i s t he n s t r e sse d

a nd a n c ho r e d u s ing the s t a nd a r d c one a nd ba r r e l . A n una v o ida b le i nc onve n ie nc e

is t ha t t he s t r a nd ha s t o be t h r e a de d th r ou gh the ba r r e l, a nd , o f c ou r se , the l e ng th

of s t r an d for the nex t pa r t o f the f loor is s to red in a co i l nea r the in te rm edia te

a nc ho r a ge . A n o r d ina r y ja c k wou ld be a lmos t imposs ib l e t o u se be c a use the

s t r a nd pa s ses t h r o ugh a ho le in i ts c en t r e a nd th r e a d ing th i s ove r a l ong l e ng th o f

s t r a nd is no t p r a c t i c a b le . A spe ci al tw in - c y l inde r j a c k w i th a n ope n th r o a t f o r

pa s s ing the s t r a nd th r o ugh is ne e de d f o r s t re s s ing a t i n t e r me d ia t e a n c ho r a ge s . A n

in t e r me d ia t e a nc ho r a ge c a n be r a the r a wkwa r d to u se a nd a c oup le r ma y be

pre fe r red in ce r ta in s i tua t ions .

C ou p le r s fo r m u l t i s t r a nd t e ndon s c ons i s t o f a n a s se mbly w i th a nu m be r o f

ho le s i n to wh ic h the a nc h o r ing we dge s f it a nd the s a me nu m be r o f s lo ts a r ou nd

the c i rc umf e r enc e . T he s t r a nds a r e st re s sed a nd a nc ho r e d in t he no r m a l m a nn e r

us ing the wedges and th en new s t ran ds w i th swaged ba r re ls a re p laced in the s lo ts .

T he who le a s se mbly i s c onc r e t e d in w i th t he ne x t s e c t ion . A ppr ox ima te

d im ens ion s of typ ica l couple r s a re g iven in Table 2 .11 .

Couple r s for use wi th f ia t tendons a re s imi la r in pr inc ip le . They a re ova l in

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54

POST-TENSIONEDCONCRETE FLOORS

Table 2 .11

Mul t i s t ra nd coup le r s

Nom ina l t end on s i z e Le ng th D ia me te r

m m ( in) mm ( in)

3 x 13 m m (3 • 0.5 in) 450 (18) 130 (5)

7 x 13 m m (7 x 0.5 in) 550 (22) 170 (7)

12 x 13 m m (12 x 0.5 in) 650 (26) 200 (8)

3 x 15 m m (3 x 0.6 in) 500 (20) 150 (6)

7 x 15 m m (7 x 0.6 in) 625 (25) 200 (8)

12 x 15 m m (12 x 0.6 in) 725 (29) 240 (10)

s h a p e , t y p i c a l l y 3 3 0 x 1 1 0 m m (13 x 5 .5 i n ) ; t h e i r l e n g t h v a r i e s f r o m 7 5 0 m m t o

1 2 50 m m ( 30 t o 5 0 i n ) d e p e n d i n g o n t h e n u m b e r a n d s iz e o f s t r a n d s .

A m o n o s t r a n d c o u p l e r c o n s i s t s o f a sp e c ia l a n c h o r a g e c a s t in g w i t h a f em a l e

t h r e a d e d p r o j e c t i o n . T h e s t r a n d i s s t re s s ed a n d a n c h o r e d i n t h e n o r m a l m a n n e r .

T h e e n d o f t h e s t r a n d t o be c o u p l e d p a s s e s t h r o u g h a m a l e t h r e a d e d b l o c k a n d t h e

s t r a n d e n d i s s w a g e d i n t h e m a n n e r o f a d e a d a n c h o r a g e . T h e b l o c k i s t h e n

s c r e w e d i n t o t h e p r o j e c t i o n o f t h e s p e ci a l a n c h o r a g e .

2 .5.4 Shea th ing and o the r ha rd wa re

S h e a t h i n g , o r d u c t , f o r u s e i n b o n d e d p o s t - t e n s i o n i n g i s u s u a l l y m a d e f r o m

g a l v a n i z e d s t ee l a n d i s c o r r u g a t e d s o t h a t i t i s e a s y to b e n d t o t h e r e q u i r e d t e n d o n

p r o f i l e w h i l e r e t a i n i n g a h i g h r a d i a l c o l l a p s e s t r e n g t h . S o m e m a n u f a c t u r e r s

s u p p l y p l a s t ic s h e a t h i n g . S h e a t h i n g fo r f i at d u c t is r a t h e r d i f fi c u lt to m a n u fa c t u r e

w i t h c o r r u g a t i o n s a n d s o m e t i m e s i t is s u p p l i e d i n r ig i d s h o r t l e n g th s . B e i n g o n l y

a b o u t 2 0 m m (0 .7 5 i n ) d e e p , i t c a n b e b e n t t o s u i t a t e n d o n p ro f i l e b u t , o b v i o u s l y ,

n o t a s e a s i l y a s a c o r ru g a t e d o n e .

T a b l e 2 . 1 2 s h o w s t y p i c a l o v e ra l l s i z e s f o r s h e a t h i n g ; t h e e x a c t d i m e n s i o n s v a ry

fo r d i f fe r e n t m a n u fa c t u r e r s . N o rm a l l y , t h e i n s id e a r e a o f a s h e a t h i n g i s n o t l es s

t h a n t w ic e t h e a r e a o f t h e t e n d o n t o b e h o u s e d i n i t. T h e c l e a r i n s id e d i m e n s i o n

m a y b e 5 m m t o 7 m m ( 0.2 to 0 .3 in ) s m a l le r t h a n t h e o v e r a ll d i a m e t e r f o r r o u n d

m u l t i s t r a n d s h e a t h i n g a n d t w o o r t h r e e m i ll i m e t re s s m a l l e r f o r s in g le s t r a n d a n d

f i a t s h e a t h i n g s .

Table 2.12

Typica l shea th in9 d iameters (ou ts ide )

13 mm (0.5 in) 15 mm (0.6 in)

Single stra nd 16

Flat 5 strand 70 x 19

Mu lt is t rand 3 45

Mu lt is t rand 4 50

Mu lt is t rand 7 60

M ultis tra nd 12 72

(0.63) 20 (0.80)

(2.8 • 0.75) 70 • 19 (2.8 • 0.75)

(1.75) 50 (2.00)

(2.00) 55 (2.17)

(2.38) 67 (2.63)

(2.83) 87 (3.42)

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M A T E R I A L S A N D E Q U I P M E N T 55

As s t ran ds in curved prof i le a re s tr e ssed , they tend to bun ch up in to the curve ,

the reb y los ing som e of the eccentr ic ity . F la t tend on s a re m ore su i tab le for use in

s l abs t ha n the i r mu l t i s t r a n d r oun d e q u iva l e n ts , be c a use o f t he i r sma l l e r l o ss o f

e c c en t r ic i ty . F o r e xa mple , c ons ide r two t e ndons , e a c h c on ta in ing f ou r 13 m m (0 .5

in) s t r ands , one in a f ia t shea th ing and the o the r in a c i r cu la r shea th ing . For a

g iven concre te cover , the he ight o f cen t ro id of s tee l above the sh ea th in g soff it i s :

S h e a th i n g R a d i u s B u n c h i n g T o t a l

Fla t 9 .5 + 2 = 11.5 m m (0.45 in)

M ul t i s t r a nd 22 .5 + 9 = 31 .5 m m (1 .24 in)

However , a f la t shea th ing i s more d i f f icu l t to bend in a hor izonta l p lane than a

r oun d she a th ing . T he l a t t e r shou ld be p r e fe r re d i f t he bond e d t e n don s ha ve to be

curved hor izonta l ly , fo r example to avoid a ho le in the f loor .

T e ndons ne e d to be suppor t e d a t r e gu la r i n t e r va l s . B onde d t e ndons a r e mor e

r ig id t h a n m o n o s t r a n d b u t t h e y w e ig h m o r e . T h e n o r m a l p r a c ti c e is t o s u p p o r t

b o n d e d t e n d o n s a t n o t m o r e t h a n 1 .2 m c e n tr es a n d m o n o s t r a n d u n b o n d e d a t

1 m (say 4 f t an d 3 ft respect ively) .

T h e m i n i m u m r a d iu s o f a b o n d e d t e n d o n is n o r m a l l y g o v e r n e d b y t h e ca p a c it y

o f t he she a th ing to be n d w i thou t su f fe ring a ny da m a ge . I t m a y be l a r ge r t ha n the

2 .5 m ( 8 f t) no r m a l ly c ons ide r e d a m in im um f o r unb on de d t e ndons . C ur va tu r e

c a pa c i ty da t a f o r t he she a th ing shou ld be ob ta ine d f r om the supp l i e r .

2.6 Equipment

T he spe ci a li s t e q u ipm e n t r e q u i r e d f o r pos t - t e ns ion ing c ons i st s o f the f o l lowing

i tems, no t a l l o f which w ould be requi red on s ite.

S t re s s ing j a c k a n d p um p

S wa g ing j a c k

S t r a n d t h r e a d i n g m a c h i n e

S t r a nd c u t t e r s o r she a r s

G r o u t m i x e r a n d p u m p

T he s t r a nd th r e a d in g m a c h ine a nd the g r ou t ing ou t f i t a r e re q u i r e d on ly fo r

bon de d t e ndons . S t r a nd m a y be th r e a de d in to t he she a th ing e i the r be fo r e p l a c ing

the shea th ing in pos i t io n , o r a f te r i t has been p laced bu t be fore concre t ing , o r a f te r

c onc r e ting . T he push ing ma c h ine w ou ld , o f c ou rse , no t be ne e de d on s it e i f t he

t e ndon l e ng ths a r e sho r t e nough to be de l ive r e d to s i t e r e a dy th r e a de d .

Jacks a re des igned to gr ip the s t r and (s ) e i the r a t the f ron t o r a t the rea r . In the

la t te r case the des ign of j ack i s m echan ica l ly s imple r bu t i t needs an ex t ra length of

s t r a nd ( o f t he o r d e r o f one me t r e bu t c he c k f o r t he pa r t i c u l a r j a c k to be u se d )

which i s cu t o f f a f te r s t re ss ing . The o lde r jacks d id n o t have the fac il ity for push ing

the we dge c one f o r wa r d a f te r s tr e s sing a nd th i s u se d to be a ma n ua l ope r a t ion ;

now a lmo s t a ll j a c ks a u tom a t i c a l ly mo ve the we dge s f o r wa r d in to t he ba r r el t o

loc k the s t r a nd a s pa r t o f t he a u to m a te d s t re s s ing ope r a t ion . J a c ks a r e e q u ippe d

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56

POST-TENSIONED CONCRETE FLOORS

wi th fac il ity fo r m easur ing ex tens ion o f s t rand and a re ca l ib ra ted wi th the i r

hydrau l i c pum ps so tha t a d i rec t read ing o f j ack ing fo rce is d isp layed .

Ap a r t f ro m m an o eu v rab i l i t y w i t h r eg a rd t o t he h an d l i n g equ i p m en t a n d s p ace

ava i l ab le a t a par t i cu la r s it e, the im po r tan t fea tu res o f a j ack a re i ts fo rce capac i ty

and the s t roke . Lon g t endo ns , i f s t res sed wi th a j ack o f shor t s t roke , m ay h ave to

be s t res sed in s t ages . The p rocess requ i res the anch or ing cones to g r ip the s t ra nd

and re lease i t s evera l t imes , which may weaken the se r ra t ions .

The s izes and weigh t s o f equ ip m ent vary in d i f fe ren t sys tems bu t those fo r

monos t rand use a re very s imi la r in shape and s i ze . These and the i r pumps a re

l igh t eno ugh to be hand led m anual ly . F la t t e ndon s a re usua l ly st ressed us ing the

m o n o s t r a n d j a c k .

Mult is t rand jacks from different suppl iers d i ffer in shape, s ize and weight .

Som e are o f a s imi la r shape to the norm al m on os t ran d jack , bu t b igger , whi le

o t h e r s a r e m u c h l a r g e r i n d i a m e t e r ~ u p t o 4 0 0 m m ( 1 6 i n ) ~ a n d s h o r t e r . A

mu l t i s tr an d j ack is mu ch h eav i e r t h an a m o n o s t r an d j ack , it ma y w e ig h a s m u ch

as 300 kg (650 lb). I t therefore need s cran e t ime d urin g s t ress ing. B ecause o f the

wide variat ion in equipment from different sources , no detai l s are g iven. I t i s

recommended tha t par t i cu la rs a re ob ta ined f rom spec ia l i s t s in the a rea .

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MATERIALS AND EQUIPMENT 57

F igu re 2 .14 Dead end anchorages

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58

POST-TENSIONEDCONCRETE FLOORS

Figure 2 .15

Flat stressing anchorage

Figure 2 .16

Flat coupler

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MATERIALS AND EQUIPMENT 59

Figure 2.17 Exploded view of monostrand anchorage

Figure 2.18

Monostrand Jack

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60 POST-TENSIONEDCONCRETE FLOORS

Figure 2 .19 Swaging Jack

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3 S L A B C O N F I G U R A T I O N

A floo r sy s te m c ons i s ts o f va r ious a r r a n ge m e n t s o f s l ab a nd be a m e l e me n t s ,

show n in F igure 3 .1. These e lements , and som e of the com m only used a r rangem ents ,

a re d iscussed in th is chapte r . Poss ib le appl ica t ions of the va r ious con f igura t io ns

a re a l so sugges ted . These a re m ean t o n ly for genera l gu idance ; in fac t , the re i s a

c ons ide r a b le ove r l a p in t he a pp l i c a t ions .

3 . 1 G e n e r a l

Co ncre te f loors have a va r ie ty of shapes . These can be sum m ar ized as :

9 f iat s labs , wi th or w i tho ut d ro p p ane ls

9 r ibb ed an d waffles f loors

9 be a m a nd s l a b c ons t r uc t ion

In re inforced concre te the te rm

flat slab

i s used to re fe r to a s lab w i th or w i tho ut

d r o p s a n d s u p p o r t e d , g e n e r a l l y w i t h o u t b e a m s , b y c o l u m n s w i t h o r w i t h o u t

c o lum n he a ds . I t i s un de r s to od tha t a fi at s la b w i ll be de s igne d spa nn in g in two

d i r e c tions , supp o r t e d on be a m s t ri p s o f t he s a me de p th a s the s l a b . It ma y be so l id

or r ibbed in bo th d i rec t ions ; the la t te r i s ca l led a waffle slab. In designing a f ia t

s lab , each pane l i s d iv ided in to c o lu m n an d m iddle s t r ips in each d i rec t ion , which

a r e a na ly se d a nd r e in f o r c e d in a c c o r da nc e w i th t he na t iona l s t a nda r ds .

In pos t - tens ione d con cre te , the re i s no genera l ly accepted de f in i t ion of a f ia t

s lab and the te rm is appl ied to any s lab which has a f ia t sof f i t , wi th or wi thout

d r op pa ne l s . T he pa ne l s ma y be de s igne d a s c o lumn a nd midd le s t r i p s ( t hough

th is is no t ve r y c om m on) , a s one - wa y spa ns su ppo r t e d on s t r i ngs o f s t r ip be a m s ,

o r a s two- w a y spa ns su ppo r t e d on a g r id o f s t r ip be a ms . A

strip

o r

band

bea m is a

s t r ip o f s l ab in li ne w i th t he c o lum ns wh ic h ha s be e n m a de s t r ong e n oug h to

suppor t the ad jacent s lab pane ls . A f la t s lab in pos t - tens ion ing does no t desc r ibe

a ny pa r t i c u l a r s t r uc tu r a l sy s t e m in t he s a me ma nne r t ha t a r e in f o r c e d c onc r e t e

f ia t s lab does . The d is t inc t ion i s pure ly v isua l . The te rm waffle i s a l so appl ied to

pos t - t e ns ione d f loo r s w i th r i b s r unn ing in two d i r e c t ions .

T he be a m a nd s l a b c on f igu r a t ion ne e ds a g r e a t e r c ons t r uc t ion de p th tha n

o the r f l oo r c on f igu r a t ions , a nd i s no t ge ne r a l ly u se d . T he a r r a nge me n t c a n be

one - wa y spa n n ing o r two- w a y , a nd the de s ign p r oc e du r e i s t he s a me a s tha t f o r

o the r one - wa y spa nn ing c on f igu r a t ions .

F r o m the s t r uc tu r a l de s ign po in t o f v i ew , it i s pe r ha p s m or e m e a n ing f u l t o

c lass i fy f loors accord ing to the i r in te rna l s t ruc tura l sys tem, such as :

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62 POST-TENSIONEDCONCRETE FLOORS

'F lat slab without drop panels

Flat s lab wi th drop panels

Beam and s lab

affle

Ribbed

Figure 3.1 Slab configurat ions

Band beam

9 o n e - o r t w o - w a y s p an n i n g

9 so l i d o r r i bbed

9 w h i ch e l em en t s a r e p o s t - t en s i o n ed

E ac h co m b i n a t i o n o f t h e ab o v e f ea t u r e s h a s i ts me r i t s an d i t s u s e s ; n o n e can b e

s a i d to b e i n h e r en t l y b e t t e r t h an t h e o t h e r s i n a ll c i r cu m s t an ces an d a l l l o ca t i o n s .

T h e c h o i c e b e t w e e n t h e m i s a m a t t e r o f t h e r e q u i r e m e n t s o f t h e p a r t i c u l a r

a p p l i c a t i o n , a n d o f e c o n o m y .

F i g u r e 3 .2 s h o w s s o m e o f t h e t y p i ca l t en d o n l ay o u t s a s s o c i a t ed w i t h t h e

v a r i o u s s l ab co n f i g u r a t i o n s . D i ag r am s ( a ) t o ( c) a r e s u i t ab le f o r o n e - w a y

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SLAB CONFIGURATION 63

3

I r 1

( a )

(b)

One-way slab panels

( c )

mm

mm

m m

+

m~

m,

II

mm

m

mm

) J

~ l l i i .m , .

" " I I~I

i i i t t t

I l l I t lm l

l l i I l i i

)

i i i t i l t

I I I I I I

I l l I l l

.:! iPT,

I I I I I

i i i i i l

I l l i i l

i i i l i i

I l l I I !

. . m . i I I l m m

I l l l im p

I I I I I I

III ill

(d) (e) (f)

Two-way slab panels

Figure 3.2 Tend on layouts (arrows indicate reinforced concrete spans)

spann ing s l abs , and (d ) to ( f ) fo r two-way s l abs . For fu r ther comments on the

associated panel configurat ions see Sect ion 3 .3 .

Ten dons in a s l ab pane l a re e i ther un i fo rmly spaced across i ts wid th , o r two o r

th ree t endons a re g rouped toge ther and the g roups d i s t r ibu ted un i fo rmly . A

grea te r f l ex ib i l i ty fo r mak ing ho les i s ach ieved by keep ing the anchorages

u n i fo rml y s p aced an d t h en b u n ch i n g t w o o r t h r ee t en d o n s t o g e t h e r aw ay f ro m

the anch orag e po in t s ; th i s , o f course , is no t poss ib le fo r shor t t end on l eng ths .

Beam tendons can a l so be g rouped toge ther in a s imi la r manner bu t the spac ing

of even the bunche d t endon s rem ains c lose and on ly smal l ho les can be p rov ided .

In Figure 3 .2(a) , only the s lab is pres t ressed, i t spans on ei ther reinforced

concre te beams o r load-bear ing wal l s ; so the t endons a re un i fo rmly spaced and

run in one d i rec t ion on ly . Th i s a r rangement i s used e i ther where the beams are

sha l low a nd span shor t d i s t ances wi th in the capac i ty o f re in fo rced concre te , o r

where wal ls or deep reinforced concrete beams are acceptable; essent ial ly i t su i ts

rec tang u lar pane l s o f abo u t 2 :1 aspec t ra t io .

In Figure 3 .2(b) the s lab is in reinforced concrete and spans on post - tens ioned

beam s t rips . The a r ran gem ent i s su i tab le for approx im ate ly squ are pane l s , where

s t rip beams are requ i red and the span i s too long fo r re in fo rced concre te . S ome of

the ax ia l com pon en t o f the p res tress is abso rbed by the ad jacen t re in fo rced

concre te s l ab . I t is necessary to assess the p rop or t ion thus los t f rom the b eam and

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64 POST- TENSIONED CONCRETE FLOOR S

to check the se rv iceabil i ty s tr e sses accord ingly . In o rde r to r educe the loss , a few

a dd i t i ona l t e ndo ns r un n ing pa r a l le l t o t he be a ms a r e o ft e n p r ov ide d in the s l a b a t

i ts c e n tr o id . T he se t e ndo ns d o n o t d i r e c t ly c o n t r ibu t e t o t he s t r e ng th o f t he s l a b

bu t a r e u se f u l i n d i s t r i bu t ing loa d c onc e n t r a t i ons a nd c on t r o l l i ng sh r inka ge .

F igure 3 .2(c ) i s a combina t ion of ( a ) and (b) , the s lab and the beams a re bo th

p o s t - te n s i o n e d . It p e r h a p s i s t h e m o s t c o m m o n l y u s e d a r r a n g e m e n t . I n o r d e r t o

r e duc e the l o s s o f a x ia l p r e s tr e s s f r om the be a m s , a f ew t e ndon s m a y be p r ov id e d

in the s lab as d iscussed above . The a r rangement su i t s a pane l a spec t r a t io of

a bo u t 1 :5 i f t he be a m s t ri p s a r e t o be c on ta ine d w i th in t he s la b de p th ; i f no t t he n

the a r r a nge me n t c a n be u se d w i th a ny a spe c t r a t i o up to a sq ua r e pa ne l .

F igu r e 3 .2 ( d ) shows the t e ndon a r r a nge me n t whe r e two- wa y r e in f o r c e d

c onc r e t e s la b pa ne l s spa n on a g rid o f pos t - t e ns ion e d be a m s . S om e o f t he be a m

axia l p res t re ss i s los t in the s lab . This a r rangement i s su i tab le for nea r ly square

pane ls of r e la t ive ly shor t spans , n o t ex ceeding say 8 m, w hich i s wi th in the

capac i ty of r e inforced concre te . This a r ran ge m ent i s foun d in waf fle f loors where

on ly the be a m s t r i p s a r e pos t - t e ns ione d .

F igu r e 3 .2 (e ) shows a n a r r a nge m e n t whe r e t he s la b spa ns i n two d i r e c tions b u t

i s p res t re ssed in on ly one d i rec t ion ; in the o the r d i rec t ion i t ac ts a s a r e inforced

c onc r e t e pa ne l. I t m a y be o f u se whe r e t he s la b spa n is longe r t ha n c onve n ie n t f o r

re inforced concre te an d so m e ass is tance f rom pos t - tens ion in g is r equi red . T his is

a t ru ly pa r t ia l ly pres t re ssed s lab and a ca re fu l a ssessme nt of c rack w id ths an d

def lec t ion i s r equi red wi th th is a r rangement .

F igu r e 3 .2 ( f ) shows the a r r a nge me n t f o r a two- wa y pos t - t e ns ione d s l a b

cons is t ing of nea r ly squ are pane ls . I t i s su i tab le for so l id s labs where a m in im um

cons t ruc t ion depth i s r equi red . The a r rangement i s a l so used in waf f le f loors . I t

r e q u i r e s t he t e ndon a s se mbly s e q ue nc e to be c a r e f u l ly wor ke d ou t be c a use the

be a m a nd the s l a b t e ndons a r e t o some e x te n t i n t e r - wove n . F o r r e a sons o f

d i f f icu l ty in ins ta l la t ion , th i s a r rangement i s no t p re fe r red .

3 .2 S t r u c t u r a l e l e m e n t s o f a f lo o r

A f loor sys tem essen t ia l ly cons is t s o f s lab and bea m e lements . A s lab can be so l id

o r r i b b e d , a n d o n e - w a y s p a n n i n g o r t w o - w a y ; i t m a y , o r m a y n o t , h a v e d r o p

pa ne l s ove r t he c o lum ns . A be a m c a n be a dow ns ta n d , a n u ps t a n d o r a s t ri p . T he

var ious e lements a re d iscussed in th is sec t ion .

3 .2 . 1 R ib s i n c on c re te f l oo rs

Ribbe d f loo r s a r e t he mos t c ommonly u se d a l t e r na t ive t o so l id f l oo r s . T he y

requi re le ss ma te r ia l and the reduced we ight leads to fur the r sav ings in the

c o lum ns a nd subs t r uc tu r e . R ibs r unn ing in two d i r e c tions , o f c ou r se , c ons t i t u t e a

waf f le f loor . This sec t ion i s equa l ly appl icab le to one and two-way spanning

f loors , i .e . , to sol id and r ibbed slabs and to sol id and waff le s labs.

In re inforced as we ll a s in pos t - ten s ion ed co ncre te , i t i s the no rm al p rac t ice to

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a vo id sh e a r r e in f o rc e me n t i n r ib s . T he se c t ion is c ha nge d ne a r t he sup por t f r om

r ibbe d to so l id a t a po in t w he r e she ar , o r t he supp or t m om e n t i n the c a se o f a

con t inu ou s f loor , i s l i ab le to be c r it ica l . This m eans th a t the r ibs a re a lm ost nev er

subjec ted to compress ive s t r e sses induced by appl ied loads . The d iscuss ion

be low , the re fore , a ssume s th a t the r ib is on the ten s ion face of the f loor .

At the u l t im a te s tage of load ing a r ibbed f loor is iden t ica l to a so l id f loor in bo th

r e in f o rc e d c onc r e t e a nd in pos t - t e ns ione d c ons t r uc t io n a s l ong a s t he de p th o f t he

c ompr e s s ion b loc k r e ma ins w i th in t he t opp ing . F r om the s e r v i c e a b i l i t y de s ign

po in t o f v iew, how ever , the s t ru c tura l func t ion of a r ib in r e inforced concre te i s

ma r ke d ly d i f f e r e n t f r om tha t i n pos t - t e ns ione d c onc r e t e .

F i r s t , cons ide r a r ib in r e inforced concre te in the pos i t ive mo m en t r eg io n of a

f loo r . A t midsp a n , w he r e t he mo m e n t i s m a x im um , the s e c tion c a rr ie s z e r o she a r

and the on ly func t ion o f the r ib is to con nec t the to pp ing concre te ( the

c om pr e s s ion zone ) a nd the r od r e in f o r c e me n t ( the t e ns ion e l eme n t ) . I t doe s no t

d i rec t ly cont r ib u te to the s t r eng th of the f loor , and can be cons ide red to se rve as a

se pa r a to r be twe e n the c ompr e s s ive a nd t e ns il e c om pon e n t s a nd a s a p r o t e c t ion to

s te e l a ga in s t c o r r o s io n a nd fire . A t a ny o the r p o in t i n t he pos i t ive m om e n t r e g ion ,

the r ib c a rr ie s some she a r f o rc e c o r r e spo nd ing to t he c ha nge in m om e n t bu t a ga in

i t makes no d i rec t cont r ibu t ion to the f lexura l s t r ength . There fore , in r e inforced

concre te the re i s no need to make the r ib wide r than s t r ic t ly necessa ry to

a c c o m m oda te t he r e in f o rc e me n t , p r ov ide d o f c ou r se t ha t t he she a r fo r ce r e ma ins

wi th in i t s capac i ty .

N ow cons id e r a r ib in a pos t - tens ione d f loor . A t the t ime o f s t r e ssing , the

pres t re ss ing force induces a comp ress ive s t r ess in the r ib . La te r , w hen the s lab i s

r e q u i r e d to c a r r y t he impose d loa d , t he f l e xu r a l t e n s ion ha s t o ove r c ome th i s

com press ion b e fore the r ib goes in to tens ion . The r ib e ffect ive ly ac ts a s a r e se rvoi r

of com press ive s t r e ss which i t re leases a s and wh en requi red . There fore , in

pos t - t e ns ion ing the r i b w id th , o r r a th e r t he r a t i o o f r i b w id th to r ib spa c ing ,

ma ke s a d i r e c t c on t r ibu t ion to t he f l e xu r a l s t r e ng th .

H o w m u c h c o m p r e s s i o n s h o u l d b e s t o r e d i n t h i s m a n n e r ? T h e m a g n i t u d e o f

the a pp l ie d m om e n t r e p r ese n t s t he r a nge o f s tr e ss va r i a t ion . A t the op t i m um

value of pres t re ss ing force in an idea l sec t ion , wh en fu l ly loaded , the r ib wo uld

have a tens i le s t re ss exac t ly equa l to the p e rmiss ib le va lue . The con cre te an d the

p r e s tr e s s wou ld p r oba b ly be mos t e ff ic ie n tly u se d i f und e r p e r m a ne n t l o a d the

s t re ss in the bo t to m f ibre of the r ib were equa l to the pe rm iss ib le limi t in

c om pr e s s ion a nd u nde r a pp l i e d loa d ing the s t re s s e q ua l l e d the m a x im um a l lowe d

tens i le va lue , though th is may not be the cheapes t des ign . In any case , such a

so lu t ion i s un l ike ly to be ach ieved in prac t ice .

T he se lf -we igh t o f t he c onc r e te f l oo r is no r m a l ly ba l a nc e d by the u pw a r d

r e a c t ion f r om the t e ndon c u r va tu r e ; t he p r e s t r e s s p r ov ide d i s o f t e n e nough to

ba lance a l l o f the dead lo ad an d p a r t o f the appl ied load . There fore the r ib

rese rvoi r need no t s to re an y com pres s ion for se lf -we ight o f the f loor and , in th is

pa r t ic u la r d iscuss ion , se l f-we ight can be cons id e red to be of no im por tanc e . I f the

a pp l i e d loa d i s l a rge the n a c o r r e spo nd ing ly l a rge r c om pr e s s ion m us t be s to r e d in

the r i b a nd th i s wou ld ne e d a w ide r r i b , a s suming tha t t he de p th i s a de q ua te .

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POST-TENSIONED CONCRETE FLOOR S

T h er e f o r e , t o a la r g e ex t en t t h e r a t i o o f r i b w i d t h t o s p ac i n g d ep e n d s o n t h e

m a g n i t u d e o f a p p l i e d l o a d .

I f a p o s t - t en s i o n ed r i b b ed s ec t i o n is f o u n d t o b e i n ad eq u a t e t h en t w o o p t i o n s

a r e a v a i l a b l e - - i n c r e a s e t h e ri b w i d t h a n d / o r t h e d e p t h . I n r e in f o rc e d c o n c r e t e , t h e

r i b w i d t h m ak e s n o d i r ec t co n t r i b u t i o n t o i ts fl ex u r a l s tr en g t h , a n d s o it i s o n l y t h e

d ep t h t h a t c an b e i n c r ea s ed .

T h e d i s t i n c t i o n b e t w een a r i b an d a b eam i s n o t c l ea r l y d e f i n ed ; i t i s o f

p a r t i cu l a r i n t e r e s t t o t h e d e s i g n e r . T h e v a r i o u s n a t i o n a l co d es o f t en s p ec i f y

d i f f e r en t f i r e co mp l i an ce an d r o d r e i n f o r cemen t co n t en t f o r t h e t w o . M o r e

c o n c r e t e c o v e r is r e q u i r e d i n a b e a m t h a n i n a ri b , a n d a m i n i m u m a m o u n t o f r o d

r e i n f o r cemen t mu s t b e p r o v i d ed i n a b eam w h i l e n o n e , o r a l e s s e r q u an t i t y , i s

s p ec if i ed f o r a r i b. M o s t o f t h e s t a n d a r d s a l s o s p eci fy an u p p e r l i m i t fo r t h e

spac ing o f r i b s .

I n a p o s t - t en s i o n ed f l o o r, it m ay b e mo r e e co n o m i ca l t o p r o v i d e r i b s a t a w i d e r

s p ac i n g t h a n t h e ma x i m u m s p eci fi ed, an d d e s i g n t h em a s b eam s . I n th i s c a s e s o m e

ad d i t i o n a l r o d r e i n f o r cemen t i s n eed ed i n t h e r i b s , b u t t h e r e w o u l d b e f ew er o f

t h em. T h e ca s e i s i l l u s t r a t ed i n E x amp l e 6 .2 .

BS 8110 spec i f i es t ha t r i b s shou ld be spaced a t cen t r es no t exceed ing 1 .5 m (

5 f t) an d t h e i r d ep t h , ex c l u d i n g a n y t o p p i n g , s h o u l d n o t ex ceed f o u r t i mes t h e i r

w i d t h . T h e m i n i m u m t h i c kn e s s o f t h e t o p p i n g s h o u l d b e o n e - t e n t h o f t h e c l e ar

d i s t an ce b e t w een t h e r ib s b u t n o t l e ss t h an 5 0 mm . T h e t o p p i n g t h i ck n e s s i s o f t en

g o v e r n ed b y f i r e co n s i d e r a t i o n s .

ACI 318 spec i f i es t ha t r i b s sha l l be no t l ess t han 100 mm in wid th ; and sha l l

h av e a d ep t h o f n o t m o r e t h an 3 89 i mes t h e m i n i m u m w i d t h o f r i b. C l ea r s p ac i n g

b e t w een r i b s s h a l l n o t ex ceed 8 0 0 mm.

T h e r e i s n o r eq u i r emen t f o r l i n k s i n t h e r i b s b u t s o me l i n k s a r e n o r ma l l y

p r o v i d e d t o s u p p o r t t h e t e n d o n s a n d t o h o l d t h e b o n d e d r o d r e i n f o r c e m e n t i n

p o s i t i o n .

3 . 2 . 2 B e a m s

I n c r o s s s ec t i o n , a b eam can h av e i t s s t em p r o j ec t i n g b e l o w t h e s o ff it o f t h e s l ab

(downstand), o r a b o v e (upstand), o r i t c an b e w i d e an d s h a l l o w , i n w h i ch ca s e i t i s

t e r m e d a band o r strip beam, s ee F i g u r e 3 .3. S t r ip b ea m s o f t h e s ame d ep t h a s t h e

s l ab a r e p r e fe r r ed , b ecau s e t h e s l ab t h en h a s a fl a t s o ff it w i t h o u t an y d o w n s t an d s .

Downstand Upstand Strip beam Strip beam

Figure 3.3

Beam sec t ions

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D o w n s t a n d b e a m s

D o w n s t a n d s a r e th e t r a d i t i o n a l f o r m o f b e a m s . B e i ng d e e p a n d n a r r o w , t h e y a r e

t h e mo s t e f fi c ien t i n mak i n g u s e o f t h e co n c r e t e an d t h e s t ee l. T h ey a l s o h av e a

h i g h m o m e n t o f i n e r t ia c o m p a r e d t o s t ri p b e a m s , a n d t h e f la n g e is a v a i l a b le f o r

co mp r es s i v e s t r e s s e s i n t h e s p an . T h e r e f o r e , a d o w n s t an d b eam h as t h e l e a s t

def l ec t i on compared wi th t he o ther two shapes . I t s h igh s t i f fness i s very usefu l on

a b u i l d i n g f acad e w h i ch r eq u i r e s a t i g h t co n t r o l o n d e f l ec t i o n s .

A f l a t s o f f i t i s g en e r a l l y p r e f e r r ed b y t h e b u i l d i n g o ccu p an t s , an d b y t h e

b u i l d e r s . D o w n s t an d s a r e s een a s i n t r u d i n g i n t o t h e s p ace b e l o w t h e s l ab s o f f i t ,

s p a c e w h i c h c o u l d o t h e r w i s e h a v e b e e n u s e d f o r a c c o m m o d a t i n g s e r v i c e r u n s .

F o r m w o r k f o r a d o w n s t a n d b e a m i s m o r e c o m p l i c a t e d t h a n t h a t f o r a s t r i p

b e a m , a n d s t r i p p i n g o f f o r m w o r k i s t i m e - c o n s u m i n g . T a b l e f o r m s a n d s i m i la r

f a s t - t r a ck i n g d ev i ce s a r e mo r e d i f f i cu l t t o u s e w i t h d o w n s t an d s .

N e a r s u p p o r t s t h e t o p o f t h e b e a m is o f te n c o n g e s t e d w i t h r e i n f o rc e m e n t a n d

t en d o n s , an d ca r e h a s t o b e ex e r c i s ed i n d e t a i l i n g b eams t o en s u r e t h a t en o u g h

s p a c e i s a v a i l a b l e f o r c o m p a c t i o n e q u i p m e n t .

F o r t h e se r e a s o n s d o w n s t a n d s a r e c u r re n t l y v e r y m u c h o u t o f f a v o u r. T h e y a r e

u s ed o n l y w h e r e u n av o i d ab l e , s u ch a s a l o n g t h e ex t e r n a l ed g es f o r a r ch i t e c t u r a l

r ea s o n s an d i n t r an s f e r s t r u c t u r e s , s u ch a s b eams w h i ch a r e r eq u i r ed t o s u p p o r t

t h e c o l u m n s a b o v e .

N a t i o n a l s t an d a r d s s p eci fy th e fl an g e w i d t h w h i ch s h o u l d b e t ak e n a s p a r t o f a

b eam . T h e w i d t h i s r eq u i r ed f o r c a l cu l a t i n g t h e m o m en t o f i n e r t i a o f t h e s ec t io n ,

wh ich i s needed fo r comput ing t he s t r esses and def l ec t i ons . BS 8110 g ives :

fo r T-b ea m s , b r + Lz /10 o r ac tu a l f l ange w id th i f l es s,

fo r L-b eam s , b r +

L z / 5

or ac tu a l f l ange w id th i f l ess .

w h e r e b r - - s t em w i d t h

Lz = d i s t an ce b e t w een p o i n t s o f z e r o mo m en t , w h i ch can b e t ak en a s 0.7

t i mes t h e e f f ec t i v e s p an f o r co n t i n u o u s b eams .

A C I 3 18 d oe s n o t m a k e a n y r e c o m m e n d a t i o n f or th e f la n g e w i d t h o f a

p o s t - t en s i o n ed T - o r L - b e am . I n s t ead , t h e d e t e r m i n a t i o n o f an e ff ec ti ve w i d t h o f

f l ang e is l ef t t o t h e ex p e r i en ce an d j u d g em en t o f t h e en g i n ee r . H o w ev e r , f o r

r e i n f o r c e d c o n c r e t e i t r e c o m m e n d s :

fo r T -be am s , t he l eas t o f b r + 8 .d s o r L/4 o r a c t u a l

fo r L-b eam s , t h e l eas t o f b r + 6 .ds o r L/12 o r ac t u a l .

w h e r e d s = s l ab t h i ck n es s

L = s p an l en g t h .

A m e t h o d o f c o n s t r u c t i n g d o w n s t a n d s , c a l le d th e shell beam s y s t em , a n d s h o w n i n

F i g u r e 3 .4 , i s g a i n i n g p o p u l a r i t y i n E u r o p e . T h e s y s t em i s d e s i g n ed f o r f a s t

co n s t r u c t i o n . I t co n s i s t s o f p r e f o r m ed c o n c r e t e U - s ec t i o n s w h i ch s e r ve a s

p e r m a n e n t s h u t t e r i n g f o r t h e b e am s . T h e p r e f o r m e d u n i ts a r e a b l e t o s p a n s h o r t

d i s t an ces , s o t h ey n eed f ew er v e r t i c a l p r o p s . T h e s y s t em , o b v i o u s l y , c an b e u s ed

o n l y w h e r e d o w n s t a n d s a r e a c c e p t a b l e .

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POST- TENSIONED CONCRETE FLOORS

' J I

, j j j j j / j / j j j j j j j j 1 - 4

Precast shell beam

Flat duct

Precast shell beam

Precast shell beam

Figure

3.4

She l l- beam a r rangem en t

The beam shel l i s prefabricated wi th holes in sui table posi t ions for the s lab

an ch o rag es w h e re n eed ed . Th e d o w n s t an d b eam can b e e it h e r of t h e co n v en t i o n a l

d eep an d n a r ro w s h ap e , o r it c an b e w id e an d s h a ll o w . Th e p e rm an en t fo rm w o rk

for the s l ab m ay cons i s t o f co r ru ga te d s tee l deck ing un i t s as com m only used in

s tee lwork cons t ruc t ion , o r o f p recas t p re ten s ioned th in p lanks , say 50 m m (2 in )

deep . These a l so need some propp ing .

The p recas t un i t s a re usua l ly p re tens ioned , made by the long l ine method .

At tem pts have been m ade to m ake s t ruc tu ra l use o f these un i ts by inco rpo ra t ing a

suff ic i en t ly h igh p res tress to car ry m os t o f the dead and super im posed loads .

This , howe ver , ca n only be of a l imi ted success , if the uni ts are no t to be too hea vy

for t ranspor t ing and hand l ing . Also , the p res t ress f rom the p recas t un i t i s no t

t rans fer red to the

in situ

concre te and sh r inkage o f the

in situ

concre te ma y cause a

di fferent ial s l ip at the in terface. Therefore, fu l ly composi te act ion between the

precas t un i t and the

in situ

con crete i s d i fficult to dev elop; p rovis io n of shea r l inks

be tween the two concre tes , however , does he lp in th i s respec t .

Th e s h e ll -beam s y s tem p ro v i d es a p e rm an en t fo rmw o rk r equ i r in g a mi n i m u m

of p rop p ing and i t is qu ick . I t can be used wi th adv an ta ge w here dow ns ta nds a re

acceptable and precas t ing faci l i t ies are eas i ly avai lable .

Upstand beams

U p s t a n d b e a m s a v o i d t h e m a i n p r o b l e m w i t h d o w n s t a n d s ~ i n t r u s i o n i n t o t h e

space below the s lab soff i t , and they al low the soff i t formwork to be f la t and

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6 9

unc ompl i c a t e d . U ps t a nds , howe ve r , c a n on ly be p r ov ide d in l imi t e d pos i t i ons ,

suc h a s a long e x t e rna l e l e va tions a nd a r ou nd some o f t he ope n ings t h r o ug h f loo rs .

A n u ps t a n d i s no t a s ef fi ci en t i n ma k ing u se o f t he ma te r i a ls a s a dow ns ta nd

be c a use i t on ly ha s t he na r r ow r ib i n c ompr e s s ion in no r ma l f l e xu r e whe r e a s a

do w ns tan d ha s the ad jace nt s lab ac t ing as a wide flange. I t , the re fore , has a h igher

de f l e c t ion tha n a downs ta nd .

T he ve r t ic a l f o r ms f o r ups t a n ds m us t be p r ope r ly r e s t r a ine d to e nsu r e t ha t t he

be a m wid th r e m a ins w i th in t o l e ra nc e , bo th a t it s ba se a nd a t t he t op , a nd tha t t he

bea m rem ains ve r t ica l . This is no t easy if the bea m is to be cas t w i th the s lab ,

because d iago na l b rac ing s a re d i f ficul t to ins tal l . A lso , i f the u ps tan d i s cas t wi th

the s lab then the ve r t ica l p ressure causes uph eava l o f the we t concre te in the

a d j a c e n t s l a b ; t o p r e ve n t t h i s, na r r ow shu t t e r s a r e r e q u ir e d on top o f t he s l ab .

U ps t a nds a r e u sua l ly c ons t r uc t e d a f t e r t he s l a b c onc r e t e ha s ha r de ne d . I n t h i s

case, s teps mu st be tak en to preve nt leakage o f f ine m a te r ia l f rom the jo in t wh ich

wou ld o the r w i se l ea ve a we a k c o ns t r uc t ion j o in t .

In sp i te of the i r d isadva ntag es , u ps ta nds a re of ten use fu l a t ve r tica l duc ts w here

se rv ice runs coming out o f the duc t c lose to the s lab sof f i t would no t a l low a

do wn s ta n d a n d the spa n is t oo long o r the sup por t t o o na r r o w f o r a s tr i p be a m.

F o r c a n t i le ve rs , t he ups t a nd is the p r e fe r re d sha pe o f t he be a m r a the r t ha n the

dow ns ta n d , be c a use the f l ange a t t he bo t tom o f t he s e c tion is i n com pr e s s ion . I n

th is case, the ex press ions g iven in 3 .2 .2 for the f lange wid th apply , bu t L z can be

take n as twice the can t i leve r span p rovid ed , o f course , tha t the f lange cont inu es

be yond the c a n t i l e ve r r oo t .

B a n d o r s tr ip b e a m s

A s t r ip beam , be ing w ide and sha l low, has the leas t ef fect ive d epth of the th ree

a l te rna t ives . I t , the re fore , uses s tee l in the m ost ine ff ic ien t m an ne r of a l l, bu t i t

doe s no t ne e d a ny a dd i t i ona l c onc r e t e o r spe c i a l f o r mwor k .

A s t ri p be a m is u sua l ly w ide r t ha n the c o lum n , so t ha t i t be c om e s ne c e s sa r y to

p r ov ide a r e in f o rc e me n t c a ge a t t he supp or t t o t r a ns fe r t he l oa d f r om the be a m to

the c o lumn .

T he r od r e in f o rc e me n t a nd t e ndon s a r e e a sy to a s semble , t he c onc r et e c a n be

com pac ted w i th the same ease a s in the s lab and the re i s m uch less chance of a

t e ndo n b e ing da m a ge d by a v ib r a to r . A s t r i p be a m i s, t he re f o r e, t he e as ie s t a nd

f a st es t t o c ons t r uc t a n d i s m uc h f a vou r e d e ve n tho ugh i t ne e ds mor e t e nd ons a nd

poss ib ly mor e r od r e in f o r c e me n t .

Mo s t na t iona l s t a nda r ds spe cify a min im um a m oun t o f bond e d r od r e in fo r c eme n t

and l inks to be provid ed for the en t ir e length o f a beam . I f bea m s t r ips a re

ana lyse d as in tegra l pa r t s o f a s lab pane l th en i t i s pe rh aps no t necessa ry to

prov ide l inks over the wh ole of the s t r ip length ; in Aus t ra l ia l inks a re provid ed

only nea r the co lumns in such cases .

I f the beam s a re of such a s ize tha t shea r , o r s up po r t m om en t , i s c r i tica l and

inc re a s ing the be a m de p th is no t a v i ab l e op t ion the n d r op pa ne l s o r p r e f a b r ic a t e d

she a r he a ds ma y be u se d , t hough the f o r me r op t ion i s be s t a vo ide d .

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3.2.3 Drop panels

A drop pane l , show n in F igure 3 .1 , is used to enh ance the shear s t reng th o f a s t r ip

beam , o r to increase i ts fl exura l s t reng th ov er the suppo r t sec t ion . D ro p pane l s ,

h o w ev e r , m ak e t h e e rec t io n an d t h e remo v a l o f f o rmw o rk r a t h e r cu mb er s o me .

Eas i ly removab le fo rms , es sen t i a l fo r fas t - t rack ing , become more invo lved and

assembly o f re in fo rcement in the d rop pa ne l s can be a time-co nsum ing o pera t ion

i f a p refabr i ca ted cage can no t be d rop ped in pos i t ion .

W herever poss ib le , d ro p pane l s shou ld be avo ided a nd use o f a p rop r ie t a ry

shearh ead co ns idered ins tead . She arhead s a re dea l t wi th in de ta i l i n C hap ter 10.

3.3 Pan el conf igurat ion

Th e elemen ts of a f loor pane l were individual ly consid ered in Sect ion 3 .2. This

sec tion d iscusses com bina t ions o f pos t - t ens ion ing and re in fo rced concre te in one

and two-way spann ing f loors .

3.3.1 One-way span

A one-way s l ab requ i res t endons runn ing on ly in one d i rec t ionmalong i t s span .

In the o ther d i rec t ion su f f i c i en t rod re in fo rcement i s p rov ided to p reven t

sh r inkage c racks , to d i s t r ibu te any load concen t ra t ion , and to sa t i s fy the

requ i rem ents fo r the u l timate c ond i t ion and those o f the re levan t s t and ards .

One-way s l abs a re o f t en p refer red because they a re eas ie r to cons t ruc t and

because the y need fewer beam s; they a re a l so eas ie r to des ign an d de ta i l. A

p a r t i cu l a r ad v a n t ag e is t h a t o f mu ch r ed u ced co n g es t io n o f t en d o n s an d

re in fo rcement near the co lumns .

Fo r a one-way s lab spann ing on beam s , there a re four poss ib le a r rang em ents o f

beams and s l abs us ing re in fo rced concre te and pos t - t ens ion ing :

9 re in fo rced concre te s lab o n re in fo rced con cre te beam s

9 pos t - t ens ioned s l ab on re in fo rced concre te beam s

9 pos t - t ens ioned s l ab on pos t - t ens ioned beam s

9 re in fo rced concre te s l ab on pos t - t ens ioned beam s

The f i rs t case, where the s lab and the beam are both in reinforced concrete , i s

ou t s ide the scope o f th is book . A pos t - t ens ioned f loor spann ing on re in fo rced

concre te s t r ip beams i s f requen t ly used . The s l ab normal ly spans in the longer

di rect ion; i f the shor ter sp an is wi thin the c apa ci ty of reinforced co ncrete then th is

a r ran gem ent shou ld be cons idered . I t would have the mer i t o f us ing the two

techn iques to the i r bes t advan tages .

Rein fo rced concre te beams genera l ly have a span- to -de p th ra t io o f the o rder o f

20 , whereas the ra t io may be abou t 40 fo r a pos t - t ens ioned s l ab . Th i s sugges t s

tha t the a r ran gem ent is su i tab le wh ere the co lumn spac ing in the two d i rec t ions is

in the ra t io 2 :1 . In fac t, a range s t a r t ing a t 1 .5 :1 m ay be wo r th inves t iga t ing .

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SLAB CONFIGURATION 71

Below the 1 .5 :1 ra t io o f long to sho r t span leng th , i t is no r m al ly b e t te r to a do p t

t h e t h i rd o f t h e ab o v e co n f i g u ra t i o n s an d u s e p o s t - t en s i o n i n g fo r t h e b eam s t r ip s

as wel l as the s lab . Al te rna t ive ly , the s lab may be spanned in two d i rec t ions .

T h e l a s t co n f i g u ra t i o n , t h a t o f p o s t - t en s i o n ed b eam s an d a r e i n fo rced s l ab , is

r a t h e r r a r e . I t wo u l d b e u s ed i n s pec ia l c i r cu m s t an ces o n l y , s u ch a s wh en a s h o r t

s p an s l ab i s s u p p o r t ed o n a b r i d g i n g b eam .

In o n e -wa y f lo o r s wh e re b eam s a re p o s t - t en s io n ed , t h e ax ia l co m p o n en t o f t h e

bea m pre s t ress sp re ads in to the ad jac en t s lab , so tha t the fu ll p res t ress ing fo rce is

no t av a i lab le in the be am sec t ion . Th is does no t a f fec t the equ iva len t loa d o f the

t en d o n s o r t h e u l t i m a t e s t r en g t h o f t h e b eam b u t i t d o es m a t t e r i n s erv i ceab i li ty

ca lcu la t ions .

D o w n s t a n d b eam s , o r s t r ip b eam s s l ig h t ly d eep e r t h an t h e s l ab , c an b e u s ed i n

an y o f t h e s i t u a t io n s m en t i o n ed ab o v e . T h e d eep e r b ea m wi ll b e b e t t e r ab l e to

co p e w i t h t h e l o ad s f ro m t h e s lab p an e l an d m ay b e m o re eco n o m i ca l t h an a b eam

wi th in the s lab th ickness , though i t i s s lower to cons t ruc t - -see Sec t ion 3 .2 .4 .

3 .3. 2 Tw o-way span

A t wo -wa y s p a n n i n g s l ab m ak es u s e o f t h e co n c re t e i n b o t h d i r ec ti o n s . I t is

s h a l l o wer t h an a o n e -w ay s l ab b ecau s e it c a r r i es o n l y a p ro p o r t i o n o f t h e l o ad i n

each d i r ec ti o n . H o wev e r , b e i n g s h a ll o wer , t h e t en d o n s w o rk a t a s m a l l e r in t e rn a l

l ev er a rm , an d , t h e re fo re , a two -w ay s lab m ay n eed m o re t en d o n s t h an a d eep e r

o n e -w ay s l ab ca r ry i n g t h e s am e ap p l i ed l o ad . A t wo -w ay s l ab is o ft en u s ed wh e re

co n s t ru c t i o n d ep t h is to b e m i n i m i z ed . I t is a ls o u s ed wh e re s o m e c o n t ro l n eed s t o

be exerc ised on the d i s t r ibu t ion o f the s lab load in the two d i rec t ions ; the

pr inc ip le i s descr ibed be low.

In a r ec t an g u l a r s l ab p an e l s u p p o r t ed o n fo u r s i de s , t h e l o ad i s t r an s fe r r ed i n

each o f t h e t wo d i r ec t i o n s ; t h e p ro p o r t i o n o f l o ad d e p en d s o n t h e e l a s ti c

p ro p e r t i e s o f t h e s l ab i n th e t wo d i r ec t i o n s, an d o n t h e s p an l en g t h s . M o re l o ad

goes a long the shor te r span and in the s t i f fe r d i rec t ion . At a span ra t io o f

ap p ro x i m a t e l y 2 :1 m o s t o f t h e lo ad g o es a l o n g t h e s h o r t s p an an d t h e p an e l

e f fec t ive ly ac t s as one-way spann ing . Therefo re , two-way spans a re mos t usefu l

fo r span ra t ios in the range o f 1 :1 to abou t 1 .5 :1 ; above th i s the advan tage i s

g ra dua l ly los t. Tw o-w ay po s t - t ens io n ing o f a f loo r i s l ike ly to be used w here the

ra t io o f span leng ths in the two d i rec t ions i s in the ran ge 1 .0 to 1 .5 and the s lab

l o ad i n g i s h eav y , s u ch a s i n a wa reh o u s e , o r wh e re b o t h t h e s p an s a r e l o n g .

In a t wo -w ay r e i n fo rced co n c re t e s o li d s lab t h e p ro p o r t i o n o f l o ad t r an s fe r r ed

i n th e t wo d i r ec t i o n s is d e t e rm i n ed b y t h e r a t i o o f s p an l en g t h s , t h e p ro p o r t i o n

a l o n g s p an L x b e i n g

L24/(L14 +

L24 ). A p a r t f ro m co n s i d e ra t i o n s o f co n t i n u i t y ,

very l i t t l e can be done to change th i s ra t io .

In a p o s t - t en s i o n ed f l o o r m u ch m o re co n t ro l c an b e ex e rc i s ed o v e r t h e l o ad

d i s t r i b u t i o n b y th e a r r an g em en t o f t h e t en d o n s . E ach cu rv ed t en d o n can b e s een

to exer t a ver t ica l load on the con cre te b y v i r tue o f i ts cu rva tu re , u sua l ly in a

d i r ec t i o n o p p o s i t e t o t h e ap p l i ed l o ad . A t t h e s u p p o r t t h e t en d o n cu rv a t u re i s

r ev e r s ed an d i t s h ed s it s l o ad o n t o t h e s u p p o r t i n g b eam . T h e re fo re , i f d e s i red , t h e

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t endo n p ro f il e in one d i rec t ion can be mad e deep er so tha t the curva tu re is m ore

acu te , wi th the resu l t tha t much more load i s car r i ed a long tha t d i rec t ion . The

same resu l t i s ach ieved if m ore t endon s a re p rov ided in one d i rec t ion . Th i s i s an

extremely useful tool , avai lable only to post - tens ioning.

Extending the idea fur ther , in a par t ia l ly pres t ressed f loor i t i s poss ible to run

the t endons in one d i rec t ion on ly . The upwards fo rce resu l t ing f rom the t endon

curva tu re can be deducted f rom the to ta l app l i ed load , and the remain ing load

d i s t r ibu ted in p ropor t ion to the s l ab geomet ry in the two d i rec t ions . Us ing th i s

a r ran gem ent , a g rea te r p rop or t io n o f the load wi ll go a long the d i rec t ion o f the

tendon s th an wou ld have been poss ib le in a simple re in fo rced concre te s lab . Such

a s lab acts as post - tens ioned in one di rect ion and as reinforced in the other . In

de te rm in ing the p rop or t io n o f load d i s t r ibu t ion in the two d i rec t ions it shou ld be

no ted tha t the m om ents of iner t ia o f the s lab in the two d i rect ions wi ll be di fferent ,

because the re in fo rced concre te sec t ion wi l l c rack whereas the pos t - t ens ioned

sec t ion wi ll rem ain in tac t . The a r ran gem ent is par t i cu la r ly usefu l where beam s in

one d i rec t ion a re capab le o f car ry ing a la rger p ro po r t ion o f the load tha n tho se a t

r igh t ang les; the t endons can then be used to d i s t r ibu te the load to su it the beams .

A two -way s lab m ay be des igned as re in fo rced in bo th d i rec tions , pos t - t ens ioned

in one d i rec t ion on ly o r pos t - t ens ioned in bo th d i rec t ions .

The f i r st comb ina t ion , tha t o f a two-w ay s l ab in re in fo rced concre te in bo th

direct ions , i s ra ther rare in a general ly post - tens ioned f loor . I t s occurrence is

l ike ly to be inc iden ta l ra th er than by cho ice , such as where the f ram ing geo m et ry

of a pos t - t ens ioned f loor l eaves a pane l som ewhere , which i s too smal l fo r

pos t - t ens ion ing . I f the ad jacen t bea ms are pos t - t ens ioned then a t t en t ion shou ld

be g iven to the sp read o f the ax ia l com pon en t o f the p res tress in to the s l ab .

Eac h o f the th ree s l ab a r ran gem ents c an be used wi th beams in re in fo rced o r

pos t - t ens ioned concre te . I f the s lab span is long enou gh to requ i re pos t - t ens ion ing

then, in al l probabi l i ty , the beams wi l l a lso need to be post - tens ioned, unless for

some reaso n they a re dow ns tan d and o f su ff ic ien t dep th to be des igned in

reinforced concrete . In the la t ter case, the reinforced concrete beams wi l l absorb

som e of the pres t ress from the s lab, leaving i t deficient in the axial com po ne nt ,

unless provis ion has been made in the des ign for th is spread.

The co m bina t ion o f a re in fo rced concre te s l ab and p os t - t ens ioned beam s is

also l ikely to be used only in very special ized ci rcumstances . I t i s feas ible but

a t t en t ion m us t be g iven to the sp read o f p res t ress in to the s lab .

A solid fla t s lab is sui table for dome st ic an d office bui ld ings w i th app rox im ate ly

equal spans in the two d i rec tions . I t is econom ica l w i th spans no t exceed ing abo u t

10 m (35 ft ). I f dr op panels are p rovid ed then the sol id fla t s lab can be used for

longer spans , say up to abou t 12 m (40 f t ) . L igh tweigh t concre te may be more

su i t ab le in the l a t t e r case than the normal dense concre te .

W h e r e h e a v ie r l o a di n g a n d / o r l o n g s p a n s ~ u p t o 20 m (6 5 f t ) ~ a r e n e e d ed , a

waff le s lab may be a good choice. I t s poss ible appl icat ions include warehouses ,

indust r ia l bui ld ings and publ ic hal ls .

Tw o-w ay s labs wi th dow ns tan d beam s can be used in e i ther o f the above

s i tua t ions , p rov ided o f course tha t the dow ns tand s a re accep tab le .

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3.3.3 Solid or ribbed

A sol id s lab can , f rom a theore t ica l p o in t o f v iew, be cons ide red as a pa r t icu la r

case of a r ibbed f loor where the r ibs a re touch ing toge the r . I t fo rms the la rges t

re se rvoi r o f com press ion for a g iven dep th and of ten a so lid f loor is used where a

min imum se c t ion de p th i s t he ma in c r i t e r ion a nd i t s a dd i t i ona l we igh t i s no t

s ign i fican t. Se l f -we ight, o f course , becom es of inc reas ing s ign if icance as the s pan

length inc reases .

Fo rm w or k for a so l id s lab cons is t s o f a s imple fla t sur face and ve r t ica l edge

boa r ds . A so lid fl oo r is he a v ie r t ha n a r i bbe d one a n d the p r ops m us t be de s igne d

for the load ing , th ou gh the w e ight o f we t concre te i s r a re ly c r it ica l for f loors of the

no r ma l spa n l e ng th . F o r mwor k f o r a r i bbe d f loo r r e q u i r e s t he f i a t p l a t f o r m a nd

the ve r t i c a l e dge boa r ds bu t i t a dd i t i ona l ly ne e ds p r op r i e t a r y o r pu r pose - ma de

forme rs shaped to the prof i le o f the r ibs , which m ust be secured in pos i t ion .

S t r ipp ing o f so ff it shu t t e r s f o r ri b s , obv ious ly , r e q u ir e s m or e wo r k a nd some t im e s

the a r ri ses ge t dam age d . T hese need ca re fu l r epa i r becau se the i r tens i le s t r ength i s

of ten re l ied on in pos t - tens ion ing .

F o r a so l id s l a b the r od r e in f o r c e me n t , t e ndon suppor t s a nd t e ndons a r e

a s se mble d on a f i a t p l a t f o r m wi thou t a ny o the r phys i c a l impe d ime n t , a nd the

c onc r e t e c an be p l a c e d a nd c om pa c te d w i th c on fide nc e in the ope n a r e a ; t he on ly

p r e c a u t ion to be t a ke n is a ga in s t t he v ib r a to r s da m a g ing o r d i sp la c ing the t e ndons .

A sse mbly o f r od r e in f o rc e me n t , t e ndon suppo r t s a nd the t e ndons i s c um be r som e

in r ibbed or waf fle f loors , because these a re ho use d wi th in the conf ines of the r ib .

S om e nom ina l r o d r e in f o r c e me n t i s a l so ne e de d in t he sha pe o f l i nks to m a in t a in

the tendons in the specif ied prof i le .

C on c r e t e ha s to be p l a c e d a nd c om pa c te d in na r r o w r ib s wh ic h m a y a l r e a dy be

cong es ted wi th re inforcem ent an d ten don s . At t imes i t i s d i ff icu l t to f ind ro om for

a poke r v ib r a to r , wh ic h m a ke s i t m or e d i ff ic u lt t o a ch ie ve a goo d c o m pa c t ion o f

concre te in the na r ro w r ibs . Thu s , whi le a r ibbed f loor ma y have a le sse r concre te

q ua n t i t y , t he e a se w i th wh ic h c onc r e t e c a n be de pos i t e d a nd c om pa c te d in a so l id

f loo r ma y ma ke the l a t t e r mor e a t t r a c t ive . I n h igh - r i s e bu i ld ings , on the o the r

ha nd , pum pin g l a r ge vo lum e s o f c onc r e t e t o t he uppe r f l oo r s m a y be a n e xpe ns ive

ope r a t ion a nd so r i b s ma y be p r e f e r r e d .

Co ns id e r ing f i re , a mu ch la rge r sur face a rea is expo sed to the h igh tem per a ture

a t the soff it o f a r ibb ed f loor tha n tha t o f a so l id f loor . A lso , the so l id f loor has a

la rge r mas s a nd a la rge r capac i ty a s a hea t s ink . There fore , in a f ir e the ra te of r ise

of tem pera ture i s s lower in a so l id f loor . For th is r eason the f ir e r equi rem ents a re

mo r e on e r ous f o r ri b s , i n t ha t m or e c onc r e t e c ove r i s p r e sc r ibe d to r e in f o r c e me n t

a n d t e n d o n s .

The am ou nt of w ork involved in the des ign o f a so l id f loor i s s l igh t ly less than

tha t for a r ibbed f loor . Because shea r i s r a re ly c r i t ica l in a so l id f loor , no

ca lcu la t ion i s r equi red for the po in t w here the sec t ion should be chan ged to so l id ,

a nd the m om e n t o f i ne r t ia r e ma ins c ons t a n t w h ic h s imp l if ie s t he c om pu ta t io n o f

def lec t ion.

I n sum m a r y , t he m a in a dva n ta ge o f a so l id s l ab i s t he e a se o f c ons t r uc t ion ,

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POST-TENSIONED CONCRETE FLOORS

be c a use o f t he e a si e r f o r m wo r k a nd r e duc e d s it e l a bou r ne e de d . I t is pa r t i c u l a r ly

use fu l where the se l f-we ight o f the f loor is no t a m a jor con s ide ra t ion and i s the

idea l so lu t ion for spans under about 10 m (33 f t ) ca r ry ing heavy loads . A so l id

f loo r is a l so l ike ly t o be c hose n whe r e a m in im um c ons t r uc t ion de p th i s t he ma in

c r i te r ion , t houg h i t m a y n o t be t he mo s t c os t e ff ec tive so lu t ion . F o r l on ge r spa ns ,

t he s e lf -we ight o f a so l id s l a b w i ll p r oba b ly be a c ons ide r a b le d i s a d va n ta ge - - a t

least in residentia l and off ice buildings.

A r ibbed f loor makes use of the concre te and the s tee l in the most e f f ic ien t

m a nne r , a n d i s l ike ly t o be a good e c onom ic a l c ho ice if c ons t r u c t ion de p th i s no t a

ma j o r c ons ide r a t ion .

3.4 Spa n- to-depth ra t io

O n e o f t he ma j o r a dva n ta ge s o f pos t - t e ns ion ing is t ha t t he f l oo r ca n be m a de

sha l low er than i t wo uld be in r e inforced con cre te . A re inforced con cre te f loor wi ll

ha ve a l a r ge r def l ec t ion tha n a pos t - t e ns ione d f loo r ( w ith e q ua l l oa d a nd de p th ) .

The reason i s tha t , to a la rge ex ten t , the de f lec t ion can be cont ro l led in a

pos t - t e ns ion e d f loo r, t hou gh th is w ou ld no t m a ke a n e f fi ci en t u se o f the p r e s tr e s s .

T h e m i n i m u m p o s si b le d e p t h f o r a n y t y p e o f f lo o r d e pe n d s o n a n u m b e r o f

f a c to r s , suc h a s c onc r e t e s t r e ng th , spa n l e ng th , i n t e ns ity o f l oa d ing , w he the r t he

sec t ion is so l id or ribbed , and r ib prop or t ion s in the la t te r case. In genera l , a so lid

s lab i s l ike ly to g ive a sha l lower depth than o the r types .

T he q ue s t ion a r i se s a s to how sha l low a pos t - t e ns ione d f loo r c a n be . T he u sua l

p r a c t i c e i s t o q uo te spa n - to - de p th r a t i o s wh ic h a r e ba se d on a c tua l p r a c t i c a l

de s igns . I t i s, pe r ha ps , w or th de vo t ing a fe w pa r a g r a p hs to t he t he o r e t i c a l a spe c t s

in c ons ide r ing the min imum de p th . A pa r t f r om the p r a c t i c a l c ons ide r a t ions o f

b u i l d ab i l it y a n d e c o n o m y , t h e m i n i m u m d e p t h o f a f lo o r is d e t e r m i n e d f r o m t w o

de s ign c r i t e r i a - - s t r e ng th a nd de f l e c t ion .

C on s ide r t he f l exu r e o f a f l oo r r eq u i r e d to c a r r y a un i f o r m ly d i s t r i bu t e d lo a d w

on a spa n o f l e ng th L . A ssume tha t t he s e c t ion r e ma ins unc r a c ke d .

M o m e n t = s t r e s s x s e c t i o n m o d u l u s

w L 2 c t t r .D 2 (3.1)

where a = f lexural s tress

D = se c t ion de p th

Express ion (3 .1) ind ica tes tha t for a cons tan t un i formly d is t r ibu ted

t o t a l

l oa d ,

dep th i s d i rec t ly pro po r t io na l to span . Ho we ver , the f loor se l f-we ight inc reases in

p r o po r t ion to t he de p th , a nd the r ef o r e t o t he spa n l e ng th , w i th t he r e su lt t ha t , t he

to t a l l oa d be ing c ons t a n t , t he c a pa c i ty a va i l a b l e f o r t he impose d loa d r e duc e s .

There fore , fo r a cons tan t

a p p l i e d l o a d ,

t he i nc r e a se i n de p th mus t be mor e tha n

l ine a r . E xa c t ly how m uc h , d e pe nds on the r a t i o o f a pp l i e d loa d to s e lf -we ight .

Cons ide r ing the de f lec t ion 6 ,

t5 oC w L 4/ D 3

(3.2)

6 / L oc w L 3 / D 3

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SLAB CONFIGURATION

75

The above re la t ionsh ip shows tha t , as suming the dep th to increase in p rop or t io n

to the span, the deflect ion is a lso pro po rt ion al ly h igher . In o th er w ords , i f the

to ta l load i s cons tan t , and span and dep th a re doub led then def l ec t ion i s a l so

doubled, i .e . ,

6 / L

remains cons tan t . Some s t andards , BS 8110 fo r example ,

s t ipulate two cr i ter ia for def lect ion; that i t must not exceed a prescr ibed

pro po r t ion o f the span l eng th an d tha t i t m us t no t exceed a g iven abso lu te va lue .

For a cons tan t def l ec t ion , dep th shou ld be p ropor t iona l to L 4/3.

Eviden t ly , the re l a t ionsh ip be tween span and dep th i s a complex one and

cannot be expressed s imply as l inear . In BS 8110, for reinforced concrete , the

complexi ty i s s impl i f ied by speci fying two di fferent span-to-depth rat ios , above

and below 10 m (33 f t) . In the case of post - te ns ion ed conc rete the s i tuat io n is even

more complex , because the ne t load caus ing def l ec t ion depends on the t endon

geomet ry , which can be used to con t ro l def l ec t ion to some ex ten t .

A n u m b er o f s p an - l o ad co m b i n a t io n s w e re d es ig n ed fo r mi n i m u m p o s si bl e

dep ths , a t a ma x im um long- te rm def lec t ion o f 20 m m for s imply suppor ted so lid

s labs . The resul ts are shown in Figure 3 .5 , where

minimum depths

fo r spans

rang ing from 6 m to 20 m (20 f t to 65 ft ) are p lot ted for impo sed loads of 2 .5 , 5.0,

10.0 and 15.0 kN /m 2 (nom inal 50, 100, 200 and 300 psf) . I t is poss ible to obta in a

di fferent set of curves from those sh ow n in Figure 3 .5 by choo sing o ther v alues for

the des ign cr i ter ia (concrete s t rength , cont inui ty , e tc .) , but the plot ted values are

co n s i d e red a s ap p ro ach i n g t h e mi n i mu m.

It i s unl ikely that a 20 m (65 f t ) span s lab would be const ructed in sol id

concrete . The curve has been extended to th is span only to indicate a base l ine

s tar t ing poin t for depth in the des ign of long sp an r ibb ed or waffle f loors. An

ac tua l des ign wi ll a lmo s t cer t a in ly have to inc lude eco nom y as one o f the

govern ing parameters which would requ i re a deeper sec t ion than ind ica ted in

Figure 3 .5 . Ec ono m y, be ing depen den t on the re l a t ive cos ts o f m ater i a l and

labour, var ies from one locat ion to the next , so that i t i s not poss ible to g ive any

mean ingfu l gu idance on economica l dep ths .

Note tha t F igure 3 .5 i s based on the fo l lowing assumpt ions :

9 f~u = 40 N / ram 2 ( fc '= 4600 ps i ), n orm al concre te

9 Te nd on cen troid 35 m m (1.375 in) above s lab soffi t

9 Serv iceab i li ty s tresses l imi ted to 2 . 3N /m m 2 t ens ion and 13 .33N /m m 2

com pres s ion (330 and 195 0 ps i respect ively)

The sp an-d ep th ra t ios in F igure 3 .5 vary f rom 32 to 54 . Ca u t ion is needed in us ing

a very s lender s lab . In i t ia l s t resses at jack ing and f inal serviceabi l i ty s t resses m ay

both approach the i r respec t ive permiss ib le l imi t s , def l ec t ion under cer t a in

cond i t ions may be excess ive , o r the f loor may have an undes i rab ly low na tu ra l

frequency, though the in i t ia l s t resses , and the creep deflect ion, can be control led

to some ex ten t th roug h app l i ca t ion o f p res tress in two s tages .

Tab le 3 .1 shows the suggested ran ge of spa n-d ep th rat ios for d i fferent f loor

configurat ions. Th e lower and the uppe r values in a range correspon d a pproxim ately

to ap pl ied loads of 10 an d 2 .5 k N /m 2 (200 and 50 psf) respect ively .

The span-dep th ra t ios fo r the beams may be more impor tan t in def in ing the

overa ll cons t ruc t ion dep th o f a s t ruc tu re . B eams are norm al ly heav i ly loaded

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7 6

POST-TENSIONED CONCRETE FLOORS

Span ( feet)

0 10 20 30 40

I I I

600 -

E

E 4 0 0 -

t -

Q . .

a

2 0 0 -

0

0

5 0

I

I i i I , I 1

2 4 6 8 10 12

Span (metres)

1

14

F i g u r e 3 . 5 M in im um dep th -app l i ed l oad f o r so li d s lab

1

16

T a b l e 3 . 1 Span-depth rat ios for f loo rs

Floo r type Span-d epth range

O n e - w a y s o l id 3 0 t o 4 5

R i b b e d s l a b 2 5 t o 3 5

S o l i d f l a t s l a b s 3 5 t o 4 5

W a f f l e f l o o r s 2 0 t o 3 0

B e a m s 1 3 t o 3 3

60

!

1

18

1

2 0

70

2 4

(I)

( ! )

16 "

r

r

D

8

T a b l e 3 . 2 Span-depth rat ios for beams

To tal load Live~Dead load ratio

kN /m 2 ks f 1 .5 1 .0 0 .5

150 3.1 13.3 14.0 14.5

100 2.1 16.3 17.0 17.8

75 1 .6 18.8 19.5 20.3

50 1 .0 23 . 0 23 . 5 24 . 5

2 5 0 . 5 - - 3 1 . 5 3 3 . 0

and, therefore, the average pres t ress level i s h igher in beams than in s labs . The

ma x i m u m l o ad t h a t a g iv en b eam s ec ti o n can ca r ry d ep en d s o n t h e p r e s t re s s in g

force , amo ng o ther fac to rs . An up per l imi t on the p res t ress ing force is imp osed by

the in tens i ty of the dea d loa d o n the b eam , i f the in i t ia l s t resses are not to exceed

the specified l imits. This is also true of slabs b ut i t is of significance o nly in high ly

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S LA B C ON F I G U R A T I ON 77

s t re s sed me m be r s , suc h a s be a ms . F o r t h i s r e a son , s imp le spa n - d e p th r a t i o s a r e

m ore d i ff icu l t to g ive for beam s. R a t ios in the ran ge of 13 to 33 can be t r ied ; the

lowe r fi gu re c o r r e spon ds to a t o t a l l oa d o f 150 kN /m 2 a nd the h ighe r t o 25 kN /m 2

o f t he p l a n a r e a o f t he be a m , w i th l i ve - to - de a d loa d r a t i o s be twe e n 0 .5 a nd 1 .5 .

Table 3.2 gives the values.

T he de p th o f a c on t inu ous s l ab o r be a m , o r a two- w a y s l a b , c a n be sma l l e r t ha n

tha t o f a s ing le span . I t is no t poss ib le to produ ce s im ple span-d ept h ra t io curves

f o r c on t inuo us m e m be r s a nd two- w a y s l a bs be c a use o f t he l a rge pos s ib le

va r i a t i on in t he r a t i o s o f a d j a c e n t spa n l e ng ths i n t he c a se o f c on t inuo us f l oo rs

a nd in t he r a t i o o f s ide l e ng ths f o r two- wa y f loo r s . I f a min imum de p th i s t he

r e q u i r e me n t i n suc h a f l oo r the n a num be r o f t r ia l c a l c u l at i ons wo u ld ha ve to be

ca r r ied ou t , pe rh aps s ta r t ing wi th a dep th of 0 .9 t imes tha t for a s imple span of the

same length .

Figure

3.6 N e w P o rt I I I - U SA

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78 POST-TENSIONEDCONCRE TE FLOORS

Figure 3.7 Monostrand f loor post-tensioning

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4 P L A N N I N G A S T R U C T U R E

T h i s ch ap t e r d ea l s w i t h s o me o f t h e fac t o r s t o be co n s i d e r ed w h en a p o s t - t en s i o n ed

s t r u c t u r e i s b e i n g p l a n n ed , b e f o r e t h e p r o ces s o f c a l cu l a t i o n s i s b eg u n . I t i n c lu d es

d i scuss ion o f a f ew des ig n - re l a t ed t op i cs , such as , the d i f fe rence be tw een the

b eh av i o u r o f co l u m n s f o r r e i n f o rced an d p o s t - t en s i o n ed co n c r e t e f l o o rs , t h e

l o n g i t u d i n a l s h o r t e n i n g o f t h e f lo o r an d i t s i n t e r ac t i o n w i t h t h e v e r ti c a l e lemen t s ,

an d t h e s p r ead o f p r e s t r e s s i n t o ad j ace n t z o n es w h i ch h a v e a l e ss e r, o r n o ,

p r e s t re s s . T h e r e l a ti v e me r i t s o f t h e a l t e r n a t i v e s - - r e i n f o r c ed o r p o s t - t en s i o n ed ,

b o n d e d o r u n b o n d e d , e t c . - - h a v e b e e n d i sc u s s ed in C h a p t e r 1 a n d a r e n o t

r ep ea t ed h e r e .

I n p l an n i n g t h e s t r u c t u r e o f a b u i ld i n g , t h e f ac t o r s t o b e co n s i d e r ed i n c l u d e :

9 bu i ldab i l i t y

9 s h ap e an d p o s i t i o n o f s h ea r w a l l s

9 co l u mn s p ac i n g

9 o n e - o r t w o - w a y s p a n n i n g

9 f l oo r dep th

9 so l i d o r r i bbed

9 co lumn s i ze

9 co n s t r u c t i o n j o i n t s

9

e c o n o m y

9 ad ap t ab i l i t y

S o m e o f t h e a b o v e h a v e a l s o b e e n d i sc u s s ed i n C h a p t e r 1 a n d a r e n o t r e p e a t e d .

E co n o my i s o f t en t h e mo s t i mp o r t an t co n s i d e r a t i o n , b u t i t i s l i s t ed l a s t , b ecau s e

t h e co st o f a s t r u c t u r e d ep en d s o n d ec i s i o n s re s u l t i n g f r o m t h e co n s i d e r a t i o n o f

t h e o t h e r p o i n t s , an d e co n o m y s u m ma r i z e s t h e v a l u e s o f t h e s e d ec i s io n s .

D e p e n d i n g o n t h e t y p e o f b u i l d in g , t h e c o st o f t h e s t r u c t u r e m a y v a r y fr o m 1 0 % o f

t h e d ev e l o p m en t co s t fo r a p r e s ti g e b u i l d i n g , t o 8 0 % f o r a u t i l i t a r i an p r o j ec t . T h i s

r e fl e c ts t h e r e l a t i v e i mp o r t a n ce o f t h e s t r u c t u r e an d i ts d i r ec t co s t.

4.1 Design object ives and bui lda bi l i ty

T h e s t r u c t u r a l d e s i g n o f a b u i l d i n g i s g u i d ed b y ce r t a i n d e s i g n o b j ec ti v e s ; t h e

o b v i o u s o n es co n ce r n s t r u c t u r a l i n t eg r i t y an d r o b u s t n e s s , w h i l e o t h e r s p e r t a i n t o

b u i l d ab i l i t y an d t h e i n t en d ed u s e o f t h e b u i l d i n g - - s ee T ab l e 4 .1 .

T h e p o s s i b l e o b j ec ti v e s f o r a h i g h - ri s e b u i l d i n g o f a p r ed o m i n a n t l y v e r t ic a l

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80 POST-TENSIONED CONCRETE FLOORS

Table 4.1 D esign object ives and post-tensioning

Objective

Effect of post-tensioning

Design

Minimum st ructural

depth

Column f ree space

Low self-weight

Small column size

Economy

Low lead t ime

Design effort

Construct ion

Simpler formwork design

Quick formwork

tu rnaround

Quick assembly of steel

Fast concreting

Min imum backpropp ing

Low labour requi rement

Crane t ime

Site presence

In use

Limited deflection

Low maintenance

Possible small holes

Large holes

Future refurbishment

Demol i t ion

Allows high span/depth rat io

Allows long spa ns w itho ut excessive self-weight

Minimum depth, possibly in l ighweight concrete

Low floor weight leads to reduce column moment

Economical above 8 to l0 m span (26 to 33 f t )

Can be faster than steelwork and reinforced concrete

May take sl ight ly more t ime

Fl at soffit floor designs easily achievable

Formwork released ear ly

Minimum steel quanti ty. Tendons in long lengths

Less concrete quanti ty, uncongested steel

Self suppo rt ing at ea rly age

Less material quan ti ty, easier assembly of steel

Less material , crane may be needed for jacking

Specialist operatives needed

Provides ini t ial camber and control

No microcracks, slow deteriorat ion

Steel spaced much further apart than in an RC floor

Need careful design study and specialist operatives

Less structural d epth and flat soffi t perm it ma xim um

flexibility for renewal of services

Needs study of structure and special ist knowledge

as p ec t , an d t h o s e f o r a lo w - r i s e b u i l d i n g o f l a r g e p l an a r ea , m ay b e s i m i l a r b u t t h e

m e a n s e m p l o y e d f o r a c h i e v i n g t h e s e , a n d t h e e m p h a s i s r e q u i r e d o n p a r t i c u l a r

o p e r a t i o n s , d if fe r. F o r ex a mp l e , w h i l e it i s d e s i r ab l e t o co n s t r u c t t h e f r ame i n a s

s h o r t a t i me a s r ea s o n ab l y p o s s i b l e , i n a t a l l b u i l d i n g t h i s i mp l i e s a q u i ck

t u r n - a r o u n d f l o o r co n s t r u c t i o n cy c l e w h e r ea s i n a l a r g e p l an b u i l d i n g t h e

co n s t r u c t i o n o f s u ccess i v e fl o o r s w il l p r o b a b l y b e ca r r i ed o u t i n s tag es , s o t h a t t h e

t i me t ak en i n co mp l e t i n g a f l o o r h a s a l e s s e r i mp o r t an ce . F o r t h e s ame o b j ec t i v e

t h e p l an n i n g o f s it e o p e r a t i o n s i s a l s o q u i t e d i f f e ren t f o r t h e t w o t y p es o f b u i ld i n g s .

T h e d e s i g n e r s h o u l d r eco g n i z e t h e b u i l d ab i l i t y r eq u i r em en t s f o r t h e p r o j ec t an d

mak e ad eq u a t e p r o v i s i o n f o r t h e s e i n t h e d e s i g n .

T h e m a i n o b j ec t i v e s u s u a l l y i n c l u d e a m i n i m u m o f s t r u c t u r a l p r e s en ce , f as t

co n s t r u c t i o n , an d r e l i ab i li t y i n s e rv i ce . T h e ad v an t ag e s o f p o s t - t en s i o n i n g h av e

a l r ead y b een d i s cu s s ed i n C h a p t e r 1 ; h o w ev e r , i t is o f i n t e r e s t t o s ee h o w

p o s t - t en s i o n i n g h e l p s i n ach i ev i n g t h e o b j ec t i v e s s e t o u t i n T ab l e 4 .1 . S o m e o f t h e

i t ems a r e s e l f - ex p l an a t o r y an d a r e n o t d i s cu s s ed f u r t h e r .

L e a d t i m e s fo r t he p r o c u r e m e n t o f m a t e r i a l s a r e p a r t o f t h e o v e r a ll b u i l d i n g

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P L A N N I N G A S TR U C TU R E 81

progra m . Bui lding elements on long lead t imes have tw o fundam ental d isadvantages .

Fi rs t ly , the mater ial has to be ordered some t ime before i t i s del ivered, which

requ i res f inanc ing in advan ce . Second ly , the des ign has to be f rozen a long t ime

before the mater i a l i s incorpora ted in to the bu i ld ing . Of ten the t enan t s '

requ i rem ents a re no t fu l ly know n un t i l a la t e s tage in the bu i ld ing p rog ram , a nd i t

becom es necessa ry to incor po rate la te chang es in the des ign. I f th is involves

mater i a l s on a long p rocurement per iod then i t becomes d i f f i cu l t to keep cos t s

und er con t ro l . S t ruc tu ra l s t ee lwork o f t en has a long de livery per iod . In re in fo rced

concre te con s t ruc t ion i t is a ques t ion o f how qu ick ly the re in fo rcement supp l i e r

can b en d an d d e l iv e r th e ro d r e i n fo rcem en t - -o n l y t h e s ma ll d i ame t e r b a r s can b e

ben t on s i t e . In pos t - t ens ion ing , s t rand and hardware can be s to red on s i t e and

the requ i s i t e l eng ths o f s t rand cu t f rom the ree l as and when needed . For l as t

minu te ho les, the t endons can s imply be dev ia ted ou t o f the way . Pos t - t ens ion ing ,

therefore, offers the shortes t lead t ime and m axim um flexibil ity for accom m oda t ing

late des ign changes .

The ease wi th which a bu i ld ing can be re fu rb i shed in the fu tu re is im por tan t in

m eas uring i ts serviceable l i fe . A bui ld ing w here services cann ot be renew ed,

becau se of insuff icient s torey height , or service zone in terru pt ion s such as by

downs tand beams , canno t be eas i ly re fu rb i shed and the on ly a l t e rna t ive may be

to pul l i t down.

In o rder to des ign an economica l s t ruc tu re , fac i l i t a t e i t s cons t ruc t ion , and

enable i t s la ter refurbishment , the des igner should consider the fol lowing points .

Ho we ver , the des ire to ach ieve op t im um eff ic iency o f des ign shou ld be t emp ered

by the e ssent ial need for s impl ici ty of ins tal la t ion.

9 The p os i t ion ing o f the co lum ns i s im por tan t in pos t - t ens ioned concre te , as it is

in o ther fo rms o f cons t ruc t ion . A regu lar rec tan gu lar g r id is l ike ly to be mo re

econom ica l than one in which the span l eng ths vary to the ex ten t tha t reversa l

o f span m om ents is poss ib le . In the au tho rs ' op in ion , the p refer red span ra t io

for a panel w ould be in the range 1:1 .3 to 1:1 .6 , and the oute r span s of a

con t inuo us f loor wo uld be abou t 80 % of the inner spans in leng th .

9 K eep t h e fo rmw o rk s i mp le . Av o i d d o w n s t an d b eams , co l u mn h ead s an d d ro p

panel s . In r ibbed and waff le floors , reduce the num ber o f m oulds by spac ing the

r i b s ap a r t . Pu rp o s e -mad e mo u l d s may p ro v e mo re eco n o mi ca l t h an t h e

p ro p r i e t a ry o n es .

9 Provide c onst ru ct ion jo ints to suit the con st ruct io n seq uence whe rever possible.

9 Kee p the t endo n l ayou t s imple . Avoid in te r l eav ing o f t endons . Space t endons

as far apart as the des ign permits , to al low holes to be cut af ter const ruct ion.

9 Bunc h t endo ns in twos and th rees . Th i s saves on the num ber o f suppo r t s an d i s

quicker to ins tal l .

9 Avoid conge s t ion o f s tee l. Th i s usua l ly means tha t the s t ruc tu ra l me m bers

shou ld no t be reduced in s i ze to the theore t i ca l min imum.

9 Deta i l ro d re in fo rcement to a l low ease o f as sembly .

9 Use p re-assemb led re in fo rcem ent cages and s t a nd ard me sh in p reference to

loose bars which have to be t ied on s i te .

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82

POST-TENSIONED CONCRETE FLOORS

4 .2 R e s t r a i n t f r o m v e r t i c a l e l e m e n t s

In a concre te s t ruc tura l f lame , the ve r t ica l e lements , such as co lumns and wa l l s ,

a re usua l ly des igned mo no l i th ic w i th the f loor . W he n such a f loor i s s t r e ssed , pa r t

of the ten do n force i s r e s i s ted b y the ve r t ica l e lemen ts an d i s los t a s f a r a s the ax ia l

s t r ess in the f loor concre te i s conce rned . The ax ia l p res t re ss in the f loor equa ls the

d i ffe rence be tween the te nd on force and the force re s i s ted by the ve r t ica l e lemen ts .

T h i s doe s no t a ff ec t t he m a gn i tude o f t he e q u iva l e n t ba l a nc ing loa d , be c a use the

force in the tendon i s no t a f fec ted . In fac t , because the ave rage s t r e ss on the

c onc r e t e is now low e r t ha n i t wou ld ha ve be e n i f t he ve rt ic a l e le me n t s h a d no

s ti ffness , e la s tic and c reep losses in pres t re ss a re a l so low er ; the re fore , the te nd on

force i s marg ina l ly h igher .

F o r a no r ma l f l oo r suppor t e d on c o lumns , t he d i f f e r e nc e be twe e n the t e ndon

f o rc e a nd the c om pr e s s ion in t he c onc r e t e m a y be o f the o r de r o f 1 t o 2% . T h i s i s

usua l ly ignored , because th is o rde r of accuracy i s ins ign i f ican t cons ide r ing the

unc e r t a in t ie s i n t he p r ope r t i e s o f t he ma te r i a l s a n d the t o le r a nc es . H ow e ve r , t he

loss can be qui te s ign i f ican t i f s ti ff wa l l s o r la rge co lu m ns a re m on ol i th ic w i th the

f loor . Such cond i t ions a re l ike ly to occ ur in the lowe r s toreys of a ta ll bu i ld ing a nd

where re ta in ing wa l l s run nin g fu ll , o r pa r t ia l , he igh t o f the co lu m ns a re cas t

monol i th ica l ly wi th the f rame . In such a case , the se rv iceabi l i ty tens i le s t r e sses

m ay exceed the pe rm iss ib le va lues in the f loor i f the loss is no t a l low ed for in the

ca lcu la t ions . The ve r t ica l e lements themse lves may ge t ove r loaded .

C ons ide r a s ing l e spa n be a m f r a ming in to two c o lumns , F igu r e 4 .1 ( a ) . F o r

s impl ic i ty , a ssum e tha t the co lu m ns a re equa l in size , a re fu l ly f ixed a t the b o t to m ,

and a re f ree to ro ta te a t the to p . I f a d i ffe ren t se t o f con di t ion s i s des i red th en the

e ff ec tive c o lum n he igh t H c a n be a d j u s t e d a c c o r d ing ly . I f t he c o lum ns p r ov id e d

only ve r t ica l suppor t and the f loor were f ree to s l ide hor izonta l ly , then the re

wo u ld de ve lop a ga p o f w id th 6 be twe e n the be a m a nd one o f t he c o lumns , F igu r e

4 .1 (b ) , bec a use o f t he sho r t e n ing o f t he be a m due to t he a x ia l c om po ne n t o f t he

prestress . In ord er for the gap to c lose , shears Pv are n eeded as sho wn in Fig ure 4.1(c).

& = sho r t e n ing in l e ng th L

= Ps L /Ar 1 6 2 (4.1)

where P~ = Axia l p res t re ss ing force in concre te , no t the ten do n force

= P t - P ~

P t = T e ndon f o r c e

Pv = Force re s i s ted by the ve r t ica l e lements

E a c h c o lumn mus t de f l e c t a d i s t a nc e

6 /2

under force P~.

6 = 2 P , , H 3 / ( 3 E c l v ) = P s L / A c E c

P,, = 1 .5P ~(L Iv) / (A r 3)

= P s / K r

= P t / ( K r +

1)

where K r = 1 . 5 H 3 A J ( I v L )

(4.2)

(4.3)

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t_. L ._j

( a )

( b )

P L A N N I N G A S TR U C TU R E 8 3

( c )

I

I _ c j I _ _ 1 _ L _ 1

v r~ y !

( d ) ( e )

F i g u r e

4 . 1 Loss o f ax i a l p res t ress due to co l umn s t if fness

I v = M o m e n t o f in e r ti a o f t h e c o l u m n

Ar = Cr oss s e c t i ona l a r e a o f t he be a m

K r

is a non - d i m e ns i o na l f a c t o r w h i c h r e la t e s the de f o r m a t i on o f a be a m i n d i r e c t

t e n s i on t o t ha t o f a c o l um n i n be nd i ng . The m od u l u s o f e l a s ti c it y Ec c a nc e ls o u t ,

a s sum i ng t ha t t he c o l um ns a nd t he f l oo r ha ve t he s a m e c onc r e t e .

F o r t he f r a m e show n i n F i gu r e 4 . 1 ( d ) , w he r e t he be a m i s suppo r t e d on a ve r y

s ti ff she a r w a l l a t one e nd a nd on a c o l um n a t t he o t he r e nd , on l y t he one c o l um n

def l ec t s and the re fore

g r =

3H3Ac/(IvL) ( 4 . 4 )

In the case of two eq ua l b ays of l eng th L each , see F igu re 4 .1 (e ), the cen t re c o lu m n

remains una f fec ted . The f rame i s , t he re fore , equ iva len t to F igure 4 .1 (d) and

Eq ua t i on ( 4. 4) is app l i c a b l e . I f t h e t w o spa n l e ng t h s d i ff er on l y by a sm a l l a m ou n t ,

so t ha t t he de f l e c ti on o f t he c e n t re c o l um n c a n be ne g le c t e d , t he n E q u a t i o n ( 4.4 )

c a n be u se d , t a k i ng a n a ve r a ge va l ue f o r L .

Equa t ions (4 .2 ) , (4 .3 ) and (4 .4) a re fa i r ly s imple to app ly to a f rame cons i s t ing

o f one o r t w o ba ys . F o r l onge r f r a m e s , i t b e c om e s ne c e s sa r y t o so l ve a se t o f

s i m u l t a ne ou s e q ua t i on s ; f u r t he r c om p l e x i ti e s a r i se i f t h e c o l um ns d if fe r i n

s ti ff nes s . F o r a un i f o r m p r e s t r e s s i ng f o rc e ove r t he w ho l e l e ng t h o f t he m e m be r ,

t he p h e n o m e n o n is a n a l o g o u s t o a d r o p i n t e m p e r a t u r e - - a p r o b l e m w h i ch m o s t

f r a m e a na l y s i s p r o g r a m s a r e c a pa b l e o f so l v i ng . The r e q u i r e d hyp o t he t i c a l

t e m pe r a t u r e d r op i s g i ve n by Eq ua t i on ( 4 . 5 ) .

T = t e m p e r a t u r e d r o p

= P t / ( A r 1 6 2 (4.5)

w he r e C t = t he rmal coe f f i c i en t fo r concre te

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84 POST-TENSIONED CONCRETE FLOORS

T h e c o l u m n m o m e n t d u e t o t h e s h o r t e n i n g i n t h e le n g t h o f a b e a m a c ts i n

oppos i t i on t o t ha t due t o t he ve r t i c a l l oa ds , a nd t he ne t e f f e c t i s t ha t i n pos t -

t e n s i one d s t r uc t u r e s t he c o l um n m om e n t s a r e ge ne r a l l y l ow e r t ha n i n r e i n f o r c e d

c onc r e t e s t r uc t u r e s . H ow e ve r , i n e x t r e m e c a se s, in a n ou t e r c o l um n o f h i gh

s ti ff nes s , t h e ho r i zo n t a l f o r c e c a n be h i gh e n ou gh t o c a use d i s tr e s s t o t he c o l um n .

The m om e n t r e su l t i ng f r om t he ho r i zon t a l f o r c e m a y c a use f l e xu r a l c r a c ks a t a

f loor leve l, o r shea r c rack s du e to the pr inc ipa l t ens i l e s t re ss exceed ing the t ens i l e

s t r e ng t h o f t he c onc r e t e . S uc h c r a c k i ng i s m or e l ike l y t o oc c u r a t t he t i m e o f, o r

soo n a f te r , s tr e s s ing w h e n t he p r e s t r e s s i ng f o r ce is h ighe s t , t h e c onc r e t e s t r e ng t h

l o w e s t, a n d t h e v e rt ic a l f o rc e o n t h e c o l u m n a m i n i m u m . I t is r e c o m m e n d e d t h a t

the e ffect o f p res t re ss ing on the ve r t i ca l e l ement s o f the s t ru c ture i s inve s t iga ted ,

pa r t i c u l a r l y i n t he c o r e w a l l s a nd t he c o l um ns a t l ow e r l e ve l s .

The f l exura l t ens i l e s t re ss can be eas i ly ca lcu la t ed f rom the normal e l a s t i c

theo ry , u s ing a t ran sfo rm ed sec t ion . The pr inc ipa l s t resses 0 -1,02 , an d 0~, the a ng le

a t w h i c h t he y oc c u r , a r e g i ve n by t he f o l l ow i ng e xp r e s s i ons . The a ng l e ~ i s i n

r a d i a ns , m e a su r e d f r om t he x - a x i s .

0-1,2 = 0.50-y +_ 0.5(0-y2 + 432) 0.5

= 0 . 5 t a n - ~ ( 2 " t '/ 0 - y )

(4.6)

wh ere try = the v er t ica l s t ress

= the shea r s t re ss

T h e a b o v e e q u a t i o n s a r e b a se d o n e la s ti c th e o r y a n d a r e n o t c o m p a t i b l e w i t h t h e

u l t i m a t e s t a t e she a r a na l y s i s i n BS 8110 . The d i s t r i bu t i on o f e l a s ti c she a r s t re s s

ove r t he c onc r e t e s e c t i on i s no r m a l l y pa r a bo l i c , be i ng m a x i m um a t t he s e c t i on

c e n t r o i d a nd ze r o a t t he e dge s .

F r om t he a bove d i sc us s i on i t i s obv i ous t ha t t he l o ss i n t he a x ia l c om po ne n t o f

t he p r e s t r e s s ing f o r ce c a n be m i n i m i ze d by pos i t i on i ng t he s ti ff ve r t ic a l m e m b e r s ,

suc h a s she a r w a l l s a nd c o r e s , ne a r t he m i dd l e o f t he f l oo r a r e a , o r i n suc h a

m a n ne r t ha t t he y o f fe r t he l ea s t r e s is t a nc e t o t he s ho r t e n i ng o f t he f loo r .

R e c t a ngu l a r c o r e s a nd c ha nne l - sha pe d she a r w a l l s , w h i l e be i ng ve r y e f f i c i e n t

f o r re s i s ti ng l a t e ra l f o r ce s , ca n a bso r b a s i gn if i c an t p r o po r t i o n o f t he p r e s t r e s s i ng

f o rc e if t h e y a r e po o r l y p os i t i one d . The w or s t c a se i s t ha t o f t w o r e c t a ng u l a r c o r e s

l oc a t e d a t e i t he r e nd o f a l ong f l oo r. S t a i rc a se s f r a m e d i n c o l um ns c a n ha v e a ve r y

h igh s t i ffness , a s they m ay for m a ve r t i ca l t russ t r i an gu la ted by the f ligh t s. F igu re

4 .2 show s som e o f t he p r ef e r a b le a n d t he unde s i r a b l e a r r a nge m e n t s .

A s i m i l a r l o s s i n the a x i a l c om po ne n t o f t he p r e s t r e s s ing f o r ce oc c u r s i n

susp e nde d g r o un d f l oo r s due t o t he s t if fne ss o f t he ba se m e n t r e t a i n i ng w a l ls a t

c o r ne r s , a nd w he r e t he t e ndons r uns a l ong t he w a l l . Aw a y f r om t he c o r ne r s , t h e

ex te rna l wa l l s o f fe r ve ry l i t t l e re s i s t ance ; they can be t rea ted in the same manner

a s the c o l um ns , a nd t he lo s s i n p r e s t r e s s c a n be c a l c u l a te d u s i ng e it he r Eq u a t i o ns

( 4 . 2 ) t o ( 4 . 4 ) , o r t he t e m pe r a t u r e d r op a na l ogy .

W he r e t he s e r v i c e c o r e s a nd she a r w a l l s c a nno t be l oc a t e d i n s t r uc t u r a l l y

de s i r a b l e pos i t i ons , t he l o s s c a n be r e duc e d by de l a y i ng t he c onne c t i on be t w e e n

the ve r t i ca l e l em ent a nd the f loor . A gap (or in f il l) s t rip , say one me t re w ide , is l e ft

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PLANNING A STRUCTURE

85

I

I I

Arrangements to be avoided

I

I I

I

Preferred arrangements

Figure

4.2 Core and shear wall arrangements

uncon cre ted be tween the s t if fver ti ca l e l ement and the s lab , fo r as long a per iod as

prac t i ca l. Th i s requ i res the s l ab edge to be suppor te d fo r an ex tended per iod . The

gap s t r ip is a lso useful in reducin g the effect of creep a t m ov em ent jo ints .

I t is a lso poss ible to des ign a s l id ing conne ct ion betw een the w al l an d the f loor ,

where the s lab is ver t ical ly supported by the wal l but i t remains free to s l ide

lateral ly . After a sui table per io d the s l id ing jo in t i s m ade ineffective.

Sui table jo ints an d f il ler s tr ips are show n in Cha pte r 12.

The lateral forces resul t ing from the res t raint offered by the vert ical e lements

m ay have a ra ther unexpe c ted e f fect in spl it -l eve l car parks , such as tha t show n in

Figure 4 .3 (a) . The f loor would no rm al ly be des igned as s imply suppo r ted , t ak ing

loca l acco un t o f the nega t ive m om ent which would deve lop a t the cen tre co lum n,

show n in F igure 4 .3 (b ) . The l a t e ra l res t ra in t o f fe red by the ex te rna l co lum ns se ts

up the sys tem of forces indicated in a simpl i fied m an ne r in Figu re 4 .3(c) , which

produce s t ens ion on the ins ide face o f the upp er f loor -co lum n co nnec t ion ,

oppos ing the t ens ion f rom the norm al ver t i ca l loads . O n the nex t f loor down , the

tens ion is on the ou t s ide face , add ing to tha t f rom the ver ti ca l loads . D epen d ing

on the m agni tu de o f the l a te ra l fo rces , the uppe r f loor may crac k nea r the cen t re

co l u mn o n t h e

underside,

and the lower f loor on the

topside~in

the first case

because no t ens ion is an t i c ipa ted on the und ers ide an d in the second case because

the t ens ion on top i s l a rger than an t i c ipa ted .

4.3 Disp ersion of the pres tressing force

In a pos t - t ens ioned f loor , the anchorages a re normal ly loca ted a long the s l ab

edge. Imm edia te ly beh ind an anchorage , the force f rom each t endon is concen t ra ted

over the a rea o f concre te in d i rec t con tac t wi th the ancho rage ; the zone be tween

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86

P O S T-TE N S I O N E D C O N C R E TE FL O O R S

( a )

Split level floor

I I "

1

(b)

Moments f rom

vertical loads

(c)

Forces due to

lateral restraint

(d) Tension

Mom ent due to ~ ~

lateral restraint Tension

F i g u r e 4.3 Ef fec t o f l a te ra l res t ra i n t on sp l i t - l eve l f l oo rs

two ad jacen t anchorages has no p res t ress . The p res t ress ing fo rce d i sperses

th rough the concre te , and , a cer t a in d i s t ance away f rom the anchorages , the s l ab

can be assumed un i fo rmly s t res sed .

AC I 318 limi ts the spac ing o f anch orage s to a m axim um of s ix times the s l ab

thickness . BS 8110 does no t speci fy any such l imi t . In a 400 m m (16 in) deep f loor

s l ab , in theory the ancho rages m ay be space d a t as m uch as 2.4 m (8 f t) cen t res. I f

the anchora ges a re spaced th i s fa r apa r t , t hen the uns t ressed wedges o f concre te

may no t be ab le to suppor t the des ign load ; they may be par t i cu la r ly vu lnerab le

to dam age f rom con cen t ra ted loads . In o rder to re in fo rce these a reas , the nor m al

prac t i ce is to p rov ide a m esh o f rod re in fo rceme nt in a s t rip a long the edge , o f

wid th equa l to the anchorage spac ing .

A d if feren t p rob le m ar ises , f rom the sam e cause , in pos t - t ens ioned beam s . N ea r

the anchorages , the beam carr ies the ful l pres t ress ing force; the force disperses

into the adjacent concrete , which is l ikely to have a lesser , or no, pres t ress .

Therefo re , awa y f rom the anchora ges , the be am sec t ion has los t some of the ax ia l

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P L A N N I N G A S TR U C TU R E 87

p r e s t re s s . I n som e c a se s t he l o ss m a y be l a r ge e no ugh t o r e su l t in t he s e r v i c e a b il i ty

t ens i l e s t re sses exceed ing the pe rmiss ib le va lues .

I t is w o r t h no t i n g t ha t o n l y t he a x ia l c om po ne n t o f t he p r e s tr e s s i s d i spe r se d i n

t h is m a n ne r ; t he e q u i va l e n t ba l a nc i ng l oa d , be i ng a f unc t i on o f t he t e nd on f o rc e ,

r e m a i ns una f f e c t e d .

Thi s loss must be a l lowed for in the se rv iceab i l i ty s t a t e ca lcu la t ions fo r the

be a m . T he se ve r i ty o f t he p r ob l e m c a n be r e duc e d by p r ov i d i n g a fe w t e nd ons i n

t he s l a b , a t i t s c e n t r o i d l e ve l , r unn i ng pa r a l l e l t o t he be a m s . S uc h t e ndons a l so

e nha nc e t he c a pa c i t y o f t he fl oo r to sup po r t l oa d c on c e n t r a t i on s by i nc r e a si ng t he

effect ive wi dth of the s lab.

T r a d i t i ona l l y , a c on c e n t r a t e d f o rc e is a s sum e d t o d i spe r se a t a n a ng l e o f 45 ~ I n

fac t, t he s t ress on a p lane , a t r igh t a ng les to the l ine of fo rce and a t a g iven d i s t ance

f r om t he po i n t o f a pp l i c a t i on , is a m a x i m um on t he a x i s o f t he fo r c e a nd q u i c k l y

dro ps aw ay e i the r s ide of i t, see F igu re 4 .4 . Th e to t a l fo rce ac t ing on a l eng th of the

p lane bo un de d by the 45 ~ angle , i .e . be tw een the l ines x / c = - 1 and x /c = + 1,

l in e s is a pp r ox i m a t e l y e q ua l t o t he a pp l i e d f o r c e; t h is i s t he ba s is o f t he c o m m on l y

a s sum e d 45 ~ d i spe r s i on .

try = ave rag e s t re ss on m id-p lane = P/2c

Figu re 4 .4 show s the s t re ss a t t he midd le o f an in f in i te ly long s l ab s t r ip sub jec t

to co nce nt ra t ed loads . Th e s t re ss try a lon g the 45 ~ l ine , a t x / c = 1, i s less than

one - n i n t h o f t ha t on t he a x i s

( x /c

= 0 ) . The d i a g r a m c a n be u se d t o a s se s s t he

s t re s s a t a ny po i n t d ue t o t he a x ia l c om po ne n t o f p r e s t r e s s ing f o r c e in a t e ndon .

I n t he c a se o f a pos t - t e n s i one d s l a b , w i t h a nc h o r a ge s a t a spa c i ng S , the s t r es s

d i a g r a m s f r om t he a d j a c e n t a nc ho r a ge s ove r l a p , a nd i t i s r e a sona b l e t o a s sum e a

un i f o r m d i s t r i bu t i on a c e r ta i n d i s t a nc e a w a y f r om t he e dge . F i gu r e 4 . 5 show s t he

s t re s s pa t t e r n s i n a n i so l a t e d l ong s t r ip o f un i t t h i c kne ss a nd w i d t h S , a t d i s t a nc e s

S/4, S/2 a nd S f r om t he a nc h o r a ge . The va l ue s show n i n d i a g r a m s ( a) , ( b) a nd ( c)

shou l d be m u l t i p l i e d by t he a ve r a ge s t r e ss P / S t o ge t the ac tua l s t re ss a t a po in t .

The i so l a t e d s t r ip i s no t a t r ue r e p r e s e n t a t i on o f a s e c ti on o f a pos t - t e n s i one d s l a b

be c a use o f t he d i sc on t i nu i t y a t t he e dge , bu t i t doe s i nd i c a t e t ha t t he s t r es s

be c om e s ne a r l y un i f o r m a t a d i s t a nc e S , sugge s t i ng a d i spe r s i on a t a g r a d i e n t o f

1 : 2 , i .e . 27 ~ ins t ead of the t rad i t io na l ly a ss um ed 45 ~

4 .4 C o l u m n m o m e n t s

The s t r u c t u r a l s iz e o f a c o l um n i s de t e r m i ne d by t he c om b i n a t i on o f d i r e c t l oa d

a n d m o m e n t r e s u lt in g f r o m t h e f ra m e a c t i o n o f a f lo o r . T h e m o m e n t b e i ng h i g h e s t

a t t he f loo r - c o l um n j unc t i o n , t h is s e c t ion o f t e n gove r n s t he c o l u m n s ize.

I n a s t r uc t u r a l f r a m e w i t h a pos t - t e n s i one d f l oo r , t h e r e a r e t w o f a c t o r s w h i c h

r e d u ce t h e m a g n i t u d e o f t h e m o m e n t a t t h e f l o o r - c o l u m n ju n c t i o n , c o m p a r e d

w i t h a f r a m e c on t a i n i ng a r e i n f o r c e d c onc r e t e f l oo r . F i r s t , t h e upw a r d a c t i ng

e q u i v a l e n t l o a d f r o m t h e c u r v e d t e n d o n r e d u c e s t h e n e t d o w n w a r d l o a d o n t h e

be a m o r s l a b , w h i c h d i r e c t l y r e duc e s t he m om e n t s . S e c ond , t he a x i a l p r e s t r e s s

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88

POST-TENSIONED CONCRETE FLOORS

X

c

2.0

O

~- 1.0..

(J

t %

I

, I

,

\

, /

/ \

-3 -2 -1 0 1 2 3

Distance x/c

(~0 = ave rage stress on mid-plan e =

P/2c

F i gu r e 4 . 4

Dispers ion o f a concent ra ted l oad (a f t e r T imoshenko an d G ood ie r, 1951 )

s12 s/4

~

P (a) ~ I

' , , ,,_

~ I ~ ~ '

~ ~ "

. . . ~o

s ( b) T - - -

( c )

!

s I

(b)

t , D

d

(c) ~

- O )

__ ,:5

P e j

O

F i gu r e 4 . 5

St ress i n a s t ri p und er concent ra ted l oads

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PLANNING A STRUCTURE

89

V

(a) Prestress equivalent load

Leg spacing

J

v I

Ip Beamlength--~'i

V

(b) Shortening in beam length

t t

(c) Applied loads

(d) Net moment

Figure 4 .6 Effect of prestress on column moment

and shr inkage s t ra ins in the f loor cause a s light shor tening in the leng th of the

f loor , which pul l s the columns inwards , induc ing moments opposi te to those

caused by the no rm al loads . The m om ents in the f loor i tse lf a re a l so a f fec ted in a

cor responding manner ; the former i s automat ica l ly inc luded in se rviceabi l i ty

ca lcula t ions , and the la t te r may be inc luded in the f ina l s tage ca lcula t ions i f

desired. Figure 4.6 i l lustrates the points. The lef t-hand sketches in Figure 4.6

show the loads and the exaggera ted shap e of the def lec ted f rame, the r ight han d

ske tches show the cor responding moment diagrams. F igure 4 .6(d) i s the ne t

moment , and a compar ison wi th the moment in Figure 4 .6(c) i l lus t ra tes the

r educ t ion in co lumn moment .

The colum ns a re norm al ly des igned for ul t imate s t rength, wi th fac tored loads

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90 POST-TENSIONED CONCRETE FLOORS

a nd mo m e n t s . I n c he c k ing the s t r e ng th o f a pos t - t e ns ione d f loo r , t he s e c onda r y

mome n t s r e su l t i ng f r om the t e ndon c u r va tu r e a r e i nc lude d , bu t t he t e ndon

eccentr ic i ty is apparently exc luded f rom the m om ents . In f ac t, the e ffec t o f the

ten do n eccent r ic i ty is impl ic i t ly a llowed for in the leve r a rm of the ten do n and i t

wou ld be dup l i c a t e d i f t he m om e n t s a l so i nc lude d i t .

T he q ue s t ion a ri se s a s to ho w the t e n don c u r va tu r e is to be c ons ide r e d in t he

des ign of the co lumns , i f a t a l l .

The equiva len t load i s a de f in i te phys ica l force wi th i t s own ident i ty ; i t i s

presen t a s long as the tend on i s in tac t , up to the po in t o f i ts fr ac ture . O nce

f rac tured , the equiva len t loa d i s los t , j us t a s the s t r ength of a r e inforced con cre te

be a m wou ld be i f t he r e in f o rc e me n t we r e s eve re d . T he u l t ima te s t r e ng th

r e q u i r e me n t s d o no t c o n te m pla t e e i t her o f t he two s i tua t ions ; t he a s sum pt ion i s

tha t p la s t ic h inges have formed a t the c r i t ica l sec t ions , and they a re capable of

r e s i s t i ng the u l t ima te mome n t s .

I n c a l c u l at i ng the m om e n t on the c o lum ns i t is , t he re f o re , r e a sona b le t o i nc lude

the mo m e n t due to t he e q u iva l e n t l oa d , bu t w i th a l oa d f a c to r o f 1 .0 . T he

a pp r oa c h is c ons i s t e n t w i th t he c a l c u l a t ion o f t he u l t ima te l oa d c a pa c i ty o f the

beam , because the ef fect o f the equiva len t load i s inc luded in bo th ca l cu la t i on s~ fo r

t h e b e a m a n d t h e c o l u m n .

4 .5 M o v e m e n t s in a c o n c r e t e f lo o r

In re inforced concre te , shr ink age causes a s l igh t r edu c t ion in the length of the

m em be r an d c reep has the lon g- te rm e ffec t o f inc reas ing the de f lec t ion of a f loor .

P os t - t e ns ione d c onc r e t e a l so unde r goe s sho r t e n ing due to sh r inka ge a nd i t s

de f lec tion i s a lso a f fected by c reep; a ddi t io na l ly , because of the ax ia l p re -

com press io n , a fur the r sh or ten ing occu rs because of e la s tic s t r a in an d c reep .

T he r e f o re , de f o r m a t ion is h ighe r i n pos t - t e ns ione d tha n in r e in f o r ce d c onc r e t e ,

because of the ax ia l com po nen t of the pres t re ss . Both re inforced an d po s t - tens ione d

c onc r e t e s a r e a f f e c t e d by c ha nge s i n t e mpe r a tu r e a nd humid i ty .

The fac tor s , which a f fec t the de fo rm at ion of concre te , a re d iscussed in Ch apt e r s

1 a nd 7 ; t h is s e c t ion de a l s w i th t he r e l a ti ve ma gn i tud e s o f t he m ove m e n t c a use d

by each .

C on s ide r t he de f o r m a t ions o f a t yp i c al pos t - t e ns ione d s l ab pa ne l . A ssum e the

fol lowing for this exerc ise :

M odu lus o f e l a st ic i ty

Cre ep coeff ic ient C c

Thermal coe f f ic ien t

C t

S hr inka ge s t r a in

Average pres t re ss

T e m p e r a t u r e c h a n g e

L e n g t h o f m e m b e r L

Eel = 2 2 k N / m m 2

= 2.5

= 8 • 10 - 6 /~

= 250

x 10 -6

= 2 N / m m 2

= 10 ~

- 1 0 m

T he c ha ng e s i n t he l e ng th o f the m e m be r a re "

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P L A N N I N G A S TR U C TU R E 91

E l as t i c s h o r t e n i n g - 2 .0 x 1 0 /2 2 = - 0 .9 1 m m (0 .0 3 6 i n )

Cr ee p - 2 .5 x 0 .91 = - 2 .28 m m (0 .090 in)

Sh r ink ag e - 10000 x 250 x 1 0 - 6 ._ _ _ 2 .50 m m (0 .098 in )

- - 5 .69 m m (0 .224 in)

T h e r m a l + 10 x 1 0 x 8 x 1 0 - 3 - - _ 0 . 8 0 m m ( 0.0 31 i n )

C h an g e i n h u m i d i t y , s ay = _ 1 .0 0 m m ( 0.03 9 i n )

+ 1 .80 m m (0 .070 in )

F o r a 1 0 m l o n g s l ab , s t re s s ed t o an av e r ag e o f 2 . 0 N / m m 2, t h e l o n g - t e r m

s h o r t en i n g o f i ts l en g t h w o u l d b e 5 .69 mm ( 0 .2 2 4 i n ). T o a l l o w f o r t h e av e r ag e

h u m i d i t y co n d i t i o n , d ed u c t 1 .0 m m f r o m t h i s f ig u re . T h en t h e ch a n g e i n l en g t h ,

i n c l u d in g t h e h u m i d i t y a n d t h e t e m p e r a t u r e c h a n g e s , w o u l d b e - 4 . 6 9 +

1 .8 0 m m , i. e. , t h e s h o r t e n i n g in a 1 0 m l e n g t h c o u l d v a r y f r o m - 6 . 4 9 t o - 2 . 8 9

m m ( - 0 . 2 5 6 t o - 0 . 1 1 4 i n) . T h e t o ta l sh o r t e n i n g , 6.4 9 m m , r e p r e s e n ts 0 . 0 6 5 % o f

the l eng th .

I t i s w o r t h n o t i n g t h a t t h e ab o v e f ac t o r s , ex cep t e l a s t i c s h o r t en i n g an d c r eep ,

a l s o a ff ec t re i n f o r ced co n c r e t e . I n t h is ex am p l e , t h e e l a s t ic s h o r t e n i n g an d c r eep

a c c o u n t f o r a p p r o x i m a t e l y 5 0 % o f t h e m o v e m e n t . A re i n fo r c e d c o n c r e t e m e m b e r

o f t h e s ame l en g t h w o u l d h av e o n l y 5 0 % o f t h e s h o r t en i n g .

I f t h e me m b er w as s t r e s s ed t o an av e r ag e co m p r es s i o n o f 6 N / m m 2 ( 90 0 ps i) , a s

a b eam can b e , t h e e l a s t i c s h o r t en i n g , c r eep an d h u mi d i t y ch an g es w o u l d b e

p r o p o r t i o n a t e l y l a r g e r . T h e n , t h e s h o r t en i n g i n a 10 m l en g t h co u l d v a r y f r o m

- 1 2 . 9 to - 9 . 3 m m ( - 0 . 5 1 t o - 0 . 3 7 in).

I n a r e i n fo r ced co n c r e t e m em b er o f t h e s ame l en g t h , t h e r e b e i n g n o ax i a l st r es s ,

n o s h o r t en i n g i n l en g t h o ccu r s d u e t o t h e e l a s t i c an d c r eep p h en o men a , an d t h e

e ff ec t o f h u m i d i t y i s a l s o s ma l l e r . T h e r e f o r e , t h e v a r i a t i o n i n len g t h m ay b e o f t h e

o r d e r o f - 2 . 5 + 1 .0 m m , s m a l l e r th a n t h a t f or p o s t- t e n s i o n e d c o n c r e te .

C r e e p , sh r i n ka g e a n d t e m p e r a t u r e c h a n g e d e p e n d o n a n u m b e r o f e n v i r o n m e n t a l

f ac t o r s , a s d i s cu s s ed i n C h ap t e r 2 , an d , t h e r e f o r e , t h e d i f f e r en ce b e t w een a

r e in f o rc e d a n d a p o s t -t e n s i o n e d m e m b e r w o u l d v a r y in a c c o r d a n c e w i t h t he l o ca l

co n d i t i o n s .

T h e e x p e c t e d r a n g e o f m o v e m e n t i n a p o s t - t e n s i o n e d f l oo r s h o u l d b e a s s e ss e d

f o r e ach p r o j ec t . T h e a r ch i t e c t u r a l a n d t h e s t r u c t u r a l d e s i g n s o f t h e b u i l d i n g mu s t

a l l o w f o r t h e l a r g e r mo v emen t ex p ec t ed i n a p o s t - t en s i o n ed co n c r e t e f l o o r , e l s e

t h e c o n c re t e m a y c r a c k s o m e w h e r e . T h i s is o f i m p o r t a n c e a t t he e x p a n s i o n j o i n t s

w h e r e t h e r e l a t i v e l y l a r g e r g ap b e t w een t w o co l u m n s , o r t h e s l i d in g j o i n t , m ay b e

u n s i g h t l y i f ex p o s e d o n e l ev a t i o n . T h e ex p a n s i o n j o i n t i n t h e f l o o r m ay n ee d a

cove r s t ri p . Th e f il l er i n t he jo in t m us t be su ff i c ien t ly res i l i en t t o wi th s t an d the

l a r g e m o v e m e n t s .

4 .6 C r ack p r ev en t ion

M o s t o f t h e f ac t o rs , w h i ch can c o n t r i b u t e t o t h e c r ack i n g o f a f l o o r, h av e b een

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92 POST-TENSIONED CONCRETE FLOORS

d iscussed in th i s and in o ther chap ters . The sub jec t i s , however , cons idered

i m p o r t a n t e n o u g h f o r p o s t - t e n s i o n e d c o n c r e t e t o b e s u m m a r i z e d h e r e .

A co n c re t e m em b er c an d ev e l o p c rack s t h r o u g h l ack o f c a re o f t h e co n c re t e a t

an ea r l y ag e , an d t h ro u g h o v e r l o ad i n g . A d d i t i o n a l l y , i n a p o s t - t en s i o n ed

m e m b er , t h e p o t en t i a l s o u rce s o f c r ack s a r e :

9 r e s t r a i n t f ro m fo rm wo rk b e t w een t wo d o w n s t an d f ace s

9 r e s t r a i n t f ro m fo rm wo rk a t a h o l e i n t h e f l o o r

9 res t ra in t f rom s t i f f ver t i ca l members

9 insu ff ic ien t ax ia l p res t ress i f i ts d i spers ion has no t been ta ke n in to acc ou n t

9 co n s t ru c t i o n j o i n t s , wh e re t h e co n c re t e m ay l ack i n ten s il e s t r en g t h

9 b r i ck o r b l o ck w a l l p an e l s e r ec t ed ea r l y an d t i g h t ag a i n s t t h e co l u m n s

In o rd e r t o p rev en t c r ack s a t t r i b u t ab l e t o fo rm wo rk , i t s d e s i g n s h o u l d a l l o w

rem o v a l o f s o m e s t r a t eg i c n a r ro w s t r ip s a s s o o n a s th e co n c re t e h a s h a rd en ed .

Fo rm wo rk i n s i d e a h o l e s h o u l d b e r em o v ed a s s o o n a s p rac t i c a l .

Re -en t ran t ang les , such as a t the co rners o f a ho le o r when a f loo r is s t epp ed in

p l an , a r e p o t en t i a l p o i n t s o f s t re s s co n ce n t r a t i o n . I t is a g o o d p rac t i c e t o p ro v i d e

s o m e ro d r e i n fo rcem en t a t t h e se p o i n t s ~ s t r a i g h t b a r s a t 45 ~ ac ro s s t h e r e - en t r an t

an g l e a r e t h e co m m o n fo rm o f r e i n fo rcem en t . T h e a m o u n t o f r e i n fo rcem e n t is a

m a t t e r o f j u d g e m e n t ; t w o 10 m m d i a m e t e r b a r s t o p a n d b o t t o m s h o u l d b e

suff ic ien t a t ho les up to abou t one met re long .

R es t r a i n t f ro m t h e v e r t ic a l e l em en t s t o f ree m o v e m en t o f t h e f lo o r , an d t h e

d i s p e r s i o n o f p re s t re s s i n t o ad j acen t co n c re t e w h i ch h a s a l o wer av e rag e p re s t r e s s,

o r i s no t p res t ressed , shou ld be a l lowed fo r in the ca lcu la t ions .

A t co n s t ru c t i o n j o i n t s wh i ch a re n o t i n co m p res s i o n d u e t o p re s t r e s s , i t i s

p ru d en t t o p ro v i d e s u f f i c i en t ro d r e i n fo rcem en t fo r t h e t o t a l t en s i o n wh i ch m ay

d ev e l o p , a s s u m i n g t h a t t h e co n c re t e h a s n o t en s i l e s t r en g t h . In t e rm ed i a t e

anc hor age s , o f course , need b urs t in g cages on the s t ressed s ide .

A p a r t i t i o n co n s i s t in g o f b r i tt l e m a t e r i a l a n d b u i l t h a rd ag a i n s t t h e s t ru c t u r a l

f r am e co n t a i n i n g p o s t - t en s i o n ed f lo o r s m ay i ts e lf d ev e l o p c rack s . A p a r t i t i o n

t r ap p ed h a rd b e t ween t wo d o w n s t a n d b eam s m a y a ls o s h o w s ig n s o f d i s tr e s s. T h e

e rec t i o n o f s u ch p a r t i t io n s s h o u l d b e d e l ay ed to a l l o w a s m u ch m o v e m en t o f t h e

f l o o r t o t ak e p l ace a s is p e rm i t t ed b y t h e co n s t ru c t i o n p ro g ram m e , an d s o ft j o i n t s

s h o u l d b e p ro v i d ed t o a l l o w s h o r t en i n g o f t h e f r am e .

T h e r e s t r a i n t t o m o v e m e n t d u e t o t h e s t if f v e r t ic a l e lem en t s can i n d u ce

h o r i z o n t a l t h ru s t s o f s u ch m ag n i t u d e t h a t t h e r e s t r a i n i n g e lem en t s t h em s e l v es

beco m e ov er load ed . As d i scussed in the S ec t ion 4 .4 , the e ffect o f p res t ress in g a

f l o o r is t o r ed u ce th e m ag n i t u d e o f co l u m n m o m en t s . H o w ev e r , in p a r t i cu l a r

c i r cu m s t an ces t h e h o r i z o n t a l t h ru s t o n a co l u m n m ay p ro d u ce t h e m o re c r i t i c a l

co n d i t i o n fo r d e s ig n . In t h is c a se , th e co l u m n wo u l d t en d t o c r ack o n t h e o p p o s i t e

face to th a t e xpec ted fo r the n orm al ver t i ca l load ing , v iz , on the ins ide face o f a

s ing le bay . M os t vu lnera b le in th i s respec t a re the co lum ns in the low er s to reys o f

t a ll b u il d i n g s, co l u m n s i n th e b a s em en t wh e re t h e r e t a i n i n g w a l l ex t en d s o n l y t o

p a r t o f t h e h e i g h t , an d t h e en d c o l u m n s i n lo n g f l o o r s p a r t i cu l a r l y i f o n e en d is

r e s t r a i n ed b y a s ti ff co re . T h e co l u m n fo rces s h o u l d b e ca l cu l a t ed an d a l l o wed fo r

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P L A N N I N G A S TR U C TU R E 9 3

in the des ign. In the case of low retain ing wal ls , i t m ay be poss ible n ot to con nec t

the wal l and the co lumns .

4 .7 T e n d o n p r o f i le

The geo me t ry and equ iva len t loads o f the var ious t end on p ro fi les used in

post - tens ioned f loors are d iscussed in detai l in Chapter 5 .

Economic cons idera t ions o f t en requ i re tha t the p res t ress ing fo rce be the

minimum which sat is f ies the serviceabi l i ty condi t ions . Any deficiency in the

s t r en g th o f a me mb er can b e mad e u p b y ad d i n g b o n d e d r e i n fo rcemen t ; t h e re fo re ,

the level of pres t ress tend s to be determ ined by the serviceabi l ity require m ents .

In a s imply supp or ted m em ber the am ou nt o f p res t ress is eas ily de te rm ined ,

b ecau s e th e r e a r e n o r ed u n d an c i e s . Th e p ro b l em o f d e t e rmi n i n g an eco n o m i ca l

t endo n p ro f il e is m ore d i ff icu lt in con t inu ous me mb ers . Th e p rocess requ i res tha t

a profi le be guessed in i t ia l ly and then ref ined in des ign by repet i t ion. The des ign

an d ref inem ent proc ess i s eas ier if the in i t ia l ly chose n profi le is a reason able

approx imat ion to the f ina l p ro f i l e .

Assuming that the f inal tens i le s t ress i s the cr i ter ion, an obvious goal i s to so

tai lor the pres t ress that the f inal tens i le s t ress equals the permiss ible value at a l l

cr i t ical sect ions . A good s tar t ing poin t i s for the pr od uc t o f the pres t ress ing force

and the eccen t r ic i ty to fo llow the shape o f the o f the dead loa d m om en t d iag ram

for the con t inu ous me mb er . A n even be t t e r pa t t e rn to fo llow is tha t o f the s tress

d i ag ram co r r e s p o n d i n g t o t h e d ead l o ad mo men t d i ag ram, b ecau s e t h i s a l l o w s

for the di fferent sect ion m odu l i a t the top an d the b ot t om of the sect ion; i t i s,

how ever, too com plicated to calculate the stresses just for the initial shape of a profi le.

I n fo ll o w in g th e d ead l o ad mo me n t d i ag ram, ad v an t ag e can b e t ak en o f t h e

prop er ty tha t i t i s the sag o f the para bo la which i s im por tan t , no t i t s abso lu te

pos i t ion ing . P ara bo la s hav ing the same sag a re equ iva len t , and they al l p rod uce

t h e s ame equ i v a l en t mo men t d i ag ram a f t e r i n c l u d i n g t h e s eco n d a ry mo men t s .

Therefo re , in each span , the sag o f the t end on p ro f il e shou ld represen t the sag o f

t h e p a rab o l a o f t h e d ead l o ad mo m en t d i ag ram . Th i s is t an t a m o u n t t o b a l an c in g

a f ixed p ro po r t ion o f the to ta l load . The p rob le m of op t imizing the p res t ress ing

force can , therefo re , be t rans fo rme d in to a cons ide ra t ion o f how m uch load

s h o u l d b e b a l an ced b y t h e ten d o n cu rv a t u re . I n o rd e r t o b a l an ce a co n s t an t l o ad

w~ in sp ans of d i ffer ing length , the rat io P.ep /L 2 shou ld be cons tan t .

In o rd er to v ary the equ iva len t load in a span , e i ther the p res t ress ing fo rce can

be changed , o r the eccen t ri c ity . I f the t endon s a re p laced a t m ax im um eccen t r ic i ty ,

then the t endon fo rce mus t vary be tween spans , requ i r ing a d i f fe ren t number o f

t endon s in each span . Al te rna t ive ly , the same t end ons m ay run th ro ugh a ll spans

an d the eccentr ici ty ma y var y from sp an to sp an. I t i s, of course, poss ible to use a

co m b i n a t i o n o f t h e t w o .

In con t inuou s m em bers o f vary ing sp an l eng ths, o r d i ffe ring loads , i t is o ft en

economica l to cu r t a i l t he t endons to su i t the p res t ress ing requ i rements . F igures

4 .7 (a) , (b ), and (c) show the poss ib le t en don cur t a i lme n t s fo r p rov id ing a l a rger

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94

POST-TENSIONED CONCRETE FLOORS

a, b ~ a,c

f

(a) Lo ng outer spans

a,b c ~ a,c

(b) Long middle spans

b d c e

f f f

(c) Combination of (a) and (b)

Figure 4.7

Curtaif ing the tendons

pres t re ss ing force in the ex te rna l spans , in the in te rna l spans , and in th ree

a l t e r na t e spa ns . I n F igu r e s 4 .7 ( b ) a nd (c ), som e o f t he t e ndons c a n r un th r o ug h a ll

o f t he spa ns . I t i s des i ra b l e t o a vo id the c om pl i c a t ion o f l oc a t ing l ive a nc ho r a ge s

away f rom the s lab edge ; th i s i s a r e s t r ic t ion which exc ludes the poss ib i l i ty of

p r ov id ing op t imum p r e s t r e s s i n a l l pos s ib l e c a se s .

A s a c u r ve d t e n don is st re s sed , t he s t r a nds t e nd to b unc h up in t he she a th ing

in to the curve . There fore , the cen t ro id of the tend ons i s no t co inc iden t wi th the

cent re of the shea th in g , wh ich resu l ts in a r edu c t ion in eccent ric i ty . The redu c t ion

is no r m a l ly o f the o r d e r o f 9 m m ( 0.35 in ) f o r m u l t i s t r a nd t e ndon s a n d 2 mm ( 0.08

in ) f o r f la t t e ndons u se d in f l oo r s . A use fu l a pp r o x im a t ion i s t o a s sum e the t e ndo n

c e n t r o id a t t he t h i r d po in t o f t he she a th ing d i a m e te r .

4.8 Ac ce ss at the l ive end

I de a l ly , a l t e r na t e a nc ho r a ge s wou ld be s t r e s se d f r om oppos i t e e nds ; t h i s wou ld

dis t r ibu te the ef fect o f f r ic tion losses , and so provid e an ap pro xim ate ly un i form

pres t re ss in a l l the spans . In so m e loca t ions , how ever , access m ay no t be ava i lab le

to one o r bo th o f t he s la b e dge s f o r s tr e s sing the t e ndons . T wo op t ion s a r e

ava i lab le in th is s i tua t ion .

9 I f access i s ava i lab le a long one edge of the s lab , then a l l o f the l ive anch orag es

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PLA N N I N G A S T R U C T U R E 95

. . . . .

Plan

1_. 450 _l

( D O

Section A

t

Section B

Previously

/ . .. Tendons Ed gebeam

,1 ,T _ _ ~ cast wall alternating or wall ~ .

]~/ Stress ing ocket ~k, ; L I

l " / / L i v e - e nd anch~ ~ ~ l ]

IrE Starterbars Dead-end anchorage

Figure 4.8 L i ve anchorage fo r s t ress ing f r om s lab su r face

are loca ted a long th i s edge . The d i sadvan tage o f th is a r ran gem ent is tha t the

pres t ress ing force is lower at the far end. I f the far span is shorter than the

others , then th is lay ou t is , in fact , preferable to th at o f a l tern ate a nch orag es

being s t ressed from opposi te ends .

9 I f the loss in p res t ress f rom the above a r ran gem ent can no t be accep ted , o r i f

access i s no t av ai lable at e i ther of the two e nds , then the live anch ora ges can be

p laced in pocke t s , away f rom the suppor t cen t re l ine , as shown in F igure 4 .8 .

Suff icient space must be avai lable behind the pocket for s t ress ing the tendons .

The d imens ions shown in F igure 4 .8 a re typ ica l fo r a 4 -s t rand f l a t t endon .

I t shou ld be apprec ia ted tha t such a t endo n does n o t t rans fer any o f the shear

fo rce to the suppor t , and tha t the concre te be tween the anchorage and the

support i s not s t ressed. The unst ressed s t r ip must , therefore, be des igned in

reinforced concrete .

When s t res sed , the e l as t i c shor ten ing , sh r inkage and c reep be tween the

s t ressed and the reinforced lengths induce tens i le s t resses in the unst ressed

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9 6 POST-TENSIONEDCONCRETE FLOORS

concre te s t r ip ; the m agn i tude o f th is t ens ion d epends on the degree o f res t ra in t

of the supp ort sys tem. Add i t ional reinforcem ent m ay be required for th is tens ion.

The s i tua t ion is imp rove d i f the t endons a re s tres sed f rom a l t e rn a te ends ,

which a l lows the dead anchorages to be p laced nearer , o r a t , t he suppor t .

4 .9 T r a n s f e r b e a m s

M ost m ode rn o f f ice and ho te l bu ild ings con ta in a co lumn-f l ee lobby a t the

ground f loor , whi l e the upper f loors have co lumns a t a much c loser spac ing .

Of ten , the c l ear lobb ies ex tend over severa l f loors . The con cen t ra ted loads f rom

the upper co lumns a re t rans fer red to su i t ab le suppor t s be low th rough a se r i es o f

transfer beams.

I f des igned in re in fo rced concre te , the t rans fer beam s wo uld

requ i re l a rge quan t i t i es o f rod re in fo rcement .

Post - tens ioning offers a very economical a l ternat ive in res is t ing the f lexure of

the t rans fer beam . P res t ress a l so enhances the shear capa c i ty o f the beam ,

becau se the axial com press ion at the ends redu ces the pr incipal tens i le s t ress . The

tendons in a t rans fer beam are normal ly harped in p ro f i l e (d i scussed in Chap ter

5 ) , the equ iva len t concen t ra ted load then d i rec t ly opposes the load f rom the

u p p e r co l u mn .

Somet imes, the span is too short for a harped profi le to be used as i t requires

the t endon to have a sharp bend , need ing an im prac t i ca l ly smal l rad ius . The s i ze

of such a shor t t rans fer beam i s govern ed by shear ra ther than bend ing . T he

tendo n , in such a shor t beam , may be p ro fi led as a para bo la . The load f rom the

u p p e r co l u m n w o u l d n o t b e o p p o s ed d i r ec tl y , b u t t h e p a rab o l a w o u l d p ro v i d e a

wor thw hi le mo m ent oppo s ing tha t f rom the load . I f the span is too sho r t fo r even

a parabo l i c p ro f i le then s t ra igh t t end ons m ay be p rov ided , so p laced as to induce

no, or very l i t t le , tens ion at the top.

The t endon s in a t rans fer beam m us t be s t ressed in severa l s tages , as the up per

f loors a re cons t ruc ted and the i r dead weigh t i s imposed on the t rans fer beam. I f

fu lly s t res sed in one ope ra t ion , the compress ive s t res s a t the bo t tom , a nd perh aps

the tensile stress at the top, is very l ikely to exceed the safe value, possibly

resul t ing in fai lure . The high level of pres t ress in a t ransfer b eam requires

par t i cu la r care in i ts dem ol i tion . A n unco ord in a ted rem oval o f dead load ,

wi thou t a co r respond ing reduc t ion in p res t ress , may cause an exp los ive fa i lu re .

A var i a t ion on the t rans fer beam is the can t i l evered cap o n the top o f the core in

a high-r ise bui ld ing, f rom which the lower f loors are hung. In th is sys tem, one

edge o f a f loor is supp or ted on the core and the o ther by the hangers . Th e p ro b lem

with reinforced c oncrete , or s teelw ork, in th is s i tuat ion is tha t of def lect ion. A s

each f loor i s cons t ruc ted , the cap def l ec t s and the hangers ex tend , thereby

increas ing the deflection of the previo usly com pleted f loors relat ive to the core.

Pos t - t ens ion ing o f the cap has the adv an tag e tha t the def lec t ion o f the cap can be

con t ro l l ed , a nd i t m ay even be poss ib le to cause the cap to def lec t upward s a t each

s t ress ing s t age , comp ensa t ing fo r par t o f the e longat ion o f the hanger .

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P L A N N I N G A S TR U C TU R E 97

4.10 Durab i l i ty

F o r t h e n o r ma l b u i l d i n g , w h i ch i s n o t ex p ec t ed t o h o u s e an y d e l e t e r i o u s

s u b s t an ces , t h e mo s t v u l n e r ab l e co n s t i t u e n t i s t h e st ee l. D u r ab i l i t y co n s i d e r a t i o n ,

t h e re f o re , r e q u ir e s a d e q u a t e p r o t e c t i o n o f t h e t e n d o n a n d t h e r o d r e i n f o r c e m e n t

t o g u a r d ag a i n s t t h e i r co r r o s i o n . T h e co n c r e t e co v e r is ex p ec t ed t o p r o v i d e t h i s

p r o t e c t i o n .

A b u i l d i n g i n p o s t - t en s i o n ed co n c r e t e c a n s u f fe r a g r ea t e r l o s s o f s t r en g t h t h an

i n r e i n f o r ced co n c r e t e , p a r ad o x i ca l l y b ecau s e p o s t - t en s i o n i n g i s m o r e e f f ic ien t. A

f l o o r co n t a i n s a mu c h l a r g e r q u a n t i t y o f s te e l i f r e i n f o r ced t h an p o s t - t en s i o n ed ,

a n d t h e m a i n r e i n f o r c e m e n t m a y c o n s i s t o f 2 0 o r 25 m m d i a m e t e r b a r s , w h e n t h e

s t r a n d c o n s i s ts o f 4 o r 5 m m d i a m e t e r w i re s. D e p e n d i n g o n t h e l o a d i n g a n d s p a n

l en g t h , t h e s p ac i n g o f s t r an d i n a p o s t - t en s i o n ed s l ab m ay b e t w o t o f o u r t i mes t h e

s p ac i n g o f r o d r e i n f o r cem en t i n a r e i n f o rced co n c r e t e s l ab ca r r y i n g t h e s ame

i m p o s e d l o a d i n g b u t 5 0 % d e e p e r. I f r e i n fo r c e m e n t is s p a c e d a t 2 0 0 m m c e n t re s

t h en t h e lo s s o f o n e b a r p e r me t r e w i d t h am o u n t s t o a r ed u c t i o n i n s t r en g t h o f

2 0 % ; a p o s t - t en s i o n ed f l o o r m ay h av e s t r an d s a t 60 0 cen t r e s an d a l o ss o f o n e

s t r an d i s t a n t am o u n t t o a r ed u c t i o n o f 6 0 % i n i ts s t ren g t h . A co r r o s i o n l o ss o f o n e

mi l l i me t r e i n t h e d i am e t e r o f a 20 m m r e i n f o r cem en t b a r r ep r e s en t s a 1 0 %

r ed u c t i o n o f a r ea , w h e r ea s i t i s eq u i v a l en t t o a 4 4 % r ed u c t i o n i n a 4 m m w i r e .

T h e s t r a n d t e n d o n s i n a p o s t- t e n s i o n e d c o n c r e t e m e m b e r n e e d a b e t te r q u a l i t y

o f p r o t e c t i o n t h a n r o d r e i n f o r c e m e n t d o e s in r e i nf o rc e d c o n c r e t e . P o s t - t e n s i o n e d

e l e m e n t s n o r m a l l y r e m a i n c r a c k - f r e e u n d e r p e r m a n e n t l o a d s , a n d t h e y c a n e v e n

b e d e s i g n ed t o b e t en s i o n -f r ee u n d e r n o r m a l l o a d co n d i t i o n s i f r eq u i r ed . T h e

c r ack - f ree co v e r p r o v i d e s a b e t t e r p r o t e c t i o n t o s te e l t h an t h a t g i v en b y th e co v e r

i n a r e i n f o r ced co n c r e t e memb er , w h i ch mu s t b e c r ack ed .

T h e d e g r e e o f p r o t e c t i o n p r o v i d e d b y t he c o n c r e t e c o v e r d e p e n d s o n t h e

p e r m eab i l i t y o f t h e co n c r e t e , t h e d ep t h o f co v e r , t h e ch emi ca l a l k a l i n i t y o f t h e

i m m e d i a t e e n v i r o n m e n t a n d t h e e x p o s u r e c o n d i t i o n s . T h e p e r m e a b i l i t y o f

c o n c r e t e c a n b e m i n i m i z e d b y h a v i n g a n a d e q u a t e c e m e n t c o n t e n t , a l o w f r e e

w a t e r - c e m e n t r a t i o , b y g o o d c o m p a c t i o n , a n d b y s uf fi ci en t h y d r a t i o n o f t h e

c e m e n t t h r o u g h p r o p e r c u r i n g .

M an y p r o ces s e s o f d e t e r i o r a t i o n o f co n c r e t e o ccu r i n t h e p r e s en ce o f w a t e r .

T h e r e f o re , a s t r u c t u r e s h o u l d b e s o d e s i g ne d a n d d e t a i l e d t h a t w a t e r a n d m o i s t u r e

h a v e a m i n i m u m a c c e s s t o c o n c r e t e . W h e r e u n a v o i d a b l e , t h e d e t a i l i n g s h o u l d

en s u r e t h a t t h e w a t e r d o es n o t p o o l o n t h e co n c r e t e s u r f ace o r g e t t r ap p ed i n

co n t ac t w i t h t h e co n c r e t e . C r ack s can p r o v i d e an acce s s r o u t e f o r t h e mo i s t u r e ,

a n d m e a s u r e s s h o u l d b e t a k e n t o a v o i d fo r m a t i o n o f c r a c k s b y th e g e n e r a t i o n o f

h ea t , o r d u e t o s h r i n k a g e . C o n c r e t e i s m o r e l i ke l y t o d e t e r i o r a t e i f t h e s ec t i o n s a r e

th in .

C a r - p a r k i n g f l o o r s an d acce s s a r ea s t o t h e b u i l d i n g s a r e n o r ma l l y ex p o s ed t o

t h e w e a t h e r ; t h e y m a y a l s o o c c a s i o n a l l y c o m e i n t o c o n t a c t w i t h d e l e t e r i o u s

ch emi ca l s . A n en c l o s ed b u i l d i n g i s l i k e l y t o h av e a co n t r o l l ed en v i r o n men t , an d

i ts f l o or s a r e n o t ex p ec t ed t o u n d e r g o f r eez i ng o r co me i n t o co n t ac t w i t h i n j u r i o u s

ch emi ca l s .

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98 POST-TENSIONEDCONCRETE FLOORS

T h e s t r a n d u s e d in u n b o n d e d p o s t - t e n s i o n i n g i s p r o v i d e d a n a d d i t i o n a l

p r o t e c t i on by t he e x t r ude d s l e e ve a nd g r e a se pa c k i ng . Th i s p r o t e c t i on i s ,

how e ve r , no t a va i l a b l e t o t he a nc ho r a ge s .

4. 10. 1 BS 8110 requ i rem ents

W h e r e c o n c r e t e m a y b e s u b j e c t e d t o f r e e z i n g a n d t h a w i n g , a i r e n t r a i n m e n t

shou l d be c ons i de r e d . F o r 28 - da y s t r e n g t h s b e l ow 50 N / m m 2 ( 7000 ps i ), BS 8110

r e c o m m e nd s t he f o l low i ng a i r c on t e n t by v o l um e o f t he fr e sh c onc r e t e a t t he t i m e

of p lac ing .

7 % f o r n o m i n a l 1 0 m m m a x i m u m a g g r e g a t e (3 i n)

6 % f o r n o m i n a l 14 m m m a x i m u m a g g r e g at e ( 9 in )

5 % f o r n o m i n a l 2 0 m m m a x i m u m a g g r e g a t e ( 3 i n)

4 % f o r n o m i n a l 4 0 m m m a x i m u m a g g r e g a t e (1 89 n )

W i t h r e ga r d t o the e xposu r e c ond i t i ons o f a c onc r e t e s t r uc t u r e , BS 8110

r e c ogn i zes f ive c a t e go r i e s o f e nv i r onm e n t , a s sho w n i n Ta b l e 4 . 2 .

F o r n o r m a l a n d l i g h t w e i g h t c o n c r e t e s w i t h 2 0 m m a g g r e g a t e , B S 8 1 1 0

r e c o m m e nd s t he m i n i m um de p t h o f c ove r t o a ll s te e l, i n c l ud i ng s t i r r ups , a nd t he

c onc r e t e c on t e n t s show n i n Ta b l e 4 . 3 .

4. 10 .2 AC l 318 requ i rem ents

W he r e t he c onc r e t e is sub j e c t to f r ee z ing a nd t ha w i ng c yc le s , AC I 318 r e c om m e nd s

a i r e n t r a i n m e n t i n n o r m a l a n d l i g h t w e i g h t c o n c r e te s i n th e p r o p o r t i o n s g i v e n in

Ta b l e 4 .4 . The spe c if ie d t o l e r a nc e i n t he a i r c on t e n t is 1 .5 % . A m o de r a t e e xp osu r e

is one w he r e i n a c o l d c li m a t e t he c on c r e t e w i ll b e on l y oc c a s i ona l l y e x po se d t o

Table 4.2

Exposure cond i t ions

Environment Expo sure cond i t ion

Mild

Concrete protected against weather or aggressive condit ions

Modera te

Surfaces sheltered from severe rain or freezing whilst wet, concrete subject

to condensation, concrete surface continually under water

Concrete in contact with non-aggressive soil

Severe

Surfaces exposed to severe rain, alternate w etting and dryin g or occasional

freezing or severe cond ensation

Very severe Surfaces exposed to sea wa ter sp ray, deicing salts, corrosive fumes, severe

freezing whilst wet

Extreme

Surfaces exposed to abrasive ac tion, e.g . sea water carryin g solids o r

flowing water with pH < 4.5 or machinery or vehicles

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Table 4.3

C o v er f o r du r ab i l i ty

PLANNING A STRUCTURE 99

E x p o s u r e N o m i n a l c o v er , m m N o m i n a l c o v er , m m

N o r m a l w e i g h t c o n c r e te N o r m a l w e i g h t c o n c r e te

Mild

Moderate

Severe

Very severe

Extreme

20

20 20 20 20 20 20 20

30 25 20 45 40 35 30

40 30 25 50 40 35

50 40 30 60 50 40

6O 5O 7O 6O

Maximum water/cement ratio

Minimum cement content, kg/m 3

28-day cube strength, N/mm 2

0.60 0.55 0.50 0.45 0.60 0.55 0.50 0.45

300 325 350 400 325 350 375 425

35 40 45 50 25 30 35 40

Table 4.4

A i r c o n t en t f o r f r o s t r e s i s tan t c o n c r e t e

M a x i m u m s i z e

o f ao gr ega t e

re'< 35 N / m m 2 (5000 psi )

fc'>_ 35 N / m m 2 (5000 psi )

S e v e r e M o d e r a t e S e v e r e M o d e r a t e

mm i n e x po s u r e e x po s u r e e x po s u r e e x po s u r e

9.5 3 7.5 % 6.0 % 6.5 % 5.0 %

8

1 7 . 0 % 5 . 5 % 6 . 0 % 4 . 5 %

2.5

3 6.0 % 5.0 % 5.0 % 4.0 %

9.0

25.0 1 6.0 % 4.5 % 5.0 % 3.5 %

37.5 1 89 5.5 % 4.5 % 4.5 % 3.5 %

moisture prior to freezing, and where no de-icing salts are used.

The ACI 318 requirements for special exposure conditions for concrete using

norma l density and low density aggregate are given in Table 4.5. The minimum

cement content of concrete exposed to freezing and thawing in the presence of

de-icing chemicals is specified as 252 kg/m 3 of concrete.

Table 4.6 shows the ACI 318 recommendation for the nominal minimum

concrete cover to all steel (including ties) for prestressing tendons and rod

reinforcement, ducts, and anchorage ends. For concrete members exposed to

earth, weather, or corrosive environments, and in which the permissible tensile

stress is exceeded, the minimum cover should be increased by 50%.

4 .11 F i re p ro tec t ion

The cover concrete acts as an insulation to the tendons and rod reinforcement,

delaying the rise of temperature of the steel in case of fire. The strength of steel is

affected when exposed to high temperature; the prestressing strand is more

vulnerable in this respect than high-strength rod reinforcement. Typically, the

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100 POST-TENSIONED CONCRETE FLOORS

Table 4 .5 Concrete for special exposure condit ions

Exposure condi t ion

M a x i m u m M i n im u m f ' c f o r

water-cement low den sity

ratio, norm al aggregate

aggregate (N /m m 2)

Concre t e i n t ended t o have l ow

permeabi l i ty when exposed to

wa te r

0.50 25

Concrete exposed to f reez ing

and t hawing i n moi s t cond i t i on

0.45 30

F or cor ros ion p ro t ec t i on fo r

re inforced concre te exposed to

deicing sal ts , brackish water ,

seawater or spray f rom these

sources

0.40* 30*

Note" If minimum co ncrete cover shown is increased by 10 mm (3 in), water-cement ratio may be

increased to 0.45 for no rm al density concrete, orfr reduced to 30 N/mm (4500 psi) for low-density

concrete.

Table 4 .6

Min imum cover ( f r om AC I 318)

Concre t e cas t aga ins t and

permanen t ly exposed t o ea r t h

70

m m

(3

in)

Exposed t o ea r t h and wea the r

Sla bs 30 (1 88

Oth er me mb ers 40 ( I 89

N o t exposed to ea r t h o r w ea the r

Slabs 20 (3)

M ain steel in be am s 40 (1 89

Ties in bea m s 20 (3)

s t r a n d s tr e n g t h d r o p s f r o m a 1 0 0 % a t 1 5 0 ~ t o n il a t 7 0 0 ~ t h e d r o p i n t h e

s t r e n g t h o f t h e h ig h - s t r e n g t h r o d r e i n f o r c e m e n t i s f r o m a 1 0 0 % a t 3 0 0 ~ t o 5 % a t

8 0 0 ~ T h e d e p t h o f c o v e r is , t h e r e fo r e , o f a g r e a t e r im p o r t a n c e i n p o s t - t e n s i o n e d

t h a n i n r e i n f o r c e d c o n c r e t e .

T h e f i re re s i s t a n c e o f a m e m b e r i s a f f ec t e d b y a n u m b e r o f f a c t o r s , i n c l u d i n g :

9 d e p t h o f c o n c r e t e c o v e r

9 d i s p o s i ti o n a n d p r o p e r t i e s o f r e i n f o r c e m e n t a n d t e n d o n s

9 t h e t y p e o f c o n c r e t e a n d a g g r e g a t e

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PLANNING A STRUCTURE

101

Table 4.7 M i n i m u m d i m e n s i o n s a n d c o v e r .

M a t e r i a l a n d e l e m e n t

F i r e pe r i o d ( h o u r s )

0 .5 1 .0 1 .5 2.0 3.0 4.0

Den s e c o n c r e t e

Simply supported beams

Continuous beams

Simple plain soffit floors

Continuous plain soffit floors

Simply supported ribbed floors

Continuous ribbed floors

width 100 120 150 200 240 280

cover 25 40 55 70 80 90

width 80 100 120 150 200 240

cover 20 30 40 55 70 80

thickness 75 95 110 125 150 170

cover 20 25 30 40 55 65

thickness 75 95 110 125 150 170

cover 20 20 25 35 45 55

thickness 70 90 105 115 135 150

rib width 80 110 135 150 175 200

cover 25 35 45 55 65 75

thickness 70 90 105 115 135 150

rib width 70 75 110 125 150 175

cover 20 25 35 45 55 65

L i g h t w e i g h t c o n c r e t e

Simply supported beams

Continuous beams

Simple plain soffit floors

Continuous plain soffit floors

Simply supported ribbed floors

Continuous ribbed floors

width 80 110 130 160 200 250

cover 25 30 45 55 65 75

width 80 90 100 125 150 200

cover 20 25 35 45 55 65

thickness 70 90 105 115 135 150

cover 20 20 30 35 45 60

thickness 70 90 105 115 135 150

cover 20 20 25 30 35 45

thickness 70 85 95 100 115 130

rib width 75 90 110 125 150 175

cover 20 30 35 45 55 65

thickness 70 85 95 100 115 130

rib width 70 75 90 110 125 150

cover 20 25 30 35 45 55

Note" the cover refers to the tendon or main reinforcement,and the thickness refers o the depth of the

topping concrete.

9 conditions of end support

9 size and shape of the member

9 stress level

Rapid rates of heating, large compressive stresses or high moistu re co ntent (over

5% by volume) can lead to spalling of concrete cover at elevated temperature s,

particu larly for thicknesses exceeding 40 mm to 50 mm. Such spalling may imp air

performance by exposing the steel to the fire or by reducing the cross-sectional

area of the concrete. Concretes made from limestone aggregates are less

susceptible to spalling than concrete made from aggregates containing a high

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102 POST-TENSIONED CONCRETE FLOORS

p r o p o r t i o n o f s il ic a , e . g ., f li n t, q u a r t z i t e s a n d g r an i t e s . C o n c r e t e m ad e f r o m

l i g h t w e i g h t ag g r eg a t e s r a r e l y s p a l l s . A ccep t ab l e meas u r e s t o av o i d s p a l l i n g a r e :

9

an ap p l i ed f i n i s h b y h an d o r s p r ay p l a s t e r , v e r mi cu l i t e , e t c .

9 t he p rov i s ion o f a f a l se ce i l i ng as a f i r e bar r i e r

9 t h e u s e o f l i g h t w e i g h t ag g r eg a t e s

9 the use of sacr i f ic ial t ensi le s teel

A l ig h t w e l d ed s te e l me s h f ab r i c , p l aced w i t h i n 2 0 mm o f t h e co n c r e t e s u r face , i s

o f t en u s ed a s s ac ri f ic i a l r e i n f o r cemen t t o p r ev en t s p a l li n g . A p r o b l e m a r i s es i f t h e

d u r ab i l i t y co n s i d e r a t i o n s r eq u i r e a co v e r o f m o r e t h an 2 0 m m t o a ll st ee l .

T h e b eh a v i o u r o f a s t r u c t u r e , w h en ex p o s ed t o f ir e, d ep en d s o n t h e av a i l ab i l i t y

o f t h e a l t e r n a t i v e s u p p o r t r o u t e f o r th e f l o o r l o ad s ; t h e co n d i t i o n o f t h e en d

s u p p o r t i s, t h e r e f o re , s i g n if i c ant . A d e q u a cy o f an a l t e r n a t i v e r o u t e r eq u i r e s t h a t

t h e s t ee l is d e t a il ed s o t h a t , i n t h e ca s e o f th e p r i m ar y r o u t e b eco m i n g i n e ff ec ti v e ,

t h e l o ad can b e s u s t a i n ed t h r o u g h t h e a l t e r n a t i v e r o u t e , a l b e i t a t a r ed u ced l o ad

fac to r .

T h e r a t e o f r is e o f t e m p e r a t u r e o f s te e l e m b e d d e d i n c o n c r e t e d e p e n d s o n t h e

r a t i o o f s u r face a r ea ex p o s ed t o t h e fi re an d co n c r e t e v o l u m e , an d o n t h e ma s s o f

the concre t e . The su r faces exposed to f i r e a re t he so f f i t fo r t he s l ab , and s ides and

s o ff it f o r b eam s an d r i bs . D o w n s t an d b eam s an d r i b s , h av i n g a l a r g e r s u r f ace a r ea

t o v o l u me r a t i o , a r e n o r ma l l y r eq u i r ed t o h av e mo r e co v e r t h an a s l ab w i t h a f i a t

s o ff it . A t en d o n l o ca t ed a t t h e co r n e r o f a me m b e r r ece iv e s h ea t f r o m t w o s u r face s ,

a n d i s , t h e r e f o r e , m o r e v u l n e r a b l e t h a n a t e n d o n a w a y f r o m a c o r n e r . N a t i o n a l

s t a n d a r d s u s u a l ly s pe c if y t h e m i n i m u m c o n c r e t e c o v e r a n d t h e m i n i m u m s iz e o f a

m e m b e r f o r v a r io u s f ir e p e r io d s . S o m e n a t i o n a l s t a n d a r d s a l so g i v e l i m i t in g

d i men s i o n s f o r t h e co n c r e t e m em b er s , w h i ch w o u l d g en e r a l l y en s u r e a s a t i s fac t o r y

s t r u c t u r a l b eh av i o u r o f t h e mem b er ; t h e s e a r e d is cu s sed i n C h ap t e r 3 . Req u i r em en t s

o f B S 8 1 1 0 w i t h r eg a r d t o t h e f ir e p r o t e c t i o n a r e g i v en i n T ab l e 4 .7 . In t h e ca s e o f

d o w n s t a n d b e a m s a n d r i b s w h e r e t h e w i d t h , m e a s u r e d a t t h e t e n d o n l e v e l , i s

g r e a t e r t h a n t h e m i n i m u m , t h e c o v er s s h o w n i n T a b l e 4 .7 c a n be r e d u c e d b y 5 m m

f o r ev e r y 25 mm i n c r ea s e o f w i d t h ; t h e r ed u c t i o n i n co v e r is l i m i t ed t o 1 5 m m f o r

d en s e co n c r e t e an d 2 0 mm f o r l i g h t w e i g h t co n c r e t e , an d i n n o ca s e s h o u l d t h e

resu l t i n g co ver be l ess t ha n t ha t spec i f ied fo r a f ia t so ff it f l oo r o f t he sa m e f i r e

r e s i s t an ce p e r i o d .

4 .1 2 M i n i m u m a n d m a x i m u m p r e s tr e s s

Ut i l i z ing on ly t he t ens i l e s t r en g th o f conc re t e , i t i s poss ib l e t o sa t i s fy t he

s e r v i ceab il i ty r eq u i r emen t s f o r a l i g h t ly l o ad ed s h o r t - s p an f l o o r w i t h o u t p r o v i d i n g

an y p r e s t r e s s . S o me b o n d e d s tee l w o u l d , o f co u r s e b e r eq u i r ed t o co m p l y w i t h t h e

u l t i m a t e s t r en g t h r eq u i r em en t s . I f s u ch a m em b e r is s u b j ec t ed t o an in c r ea s i n g

l o ad , t h e r e i s a s u d d en i n c r ea s e i n t h e d e f l ec t i o n a s t h e co n c r e t e c r ack s , b ecau s e

t h e m o m e n t o f i n e r ti a o f t h e s e ct i on d r o p s s u d d e n l y f r o m t h a t o f a n u n c r a c k e d

s ec t i o n t o t h a t o f a c r ack ed s ec t i o n.

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PLANNING A STRUCTURE 103

W ith a smal l am ou nt of pres t ress , i t is conce ivable for the crack ing s tage to be

the cr i t ical condi t ion rather than the serviceabi l i ty or the ul t imate s tates . Such a

member would fa i l i n an exp los ive manner wi thou t warn ing .

At the o th er ex t rem e, a m em ber wi th a h igh average p res t ress is a lso l ike ly to

fail in an exp los ive m ann er , wi thou t m uch w ar n i ng , i f i t is sub jec ted to an

increas ing load. In th is case, the ul t ima te s t ren gth is gov erne d by the compre ss ive

s t rength of the conc rete rath er tha n the tens ile s t rength of s teel . This s tate i s l ikely

to be app roa che d i f the s ize o f a mem ber is reduced to the m in im um poss ib le .

Bo th o f the above ex t remes a re undes i rab le . Fa i lu re o f a p roper ly des igned

me mb er s h o u ld o ccu r b y t h e y ie ld in g o f t h e s t ee l ~ t en d o n s an d ro d r e in fo rcemen t ,

so that ample warning is g iven by the increas ing deflect ion.

N o m inim um l imi t is specif ied in BS 8110 for the av erage pres t ress in concrete .

The upp er l imi t is consid ered to be satis fied if the ul t im ate m om en t of res is tance

exceeds the moment necessary to produce a f lexural tens i le s t ress in the concrete

at the ex t reme tens ion f ibres equal to 0 .6(f~u)~ In th is calcu lat ion the pres t ress

may be taken as the value af ter a l l losses have taken place.

AC I 318 s t ipu la tes a m in im um average p res t ress o f 1 .0 N / ram 2 on the g ross

concre te sec t ion a f t e r al l l osses, and i t l imi t s the m axim um area o f s t e e l~ ten do ns

plus rod reinforcement as d iscussed in sect ion 8 .2 .

4 . 1 3 A d d i t i o n a l c o n s i d e r a t i o n s fo r s t r u c t u r e s in s e i s m i c z o n e s

I t is well kno wn tha t e ar thq uak es imp ar t very la rge hor izon ta l loads to s t ruc tu res

in add i t ion to the vert ical grav i ty loads . The pr im ary o bject ive of seismic-res is tant

des ign is , t herefo re , to p rov ide a dequ ate hor izon ta l loa d res i s tance in the frame

me mb ers , toge ther w i th m om ent res is t ance a nd duc t i li ty in the jo in t s . Se i smic

res i s t ance i s normal ly p rov ided by a beam-co lumn f rame, o f t en s t reng thened by

shear wal ls . Al thou gh no t fo rb idden by ear th qua ke codes , the use o f f ia t sl abs,

wi tho u t be am s , as par t o f a se ismic mom ent res is t ing f rame i s qu i te ra re ,

espec ia l ly in a reas o f very h igh se ismic ri sk . No rm al ly the on ly req u i rem ent fo r

the s labs themselves , therefore, i s that they be able to act as horizontal

d iaphragms , d i s t r ibu t ing the se i smic fo rces across the s t ruc tu re . Th i s func t ion

can usual ly be ful f i l led wi thout d i ff icul ty by s labs des igned for normal ver t ical

grav i ty loads . F or th is reaso n, a detai led acc oun t of seismic-res is tant des ign is

outs ide the scop e of th is book . S om e issues relat ing to the plan ning of the

s t ructure, wi th part icular reference to f loors , are d iscussed in th is sect ion, whi le

detai l ing req uirem ents are br iefly d iscussed in Sect ion 12.6 . Fo r a fu l ler t rea tm ent

of seismic-res is tant de s ign, the read er i s referred to special is t texts (D ow rick,

1987; Key, 1988; Naeim, 1989).

The response of a bui ld ing to seismic loads can be s ignif icant ly affected by i t s

s t ruc tu ra l conf igura t ion bo th in p lan and in e l eva t ion , mak ing i t impor tan t tha t

seismic beha vio ur i s fu lly consid ered at the conc eptu al d es ign s tage. As a general

rule , des i rable aspects of bui ld ings in seismic zones are s impl ici ty , regula r i ty and

sym me t ry . These p roper t i es he lp to ensure tha t the se ismic fo rces a re d i s t r ibu ted

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104 POST-TENSIONED CONCRETE FLOORS

M,K

4-

M,K

-7

M - - K

(a) Desirable plan layouts

+.

(b ) Layouts to be avoided

K = centre of st i f fness

M = cent re of mass

Figure 4 .9

Plan layouts o f bu i ld ings in se ism ic zones

evenly , whereas any variat ions in mass or s t i f fness d is t r ibut ion are l ikely to lead

to an increase in the dynamic response .

Prov i s ion o f hor izon ta l loa d res i s t ance in a bu i ld ing can be ach ieved by us ing a

m om ent - res i s t ing f rame ( i .e . one where the beam -co lum n jo in t s a re des igned to

wi ths tan d the se i smica lly - induced m om ents ) , by p rov id ing b rac ing , by the use o f

shear wal l s, o r by some co m bina t ion o f these me thods . The use o f shear w al ls is

popular and effect ive, but can be inconvenient in off ice bui ld ings , s ince a h igh

dens i ty o f shear wal l s m ay in te r fe re wi th the need fo r la rge , unob s t ruc ted a reas . In

loca t ing the hor izon ta l load- res i s t ing e lements , ca re shou ld be t aken to ensure

tha t the cen t res o f mass and s tif fness o f the bu i ld ing rem ain a ppr ox im ate ly

coincident . Any eccentr ici ty between the mass and s t i f fness centres wi l l resul t in

t h e g en e ra ti o n o f ad d i t io n a l t o r s io n a l m o m en t s an d d i s pl acemen t s u n d e r

ear thqu ake load ing . F igure 4 .9 shows some des i rab le p lan l ayou t s o f hor izo n ta l

load- res i s t ing e lements fo r se i smic a reas , and some which shou ld be avo ided .

I r regu lar it i es in the p lan shape o f a bu i ld ing shou ld a l so be avo ided where

poss ib le . For example , the L-shaped bu i ld ing shown in F igure 4 .10 has two

proble m s. Fi rs t ly , the la teral s t if fnesses of the tw o legs of the bu i ld ing are

substant ial ly d i fferent , resul t ing in d i fferent ial movements between the two legs ,

and hence very large s t resses at the in terface. Secondly, as d iscussed above, the

layo ut causes a n eccentr ici ty betwee n the centres of m ass an d s ti ffness, w hich is

l ikely to resul t in s ignif icant tors ion al m ot ion . V ery s imi lar pro blem s are l ikely to

be encoun tered in bu i ld ings which a re t apered in p lan .

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Stiff in direction

of motion

- j

J

~ . . y ,

/

/

/

/

PLANNING A STRUCTURE

105

High risk of damage

in this region

Flexible in direction

of motion

I 1

i I

F i g u r e 4.10 Seismic m ovem ent o f bu i ld ing w i th i r regu lar f loor p lan

Simi la r ly , the ve r t ica l con f igura t io n of the bu i ld ing s hou ld be kept a s r egula r a s

poss ib le . Idea l ly , f loors should have the same p lan conf igura t ion a t each leve l ,

wi th no se tbacks , s ince the sud den change in s ti ffness a t the leve l o f the se tba ck

c a use s a c onc e n t r a t i on o f l oa d a t t ha t l evel . F loo r s sh ou ld a l so be e q ua l ly spa c ed

as fa r a s is p rac t icab le . S toreys w hich a re h igher th an the o the r s in a bu i ld ing a re

nor m al ly le ss st if f, and so sus ta in v e ry la rge d isp lacem ents , caus ing a soft storey

fa i lu re . I f , a s i s common, i t i s des i red to have a h igh ground s torey , the co lumns

mus t be ve r y c a r e f u l ly de s igne d to e nsu r e t he y ha ve a de q ua te s t r e ng th a nd

stiffness to resist such a failure.

I n r e s is t ing the ho r i zon ta l f o rc e s c a use d by a n e a r thq u a ke , i t is im po r t a n t t ha t

a ll the load- res i s t ing e lements a re secure ly t ied toge th e r , ensur ing g oo d com pos i te

be ha v iou r a nd p r ov id ing a dd i t i ona l r e dund a nc y . I n m a ny in s t a nc e s , t h is r e q u ir e s

the fl oo r s l abs t o a c t a s d i a ph r a g m s , d i s t r i bu t ing the ho r i zon ta l l oa ds a c r os s t he

s t r uc tu re . P os t - t e ns io ne d f loo rs c a n be r e ga r de d a s r i g id d i a ph r a g m s , c a us ing the

loa ds t o be d i s t r i bu t e d be twe e n the ho r i zon ta l l oa d - r e s i s ti ng e l e me n t s i n d ir e c t

propor t ion to the i r s t i f fnesses .

T he d i a ph r a gm a c t ion c a use s ho r i zon ta l she a rs t o be s e t up in t he fl oo r ; u sua l ly

these wil l be smal l and can be neglec ted , bu t occas ion a l ly conf igu ra t ions m ay be

used which g ive r i se to qu i te h igh shea rs , such as tha t sh ow n in F igure 4 .11 . Even

whe n the a ve r a ge she a r i s l ow , c a r e shou ld be t a k e n to e nsu r e t ha t ope n ings i n t he

f loor a re adequ a te ly re inforced , a re no t so num ero us or c lose ly spaced as to cause

lo ss o f d i a p h r a g m a c t ion , a nd do n o t i n t er fe r e w i th t he a t t a c hm e n t o f t he f loo r t o

wa l l s o r f r ames .

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106

POST-TENSIONED CONCRETE FLOORS

Distr ibuted seismic load

Load-resist ing

element I

\

High diaphragm shear

stresses

Load-res is t ing

element

F i gu r e 4 . 11

Shear s t resses due to d iaphragm ac t ion

In general , the above considerat ions are equal ly appl icable to post-tensioned

and reinforced concrete f loor slabs. However, there are some respects in which

the performan ce of post-tensione d floors is preferable . Firstly, the con t inuo us

post-tensioning tendons are excel lent for ensuring st ructural cont inui ty. Also,

when u nb on ded tendo ns are used, they are able to redist r ibute peak forces along

their length, and so have a lower r isk of failure than bon ded reinforcement . Th e

only add i t ional issue to be considered for post-tensioned floors is the locat ion of

the anchorages. These should be spaced as evenly as possible so as to avoid

congest ion and st ress-raising in highly st ressed zones. Anchorages should be

posi t ioned as far as possible from potent ial plast ic hinge locat ions.

Post-tensioned f loor

Core / Ritainingwall

! ! 1 ~ I I ! i I

i_ 6.5m _~_ 7.5m _~_ 7.5m .. l_

| @ @ |

7.5m

_1

- I

|

(a) Elevat ion

J " -~ e b ~ P c ~ ,~ P d ~ L e e

@ 9 | |

(b) Line diagram

Figure 4.12

Example 4 .1

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PLANNING A STRUCTURE

107

Example 4 .1 Res traint f rom columns

F i g u r e 4 . 1 2 s h o w s p a r t o f a b u i l d i n g w i t h p o s t - t e n s i o n e d f l o o r . A t o n e e n d i s a s ti ff

c o r e , a n d a s h o r t r e t a i n i n g w a l l is c a s t m o n o l i t h i c w i t h t h e c o l u m n s i n al l t h e b a y s .

C a l c u l a t e t h e t h r u s t s w h i c h w o u l d d e v e l o p i n t h e c o l u m n s w h e n t h e f l o o r is s t re s s e d .

A s s u m e t h a t t h e c o r e is i n fi n i te l y st if fe r t h a n t h e c o l u m n s , t h e c o l u m n s a r e p i n n e d a t

t o p , a n d a r e f ix e d 1 .5 m b e l o w t h e c e n t r o i d o f t h e f lo o r . T h e o t h e r r e l e v a n t d a t a a r e :

T e n d o n f o rc e , c o n s t a n t f r o m ( a) t o ( d) = 1 0 00 k N

T e n d o n f o r c e f r o m ( d ) t o ( e) = 1 2 0 0 k N

C r o s s s e c t io n a l a r e a o f t h e a s s o c i a t e d f l o o r ( A r 0 . 50 m 2

M o m e n t o f i n e r t i a o f e a c h c o l u m n ( Ir = 0 .0 1 m 4

E f f e ct iv e c o l u m n h e i g h t ( H ) = 1 .5 0 m

Solution

L e t g 12 r e p r e s e n t c h a n g e i n le n g t h b e t w e e n p o i n t s 1 a n d 2 , a n d P x t h e t h r u s t o n

c o l u m n x . F o r c o n v e n i e n c e , u s e k N / m 2 a s th e u n i t s f o r Er t h e m o d u l u s o f e l a s ti c i ty o f

c o n c r e t e .

I n a n u n r e s t r a i n e d f l o o r , t h e e l a s t i c s h o r t e n i n g w i l l b e :

gab = 1000 X 6.5 / (0 .5 X E~) = 1300/Er

gac = 1000 x 14.0/ (0 .5 x Ec) = 2800 0/Er

gind = 1000 X 21.5 / (0 .5 X Er = 430 00/E~

ga~ = 1200 x 7 .5/ (0 .5 x E~) + gad = 6100 0/Er

I n t h e p r e s e n c e o f t h e c o l u m n t h r u s t s , a t e a c h c o l u m n p o s i t i o n t h e s u m o f f l o o r

e x t e n s i o n a n d c o l u m n d e f l e c t i o n m u s t e q u a l t h e a b o v e s h o r t e n i n g s . T h e r e f o r e

a t b , ( 6 .5P b + 6 .5P~ + 6 .5P d + 6.5P~)/(Ar162 + Pb.H 3/(3Er = gab

a t c ,

(6.SPb + 14P~ + 14.0P~

+ 14.0Pe)/(A~Er P~ .n3 /(3E d c) = 6ar

a t d ,

(6.5Pb + 14Po

+ 21 .5Pd + 21.5Pe)/(A~Er P d . n 3 / ( 3E j c ) = gad

at e , (6 .5Pb + 14Pc +

21.5P~

+

29.0Po)/(A~Ec)+ P ~.H3/(3E~I~) --ga~

E r c a n c e ls o u t f r o m t h e t w o s id e s o f t h e a b o v e e q u a t i o n s . S u b s t i t u t i n g v a l u e s o f A r I r

a n d H g i v e s :

125 .5P b + 13 .0Pc + 13 .0P d + 13 .0Pe = 13000

13 .0P b + 140 .5Pc + 28 .0 P d + 28 .0P~ = 2 8000

13 .0P b + 28 .0P~ + 155 .5P d + 43 .0P~ = 43000

13 .0P b + 28 .0P~ + 43 .0P d + 170 .5P~ = 6100 0

S o l v i n g t h e a b o v e s i m u l t a n e o u s e q u a t i o n s g i v e s t h e f o l l o w i n g c o l u m n s h e a r s .

P b - - 4 4 . 6 k N P c = 10 2.1 k N P d = 17 3.1 k N P e = 2 9 4 .0 k N

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5 T E N D O N P R O F IL E S A N D

E Q U I V A L E N T L O A D S

In pos t - tens ioned f loors , the tendons se ldom run in s t ra ight l ines . They are

norm al ly curved in the shape of shal low parab olas ; occas ional ly , a profi le ma y

cons i s t o f shor t p arab olas jo ined by s t ra ight l engths .

In ord er to ca lculate the effect of a tend on on the s t ructure , i t i s nece ssary to

kno w i ts height a long the length of the memb er . Te ndo n heights a t var ious po in t s

are a l so needed la ter for their assembly on s it e. The geom et ry of the ten don

profi les is discussed in this chapte r , the idea of equ ivalen t loads is intro duc ed, an d

equat ions def in ing the var ious prof i l es in common use , and thei r equivalen t

loads , are g iven . The appl ica t ion of the equivalen t load is d i scussed in Ch ap ter 6 .

5 . 1 G e n e r a l

As an exam ple of a t end on profi le con ta in ing the var ious shapes in genera l use ,

cons ider the con t inuou s beam show n in F igure 5 .1(a) . I t cons is t s of three

cont inuous spans wi th a cant i l ever a t one end . The loading cons i s t s of

p redom inan t l y conce n t ra t ed l oads on t he can t il eve r and t he nex t span 1 -2 , and o f

uni formly d i s t r ibu ted loads on the o ther two spans 2-3 and 3-4 .

Figu re 5.1(b) show s a possible tend on profi le for such a beam . In the cant i lever

and in span 1-2 , where the loadings are pred om inan t ly con cent ra ted , the prof i le

consis ts of s t raight l ines; the t r iangular shape in span 1-2 is cal led

harped.

In

spans 2-3 and 3-4 , where the load i s un i formly d i s t r ibu ted , the prof i l es are

parabol ic . The prof i le i s symm etr ica l in span 2-3 and i ts lowes t po in t is obvious ly

at m idspan . In spa n 3--4 , the profi le i s unsymm etr ica l , and the length b , the

dis tance f rom the suppor t to the lowes t po in t , i s no t known in th i s case .

F igure 5 .1(c) shows the equivalen t load c orresp onding to the prof il e shown at

(b) . A tendon can not be bent a t sha rp angles , a ll corners in the prof il e are roun ded

off wi th shor t parabolas , as over suppor t s 2 and 3 , and a t midspa n 1-2 . W i tho ut

these shor t curves the prof il e wou ld look l ike F igure 5 .1(d) whose e quivalen t loa d

is sho w n in Figu re 5.1(e) . At prel im inary s tages of design, the profi le is often

simplif ied to the form shown in Figure 5.1(d).

The prof i le in F igure 5 .1(b) can be broke n dow n in to the three bas ic e lemen ts

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TENDON PROFILES AND E QUIVALENT LOAD S

1 0 9

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2 3 4

(a) Elevation

co ) ITm l ~ ~ I111

I1 11 i1 1 1 11 1 . . . . . . . . . . . . . . " . . . . . . . . . . .

t t

I I I i

1 2 3 4

- \ - / \ - | ( d )

' , i ,

I I I I

; ~ ' ~ ~ ( e )

I I I I I 1 1 1 1 1 I I 1 1 1 1 1 1 1 1 1

f ~

I I I I

i ,

l

.

I I I I

1 2 3 4

F i g u r e

5 . 1 Tendon prof i le in a cont inuous beam

( a ) S t r a ig h t ( b ) H a r p e d ( c ) P a r a b o l ic

F i g u r e 5 . 2 Basic tendon prof i les

show n in F igure 5 .2 - -a s t ra igh t line , a tr i ang le (harp ) and a parabo la . T he uppe r

d iagrams show symmet r i ca l p ro f i l es and the lower d iagrams show the i r

u n s y mm et r i ca l f o rms .

5 . 2 E q u i v a l e n t l o a d

M ost o f the com pute r so f tware pack ages fo r the des ign o f pos t - t ens ioned f loors

have the capac i ty to ca lcu la te t endon geomet ry , equ iva len t loads and secondary

mo me nts f rom a s imple inpu t . Never the less , an un ders t a nd in g o f the equ iva len t

loads generated by di fferent profi les and thei r combinat ions i s very useful in the

des ign o f pos t - t ens ion ing .

In or der to app recia te the ac t ion of a tend on i t i s helpful to con sider i t as a

l eng th o f rope s t ru ng be tw een two f ixed po in t s rep rese n t ing the end anch orages .

In the absence o f any o ther load , and assum ing the rope to be weigh tl ess , it would

rem ain in a s t ra igh t l ine as show n in F igure 5 .2 (a) . I f a conc en t ra te d loa d w ere

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110

POST-TENSIONEDCONCRETE FLOORS

now suspe nde d some whe r e i n t he midd le t he n the r ope wou ld s a g a nd a s sume a

t r iang ula r , o r ha rpe d , p rof i le a s in F ig ure 5 .1(b). I f the load w ere un i form ly

d i s t r i bu t e d be twe e n the suppor t po in t s t he n the r ope p r o f i l e wou ld r e se mble

F igure 5 .1(c ) , which i s pa rabol ic .

In each case the rope wo uld exe r t a hor izo nta l force on the two f ixed poin ts an d

in the la t te r two cases the re w ould a l so be a ve r t ica l force on each f ixed poin t . The

sum of the two v e r t ica l forces wil l equa l the ma gn i tud e of the susp end ed lo ad .

These loads a re a l so sho wn in F igu re 5 .2 . The d i rec t ion of load on the rop e i s

dow nw a r ds a nd the a r r ows a t t he two f ixe d po in t s i nd i c a te t he d i r e c t ion o f f o rc e

e xe r te d by the f ixe d po in t on the r ope ; t he y do no t r e p r e se n t t he d i r e c t ion o f t he

f o rc e a c t ing on the fi xe d po in t s w h ic h , o f c ou r se , wou ld be opp os i t e t o t ha t

shown . T he uppe r d i a g r a ms ha ve the suppor t s a t t he s a me l e ve l a nd the l owe r

ones a t d i f fe ren t leve ls , bu t th i s does no t a f fec t the bas ic load pa t te rn .

F r om the a bove a na lo gy i t is e v ide n t t ha t a un iq ue lo a d pa t t e r n is a s soc i a t e d

wi th each prof i le . A s t ra igh t tendon does no t exe r t any t r ansverse force on the

c onc r e t e me mbe r , a ha r pe d t e ndon e xe r t s a c onc e n t r a t e d f o r c e a nd a pa r a bo l i c

tend on , o f the bas ic y =

A x 2

form, exe r t s a un i form ly d is t r ibu ted load . In fac t , the

t e n d o n p r of il es n o r m a l l y a p p r o x i m a t e t o t h e s h a p e s o f t h e b e n d i n g m o m e n t

d i a g r a m s c o r r e spon d ing to t he a pp l i e d loa ds . A d d i t i ona l ly , e a ch p r o f i le e xer t s a n

axia l force a long the member ax is .

I t i s o f t e n c onve n ie n t t o s e e a t e ndon p r o f i l e a s a n impose d be nd ing mome n t

d i a g r a m . I t shou ld , howe ve r , be r e me m be r e d tha t t he t e ndo n r e p r e se n t s a li ne o f

c o m p r e s s i o n a n d , t h er ef o re , t h e b e n d i n g m o m e n t d i a g r a m is o n t h e c o m p r e s s i o n

f ac e o f t he me mb e r , i .e . oppo s i t e a nd a mi r r o r ima ge o f t he c onve n t ion in c onc r e t e

de s ign whe r e t he mome n t d i a g r a m i s d r a wn on the t e ns ion f a c e .

A c om pos i t e p r o fi le , a c om bina t ion o f one o r m or e o f t he ba s i c p ro f il e s,

c o r r e spo nds to a l oa d d i s t r i bu t ion wh ic h i s a c om bina t ion o f t he i nd iv idua l l oa d

pa t t e r n s f o r the c om pon e n t s o f t he p ro f il e. F o r e xa mple , t he p r o fi le o f a r ope w i th

a s t r a igh t l e ng th ne a r one e nd a nd a pa r a bo l i c sha pe f o r t he r e ma in ing l e ng th

c o r r e sponds to a un i f o r mly d i s t r i bu t e d loa d a long the pa r a bo l i c l e ng th a nd no

loa d a long the s t r a igh t po r t i on , F igu r e 5 .3 .

A pp l i e d loa ds a re some t ime s t r i a ng u la r i n sha pe , suc h as on a be a m sup po r t ing

a two way s lab ; th i s load pa t te rn i s a ssoc ia ted wi th a cubic curve of the form

y = A . x 3 . H owe ve r , t he c ub ic p r o f i l e i s a lmos t ne ve r u se d f o r t e ndon d r a pe , i t

be ing su f f i c i e n t t o u se a pa r a bo la c o r r e spond ing to a un i f o r m loa d ing .

In ca lcu la t ions , the pres t re ss ing force in a tendon and i t s p rof i le can be

cons ide red in two a l te rna t ive ways , see F igure 5 .4 .

Straight Parabolic

v I

Figure

5.3

A composite profi le

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TE N D O N P R O F IL E S A N D E Q U I V A L E N T L O A D S 111

(a) Parabol ic tendon

t

Y

v T

x

(b ) Equivalent moment

t t t t t t t t

1 1

i _ L

_1

i~ "-i

(c) Equivalent load

F i g u r e 5 . 4

Equivalent alternatives for a parabolic tendon

9 e i ther as an ax ia l fo rce and a moment d iagram represen ted by the p roduct o f

the pres t ress ing force and i ts eccentr ici ty at each po int a long i t s length , F igure

5.4(b)

9 o r , as an ax ia l fo rce and an equ iva len t load ac t ing a t r igh t ang les to the me m ber

along i ts length , F igure 5 .4(c) . The e quiva lent lo ad wi ll , of course, g ive ri se to

t h e b en d i n g mo men t d i ag ram s h o w n i n F i g u re 5 .4 (b ) .

Con s ider the f i r st a l t e rna tive , tha t o f a t endo n p ro fi le be ing represen ted by an

ax ia l fo rce and a mo m ent d iagram . L e t Ym be the t endon o rd ina te a t midspan ,

measured f rom the sec t ion cen t ro id .

Th en y = Ax 2 + Ym, an d the m om en t a t x i s g iven by

Mx = PY = P( Ax2 + Ym)

( 5 . 1 )

The d ia gram fo r the equ iva len t m om ent M x is shown ab ove the beam cen t re line

in F igure 5 .4 (b ) fo l lowing the nor m al c onven t ion used in concre te des ign where

the d iagram i s d rawn on the t ens ion face . Note tha t by conven t ion the

eccen t r i c i ty , deno ted by e , i s t aken to be pos i t ive when the t endon i s loca ted

be low the sec t ion cen t ro id . However , in th i s case the s t andard geomet r i c

con ven t ion is being fol lowed w here posi t ive i s upw ard s , an d he nce the sym bol Ym

is used rather than ep.

In the case o f a simply suppor te d span th i s m om ent m us t be super imposed on

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112 POST-TENSIONED CONCRETE FLOORS

moments f rom the load ing , and tha t i s as fa r as the f l exure o f the member i s

a f fec ted . The me m ber can then be des igned fo r the com bina t io n o f the ax ia l fo rce

an d t h e n e t mo men t . I n t h e ca s e of a co n t i n u o u s mem b er , t h e d e fo rma t i o n o f t h e

m em ber unde r the in fluence o f the mo m en t wi l l genera te som e cor rec t ive fo rces a t

the supports . These are d iscussed in Sect ion 5 .3 .

No w con s ider the second a l t e rna t ive , tha t o f the tendo n be ing represen ted by

an axial force and a t ransverse load. Figure 5 .4(c) shows the load which is

equ iva len t to the s t res sed parabo l i c t end on . The t en don i s anc hore d a t the sec t ion

cen t ro id a t each end and the eccen tr i c ity a t midsp an is - -Ym" Fo r s t a t ic

equ i l i b r i u m, t h e mo men t a t mi d s p an p ro d u ced b y t h e t en d o n eccen t r i c i t y mu s t

equ a l t h e mo m en t p ro d u ced b y t he u n i fo rml y d i s t ri b u t ed

equi vale nt load w e,

d u e

t o t en d o n cu rv a t u re .

M = - PYm = W e .L 2~ 8

w e =- 8PYm/ L 2

(5.2)

Both the abo ve a l t e rna t ives a re m athem at ica l ly cor rec t and e i ther can be used to

an a l y se a p o s t - ten s i o n ed m emb er . Ho w ev e r , t h e equ iv a l en t lo ad ap p ro a ch is v e ry

m uch the favoured m etho d in the design o f pos t - t ens ioned f loors and is fu r ther

d i scussed in Chap ter 6 .

A t endo n i s d rap ed so tha t i ts equ iva len t load ac t s in a d i rec t ion o ppos i t e to the

dead a nd su per im posed loads , i. e. , t he equ iva len t load is a r ra nge d to ac t upw ard s

on a span in a norm al f loor . I t t hen ba lances par t o f the des ign load ; hence the

equ iva len t load is al so t e rmed the balanced load and the ana lys i s as soc ia ted wi th

th i s approach i s known as the

load balancing method.

In F igu re 5 .1 , d iagr am (d ) shows the equ iva len t loads fo r the t endo n p ro f il e (b ).

Di agr am (b) rep resen ts the p rac t i ca l shape o f the t endo n p ro fi le as it m igh t be

used , and i t s equ iva len t load d iagram represen t s the t rue load ing which the

t en d o n ex e r ts o n th e co n t in u o u s b eam. Mo s t o f t h e co m p u t e r p ro g ram s a r e

des igned to work wi th the t rue equ iva len t load d iagram, such as tha t shown in

Figure 5 .1(d ). H owe ver , a t the in i ti a l des ign s t age and fo r m anu al ca lcu la t ions , i t

i s expe dient to use a s impl i f ied profi le and i ts s impler equiva lent loa d d iag ram , see

Figures 5 .1(c) and (e) . The inaccuracy resul t ing from the s impl i f icat ion is

negl ig ible in mo st cases . I t is wo rth not in g th at the p rofi le in d iag ram (c) projects

ou t s ide the ou t l ine o f the beam ; th i s is qu it e accep tab le , kn ow ing tha t the curves ,

to be in t roduced later to round the corners off , wi l l br ing the profi le wi thin the

desi red envelope of the require d co ncrete o ut l ine al lowing for the necessary covers .

5 .3 Se c o n d a ry mo m e n ts

Cons ider a two-span con t inuous pos t - t ens ioned beam. Ignore the se l f -weigh t o f

the beam. Before pres t ress ing, the beam soff i t i s in contact wi th the three

suppor t s , F igure 5 .5 (a) . When pos t - t ens ioned wi th a s t ra igh t eccen t r i c t endon , a

un i fo rm m om en t i s induced a long the l eng th o f the beam. I f the beam were no t

he ld down a t the suppor t , i t would def lec t upw ards , c rea t ing a gap 6 be twee n the

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TENDON PROFILES AND EQUIVALENTLOADS

113

e

I - - - - ' - = - I

(a) Before stressing

(d) Primary moment

(b) Stressed, withou t seco nda ry effect (e) Seco nda ry effect

~ L

t t

1.5Pe/L +0.5 Pe

(c) With secondary effect

(f) Net f inal mom ent

Figure 5.5 Secondary orces and moments

middle suppor t and the beam sof f i t , F igure 5 .5 (b ) . The bend ing moment due to

the t endon eccen t r i c i ty , PY x at any point x , i s cal led the p r i m a r y m o m e n t .

In o rder to main ta in con tac t wi th the midd le suppor t , a fo rce i s genera ted

b e t w een t h e s u p p o r t an d t h e memb er , w h i ch p ro d u ces a b en d i n g mo men t

d iag ram a long the m em ber o f such a shape tha t the ga p 5 closes , F igure 5 .5 (c).

This res tor ing force is cal led the s e c o n d a r y f o r c e an d i t s co r r e s p o n d i n g mo men t

the s e c o n d a r y m o m e n t . I t s value at a po int x is g iven by M X - PYx, where Mx is the

mo men t p ro d u ced b y t h e p r e s t r e s s o n t h e i n d e t e rmi n a t e s t ru c t u re a t p o i n t x .

The p r im ary sys tem i s in te rna l to the beam s t ruc tu re , in tha t i t causes fl exure o f

the beam bu t h as no d i rec t e ffec t on the sup por t reac t ions ; the shears a re ba lanced

by the t endon s lopes over the l eng th o f the bea m and there is no res idua l shear .

F l ex u re o f t h e mem b er d u e t o t h e p r i mary s y st em may g en e ra t e s eco n d a ry

res to r ing fo rces as descr ibed above , and i t i s these secondary fo rces and the i r

co r r e s p o n d i n g s eco n d a ry mo men t s w h i ch amen d t h e s u p p o r t r eac t i o n s .

In t h e ab o v e ex amp l e , t h e t w o - s p an b eam h as o n l y o n e r ed u n d an cy an d ,

therefo re , on ly one secondary fo rce i s genera ted . In an inde te rmina te s t ruc tu re

t h e n u m b er o f s eco n d a ry fo rce s equ a l s t h e n u m b er o f i n d e te rmi n an c i e s , an d t h e

secondary moment i s the moment due to a l l such secondary fo rces ac t ing

s imul taneous ly .

Obv ious ly , the ne t e f fec t o f a t end on i s the sum of the p r im ary and the

seco nda ry m om en ts , F igure s 5 .5(d) to (f). Effectively , the vi r tual posi t ion of the

tendon di ffers f rom i ts actual posi t ion. This v i r tual posi t ion is referred to as the

l ine o f pre s sure .

The secondary moments , be ing caused by the concen t ra ted fo rces a t the

supp or t po in t s , a re a lways l inear, vary ing un i fo rm ly over the l eng th o f a span ; the

secondary shear fo rces a re cons tan t over the span l eng th . The secondary

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114 POST-TENSIONEDCONCRETE FLOORS

m om ents can eas ily be wo rked ou t f rom e last ic theory . H owe ver , fo r each o f the

three bas ic profiles , th is aspect i s a lso discussed b elow, because a n ap pre cia t ion of

the effect of a change in tend on profi le i s very useful in choo sing the dr ap e of

t endon p ro f i l es in con t inuous members a t the des ign s t age .

5 . 4 C o n c o r d a n c e

In an i n d e t e rmi n a t e s t ru c t u re , t h e r e s t o r i n g s eco n d a ry mo men t s a r e g en e ra t ed

because the t endo n eccen t r ic i ty causes f lexura l defo rm at ion o f the me m ber in

such a man ner tha t i t wo uld lose con tac t wi th one o r m ore sup por t s . I t is poss ib le

to dev ise a pro fi le such tha t the be am rema ins in con tac t w i th a l l i ts suppo r t s and

no seconda ry mom ents deve lop . Te ndo n p ro fi les which do no t p roduce s econd ary

mo men t s a r e t e rmed

conco rdan t .

I f a t endon is d rap ed in the exac t shape o f the mom ent d iagra m which w ould

develop i f a cer ta in load ing p a t t e rn were app l ied to the s t ruc tu re und er

cons idera t ion , then no secondary moments wi l l be genera ted , because the d rape

is a l read y based on a shape which inc ludes the e f fect o f inde te rm inancy . The

load ing need no t rep resen t the ac tua l load ing to be app l i ed . For example , in a

th ree-span s l ab , no second ary m om ent resu lt s if a t endo n is p ro f il ed in the shape

of the bend ing m om ent d iagra m due to a single po in t load on any o f the spans . Of

course , such a p ro f il e is unsu i t ab le i f the bea m car r i es a un i fo rmly d i s t r ibu ted load

and i t wou ld be m uch be t t e r i f the p ro fi le were parabo l i c , rep resen t ing the

un i fo rm load ing .

5 .5 T e n d o n p r o f il e e l e m e n t s

The th ree e l ements o f t endo n p ro fi les ( the s t ra igh t l ine , the ha rp and the

parabo la) a re d i scussed be low. Geomet r i ca l equa t ions def in ing the curves and

the i r equ iva len t loads a re g iven where appropr ia t e . In each case i t has been

assumed tha t the t endon s lope 0 a t the anchorage i s smal l so tha t

s i n O ~ t a n O g O , an d c o s O g l

5.5.1 S t ra igh t tendon

St r a i g h t t en d o n s o n t h e i r o w n a r e mo s t co mmo n l y u s ed i n g ro u n d s l ab s . I n

pos t - t ens ioned suspended f loors a shor t s t ra igh t l eng th i s usua l ly p rov ided

immedia te ly beh ind a l ive o r a p re- locked dead anchorage , the end f rom which

the t endon is to be s tres sed ; a shor t s t ra igh t l eng th m ay a l so be p rov ided to b r idge

any gap be tween two curves . A s t ra igh t t endon does no t have any load shape

direct ly associated wi th i t but i t may be useful for t ransferr ing shear between

ad j acen t s u p p o r t s o f a co n t i n u o u s m emb er .

Tw o t y p es o f eccent ri c s tr a i g h t t en d o n s a r e co n si d e red b e l o w - - ru n n i n g

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TENDON PROFILES AND EQUIVALENT LOADS

115

para l l e l to the me m ber ax is wi th a cons ta n t eccen t ri c ity , and wi th the eccen t r ic i ty

vary ing l inear ly a long the member l eng th .

A s t ra igh t eccen t ri c t endon , ru nn ing para l le l to and be low the ax i s o f the

me mb er , g en e ra t e s a co n s t an t p r i mary m o m en t o f m ag n i t u d e

P e ,

where P is the

pres t ress ing force and e the eccentr ici ty , see Figu re 5 .5(d) . F or the ten do n profi le

s h o w n , t h e s eco n d a ry fo r ce ac t s d o w n w ard s o n t h e b eam an d u p w ard s o n t h e

midd le suppor t , pu l l ing the two toge ther . Therefo re , under app l i ed load ing , the

midd le su ppo r t i t se l f has i ts load reduced b y the am ou nt o f the pu ll and the two

outer suppor t s have a co r respond ing increase in the i r reac t ions . The resu l t i s a

t rans fer o f shear f rom the m idd le supp or t to the ou te r su ppor t s . In th is case a load

o f

3 P e / L

is t rans fer red f rom the cen t re supp or t , ha l f the va lue to each o f the two

outer su ppor t s . I f the t endon is p laced above the neu t ra l ax i s then the t rans fer o f

load wi l l be in the opp osi te d i rect ion, i .e ., from the oute r su ppo rts to the centre one.

For F igure 5 .5 ,

R 2 = - 3 P e p / L

R 1 = R 3 -- - R2 /2 - -~ 1 . 5 P e p / L

(5.3)

Fo r uneq ual spans , L 1 and L 2,

R 2 = - 1 . 5 P e p ( L 1 + L 2 ) / L 1 L 2

R 1 = + 1 . 5 P e p / L 1

R 3 -- -F

1 . 5 P e p / L 2

(5.4)

In th e se equ a t i o n s , th e r eac t io n ac t i n g u p w ard s o n t h e b eam, an d d o w n w ard s o n

the support , i s taken as posi t ive. Eccentr ici ty ep is posi t ive when the tendon is

below the sect ion centroid .

N ow cons ider the case o f a two -span b eam wi th end anchorag es a t the sec tion

cen t ro id bu t the t endon ra i sed a t the cen t re suppor t , F igure 5 .6 . Note tha t the

eccentr ici ty e in th is case is negat ive. Te nd on eccentr ici ty var ies l inearly along the

s p an l en g t h. Th e s h ap e o f t h e p r i mary m o m en t d i ag ram is t h a t o f t h e t en d o n

prof il e , and i ts ma gn i tud e a t each po in t equa l s the p ro duc t o f the p res tress ing

force and the eccen t r i c ity . The top o f the beam is in compress ion and the bo t to m

in t ens ion , the beam tends to def l ec t downwards . The cen t re suppor t , however ,

does no t a l low any def lec t ion a t tha t po in t and i t exer ts an upw ard fo rce which , in

t u rn , g en e ra t e s d o w n w ard r eac t i o n s a t t h e t w o o u t e r s u p p o r t s . A s eco n d a ry

bend ing moment i s thereby induced , exac t ly oppos i t e to tha t p roduced by the

eccen t r ic i ty o f the tendo n . The two o ppos ing m om ents , the p r ima ry and the

secondary , cance l each o ther ou t and as a resu l t the member has on ly the ax ia l

fo rce ac t ing on i t , and no ne t bend ing moment a t a l l .

The re is , howe ver , a t rans fer o f load f rom the ou ter supp or t s to the cen t re

suppo r t . I f the t endo n a t the cen t re su ppo r t i s be low the sec t ion cen t ro id then the

t rans fer o f load is f rom ce n t re supp or t to the ou te r ones .

For F igure 5 .6 ,

R 2 = 2 P e / L

= 2 P tan 0

R ~ = R 3 = - R 2 / 2 = - e e / L

(5.5)

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116 POST- TENSIONED CONCR ETE FLOO RS

p 1 0 l ~

- - - ~ -

I - " Vl - "

1 2

(a) Tendon profile

f

I

3

(b) Primary moment

A t 5

(c) Deflected mem ber with centre support removed

L Pe

(d) Reactions from reinstated centre support

R3

(e) Net effect- no moment

Figure 5.6

Stra ight tendon in cont inuous spans

I f t h e t w o s p a n s a r e u n e q u a l , o f l e n g t h s L 1 a n d L 2 a s s h o w n i n F i g u r e 5 .7 t h e n t h e

n e t r e s u l t s t i l l a m o u n t s t o a n a x i a l f o r c e a n d n o m o m e n t . T h e r e a c t i o n s ,

i n d i c a t i n g t h e t r a n s f e r o f l o a d i n t h is c a s e a r e"

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TENDON PROFILES AND EQUIVALENT LOA DS 117

0 2 0 1 0 2

I ~ . , , , I , , ,,

~ 1 ~

1 2

Figure 5.7 Straight tendon over two unequal spans

I p

=I

3

F or F i gu r e 5 . 7 ,

R1 =

P e 2 / L 1

= P t a n 0 1

R 3 = - P e 2 / L 2 = - P t a n 0 2

g 2 = - ( g 1 + g a ) = P e 2 . ( L 1 + L 2 ) / L 1 L 2

(5.6)

w he r e 01 a nd 0 2 a r e t he ang les o f t end on s lope in s pan s L 1 and L 2 re spec t ive ly .

Th i s p r op e r t y o f a s tr a i gh t t e ndo n , t ha t i t c a n b e q u i t e a r b i t r a r i l y d i s p l a c e d a t a n

i n t e r n a l s u p p o r t b u t t h e t e n d o n r e m a i n s e q u i v a l e n t t o a n a x i a l l o a d o n l y , i s very

s ign i f i can t . I t e f fec t ive ly means tha t a s t ra igh t l i ne p rof i l e , such as tha t shown in

F i gu r e 5 .7 , c a n be sup e r i m pos e d on a ny o t he r p r o f il e ( ha r pe d , p a r a bo l i c , e t c .) in a

c on t i nuous m e m be r w i t hou t a f f e c t i ng i t s e q u i va l e n t l oa d .

C o n v e r s e l y , i n t h e c a s e

o f a c o n t i n u o u s s p a n w i t h t h e t e n d o n d r a p e d a s a h a rp o r a p a r a b o l a , t h e

e c c e n t r i c i t i e s a t t h e i n t e r i o r s u p p o r t s c a n b e i g n o r e d i n c a l c u l a t i n 9 t h e e q u i v a l e n t

l o a d , p r o v i d e d t h a t t h e t o t a l s a 9 o f t h e p r o f il e i s t a ke n i n th e c a l c u l a t i o n s . This i s

a no t he r ve r y u se f u l p r ope r t y a nd w i l l b e u t i l i z e d i n f u r t he r d i s c us s i ons .

I t i s i m p o r t a n t t o r e m e m b e r t h a t t h e s e c o n d a r y r e a c t i o n s R 1 , R 2 and R 3 in

Equa t ions (5 .3 ) , (5 .4 ) , (5 .5 ) and (5 .6) a re ac tua l phys ica l fo rces which ac t on the

suppor t s . They a re the cor rec t ive forces re su l t ing f rom the eccen t r i c i ty o f

p r e st re s s . T he s e s e c o n d a r y f o r ce s a n d t h e c o r r e s p o n d i n g s e c o n d a r y m o m e n t s a r e

t o be a dde d t o o t he r e x t e r na l f o r c e s a nd m om e n t s , suc h a s t hose r e su l t i ng f r om

a pp l i e d l oa ds , t o a r r i ve a t t he ne t m om e n t s a nd she a r f o r c e s .

5 . 5. 2 H a r p ed p r o f i l e

A ha r pe d p r o f i l e g i ve s r i s e t o a n e q u i va l e n t c onc e n t r a t e d l oa d . Th i s p r o f i l e i s

s u i ta b l e fo r m e m b e r s w h i c h c a r ry d o m i n a n t c o n c e n t r a t e d l o a d s , s u ch a s t ra n s f er

b e a m s w h e r e a c o l u m n c a n n o t b e c a r r i e d d o w n t o i t s f o u n d a t i o n a n d m u s t b e

suppo r t e d by a be a m , o r a s l a b w h i c h c a r r i e s a s e t - ba c k f a c a de a bove . The

p r i m a r y sy s t e m i n t h is c a se c ons is t s o f t he t r i a ngu l a r m om e n t d i a g r a m r e p r e se n t ing

Pep, a nd t he a s soc i a t e d e q u i va l e n t po i n t l oa d W a nd t he she a r s V , a nd Vb in the

b e a m .

I n F i gu r e 5 . 8 ( a ) t he t e ndon i s a t t h e s e c t i on c e n t r o i d a t bo t h e nds . I n F i gu r e

5 . 8 (b ) a t one e nd t he t e n do n i s a t t h e s e c ti on c e n t r o i d bu t a t t he o t he r e nd i t h a s a n

eccen t r i c i ty e r (nega t ive ) and the spa n eccen t r i c i ty i s e m (pos i t ive ) . F or the genera l

case of F igure 5 .8 (b) ,

V~ = P ta n 01 = + Pem /a

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118 POS T- TENSIONED CON CRETE FLOORS

P

S

P

_1

"- I

v , T w

a

L ~1_....

p ,, V l ~

( a ) Anchorages a t cen t ro id

S

01 e m

0 2

v . l w v b

L. a _ j_ b .._a

I - " - I

(b) One ancho rage eccen tric

Figure 5.8

Harped tendon

V b = P

t an 0 2

= +

P ( e m -

e r ) / b

W = - V a - V b = - P s L / a b

(5.7)

whe re s = the sag of the prof i l e

L = span l eng th = a + b

S a g s i s m e a su r e d be l ow t he s t r a i gh t l i ne j o i n i ng t he t w o e nds o f a p r o f il e a nd i ts

va l ue is ne ga t i ve i n t he no r m a l d r a pe w he r e t he t e nd on i s l ow e r in t he spa n t ha n a t

t he suppo r t s .

T h e e q u i v a l e n t c o n c e n t r a t e d l o a d W d e p e n d s o n t h e t o t a l s ag o f t h e pr o fi le , a n d

is i nde pe n de n t o f e , t h e e c c e n t ri c i ty a t i nd i v i du a l sup po r t s . The r e f o r e , F i gu r e s

5 . 8 (a ) a nd ( b ), i n w h i c h bo t h t e ndo ns ha ve t he s a m e t o t a l s a g s , r e p r e se n t t he s a m e

e q u i va l e n t l oa d . How e ve r , i n ( b ) t he r e i s a m om e n t a t one e nd a nd t he suppo r t

eccen t r i c i ty a f fect s the v a lues o f 0~ and

0 2.

I f t he she a rs V~ and Vb a re ca lcu la t ed

for the two d iagrams, the i r va lues wi l l be found to d i f fe r .

I t is , o f c ou r se , no t po s s i b l e to p r ov i de a sh a r p k i nk i n a t e nd on a s the ha r pe d

prof i l e impl i e s ; t he t en do n , in fac t, is a rced wi th a rad ius o f ab ou t 2 .5 m (8 f t) and ,

the re fore , t he reac t io n i s a d i s t r ibu ted loa d ov er the l eng th o f the a rc , bu t i t i s

c onve n i e n t t o t h i nk i n t e r m s o f a c onc e n t r a t e d l oa d . The sho r t c u r ve i s i n p r a c t i c e

t r e a t e d a s a pa r a bo l a .

5 .5 .3 Pa ra bo l i c p ro f i l e

M os t o f t he su spe n de d f l oo r s in bu i ld i ngs a r e de s i gne d f o r a un i f o r m l y d i s t r i bu t e d

l oa d w h i c h c o r r e sponds t o a pa r a bo l i c p r o f i l e .

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T E N D O N P R O F I L E S A N D E Q U I V A L E N T L O A D S

1 1 9

y /

c

(a) y = Ax 2 + B x + c

c

~ x

t

( b ) y = A x 2 + c

t J

Figure 5.9 P a r a b o l a s

(c) y = Ax 2

A s eco n d d eg ree p a rab o l a is r ep re s en t ed b y t h e g en e ra l E qu a t i o n (5 .8 a ). I n t hi s

equa t ion A rep resen ts the cu rva tu re , so tha t the smal les t rad ius fo r the cu rve i s

1/(2A); B equ als tan /3, the s lope of the cu rve at the orig in (x = 0); and C

represen ts the h e igh t a t the o r ig in , see F igu re 5 .9(a) . If the o r ig in i s loca ted a t a

po in t w here the t an gen t to the cu rve i s ho r izo n ta l , i .e . a t the lowe s t o r the h ighes t

p o i n t , F i g u re 5 .9 (b) , t h en t h e t e rm B b eco m es z e ro a n d t h e equ a t i o n r ed u ces t o a

mo re conv en ien t fo rm (5 .8b) , and i f the zero t angen t po in t co inc ides wi th the

or ig in , F igure 5 .9 (c) , then the equa t ion reduces to i t s s imples t and mos t

conven ien t fo rm (5 .8c) .

y = a x 2 Jr

Bx + C

(5.8a)

y - a x 2 + C

(5.8b)

Y = A x 2 (5.8c)

I t i s no t , however , conven ien t to express a l l poss ib le parabo las in a p ro f i l e in

te rms o f Equa t ion (5 .8c). Equa t ion s (5 .9a) and (5 .9b) a re used in such cases as

more conven ien t rep lacements fo r Equat ion (5 .8b) and (5 .8c) respec t ive ly .

y = A( x - x o)2 + Co

(5.9a)

y = A( x - Xo) 2 (5.9b )

where x o = d i s tance to the low es t po in t

C O = ten do n he igh t a t the lowes t p o in t ( see F igure 5 .10).

A para bo la ma y be req u i red to pass th rou gh th ree k no w n po in t s x I ,Yl ; x2 ,Y2 and

x3 ,y 3 . Fo r the gene ra l E qua t ion (5 .8a) , va lues o f A, B and C are g iven by :

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120 POST-TENSIONED CONCRETE FLOORS

l_ x~

I "

(a)

x

[ _ X o

1

_ 1

I

( b )

X

F i g u r e

5.10

Alternatives to Figures 5.9(b) and 5.9(c)

A - - ( m 1 - m 2 ) / ( x I - x 3 )

B = m ~ - A ( x ~ + x2) (5.10)

C = Y l - A x l 2 - - B x l

w h ere

m ~ = ( Y l - y 2 ) / ( x l - x 2 )

m2 -- (Y2

- - Y a ) / ( X 2 - - X 3 )

Pa rab o las o f the fo rm of Eq uat ion s (5.8) and (5 .9 ) exh ib it a few proper t i es which

are conven ien t fo r wo rk ing ou t the geom et ry o f a t endo n and i t s equ iva len t loads .

As s ta t ed ear l i e r in th is chap ter , the m in im um rad ius o f cu rva tu re o f a para bo la

is 1/(2A).

For Equat ions (5 .8b) and (5 .8c) the s lope a t any po in t i s

2 A x ,

which equa l s

2 y / x .

The refore, the ta ng ent at any d is tance Xl b isects the dis tance Xx as show n in

Figure 5 .11 . Also, the t angen t a t a po in t (Xl , yx) in te rsec t s the y -ax is a t - y ~ ,

which m eans tha t the s lope o f the t angen t i s twice the s lope o f the l ine jo in ing the

origin and the poin t (x I ,y l ) .

For a genera l parabo l i c p ro f i l e , F igure 5 .9 (a) and Equat ion (5 .8a) ,

M~ = moment a t d i s t ance x f rom le f t

= Py~ = P(A x 2 +

B x + C )

w, = equ iva len t load

= - d 2 ( m x ) / d x 2

= - 2 P A

(5.11)

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TENDON PROFILES AND EQUIVALENT LOADS 121

o

Tangent at ( x 1 Yl)

r --r- n

F igure 5 .11 Tangent to a pa rabola

Besides the pres t re s s ing force P , the va lue o f w e dep end s on ly on the coe f f i c ien t A;

i t is i nde p e nd e n t o f B a nd C i n Eq ua t i on s ( 5. 8) a nd ( 5.9 ). The r e f o r e , a ll p a r a bo l a s

sha r i ng t he s a m e va l ue o f A a r e e q u i va l e n t so f a r a s the i r e q u i va l e n t l o a d i s

conce rned . Thi s re su l t can , in fac t , be in fe r red f rom Sec t ion 5 .4 .1 , where i t was

show n t ha t a s t r a i gh t p r o f i l e c a n be supe r i m pose d on a ny o t he r p r o f i l e w i t hou t

a f fec t ing i t s equ iva len t load .

T h i s m e a n s t h a t a t e n d o n c a n b e d r a p e d t o f o l l o w a n y s e c t i o n o f a gi v e n p a r a b o l a

w i t h o u t c h a n g i n g i ts e q u i v a l en t l o a d .

Of c ou r se , e a c h se c t i on w i ll p r od uc e i ts ow n

s e c o n d a r y m o m e n t s i n a n i n d e t e r m i n a t e s t r u c t u r e b u t t h e f in a l m o m e n t d i a g r a m ,

t he sum o f t he p r i m a r y a nd t he s e c ond a r y m om e n t s , w i ll b e e xa c t l y the s a m e f o r

each sec t ion .

The r e l a t i onsh i p be t w e e n s , t h e t o t a l s a g o f t he pa r a bo l a , t he c oe ff ic i en t A a nd

t he e q u i va l e n t l oa d c a n be c a l c u l a t e d a s f o l l ow s .

M s =

m i d s p a n m o m e n t

= - P s = - w e . L 2 / 8 = - 2 P A L 2 ~ 8 which g ives

A = + 4 s / L 2

(5.12)

w e = - 2 P A

or w e = - 8 P s / L 2 (5.13)

5 .5 .4 Ha rped versus para bo l i c p ro f i l e

The d i sc us s i on so f a r ha s i m p l i e d t ha t f o r a un i f o r m l y d i s t r i bu t e d l oa d t he

su i t a b l e t e ndon p r o f i l e i s pa r a bo l i c a nd f o r a c onc e n t r a t e d l oa d i t i s ha r pe d . I n

f a ct , g i ve n t he f i gh t c ir c um s t a nc e s , a ha r p e d p r o f il e m a y be u se d w i t h a dv a n t a g e

f o r a u n i f o r m l y l o a d e d m e m b e r .

Co ns i de r t he t w o p r o fi le s in a be a m e l e m e n t w i t h f ixe d e nds a n d w i t h t he s a m e

s a g . T h e h a r p e d t e n d o n c o r r e s p o n d s t o a c o n c e n t r a t e d e q u i v a l e n t l o a d W e of

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122 POST-TENSIONED CONCRET E FLOORS

- 4 P s / L ; i ts f ixed-end m om en t is + P s / 2 , a n d t h e s p a n m o m e n t - P s i 2 . Th e

parab o l i c t end on i s equ iva len t to a to ta l un i fo rmly d i s t r ibu ted load We of

- 8 P s / L ;

i ts f ixed-end m om en t is +

2 P s / 3

an d , t h e s p an mo men t

- P s / 3 .

T h e

f ixed-end moment fo r the parabo l i c t endon i s 33% h igher than tha t fo r the

h a rp ed t en d o n , an d i ts s p an m o m en t is 3 3 % l o w er. I n a rea l co n t i n u o u s me mb er

the p rop or t ion s wi ll di ffe r f rom the abo ve percen tage s because o f the in f luence o f

the ad jacen t spans , bu t the parabo l i cp rof i l e may s t i l l show a smal l e r span

moment due to the equ iva len t load than the harped p ro f i l e .

In a con t inu ous me m ber o f uneq ual spans , o r o f d i ffe ring in tens i ti es o f

un i fo rmly d i s t r ibu ted load , a har ped p ro fi le is wo r th t ry ing in one o r mo re sp ans

if the span stresses exceed the l imits.

5.6 Com pos ite profi les

Ap ar t f rom gro und s labs , s t ra igh t t endons a re ra re ly used . S imple second-degree

para bo la s a re used in suspended f loors on ly whe n an an cho rage i s se t in a poc ke t

fo r st ress ing f rom the top o f the s lab . Norm al ly , the ancho rages a re pos i t ioned

with thei r axes paral le l to tha t of the s lab , or the be am , and , therefore, even in the

case o f a s imply suppor ted span , the t end on p ro f il e m ay cons i s t o f th ree para bo la s

an d t w o s t r a i g h t l i n e s - - a l a rg e p a rab o l a i n t h e mi d d l e an d t w o s h o r t o n es n ea r

the anchorag es , and two sho r t s t ra igh t l eng ths a t each anch orage . I f the load to be

s u p p o r t ed i s p r ed o mi n an t l y co n cen t r a t ed t h en t w o s t r a i g h t l en g t h s may b e

in te rposed be tween the th ree parabo las .

Th i s sec t ion looks a t a few of the com pos i t e p ro fi les ; the i r geom et ry a nd the i r

equ iva len t load pa t t e rns a re d i scussed where appropr ia t e .

5 .6 . 1 G enera l harp ed p ro f i l e

Cons ider a genera l harped p ro f i l e su i t ab le fo r a p redominan t ly concen t ra ted

load , such as sho wn in F igure 5 .12 . The p ro f il e cons is t s o f th ree shor t par abo las

jo ined toge ther by two s t ra igh t l ines . The l eng ths a , b and d , and the he igh t s a t the

suppo r t s a re know n. The g eom et ry o f the p ro fi le is ca lcu la ted be low fo r the

d i s t ance b ; fo r the rem ain ing d i s t ance (L - b ) a s imi la r p roc edure can be fo llowed .

For Figure 5 .12, the fol lowing three equat ions represent the three curves , 1-3 ,

3-4 and 4-2 . In the se cond eq ua t ion m~ is the s lope o f the s t raigh t l ine, tan 0 .

Cu rve 1-3: for x < a y =

A1 X2 -~-

Y

L en gth 3 -4 : f o r a < x < b - d y = m x x + C

C u r v e

4 -2 : f o r b - d < x < b y = A 2 ( b - x ) 2

Equat ing the s lopes and o rd ina tes a t po in t s 3 and 4 , g ives

A 1 = - Y l / a ( 2 b - a - d )

A 2 = + Y x / d ( 2 b - a - d )

(5.14)

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TENDON PROFILES AND EQUIVALENT LOAD S 123

t t t t t

Curve 1 Length 2 Curve 3

_J_ _l

L

- r -I r - ~

1 I I I ~

I

F i gure 5.12 G eneral ha rped p ro f i le

A p a r t o f th e c o n c e n t r a t e d l o a d i s p i c k e d u p b y t h e t e n d o n b e t w e e n p o i n t s 4 a n d 2 ,

a nd de pos i t e d a t t he l e f t suppo r t a s a un i f o r m l y d i s t r i bu t e d l oa d ove r l e ng t h a .

V~ = - 2A I Pa = + Pm l

(5.15)

The r e is a s i m i l a r she a r a t e nd 2 o f m a gn i t ude

P m 2 ,

w he r e m 2 is t he t e nd on s l ope

in length 5-6.

I n t r a n s f e r s tr uc t u r e s w he r e m o r e t ha n o ne c onc e n t r a t e d l oa d i s t o be c a r r ie d ,

t he ha r pe d t e ndon c a n be g i ve n a s m a ny k i nks i n i t a s i s p r a c t i c a l . F i gu r e 5 . 13

show s a p r o f i l e w h i c h m a y su i t t w o c onc e n t r a t e d l oa ds .

F o r F i gu r e 5 . 13 ,

Va = P tan Ox

Vb = P tan 03

W1 = P ( ta n 0~ - tan 02)

I412 = P (t a n 03 + ta n 02)

(5.16)

N o t e t ha t t he m i n us a n d t he p l u s s igns i n e q ua t i ons f o r W 1 a nd W E r e p r e se n t t he

l T

F i gure 5.13 Harped prof i le for two loads

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124

POST-TENSIONEDCONCRETE FLOORS

,Anchorage Sec tion entroid a x i s Anchorage,~_,.

Reverse parabolas

Straight lengths

Figure 5.14 Symm etrical parabolic profile

pr o fi le a s shown in F igu r e 5 .13 , whe r e t he t e ndo n is h ighe r unde r W 1 tha n und e r

W 2 . I f th i s s lope w as reve rsed then the s igns w ould a l so reve rse.

5.6 .2 Sym metr ica l parabol ic prof i le

F igu r e 5 .14 shows a p r o f i l e f o r a s imp ly suppor t e d spa n c a r r y ing a un i f o r mly

dis t r ib u ted load . In f loors , a s aga ins t b r idges , the long i tud in a l ax is o f the

anc ho rage i s no rm al ly p laced pa ra l le l to the p lane o f the s lab . This a l lows a

s imp le sha pe f o r the po c ke t f o r me r s , a nd du r ing s t re s s ing the j a c ks a r e a t r i gh t

a ng le s t o t he s l a b edge . A sho r t l e ng th o f t e nd on in a nd im me d ia t e ly be h ind a n

a nc ho r a ge a vo ids a sha r p be nd in t he t e ndon wh ic h wou ld o the r w i se a f f e c t t he

s t reng th o f the s t r an d and the e ff ic iency of the ancho rage . The le ngth of the

s t r a igh t s e c tion de pe nds on the num be r a n d c on f igu r a t ion o f s t r a nds i n a t e nd on

an d va r ie s s l igh t ly f rom sy s tem to sys tem . The m ain p a r t o f the prof i le be ing a

pa r a bo la , t he t r a ns i t i on f r om a ho r i zon ta l t o t he pa r a bo l i c sha pe in t he spa n

r e q u i r e s sho r t r e ve r se pa r a bo l i c c u r ve s ne a r t he suppor t s .

A symm e t r i c a l pa r a b o la , o f c ou r se , is a l so u se d in c on t inu ous me m be r s ,

pos s ib ly in i n t e r io r spa ns ; i n t h is c a se t he t e nd on a t t he su ppo r t s i s u sua l ly no t a t

the sec t ion cen t ro id , see F igu re 5 .1 span 2-3 . F igu re 5 .15 show s ha l f o f such a

prof ile , d ra w n to an e xagg era ted scale. The prof i le cons is ts o f two p a ra bo las

t a nge n t i a l l y me e t ing a t p o in t 4 . T he to t a l s a g o f t he p r o fi le s a nd l e ng ths a a nd b

a r e k n o w n .

F o r t he sup por t pa r a b o la , x ~< a

he ight y = A1 x2 -~-S

slope 0 =

2Ax

(5.17a)

and a t po in t 4 , x = a ,

heig ht Y4 =

A1 a2 + s

slope 04 =

2A~a

Fo r th e sp an pa rab ola , x >~ a

h e ig h t y = A 2 ( b - x ) 2

slope 0 = - 2A2(b - x)

(5.17b)

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TENDON PROFILES AND EQUIVALENT LOADS 125

w 1

per unit length

1 1 1 1 t

w2 per unit length

t t t t t t t t

Y ~ y = A l x 2 + s

~ ~ y = A 2 ( b - x) 2

2

I

[_. a

"-I

Figure 5.15 Half of a sym me trical prof i le

an d a t p o i n t 4 , x = a

height Y4 =

A 2 ( b -

a) 2

s l op e 0 4 = - 2 A z ( b - a )

Eq ua t in g t he tw o se t s o f va lues o f Y4 and 04 g ives

S u p p o r t p a r ab o l a : x < a , A x =

- y l / ( a b )

(5.18)

S p a n p a r a b o l a : x > a , A 2 - + y l / ( b

2 - ab )

(5.19)

an d a t p o i n t 4 :

x = a , Y 4 = Y l . ( b - a ) / b

(5.20)

0 4 = - 2 y x / b

Equat ions (5 .18 ) , (5 .19 ) and (5 .20 ) can be used where t he t endon p ro f i l e i s

s y mmet r i ca l i n a s p an , s u ch a s i n a s i n g l e s p an memb er o r i n t h e s y mmet r i ca l

i n t e r io r s p a n s o f a c o n t i n u o u s m e m b e r .

A us e fu l g e o m e t r i c a l p r o p e r t y c a n b e d e d u c e d f r o m t h e a b o v e e q u a t i o n s , t h a t

p o i n t 4 , w h e r e t h e t w o p a r a b o l a s mee t , l ie s o n t h e s t r a i g h t l i n e j o i n i n g p o i n t s 1

an d 2 . I t fo l l ows tha t t he p a r a b o l a 1 - 4 i s a s c a l e d d o w n v e r s i o n o f t h e p a r a b o l a 4 - 2 .

N o w co n s i d e r t h e eq u i v a l en t l o ad f o r t h e t en d o n p r o f i l e i n F i g u r e 5 .1 4 . T h e

s u p p o r t p a r a b o l a e x e r ts a d o w n w a r d a c t i n g t o t a l f or c e V x d i s t r i b u t e d u n i f o r m l y

o v e r l en g t h a , an d t h e s p an p a r a b o l a ex e r t s an u p w ar d a c t i n g t o t a l f o rce - W e

u n i f o r m l y d i s t r i b u t ed o v e r l en g t h ( b - a ) . F o r a t en d o n f o r ce P , t h e i n t en s it i e s o f

t h e t w o l o ad s ca n b e ca l cu l a t ed f r o m E q u a t i o n s ( 5.13 ). F o r t h e h a l f s p an s h o w n i n

F igure 5 .15 ,

V 1 = w l a = + 2 P y l / b

W e = w 2 ( b - a ) = - 2 P y x / b

(5.21)

I t is n o t s u r p r i si n g t h a t t h e t o t a l l o a d f r o m t h e s p a n h a l f p a r a b o l a is e q u a l a n d

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1 2 6 POST-TENSIONED CONCR ETE FLOORS

Y

Par ab ola 1 , y = A 1 x 2 + c 1 Pa rab ola 3 , y = A 3 ( L - x ) 2

Parab o la 2 , y = A 2 ( b - x ) 2 I _ c I

E "-1 L

r " ' ~1

Figure

5 . 1 6

Unsymm etrical parabo lic prof i le

oppos i t e to tha t f rom the suppor t parabo la . Ef fec t ive ly , the t endon p icks up i t s

load f rom the span and sheds i t on the suppor t .

The o ther usefu l and in te rest ing fea tu re is tha t t h e t o t a l l o a d W e is a f u n c t i o n o f

t h e s ag s a n d s p a n l e n g t h ; i t is i n d e p e n d e n t o f th e p o s i ti o n w h e r e t h e t w o p a r a b o l a s

m e e t .

This load i s d i s t r ibu ted un i fo rmly over l eng th a ac t ing downwards , and

over leng th (b - a ) ac ting upw ards .

5.6.3 Unsymmetricalparabolic profile

N ow cons ider an unsym m et r i ca l p ro fi le , such as tha t in span 3 -4 o f F igure 5 .1 ,

wh ere the tend on profi le is d i fferent over the two su ppo rts . This i s the m ost of ten

employed p ro f i l e and , therefo re , i t i s dea l t wi th in some de ta i l . Equat ions a re

g iven fo r i ts geom et ry and ce r t a in param eters o f the geome t ry a re t abu la ted in

non-d im ens iona l fo rm; these can be used to ca lcu la te the t endon p ro f il e he igh ts

wi thou t hav ing to so lve the equa t ions .

In th is case, the leng th b is no t kno w n, see F igure 5 .16. Eq uat ions fo r parabo las

1 and 2 a re the same as those fo r the symm et r i ca l p ro f il e shown in F igure 5 .15. L e t

parabo la 3 , over the r igh t -hand suppor t in F igure 5 .16 , be represen ted by"

Y = A 3 ( L - x )

2 +Y3

Eq uat ing the defl ec tion and the s lope o f para bo la s 2 and 3 a t the i r co m m on po in t

5 g ives a quadra t i c equa t ion fo r b / L , whose solut ion is"

b / L = [ - Y - (y2 _ 4 X Z ) O . S ] / ( 2 X )

(5.22)

where

X = (1 - y 3/Y l )

Y = a Y 3 / y l L + c / L - 2

Z = 1 - c / L

The equ a t ions o f the th ree parab o las and the i r equ iva len t loads a re then :

Fo r 0 < x < a , y = A l x 2 + y l

0 = 2 A ~ x

(5 .23a)

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TENDON PROFILES AND EQUIVALENT LOA DS 127

F o r a < x < L - c ,

y = A 2 ( b - x) 2

0 = 2A2(x - b) (5 .23b)

F o r L - c < a < L , Y =

A 3 ( L

- -

X ) 2 "~-

Y3

0 = - - 2 A a ( L - - x )

(5.23c)

w he r e A 1 = - y l / ( a b )

A 2 = + y x / [ b ( b - a ) ]

A3 = - Y 3 / [ c ( L - b ) ]

I n t h e a b o v e e q u a t i o n s v a lu e s o f A 1 a n d A 3 a re expec ted to be nega t ive for reve rse

p a r a b o l a s o v e r t h e s u p p o r t s . N o t e t h a t :

I f a = 0 then A 1 -- 0

I f C = 0 th en A 3 -- 0

If Yl = Y3 the n b = ( L - c ) L / ( 2 L - a - c )

I f yx = Y3 an d a = c th en b = L / 2 a nd A x = A 3

T h e l o a d o n p a r a b o l a 1 is e q u a l a n d o p p o s i t e t o t h e l o a d f r o m t h e s p a n p a r a b o l a

b e t w e e n p o i n t s 4 a n d 2 , a n d s im i l a rl y th e l o a d o n p a r a b o l a 3 is e q u a l a n d o p p o s i t e

t o th e l o a d f r o m t h e s p a n p a r a b o l a b e t w e e n p o i n t s 2 a n d 5 . P o i n t s 4 a n d 5 , w h e r e

t h e s p a n p a r a b o l a m e e t s t h e s u p p o r t p a r a b o l a s , a r e n o r m a l l y a r r a n g e d s o t h a t

t he supp o r t p a r a b o l a s she d t he i r l oa ds i n s ide t he c r it ic a l she a r zone s . A c o m m on

r u l e o f t hu m b is t ha t d i s t a nc e s a a nd c a r e a bo u t 10% o f t he spa n l e ng t h .

R e m e m b e r i n g t h a t 1 /( 2A ) r ep r e s e n ts t h e m i n i m u m r a d i u s o f c u r v a t u r e , t o l im i t

t he m i n i m um r a d i u s ove r a sup po r t t o 2 .5 m (8 f t) , t h e va l ue s f o r A ~ a nd A 3 shou l d

no t e xc e e d 0. 2 m ( 0 .0625 f t) . No t e t h a t t he m i n i m um r a d i u s o f a bon de d t e nd on is

o f te n g o v e r n e d b y t h e c a p a c i t y o f t h e s h e a t h t o b e n d w i t h o u t d a m a g e .

F o r c o n v e n i e n c e , n o n - d i m e n s i o n a l v a l u e s o f b / L a n d o t h e r p a r a m e t e r s a r e

show n i n Ta b l e 5 . 1 f o r a = c = 0 . 1L a nd a = c = 0 . 05L , w h i c h c a n be u se d f o r

c a l c u l a t ing he i gh t s o f a p r o fi le a t va r i ous po i n t s , g i ve n it s pos i t i on a t sup po r t s .

The no t a t i ons a r e de f i ne d i n F i gu r e 5 . 17 .

I n Ta b l e 5 .1 , b i s m e a su r e d f r om t he po i n t w he r e t he t e nd on he i gh t i s y~ . F o r

e a c h va l ue o f Y3/Y 1, t h e r a t i o s Yo/Y ~ a n d Y4/Y 1 a r e t he i nve r ses o f e a c h o t he r , a nd

s i m i l a rl y t he r a t i o s Y J Y 3 a n d Y6/Y3 a re inve rse pa i r s . F igure 5 .18 i s a g raphica l

r e p r e s e n t a t i on o f Ta b l e 5 .1 f o r a l L = e l L = 0 .1 on ly . The curves g ive a be t t e r

i nd i c a t i on o f t he r a nge a nd t he s e ns i ti v it y o f t he va r i ou s r a t i o s t h a n t he t a b le .

Eq ua t i on ( 5 .22 ) f o r b / L s impl if i es to Eq ua t io n (5 .24) i f a / b is a s su m e d t o e q ua l

c / ( L - b ) , i .e ., t h e l e ng t h s o f t he t w o su pp o r t p a r a b o l a s a r e a s sum e d i n

p r o p o r t i o n t o t h e l e n g t h s o f t h e t w o p o r t i o n s o f t h e s p a n p a r a b o l a e i th e r s id e o f i ts

l ow e s t po i n t . The va l ue s o f b / L f r o m E q u a t i o n ( 5.2 4 ) a r e w i th i n 5 % o f t h o s e f ro m

Eq ua t i on ( 5 . 22 ) f o r

Y 3 /Y l

in the range 0.5 to 1 .0.

b / L

= 1/[1 +

( y 3 ( Y 3 / y l ) ~

(5.24)

5 .6 .3 .1 App roximat ing an unsym metr ical prof ile

F or p r e l i m i na r y c a l c u l a t i ons , be f o r e t he t e ndon p r o f i l e ha s be e n f i na l i z e d , i t i s

r a t h e r c u m b e r s o m e t o h a v e t o e v a l u a te t h e u n k n o w n s f r o m t h e s et o f E q u a t i o n s

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128 POST-TENSIONEDCONCRETE FLOORS

Table 5.1 Ord inate ratios or Figure 5.17

a/ L = c/L Y3/Y, b/L Yo/Y, Y,/Y, Ys/Y3 Y6/Y3

0.1

0.1 0.730 1.159 0.863 0.630 1.588

0.2 0.669 1.176 0.851 0.698 1.434

0.3 0.630 1.189 0.841 0.730 1.370

0.4 0.600 1.200 0.833 0.750 1.333

0.5 0.576 1.210 0.826 0.764 1.309

0.6 0.556 1.219 0.820 0.775 1.291

0.7 0.540 1.228 0.815 0.783 1.277

0.8 0.525 1.235 0.809 0.790 1.267

0.9 0.512 1.243 0.805 0.795 1.258

1.0 0.500 1.250 0.800 0.800 1.250

0.05

0.1 0.746 1.072 0.933 0.803 1.245

0.2 0.681 1.079 0.927 0.843 1.186

0.3 0.638 1.085 0.922 0.862 1.160

0.4 0.607 1.090 0.918 0.873 1.146

0.5 0.581 1.094 0.914 0.881 1.136

0.6 0.560 1.098 0.911 0.886 1.128

0.7 0.542 1.102 0.908 0.891 1.123

0.8 0.526 1.105 0.905 0.894 1.118

0.9 0.512 1.108 0.902 0.897 1.114

1.0 0.500 1.111 0.900 0.900 1.111

1

U 2 U 2

o

_ _ _ _ _ _ _ _ _ . _

._JJ

, 2

L I

vl L

6

F i g u r e 5.17

Notations for Table 5.1

(5.22) and (5.23) for each trial profile. At this stage the designer is interested only

in an approximate value of the equivalent load, and this can be calculated from

the total sag s, Figure 5.16.

In an unsymmetrical profile, the maximum sag should be measured at the

point where the tangent to the parabol a is parallel to the line joining points 1 and

3 in Figure 5.16. However, it is sufficiently accurate for preliminary purposes to

measure the sag at midspan.

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TENDON PROFILES AND EQUIVALENT LOA DS 129

1.5

8

1.0

5

0.5

- I ~ 1

0.2 0.4 0.6 0.8 1.0

Ratio y3/yl

Figure

5.18

G raphical representation of Table 5.1 for alL = c/ L =0. 1

s ~ (Yl + Y3)/2 (5.25 )

W e = 8P s/ L

The t o t a l e q u i va l e n t l oa d W e is a s sum e d un i f o r m l y d i s t r i bu t e d ove r t he spa n

length L . A s l igh t ly be t t e r a ccur acy i s ob ta in ed i f We i s d i s t r ib u ted ove r the l eng th

(L - - a - c ), t he l eng th o f the spa n p a ra bo la .

5 .7 T e n d o n d e v i a t i o n i n p l a n

S l a b t e ndo ns a r e o f t e n m ove d ou t o f l ine t o a l low ho l e s t o be f o r m e d , a s show n i n

F igu re 5 .19 . Such a de v ia t io n i s no t s t r i c t ly re l a t ed to i t s p rof i l e in the sense tha t i t

does n o t a f fect i t s equ iv a len t load , a l th ou gh i t m ay a f fect ca l cu la t ion of losses.

The c u r v a t u r e , a s sum e d c i r c u l a r o f r a d i u s R , p r odu c e s r a d i a l fo r c es o f m a gn i t ude

P /R

pe r un i t l e ng t h t o be r e s is t e d by the c onc r e t e i n c on t a c t w i t h t he t e ndo n , a s

i nd i c a t e d . A sm a l l r a d i u s g i ve s a h i gh r a d i a l f o rc e pe r un i t l e ng t h , a nd t he t e nd on

m a y be m or e d i ff ic u lt t o ho l d i n pos i t i on t ha n i n the c a se o f a l a r ge r r a d i u s .

S u f fi ci en t r e i n f o r c e m e n t sho u l d be p r ov i de d t o p r e ve n t t he t e ndo n bu r s t i ng ou t o f

t he c onc r e t e ; a l ow s t re s s i n the r e i n f o r c e m e n t , o f 200 N / m m 2 ( 30000 p s i) , w o u l d

ensure tha t the s t ee l is we ll w i th in the e l a s t ic range . I t is , o f course , p re fe rab le to

h a v e t h e cu r v e s a w a y f r o m t h e h o le . R e c o m m e n d a t i o n s o f A C I 3 1 8 a re s h o w n i n

F i gu r e 5 .19 w i t h r e ga r d t o t he p r ef e r r ed c l e a r a nc e s be t w e e n ho l e a nd t he t e ndons .

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/

/

F ig u re 5 . 1 9

Tendon dev ia t i on a t a ho le

600 12A

I _ .. , - I ~ ,

-

ra in - I -

I

1 _

o o (3 o o

Slab tendons

o o

(a) Beam tendons be low s lab tendons

(b) Beam and slab tendons interlaced

t_.. Straight _l

oo oo o oo O,3

130 P O S T-TE N S IO N E D C O N C R E TE FL O O R S

(c) Slab tendo ns with a straight length

F i g u r e 5 . 2 0 Slab tendons ove r s t r i p beam

5 .8 C l a s h o f b e a m a n d s l a b t e n d o n s

I t i s d e s i r a b l e t o p l a c e t h e t e n d o n s a t t h e m a x i m u m p o s s i b l e e c c e n t r i c i t y i n a

s ec t i o n , i n t h e s l ab s a s w e l l a s in t h e b eam s . A t t h e co l u m n i n a co n t i n u o u s s l ab

s p a n n i n g o v e r a c o n t i n u o u s b e a m , t e n d o n s f r o m b o t h t h e m e m b e r s a r e a t th e t o p .

B o t h o f t h e m c a n n o t r u n a t t h e i r m a x i m u m e c c e n tr i ci t ie s , a n d t o a v o i d t h e c l as h ,

o n e s e t o f t en d o n s m u s t r u n a t a s m a l l e r e ccen t r i c i t y .

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T E ND ON PR OF ILE S A N D E QU I V A LE N T LOA D S 131

The s l ab t endons a re o f ten a r rang ed to run on top o f the beam cage a t the

sup po rt ; th is s impl if ies f ix ing an d there i s , of course, no poss ibi l i ty of a clash in the

span zone o f the beam. O ver the co lum n sup por t , e i ther the reverse par abo la o f

the s l ab t endons would requ i re the beam tendons to be pushed down, F igure

5 .20(a), o r a cer t a in am ou nt o f weav ing o f the t endons wo uld be needed , F igure

5.20(b); bo th arr ang em en ts resul t in an effective loss of eccen tr ici ty of the be am

pres t ress .

Th e p ro b l em can b e av o i d ed b y b u n ch i n g t h e b eam t en d o n s t o g e t h e r , an d

prov id ing a shor t s t ra igh t ho r izon ta l l eng th in the s lab t endons over the wid th o f

the beam , as show n in Figure 5 .20(c). O f course, the she ar force carr ied by the

s lab tendo ns is shed at the b eam edges in th is case and th is m ust be al low ed for in

the des ign of the b eam s t r ip .

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6 F L E X U R E IN T H E

S E R V I C E A B I L I T Y S T A T E

Th e serviceabi l i ty s tate covers the f lexural s t resses in a m em ber, and i ts def lect ion

and v ib ra t ion under se rv ice loads . Th i s chap ter dea l s wi th the f l exura l aspec t s

on ly ; def l ec t ion and v ib ra t ion a re d i scussed in Chap ter 9 .

6 .1 T h e d e s ig n p ro c e s s

Figu re 6 .1 il lus t rates the sequenc e of steps in the des ign of a po st - ten s ione d

member . For comple teness , i t i nc ludes the ca lcu la t ions re l a t ing to def l ec t ion ,

shear s t reng th and u l t imate f l exure , thoug h these a re d i scussed in o ther chap ters .

Analys i s and des ign o f a concre te m em ber is a longer p rocess in pos t - t ens ioned

than in re in fo rced concre te , because the pos t - t ens ion ing des ign requ i res severa l

add i t iona l s teps no t needed in re in fo rced concre te , an d a wider cho ice is ava i lab le

for opt imizing the des ign. The process tends to be i terat ive, in that cer tain

assumpt ions a re made abou t the sec t ion and the p res t ress , which a re ad jus ted in

the l ight of success ive calculat ion resul ts .

In re in fo rced concre te , se rv iceab il ity ca lcu la tions a re norm al ly no t needed ; the

usua l p rocedure i s to choose a concre te sec t ion and ca lcu la te the re in fo rcement

requ i red to car ry the fac to red loads cor respond ing to the u l t imate s t a t e ;

def l ec t ion c r i t e r i a a re normal ly deemed to be sa t i s f i ed by adop t ing p rescr ibed

span - to -de p th ra t ios . I f ca lcu la t ions f ind the in it ia l sec t ion to be inad equa te , then

i t i s increased , there b eing no rea l a l ternat ive in the m at ter . I f i t i s fou nd t ha t the

ini t ia l ly chosen sect ion is larger th an i t needs to be, then i t i s no rm al ly a cce pted; i t

i s ra ther rare to repeat calculat ions for a smal ler sect ion.

In pos t - t ens ion ing , un t il ca lcu la t ions a re car r i ed ou t , it is no t obv ious w hethe r

the serviceabi l i ty or the ul t imate s t rength s tate , or both , are cr i t ical . Therefore,

the serviceabi l i ty checks for s t resses and deflect ion are an essent ial par t of the

design. At the serviceabi l i ty s tate , the s t resses in the concrete are required to be

within speci f ied l imi ts in compress ion and in tens ion, both at the t ime of

p res t ress ing wi th a m in im um of load ( i n i t i a l s t a g e ) and in the long t e rm und er fu ll

app l i ed load ( f i n a l s t a g e ) . If the serviceabi l i ty s tate i s sat i s factory then the

ca lcu la t ions p roceed to cons idera t ions o f the u l t imate s t reng th .

Serviceabi l i ty calculat ions fol low the class ical e las t ic theory, where s t ress i s

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FL E X U R E I N TH E S E R V I C E A B I L ITY S T A TE 133

,L I I . . . . . i

hoose concrete + Choose endon force

I| se ct ion ~ and profile

Check initial stresses

....

....

O . e c O e o c o n s I

Add bonded steel

[ -

I

Add shear steel

,,

OK

, r

" , ~ t ~ C h e c k f~ e xu ra ~ t re n g t h i

~ 1 ~ - - ~ C h e c k s h e a r s t re n g t h i

t o K

[Design anchorage steel 1

, , t

"1[__ Design c~ i

T

E

R

V

I

C

E

U

L.

T

I

M

A

T

E

Figure

6.1 The des ign p rocess

pr o po r t ion a l t o s t r a in a nd the c ompr e s s ion b loc k i s t r i a ngu la r . B o nde d s te el c a n

be a l lowed for in ca lcu la t ing the m om en t o f ine r t ia o f the sec t ion , by rep lac ing i t

w i th a n e q u iva l e n t a r e a o f c onc r e t e a t a n a pp r op r i a t e va lue o f t he mo du la r r a t i o .

I n de s ign ing pos t - t e ns ione d f loo r s , howe ve r , t he no r ma l p r a c t i c e i s t o t a ke the

g r os s c onc r e te s e c t ion a nd igno r e bon de d s te el in c a l c u l a ti ng the s e c t ion mod u l i .

T he a r e a o f t e nd on duc t s i s no t no r m a l ly de duc te d in the de s ign o f f l oo rs , t houg h

AC I 318 requi res the e ffec t o f loss of a rea d ue to ope n d uc ts to be cons ide red .

Du c t s m a y be op e n w he r e a cc es s to a t e n don is re q u i r e d a f t e r c onc r e t ing , suc h a s

a t c oup le r s o r whe r e t e ndo ns a r e t o be s t r es se d f rom ope n p oc ke t s a t t he t op o f a

f loor .

In t r ansfe r beam s, the am ou nt of p res t re ss r equi red i s h igh; usua l ly , the

me m be r c a n no t be s t re s sed in one ope r a t io n be c a use the de a d loa d f r om the se lf-

we ight a lone i s insuf f ic ien t to conta in the in i t ia l t ens i le s t r e sses tha t would

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1 3 4

POS T- TENSIONED CON CRET E FLOOR S

( 1 ) (2 ) ( 3 ) (4 )

I n i t i a l s t r e s s e s

17

S I

( a ) (b ) ( c ) ( d )

F i n a l s t r e s s e s

Figure

6 , 2 Poss ib l e i n i t i a l and f i na l s t ress l eve l s

develop f rom pres t ress ing . T herefo re , in trans fer beam s , the t endon s a re s t res sed

in severa l s t ages as cons t ruc t ion p roceeds . In such s t ruc tu res , and where open

duct s a re used , the loss in concre te sec t ion due to the duc t s shou ld be t ake n in to

acco u n t .

6 .2 O p t ions in a des ign

An un ders tan d ing o f the des ign sequence , and the cho ices ava i l ab le a t va r ious

s tages , wil l be useful in deciding o n the s tep to be ta ke n i f the in i tia l an d/ or the

final stress(es) exceed the specified l imits. The possible stress levels, being

accep tab le o r no t , a re shown in F igure 6 .2 . The a r rows ind ica te the accep tab le

levels ; the diagrams are shaded where the s t ress exceeds the l imi t .

Tab le 6 .1 l is ts the s ix teen poss ible co m bin at ion s of in i tia l and f inal s t resses at

the top a nd bo t to m of a sec tion . The second co lum n re fers to the in it ia l and f ina l

s t res s combina t ion f rom Figure 6 .2 . I t has been assumed tha t the moment i s

sagging. A zero (0) denotes a case where the s t ress exceeds the l imi t and some

act ion i s requ i red on the par t o f the des igner .

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Table 6 .1 Design options

FLEXURE IN THE SERVICEABILITY STATE

135

Case Stress Initial

diagram Top Btm

Final Possible step

Top Btm to take

1 l a 9 9 9

2 4 a 0 0 9

3 2a 0 9 9

4 3a 9 0 9

5 ld 9 9 0

6 lb 9 9 0

7 lc 9 9 9

8 2 c 0 9 9

9 3b 9 0 0

10 3c 9 0 9

11 4 b 0 0 0

12 2b 0 9 0

13 4c 0 0 9

14 2d 0 9 0

15 3d 9 0 0

16 4 d 0 0 0

9 Accept

9 Redu ce p res t r e s s

9 an d /o r eccen t r ic i t y

9 Inc rease p res t r e s s

0 an d /o r eccen t r i c it y

Inc rease sec t ion

I n t h e f i r s t case, w h e n a ll s tr e s s e s a r e w i t h i n t h e l i m i ts , th e d e s i g n c a n e i t h e r b e

a c c e p t e d o r a s m a l l e r s e c t i o n m a y b e c o n s i d e r e d i f s o d e s i re d .

W h e n p e r m i s s i b l e s tr e ss e s a r e e x c e e d e d in t h e s a m e p o s i t i o n a t i n i t ia l a n d a t

f in a l s ta g e s t h e n t h e s e c t i o n m u s t b e i n c r e a s e d . I n C a s e 5 , i f t h e s e c t i o n i s a T e e

t h e n i n c r e a s i n g t h e r i b w i d t h i s a n o t h e r o p t i o n . T h i s o p t i o n i s a l s o a v a i l a b l e

w h e n e v e r t h e b o t t o m is o v e r s tr e s s e d in a T - s e c t io n .

T h e o p t i o n s i n T a b l e 6 .1 a r e d ir e c tl y a p p l i c a b le t o a s i m p l y s u p p o r t e d m e m b e r .

F o r c o n t i n u o u s m e m b e r s , i t is e a s ie r to t h i n k i n t e rm s o f t h e e q u i v a l e n t l o a d s . A n

i n c r e a s e o f e c c e n t r i c i t y a n d / o r p r e s t r e s s i n g f o r c e i n a s p a n i n c r e a s e s t h e

e q u i v a l e n t l o a d , a n d i t is t a n t a m o u n t t o a r e d u c t i o n o f t h e a p p l i e d l o a d .

C o n s i d e r t w o a d j a c e n t s p a n s o f a c o n t i n u o u s s t r i n g , s h o w n i n F i g u r e 6 . 3. A n

i n c r e a s e o f p r e s t r e s s i n g f o rc e , o r e c c e n t r i c i ty , in s p a n A B i n c r e a s e s t h e u p w a r d

e q u i v a l e n t l o a d , w h i c h r e s u l ts in a r e d u c t i o n i n th e m o m e n t a t s u p p o r t B , b u t i t

i n c r ea s e s th e m o m e n t s i n s p a n B C a n d a t s u p p o r t C . T h e e f fe ct a t s u p p o r t C i s, o f

' i

I Moment diagram

Equivalent UDL

t t ' t t ' f , t I I t

Figure 6 .3 Effect of tendon equivalent load in one span of a continuous b e a m

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POST-TENSIONED CONCRETE FLOORS

c ou r se, sma l l er tha n tha t a t sup po r t B - - i n t he c ase o f a p r isma t i c me m be r , ha l f a t

C a nd a q ua r t e r i n spa n B C . T he r e f o r e , t o r e duc e the mome n t a t suppo r t B , t he

e q u iva l e n t l oa d c a n be inc r e a sed in e it he r o f t he two spa ns A B a nd B C . T o r e duc e

the mo m e n t i n spa n B C , t he e q u iva l e n t l oa d m us t be ei t her r e duc e d in spa n A B ,

or inc reased in span BC.

M os t o f t he f loo r s ar e de s igne d f o r un i f o r mly d i s t r ibu t e d loa ds a n d ha ve

pa r a bo l i c p ro f il e s. I n a c on t in uou s m e m be r o f une q ua l spa ns , o r whe r e the l oa d

in tens i t ie s d i f fe r in ad jacent spans , a ha rped prof i le may be wor th t ry ing , a s

discussed in 4.5.4.

6 .3 C omputer p rograms

T he m e tho ds o f a na ly s is u se d f o r a pos t - t e ns ione d f loo r a r e t he s a me a s f o r

r e in f o r ce d c onc re t e . Ma nu a l de s ign c an be c a r r ie d o u t u s ing m om e n t d i s t r i bu t ion

o r s imi l a r me thods . H owe ve r , t he de s ign p r oc e du r e f o r a pos t - t e ns ione d f loo r ,

except in the s imples t cases , r equi res i te ra t ive ca lcu la t ions , o f ten wi th a chan ge in

sec t ion , p res t re ss or eccent r ic i ty , to ob ta in a sa t i s fac tory des ign . A compute r

p r og r a m , t ho ugh no t e s se n t ia l , a l lows the de s igne r t o t ry a num be r o f so lu t ions

q u ic k ly , a nd pos s ib ly mor e a c c u r a t e ly , t o a r r ive a t t he op t imum de s ign .

A pos t - t e ns ione d f loo r c a n be a na ly se d w i th a ny o f t he no r m a l a na ly s is

programs, such as p lane f rame , f lex ib i l i ty ma t r ix , s t i f fness ma t r ix , o r g r id

p r og r a ms . Mor e c ompl i c a t e d ge ome t r i e s ma y be a na ly se d mor e e a s i l y w i th a

f in i te e l e me n t p r o g r a m . A f te r t he a na ly s is o f l oa ds a nd t e ndo n f o rc e s, c a l c u l a t i on

o f s tr e sse s m a y ne e d the u se o f a no the r p r og r a m , o r i t c a n be c ar r i ed ou t m a nua l ly .

S pe c ia l p r og r a m s h a ve be e n de v e lope d f o r t he de s ign o f pos t - t e ns ion e d f loo rs ,

wh ic h c a r r y ou t t he a na ly s i s o f t he s t r uc tu r e a n d the n de s ign e a c h me m be r a t

c r i tica l sec t ions . M os t o f these a re wr i t ten in the Uni ted S ta te s and they pr im ar i ly

c omply w i th t he A me r i c a n r e gu la t ions ; ve r s ions a r e a va i l a b l e f o r u se i n o the r

c oun t r i e s , w i th t he l oc a l r e gu la t ions i nc o r po r a t e d in to t he p r og r a m. H owe ve r ,

com pl ianc e wi th the loca l r egula t ion s an d prac t ices is o f ten of fe red as a choice for

mo d i f y ing the de s ign pa r a me te r s , a nd the ba s ic fe el o f t he p r og r a m r e ma ins U S

biased . Never the less , the y a re ve ry use fu l , pa r t ic u la r ly for mu l t i sp an f loors wh ere

ma nua l c a l c u l a t i ons c a n be t e d ious a nd l e ng thy .

T he p r og r a m s a r e c a pa b le o f ha nd l ing a w ide va r i e ty o f ge om e t r ic a l sha pe s : t he

sec t ion can be rec tangula r o r r ibbed; a f loor can be ana lysed as a f r ame or a s a

be a m s t r ing ; it ca n ha ve d r o p pa ne l s o r a c ha nge o f se c t ion ne a r t he su ppo r t s ;

m om e n t s c a n be c u rt a i le d a t sup po r t f a ce s ; a nd r e d i s t r i bu t ion o f m om e n t s c a n be

ca r r ied ou t .

The three tendon prof i le s d iscussed in Chapte r 5 a re ava i lab le in these

p r og r a ms . A pp l i e d loa d ing c a n be un i f o r m, l i ne a r ly va r y ing o r c onc e n t r a t e d .

T e n d o n s c a n b e b o n d e d o r u n b o n d e d . T h e p r o g r a m s u s u a l l y w o r k o n t h e l o a d

ba la nc ing m e thod , whe r e t he t e nd on p r o fi le is tr a ns f o r m e d in to a n a x i a l fo r ce a nd

a n e q u iva l e n t l oa d a c t ing no r ma l t o t he me mbe r a x i s .

T he p r og r a m s a r e u sua l ly i n t er a c t ive , i n t ha t t he y m a y sugge s t a p r e s tr e s s ing

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FLEXURE IN THE SERVICEABILITY STATE 137

fo rce an d a t en d o n p ro f i l e fo r u s e a s a s t a r t i n g p o i n t . A m o re eco n o m i ca l

a r r a n g e m en t is n ea r l y a l way s a r r i v ed a t b y v a ry i n g t h e m a g n i t u d e o f t h e

p re s t r e s s i n g fo rce an d ad ju s t i n g t en d o n d rap e i n each s p an . T h e m ag n i t u d e o f

p res t ress ing fo rce can be var ied f rom span to span .

The p rograms ca lcu la te p res t ress losses , se rv iceab i l i ty s t resses , u l t imate

f l ex u ra l s t r en g t h an d s h ea r s t r en g t h , t h o u g h n o t n eces s a r i l y i n t h i s o rd e r . T h e

requ i r ed am o u n t s o f b o n d ed ro d r e i n fo rcem en t an d s h e a r r e in fo rcem e n t a r e a l so

ca l cu l a ted . L o s s e s a r e ca l cu l a t ed a f t e r t h e ad ju s t m en t s o f p re s t r e s si n g fo rce an d

the t end on p ro f i le have been car r ied o u t , an d i f the losses a re found to be

subs tan t ia l ly d i f fe ren t f rom the in i t i a l as sumpt ions then the who le p rocess has to

be repea ted , th i s t ime w i th a be t te r as sessm en t o f the p res t ress ing fo rces .

Gen e ra l l y , t h e p ro g ra m s a re n o t ab l e to co p e w i t h th e d i s p e r s i o n o f t h e ax i a l

co m p o n en t o f p re s t re s s wh i ch , fo r ex am p l e , m ay o ccu r wh en a h i g h l ev e l o f

p res t ress i s p rov ided in a beam, wi th the ad jacen t s lab pane l s l igh t ly p res t ressed

or in re in fo rced conc re te , see Cha p te r 4 . In such a case the s t resses a t the c r i t ica l

s ec t i o n s , a l l o wi n g fo r t h e d i s p e r s i o n , m ay h av e t o b e ch eck ed m an u a l l y , o r t h e

a l l o wab l e s t r e s s e s fo r a p ro g ram m ay b e m o d i f i ed t o s u i t t h e p a r t i cu l a r c a s e .

Des i g n ex am p l e s g i v en a t th e en d o f t h is ch ap t e r u s e h an d c a l cu l a t i o n m e t h o d s

i n o rd e r t o g i ve a b e t t e r u n d e r s t an d i n g o f t h e p ro ced u re s i n v o l v ed .

6 .4 Part ia l p rest ress ing

In re in fo rced concre te , the tens i le s t ra ins a re o f such a mag n i tu de tha t the

concre te on the t ens ion face c racks a t se rv iceab i l i ty load ings ; wi thou t th i s

c rack i n g , t h e r e i n fo rcem en t can n o t d ev e l o p s u f f i c i en t t en s i o n t o m ak e an y

s i g ni f ic an t co n t r i b u t i o n t o t h e s t r en g t h o f t h e m em b er . In t h e ea r l y p e r i o d o f

dev e lop me nt o f p res t ress ing , the p rac t ice w as to app ly su f fic ien t p res t ress to

e l imina te f l exura l t ens ion f ro m the conc re te sec t ion und er se rv iceab i l i ty load ings .

W i t h ex p e r i en ce , i t was r ea l i z ed t h a t an i n t e rm ed i a t e s t r a t eg y m ay b e m o re

econ om ica l w h i le st il l be ing techn ica l ly sa t i s fac to ry . The leve l o f p res t ress w here

tens ile s tresses a re no t a l lowed to deve lop i s now

term ed full prestressing.

The re i s

no ag reed def in i t ion to d i s t ingu ish be tween fu l l and par t i a l p res t ress ing . In th i s

b o o k , t h e t e r m

partial prestressing

re fers to the sys tem in which tens ion i s a l lowed

to deve lop in the concre te under fu l l se rv ice load ing .

In cu r ren t p rac t ice , the a l lowab le t ens i l e s t ress in concre te co r responds to a

s t ra in o f the o rd er o f 2 .5 x 10- 4 in pos t - t e ns ion ing ; in re in fo rced conc re te the

av e rag e s t r a i n i n co n c re t e ad j acen t t o t en s io n r e i n fo rcem en t m ay b e 1 2 .5 x 1 0 -4 .

I t f o ll o ws th a t a co n c re t e m em b er m ay h av e a s t r a i n o f e i t h e r u n d e r 2 .5 x 1 0 -4 a s

i n p o s t - t en s i o n i n g , o r a s tr a i n ap p ro ach i n g 1 2. 5 x 1 0 -4 a s in r e i n fo rced co n c re t e .

The rang e o f t ens il e s t ra in in conc re te be tw een 2 .5 x 10 -4 and 12.5 x 10 -4 is a t

p re s en t n o t u s ed .

I t s h o u l d b e ap p rec i a t ed t h a t fu l l p r e s t r e s s i n g d o es n o t n eces s a r i l y e l i m i n a t e

t en s i o n o r m i c ro -c rack s i n co n c re t e ; t o w ard s t h e s u p p o r t t h e p r i n c i p a l s t re s s d u e

to the com bin a t io n o f p res t ress a nd shea r is st il l t ensi le . Also , m os t f loo rs a re

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138 POST-TENSIONEDCONCRETE FLOORS

s t re ssed in one d i rec t ion only and a re c racked in the non-pres t re ssed d i rec t ion .

O b se r va t ions o f t he l oa d ing h i s to r y o f no r m a l f l oo rs show tha t t he a c tua l l oa d o n

a f loor ra re ly app roach es i ts fu l l des ign va lue , so tha t in prac t ice even the co ncre te

in a pa r t ia l ly pres t re ssed f loor i s no t usua l ly subjec ted to any tens ion .

The d is t inc t ion be tween fu ll and pa r t ia l p res t re ss ing re fe r s to the se rv iceabi l i ty

s ta te on ly ; the des ign procedure for u l t ima te s t r ength fo l lows the same l ines for

both . There fore , the u l t im a te s t r en gth of a mem be r is no t a f fec ted by th is

c lassif ica t ion.

A f u lly p r e s tr e s se d m e m be r m a y de ve lop una c c e p ta b le c a m be r i f i t no r m a l ly

ca r r ie s on ly a smal l p r op or t io n o f i ts fu l l des ign load . Pa r t ia l p res t re ss ing i s ,

t he r e f o r e , a be t t e r a nd mor e e c onomic a l so lu t ion f o r f l oo r s de s igne d f o r

oc c a s iona l he a vy loa ds o r whe r e t he no r ma l l oa d in t e ns i ty i s sma l l c ompa r e d

wi th the des ign load .

In the cur ren t p rac t ice , except for spec ia l ized s t ruc tures such as r e se rvoi r s ,

a lmos t a l l f l oo r s a r e pa r t i a l l y p r e s t r e s se d . T he q ue s t ion a r i s e s a s t o how muc h

te ns ion shou ld be a l l owe d in c onc r e t e . Va r ious na t iona l s t a nda r ds spe c i f y

l imi t ing s tr es ses wh ic h c o r r e sp ond to mu c h lowe r s t r a in s t ha n those in r ein f o rc e d

concre te . Th ere a re tw o a pp roac hes to spec i fy ing the l imi t : the tens i le s t r ess i tse l f

m ay be l imi ted to a sa fe va lue so tha t the con cre te does n o t c rack , o r the co ncre te

ma y be a l l owe d to c r a c k bu t t he c r a c k w id th ma y be c on t r o l l e d . T he c r a c ke d

sect ion of the la t te r case wil l hav e a large r def lec t ion.

The Br i t i sh S tan da rd 8110 recognizes th ree c la sses for the level o f p res t re ss in

s t ruc tures a t the se rv iceabi l i ty s tage .

1 . W here tens ion i s never a l low ed in concre te . This c la ss is m ean t for spec ia lized

uses such as l iqu id re ta in ing s t ruc tures , which a re no t covered in th is book .

2 . A l imi ted tens ion i s a l lowed but wi th no v is ib le c rack ing . However , in

se rv iceabi l i ty ca lcu la t ions the concre te i s a ssumed to remain uncracked .

3 . Ten s ion i s a l lowed to a h igh er level and the concre te sec t ion is a ssu m ed to be

c racked . Sur face c racks must no t exceed 0 .1 mm for members in ve ry seve re

e nv i r onme n t s a nd 0 .2 mm f o r a l l o the r me mbe r s . T he 0 .2 mm l imi t i s

a pp l i c a b le t o pos t - t e ns ione d f loo r s i n no r ma l bu i ld ings .

In C lass 3 , though the sec t ion i s c racked , BS 8110 requi res no t iona l tens i le

s t re s se s t o be c om pu te d , a s sum ing a n un c r a c ke d se c tion , f o r c om ply ing w i th

the spec i f ied l imi ts . Some bonded rod re inforcement i s r equi red to take the

theoret ica l tensi le s tresses in a cracked sect ion.

I n t he U K , m or e o f t he f loo r s in bu i ld ings a re de s igne d to c om ply w i th t he C la s s 2

r e q u i r e me n t s t ha n C la s s 3. T he l a t t e r, how e ve r , ha s t he a dva n ta ge tha t t he e x t r a

rod re info rcem ent o n the tens ion face i s use fu l in cont ro l l ing ea r ly shr ink age a nd

c r a c k d i s t r i bu t ion .

6 . 5 P e r m i s s i b l e s t r e s s e s i n c o n c r e t e

In the UK the re a re some d i f fe rences in the pe rmiss ib le s t r e sses spec i f ied in BS

8110 for C lass 2 and Class 3 s t ruc tures . No such d is t inc t ion i s made in ACI 318 ,

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FL E X U R E I N TH E S E R V I C E A B I L ITY S TA TE 139

Tab le 6 .2

BS 8110 Class 2. Permissible stresses (N /m m 2)

In i t i a l s t r e s ses

t e n s i o n

0.36

N/fci

com pre ss io n 0 .50 fc i

F ina l s t r e s ses

ten sio n 0 .36 x /fcu

com pre ss io n : span 0 .33 fc u

c o m p r e s s i o n : s u p p o r t 0 .4 0fr

Note: The 0.4fcu allowable compression stress at support does not apply

to cantilevers.

which does not use the class i f icat ion sys tem.

The tens i le s t resses g iven in Table 6 .2 are for normal pres t ressed concrete ,

wi thou t a ny enh ancing ingred ien t such as s tee l f ib res . Where such an ingred ien t is

used in the concrete , i t s tens i le propert ies should be determined by tes ts .

6.5 . 1 BS 811 0

BS 8110 specifies the following l imits for Class 2 structures at the serviceabil i ty

l imi t s ta te . At the in i t ia l s tage, if the s t ress d iag ram is nea r-rec tang ular , th en the

compressive stress is l imited to 0.4fr

At the f ina l s t age , compress ion in con t inuous members in the suppor t reg ion

can be a m ax im um of 0 .4fr in spans i t is l imi ted to 0 .33fr I f the im pose d load is

o f a t emp orar y na tu re and i s excep t iona l ly h igh in comp ar i son wi th the norm al

load then the a l lowable t ensi le s tress m ay be increased by 1 .7 N / ram 2 p rov ided

that the s t ress i s normal ly compress ive.

These s t res s l imi t s a re used fo r bo th bonded and unbonded cons t ruc t ion . The

allowa ble tensile stresses show n in the Ta ble 6.2 include a pa rt ial safety factor of 1.3.

In Class 3 m em bers the al lowable in i t ia l tens ile and co mp ress ive s t resses are

the same as those for Class 2 . For the f inal tens i le s t ress , however , a l though

crack ing i s a l lowed , i t is as sum ed tha t the concre te sec t ion is uncrack ed and tha t

des ign hypo thet ic al s t resses exist a t the l imi t ing crac k widths . T he al low able f inal

s t ress i s related to the concrete s t rength , crack width and to the sect ion depth , as

shown in Table 6 .3 .

BS 8110 al lows the tens i le values given in Tables 6 .2 (modif ied for except iona l

load ing where app l i cab le ) and 6 .3 to be exceeded under two cond i t ions .

F i r s t ly , i f add i t iona l re in fo rcement is con ta ined wi th in the t ens ion zon e , and i s

posi t ion ed close to the ten s ion faces of the concre te , these mo dif ied des ign s t resses

m ay be increased by an am ou nt tha t i s in p ro por t ion to the c ross -sec t iona l a rea o f

the add i t iona l re in fo rcement (expressed as a percen tage o f the c ross -sec t iona l

a rea o f the concre te in the t ens ion zone) . F or 1% of add i t iona l re in fo rcement , the

s tresses m ay be increased by 4 .0 N / ram 2. Fo r o ther percen tages o f add i t iona l

reinforcem ent , the s t resses m ay be increased in pr op ort ion up to a l imi t of 0 .25fr

Second ly , when a s ign if ican t p ro po r t ion o f the des ign serv ice load i s trans i to ry

so tha t the whole sec t ion i s in compress ion under the permanen t (dead p lus

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POST-TENSIONEDCONCRETE FLOORS

Table 6.3 BS 8110 C lass 3, bonded tendons . Perm iss ib le f ina l tens i le s t ress

C r a c k M e m b e r

w id th d e p th

m m m m

De s ig n s t r e s s f o r conc re te g ra d e

3 0 4 0 50 a nd ove r

0.2

0.1

200 4.18 5.50 6.38 N /m m 2

400 3.80 5.00 5.80

600 3.42 4.50 5.22

800 3.04 4.00 4.64

> 1000 2.66 3.50 4.06

200 3.52 4.51 5.28 N /m m 2

400 3.20 4.10 4.80

600 2.88 3.69 4.32

800 2.56 3.28 3.84

> 1000 2.24 2.87 3.36

f r e q ue n t ly oc c u r r ing impose d ) l oa d , t he hypo the t i c a l t e n s i l e s t r e s se s ma y be

exceeded under the fu l l se rv ice load .

BS 8110 does n o t g ive any l imi ts for tens i le s t re sses in C lass 3 s t ruc tures whe re

unbonde d t e ndons a r e u se d ; i n p r a c t i c e , i t i s a s sume d tha t t he l imi t s shown in

T a b l e 6 .3 a p p ly . A C l as s 3 m e m b e r w i th u n b o n d e d t e n d o n s i s p r o n e t o m o r e

c r a c k ing und e r f ul l de s ign loa d i f a de q u a te b ond e d r od r e in f o r c e me n t ha s n o t

be e n p r ov ide d .

6.5 .2 Con crete Soc ie ty

T he C onc r e t e S oc iety ( 1994 ) a l lows the 0 .1 m m c r a c k w id th C la s s 3 pe rmis s ib l e

s tr e sse s to be u se d fo r unb on de d t e ndo ns p r ov ide d tha t t he t e ns ion i s c a r r i e d on

bonde d r od r e in f o r c e me n t .

F o r tw o- wa y sp a nn ing f l a t s l abs , C on c r e t e S oc ie ty r e c om m e nds the pe rmis s ib l e

s t resses g iven in Table 6 .4. The a l lowa ble tens ile s t r e sses a re s ign i f ican t ly lower in

the suppo r t r e g ions , c om pa r e d w i th T a b le 6 .3 , due to t he pe a k in g o f t he mom e n t s .

I n T a b le 6 .4 , the sup por t zone i s a s sum e d to e x t e nd f o r a d is t a nc e o f 0 .2L f r om

the supp or t ; a n y se c t ion be y ond th i s po in t i s c ons ide r e d to be i n the spa n zone .

B on de d r e in f o r c e me n t m a y c ons i s t o f e i t he r r od r e in f o r c e me n t o r t he t e nd ons

themse lves .

6.5 .3 AC l 318

In the US A tw o au tho r i t ie s spec ify a l lowa ble s t r e sses in pres t re ssed concre te : T he

A m e r i c a n A ssoc i a t i on o f S t a t e H igh wa y a nd T r a nsp o r t a t i o n O f fi ci al s ( A A S H T O )

a nd T he A me r i c a n C onc r e t e I n s t i t u t e ( A C I ) . T he A A S H T O spe c i f i c a t ion i s t he

mor e c onse r va t ive a s i t a pp l i e s t o b r idge s , whe r e t he e xposu r e c ond i t i ons a r e

muc h mor e s e ve r e t ha n in bu i ld ings . A C I 318 : 1989 g ive s t he l imi t s f o r

se rv iceabi l i ty s t r e sses shown in Table 6 .5 .

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FLEXURE IN THE SERVICEABILITY STATE

141

Table 6.4

Permissible s tresses in tw o-w ay f lat s labs (Conc rete So cie ty , 1994)

Loca t ion Compress ion

Tension Tension

wi th bonded wi th unbonded

reinforcement reinforcement

Suppo rt 0.24 fr

Span 0.33 fr

0.45 ~ / f ~ 0

0.45 0.1 5 x/f~u

Table 6.5

A C I 318 Permissible s tresses in conc rete

S tage M od e M e t r i c Impe r ia l

N / m m 2 p si

Initial tensio n 0.25 ~/f'ci 3.0 ~/f'ci

com pression 0.60f'r 0.60f'r

Fina l tension 0.50 x/f'c 6.0 x/f'r

compression 0.45f'r 0.45fc

AC I 318 al lows the perm iss ible in it ia l tensi le s t ress at the e nds of s imply

sup po rted m em ber s to be increa sed to doub le the tabu late d va lue, i .e . to 0.5x/(fci)

N /m m 2 (6x/fc ps i ) . A n incre ase in the f inal perm iss ible tens i le s t ress to x /fr

N /m m 2 (12x/f r ps i) is a l lowed in one-wa y sp ann ing m em bers sub jec t to

compl iance wi th the def l ec t ion and cover requ i rements .

Where the tens i le s t resses exceed the permiss ible values , the to tal force in the

tens i l e s t res s zone may be ca lcu la ted assuming an uncracked sec t ion and

reinforc em ent pro vide d on the bas is of th is force at a s t ress of 0 .6fy but not m ore

than 200 N /m m 2 (30000 ps i) .

S t resses in the conc re te a re ca lcu la ted on the bas i s o f an unc rack ed sec tion . A

m in im um average p res t ress o f 1 .0 N /m m 2 (150 ps i) i s requ i red on the g ross

concrete sect ion, af ter a l lowing for al l losses in the pres t ress ing force.

6 .6 P e r m i s s i b l e s t r e s s e s i n s t r a n d

BS 8110 specifies that:

The jack ing fo rce shou ld no t norm al ly exceed 75% of the charac te r i s t ic

s t r en g th o f t h e t en d o n b u t m ay b e in c r ea s ed t o 8 0 % p ro v i d ed t h a t ad d i t i o n a l

cons ide ra t ion is g iven to sa fe ty and to the load /ex tens ion charac te r i s t i c o f the

tendon . At t rans fer , the in i ti a l p res t ress shou ld no t norm al ly exceed 70% of the

charac te r i s t i c s t reng th o f the t endon , an d in no case shou ld i t exceed 75%.

AC I 318 specif ies the fol lowing ma xim um values of s t ress in low relax at ion s t rand :

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142 POST-TENSIONEDCONCRETE FLOORS

During s t res s ing : 0 . 9 4 f p y b u t n o t mo re t h an 0 .8 fp , an d t h e max i mu m

reco mmen d ed b y t h e man u fac t u re r .

Immediately af ter s t ress ing: 0 .70fp, .

A s t rand carr ies the highest force during s t ress ing; thereaf ter the force reduces

when the anch orage is locked , an d as c reep an d sh r inkage t ake e ffec t. The in i t ia l

s t rand fo rce is norm al ly l imi ted to a ma xim um of 80% of i ts b reak ing load . In

prac t i ce a fo rce o f 70% to 75% of the b reak ing s t reng th is o f t en a imed fo r when

the s t rand is s t ressed; th is leaves a useful smal l reserve for cont ingencies .

6.7 Ana lysis

F or analys is , a floor is no rm al ly d ivided in to a series of s t r ips which are then

ana lysed , genera l ly in the sam e m ann er as fo r re in fo rced concre te . Eac h o f the

cri t ical sect ions i s then checked and des igned for adequacy at the serviceabi l i ty

and u l t imate s t a t es .

In the case o f s imply suppo r ted spans , the ana lys is cons i st s m ere ly o f

ca l cu l a t i n g t h e mi d s p an mo men t s an d en d s h ea r s fo r t h e max i mu m l o ad . Th e

requ i red p res t ress an d rod re in fo rcement a re then ca lcu la ted i f the sec t ion i s

ad equ a t e .

Co nt inu ous beam s , be ing one-w ay e lements , can be ana lysed e ither as s t r ings ,

o r as f rames in tegra l wi th the co lumns . In the l a t t e r case , the co lumns a re

assum ed to be fixed a t the i r fa r ends. The des ign o f con t inu ous m em bers i s

a f fec ted by the long i tud ina l shor ten ing o f the mem ber m uch m ore in pos t - t ens ioned

tha n in reinforced concre te . In reinfo rced co ncre te des ign, the effect of creep is to

increase the def l ec t ion , and sh r inkage i s normal ly ignored . In pos t - t ens ioned

members , e l as t i c shor ten ing due to ax ia l p res t ress , c reep and sh r inkage cause a

shor ten ing in the l eng th o f the m em ber , which resu l ts in l a t e ra l def l ec t ion o f the

co lumns . The fo rces deve loped by the shor ten ing o f the m em ber sho u ld be

cons idered in the design o f pos t - t ens ioned m em bers a nd the ver t i ca l e l ements o f

the f rame.

In s t rip beams , t ransverse m om ents deve lop across the wid th o f the beam, a nd

are s ign if ican t if the s lab is con t inuo us . Such m om ents a re norm al ly t rea te d as

par t o f the s lab des ign . They m ay be h igher a long the co lum n l ines , where a shor t

l eng th o f the beam co ll ect s and t rans fers the loads to the co lum n. These m om ents

shou ld be t aken in to acco un t in the des ign o f pos t - t ens ion ing o r re in fo rcem ent

a long the co lumn l ines .

On e-w ay s labs a re norm al ly ana lysed in un i t wid ths , suppo r ted on bea m l ines .

A s lab sec tion in line wi th the co lum ns an d s upp or ted on a s t r ip beam m ay need

t o b e d es ig n ed fo r t h e ad d i ti o n a l n eg a t iv e m o m en t f ro m t h e b eam as men t i o n ed

above . A f loor s l ab and i ts suppo r t ing co lumn s m ay a l so be ana lysed as a f ram e o f

wid th equa l to the pane l wid th , o r ha l f the pane l wid th fo r an ou ter f rame; th i s

m e t h o d , h o w ev e r , l e ad s t o th e p ro b l em o f ap p o r t i o n i n g t h e to t a l mo m en t

be tween the co lumn and midd le s t r ips .

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FLEXURE IN THE SERV ICEABIL ITY STATE 143

Two-way f loors may be ana lysed in severa l d i f fe ren t ways . The methods

usua l ly men t ioned in the con tex t o f pos t - t ens ioned f loors a re l i st ed be low.

9 A s a

two -wa y s l ab spann ing on to a g r id o f beam s . In each d i rec t ion the s l ab

pane l i s cons idered in s tr ips o f un i t wid th , w i th a p rop or t io n o f the ne t load

ac t ing in each d i rec t ion , the p rop or t io n a long sp an L 1 be ing

L24/(L14 +

L24)

as discussed in Sect ion 3 .3 .2 . The sum of al l loads on the s lab pane ls and the

beams in each d i rec t ion shou ld no t be l es s than the to ta l load .

9 As a g rid . Th i s is a useful m etho d wi th com pl ica ted load ing pa t t e rns , o r

conc en t ra ted loads . I t requ i res the use o f a com pute r because o f the l a rge

number o f var i ab les invo lved .

Th e mo men t s a r e ex p ec t ed t o p eak a ro u n d t h e co l u mn s , an d i n o rd e r t o

ach ieve an accu ra te va lue o f the peak m om ent , the s t r ips in the v ic in ity o f the

co lum ns sho u ld be as na r row as p rac t ica l ly poss ib le . Th i s , however , a l so g ives

an unnecessar i ly l a rge nu m ber o f po in t s a long the co lum n l ines .

9 Using f in i te e lement programs. The ext ra effor t and expense involved in us ing

f in i te e l ements m ay be jus ti f ied i f the p lan shape o f the f loor and /o r the load ing

p a t t e rn can n o t b e acco m m o d a t ed b y an y o f t h e o th e r me t h o d s .

9 As frame s in two di rect ion s , tak ing the full load in each d i rect ion. The w idth o f

a frame is chose n so tha t i t s edges coincide wi th the l ines of zero shear in the

o ther d i rec t ion .

Th i s me thod g ives the to ta l mo m ent fo r the whole wid th o f the f rame; the

m om ent per un i t wid th , in fac t , var ies over the f rame wid th , be ing h igher on the

column l ines .

9 As a f ia t s lab , d ivided in to m iddle an d c olum n s t r ips , fo l lowing the pra ct ice for

re in fo rced concre te . Th i s i s an em pi r ica l m etho d fo r the d i s t r ibu t ion o f loads

on co lumn and midd le s t r ips in a pane l . There a re l imi ta t ions on the span

ra t ios , nu m ber o f pane l s on each d i rec t ion , and the loads .

The ru les fo r the d i s t r ibu t ion o f the load spec i fi ed in the na t ion a l s t and ards

y ie ld on ly the u l t imate moments ; no gu idance i s g iven fo r moments a t

serviceabi l i ty s tate . I t , therefore, becomes di ff icul t to check the des ign for

compl iance wi th the se rv iceab i l i ty requ i rements .

The f ir st m en t ione d is the mos t co m m only used m etho d and the las t is used on ly

in very spec ia l c i rcums tances . O ther m ethod s o f ana lys is which g ive on ly the

u l t imate moments , such as the y ie ld l ine method , a re no t su i t ab le fo r des ign ing

pos t - t ens ioned f loors , because adequ acy o f the s t ruc tu re a t the se rv iceab i li ty s t a t e

cannot be veri f ied; the yield l ine method can be used in the calculat ion of

s t reng th , p rov ided tha t the se rv iceab il ity s t a te requ i rem ents a re checked us ing a

m etho d o f e l ast ic ana lys i s .

6 .7 . 1 L o a d c o m b i n a t i o n s

As ind ica ted ear l i e r , t he se rv iceab i l i ty check compr i ses two s t ages - - in i t i a l and

f ina l. The in it ia l s t age represen t s the s t a t e o f the m em ber imm edia te ly a f t e r the

app l i ca t ion o f the p res t ress , befo re any o f the long- te rm pres t ress losses have

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144 POST-TENSIONED CONCR ETE FLOOR S

t ak e n p l ace . T h e p r e s t r e s s i n g f o r ce is a t i ts h i g h es t , an d t h e w o r s t co n d i t i o n a t t h i s

s t ag e is t h a t o f m i n i m u m a p p l i e d l o a d . N o e x t e r n a l l o a d s a r e , th e r e fo r e , i m p o s e d

a t t h is s t ag e , an d t h e l o ad i n g co n s i s t s o f o n l y t h e d e ad l o ad an d t h e e ff ec t o f t h e

i n i ti a l p r e st r e s s . T h e f i n a l s t ag e r ep r e s en t s t h e l o n g - t e r m s t a t e , w h en a l l l o s se s i n

p r e s t r e s s h av e t a k en p l ace . T h e s e l f -w e i g h t o f t h e me m b er an d t h e f in a l p r e s t r e s s

a r e c o m b i n e d w i t h o t h e r a p p l i e d l o a d s i n t h e m o s t a d v e r s e m a n n e r f o r e a c h o f t h e

c r i ti c a l s ec t io n s o f t h e m em b er .

I n t h e ca s e o f a s i mp l y s u p p o r t ed m em b er , t h e l o ad t o b e co n s i d e r ed i n th e f in a l

s t ag e co n s is t s o f a l l i mp o s ed l o ad s a c t i n g s i m u l t an eo u s l y . I n t h e an a l y s i s o f

c o n t i n u o u s m e m b e r s , t h e n o r m a l s e r v i c e l o a d s a r e c o m b i n e d t o y i e l d t h e

m a x i m u m v a lu e o f t he t o t al m o m e n t a t e ac h s u p p o r t a n d i n e a c h s p a n. S u p p o r t

m o m e n t s m a y b e t a k e n a t t h e s u p p o r t f a c e s r a t h e r t h a n a t t h e i r c e n t r e l i n e s .

N o r m a l l y , o n l y t h e d e a d a n d l i ve l o a d s a r e i n v o lv e d , in w h i c h c a s e t h e m o m e n t

a t a s u p p o r t i s m ax i m u m w h e n fu ll l o ad i s ap p l i ed o n ad j ace n t s p a n s e i t h e r s i d e o f

t h e s u p p o r t a n d o n a l t e r n a t e s p a n s t h e r e a f t e r . T h e m o m e n t i n a s p a n i s a

m a x i m u m w h e n f u l l l o a d i s a p p l i e d o n a l t e r n a t e s p a n s . T h e m o m e n t s i n a f l o o r

m ay a l s o b e a f fec t ed b y t h e i mp o s i t i o n o f l iv e l o ad o n t h e f l o o r s ab o v e an d b e l o w ,

b u t t h i s i s n o r ma l l y i g n o r ed .

T h e a b o v e c o m b i n a t i o n s a r e , h o w e v e r , g e n e r a l l y c o n s i d e r e d u n r e a l i s t i c a n d

t o o s ev e r e , an d n a t i o n a l s t an d a r d s a l l o w d es i g n t o b e b a s ed o n l e s s o n e r o u s

co n f i g u r a t i o n s . B S 8 11 0 an d A C I 3 1 8 a l l o w t h e a r r a n g e m en t o f l iv e l o ad t o b e

l i m i t ed t o :

9 d ead l o ad o n a l l s p an s w i t h l iv e l o ad o n t w o ad j ac en t s p an s , an d

9 d ead l o ad o n a l l s p an s w i t h l i ve l o ad o n a l t e r n a t e s p an s .

6 .8 S i m p l y s u p p o rt ed s p a n

T h e r e b e i n g n o r ed u n d a n c i e s , c a l cu l a t i o n s f o r a si mp l e s p an a r e ea s y t o ca r r y o u t

m a n u a l l y . F o u r s t r e s s c o n d i t i o n s n e e d t o b e c h e c k e d i n a s i m p l y s u p p o r t e d

memb er , v i z . , i n i t i a l an d f i n a l s t r e s s e s a t t h e t o p an d b o t t o m f i b r e s .

L e t M o = m o m e n t d u e t o o w n w e i g h t o f c o n c r e t e s e c ti o n

M s - m o m e n t d u e t o o t h e r a p p l i e d l o a d s

T h en t h e s t r e s s a t t h e t o p d u e t o t h e i n i t i a l p r e s t r e s s i s

O ' p t i

- - "

Pi /Ac - - P iep /Z t

= ( e i / A c ) . (1 - e p A c / Z t ) = s t e i / a c

wh ere S t = 1 -

epAe /Z t

a n d s i m i l a r l y a t t h e b o t t o m ,

(6 .1a)

O ' p b i

- -

(P i /Ac ) . (1 + e p A c / Z t ) = S b P i / A c

wh ere S b = 1 + e p A J Z b

Sim i l a r ly t he s t r esses due t o t he f i na l va lue o f t he p res t r e ss ing fo rce a re"

(6 .1b)

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FLEXURE IN THE SERVICEABILITY STA TE

145

O'pt = ( P f / A ~ ) S t ( 6 . 1 c )

O'pb =

(Pf/Ar

( 6 . 1 d )

The four l imi t ing s t resses can now be formulated. The in i t ia l s t resses in the top

and bo t tom f ib res mus t sa t i s fy :

O'ti -" O'pti "[- M o / Z t > Pti

O ' b i - - O ' p b i - M o / Z b < Pci

and the f inal s t resses must sat i s fy:

O'if- - O'ptf + (Mo +

M s ) / Z t

< Pcf

O'bf -- O'pb

-- ( M o "JI-

Ms)/Z b

> Ptf

where PIi " - permissible init ial tensile stress

Pea

" -

permiss ible in i t ia l compress ive s t ress

Ptf " - permissible final tensile stress

Pef-- permiss ible f inal compress ive s t ress

(6.1a)

(6.2b)

(6.2c)

(6.2d)

( - v e ) S e e

( + ve) Sec tion

( - v e ) 6 . 5

(+ve)

Normal ly , the f inal tens i le s t ress condi t ion, Equat ion (6 .2d) , i s expected to

govern . Rea r rang ing g ives the m in im um requ i red va lue o f Pf .

efmin--

[Ptf +

(Mo

+ M s ) / Z b ] a e / S b ( 6 . 3 a )

The maximum pres t ress ing fo rce which can be imposed a t a sec t ion i s usua l ly

gove rned by the initial compressive stress an d c an be calculated from equ ation (6.2b).

P ~m ~, = ( P r + M o / Z b ) A J S b

(6.3b)

If Rp is the r at io of final to init ial p restre ss,

P f / P i ,

t hen the m axim um f ina l fo rce is

given by

e f m a x - - R p ( P c i q -

M o / Z b ) A c / S b

(6.3c)

Obviously , for a des ign to be feas ible the value of

P f m a x

must be l a rger than the

P f m i n

value. I f the v alue of P f m a x i s found to be l es s than P f m i n t hen the concre te

sec t ion i s inadequate and i t mus t be increased .

6 .8 .1 Sh ape fac tors

Th e n o n -d i men s i o n a l t e rms S t and S b, Eq uat io n (6.1) , depe nd on the shape o f the

sec t ion and on the t endon eccen t r i c i ty . For a t endon p laced be low the sec t ion

cen troid (posi t ive eccentr ici ty) , St i s expe cted to be negat ive, be cause the pres t ress

wo uld te nd to induc e ten s ion at the to p f ibre, an d Sb to be posi t ive. S an d S b are

rat ios of ext reme f ibre s tresses to the average s t ress and, therefore, are representat ive

of the efficiency of the section.

S t = 1 - e p A c / Z t

(6.1a)

S b = 1

+ e p A c / Z b (6.1b)

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146 POST-TENSIONEDCONCRETE FLOORS

tT

t t t t t

I l l

T I T T

P

P

p

(a)

f

0 1 s e m

_ _

, L ' " l

Figure 6.4

Simply supported beam

(b)

From Equat ion (6 .1a) , i t can be seen tha t S t becomes zero a t an eccen t r i c i ty o f

Z t / A e .

At th is eccentr ici ty the s t ress in the top f ibre remains zero for a l l

m ag ni tu des of the pres t ress ing force. I f the eccentr ici ty exceeds th is value then the

top s t ress i s negat ive ( tensi le) ; the to p f ibre s tress i s posi t ive (com press ive) i f the

eccentr ici ty i s less than th is value. For a rectangular sect ion, Z t / A r - - ( t / 6 .

An eccen tr i ci ty o f -Z b / A

c

has a s im i lar effect on the s t ress in the bo t to m f ibre.

A negat ive eccentr ici ty i s , however , unl ikely to be used.

6.8 .2 Equ iva lent load m ethod

The se ts o f Eq uat ion s (6.2) and (6.3) t ake d i rec t ac coun t o f the eccen t ri c i ty o f

p res t ress ra the r than us ing the load ba lan c ing m etho d ; in the case o f s imply

s u p p o r t ed memb er s t h e fo rmer ap p ro ach i s e a s i e r . Fo r co mp ar i s o n , t h e

load-ba lanc ing approach i s shown be low fo r the in i t i a l cond i t ion a t the top ,

Equ a t i o n (6.2a ). I t d emo n s t r a t e s t h e equ iv a l en ce o f t h e lo ad -b a l an c i n g m e t h o d

and the meth od used in der iv ing the se ts o f Eq uat ion s (6.2) and (6.3) , and m ay be

sk ipped i f the de m ons t ra t ion is no t o f in te res t .

Assume tha t the t e ndo n p ro f il e i s parabo l i c , e m is the eccen t r i c ity a t m idsp an

and the t endons a re anchored a t each end a t the sec t ion cen t ro id , F igure 6 .4 (a) .

Then f rom Equat ion (5 .13) ,

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FLEXURE IN THE SERVICEABILITY STATE

147

Wei

=

i n i t i a l e q u i v a l e n t l o a d

= - 8 P i e m / L 2

(6.4)

M i = m o m e n t d u e t o t h is l o a d

= w e i L 2 / 8 = _ P i e m

O ' t i - - P i / a e + ( M o

+ M i ) / Z t >

Pti

(6.5)

T h e t e r m

P ie m

is u s e d h e r e a s a n a p p l i e d m o m e n t , w h e r e a s i n E q u a t i o n ( 6.1 ) t h e

e q u i v a l e n t t e r m

Pi e

i s u s e d i n c a l c u l a t i n g t h e s t r e s s d u e t o p r e s t r e s s .

T h e e q u i v a l e n t l o a d w~i is s ta t i ca l l y b a l a n c e d b y t h e v e r t i ca l c o m p o n e n t s o f t h e

t e n d o n f o rc e s a t t h e e n d , e a c h o f m a g n i t u d e P i . t a n 0 , s o t h a t

f o r F i g u r e 6 . 4 ( a ) ,

WeiL

- - 2P i t an 0

f o r F i g u r e 6 . 4 ( b ) ,

w ~ i L

= P~(tan 01 + ta n 02).

T h e e q u i v a l e n c e o f e q u a t i o n s ( 6 .2 a ) a n d ( 6.5 ) i s e a s il y d e m o n s t r a t e d .

O ' t i - - P i / A r + ( M o + M i ) / Z t ( 6 . 5 )

= P i / A e + M o / Z t - P i e m / Z t

= ( P i / A e )( 1 - e m . A e / Z t ) + m o / Z t

= apti +

M o / Z t

(6 .2a)

I f t h e t e n d o n w a s n o t a n c h o r e d a t th e s e c t i o n c e n t r o i d b u t t h e e c c e n t r ic i t i e s a t le ft

a n d r i g h t w e r e e~ a n d e r a s s h o w n i n F i g u r e 6 . 4 (b ) , t h e n t h e s a g o f th e p a r a b o l a w i ll

b e l a r g e r t h a n em , t h e m i d s p a n e c c e n t r i c i t y , b u t t h e e n d e c c e n t r i c i t i e s w il l n e e d t o

b e c o n s i d e r e d t o a r r i v e a t t h e c o r r e c t e q u i v a l e n t l o a d a n d m o m e n t .

s = sag o f pa r ab o l a = e m - ( e 1 + e r ) /2

w~ = e q u i v a l e n t l o a d = -

8 P i s / L 2

M ~ = M o m e n t a t m i d s p a n d u e t o e n d e c c e nt r ic i ti e s

= - P i ( e 1 + e , )/ 2

M = N e t m i d s pa n m o m e n t

= w e L 2 / 8 +

M ~

= - P i s - P i ( e l +

er)/2

= - - P i [ e m - (e 1 +

er) /2 ] - -

P i ( e l +

er)/2

= _ P i e m

I t c a n b e s e en t h a t t h e t w o a p p r o a c h e s l e a d t o e x a ct l y th e s a m e r e s u lt a n d t h a t f o r

a s i m p l y s u p p o r t e d s p a n t h e l o a d b a l a n c i n g m e t h o d i s u n n e c e s s a r i ly c u m b e r s o m e .

6 . 9 C o n t i n u o u s s p a n s

T h e d e s ig n o f a p o s t - t e n s i o n e d m e m b e r c a n b e c o n s i d e r e d t o c o m p r i s e t w o

d i s t in c t o p e r a t i o n s - - c a l c u l a t i o n o f m o m e n t s , a n d c a l c u l a t i o n o f s t re s se s . T h e

c a l c u l a t i o n o f m o m e n t s i s m u c h m o r e i n v o l v e d i n t h e ca s e o f c o n t i n u o u s s p a n s ;

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148

POST-TENSIONED CONCR ETE FLOORS

once the moments have been ob ta ined , the s t res ses a t c r i t i ca l sec t ions a re

ob ta ined f rom the s imple equa t ion

tr = P /A r + M / Z

(6.6)

where the moment M inc ludes the t endon eccen t r i c i ty , the moment due to se l f -

w e i g h t an d ap p l i ed l o ad s , an d t h e s eco n d a ry mo men t .

The requ i red p res t ress ing fo rce canno t be d i rec t ly ca lcu la ted fo r a s t r ing o f

spans in the m ann er o f a s imple span . In s tead , an in it ia l guess mus t be mad e ,

a l lowing fo r losses a long the t e ndo n l eng th , and the s t res ses can the n be che cked

for compl iance w i th the permiss ib le va lues . An a d jus tm ent to the in i ti a lly chosen

pres t ress , fo l lowed by re-calculat ion of s t resses , may be necessary .

6 .9 . 1 Equ iva len t load m ethod

Th e t en d o n eccen tr i ci ty r ep re sen t s a mo m en t o f mag n i t u d e

P , e ,

a t a po in t x a lon g

the tendon, where Px is the pres t ress ing force and e , the eccentr ici ty . I f the

cu rv a t u re o f a t en d o n i s t r ea t ed a s a mo m en t d i ag ram ap p l ied t o a mem b er , t h en

the ca lcu la t ions wi ll i nvo lve in tegra t ion o f m om en t d iag ram s to se t up an d so lve a

n u m b er o f s i mu l t an eo u s equ a t i o n s . Ca l cu l a t i o n o f i n d e t e rmi n a t e fo rce s can t h en

be car r i ed o u t us ing any o f the es t ab l i shed m etho ds o f e l ast ic ana lys i s , such as

inf luence coeff icients or f in i te e lements . This process , however , i s unnecessar i ly

complex , espec ia l ly fo r manual use .

Th e equ i v a l en t l o ad , o r l o ad b a l an c i n g , me t h o d p ro v i d es a mu ch s i mp l e r

approach . Us ing the equa t ions deve loped in Chap ter 5 , the t endon p ro f i l e i s

t r an s fo rmed i n to an ax i al l o ad an d an equ i v a l en t l o ad o n each o f t h e mem b er s .

Th es e equ i v a len t l o ad s a r e t h en t r ea ted i n t h e s ame ma n n e r a s an y o t h e r ap p l i ed

load in the ana lys i s and ca lcu la t ion o f m om ents and shears '.

P r i mary an d s eco n d a ry mo men t s h av e b een i n t ro d u ced i n Ch ap t e r 5 . Us i n g

t h e equ iv a l en t lo ad m e t h o d , t h e s eco n d a ry m o m en t s a r e au t o ma t i ca l l y i n c lu d ed

in the se rv iceab i l i ty ca lcu la t ions . The secondary moments a re , however , needed

for the ul t imate s t ren gth a nd m ust be ext racted from the serviceabi l ity calculat ions .

Seco n d a ry mo men t s a r e cau s ed b y t h e r ed u n d an c i e s p r e s en t i n co n t i n u o u s

s t ruc tu res and so vary l inear ly be tween su ppor t s . Som e meth ods o f ana lys i s, such

as m om ent d i s t r ibu t ion , d i rec tly g ive the final m om ents a nd the red un da n t shears

are der ived f rom the m om ents . I f the final mo m en t a t a po in t x resu l t ing f rom the

i mp o s i t io n o f t h e equ iv a l en t lo ad i n g is M, , t h en t h e s eco n d a ry mo m en t M = is

given by

M s , = M , - P , . e ,

(6.7)

The eq ua t ions deve loped in Cha p ter 5 m ay resu l t in two d i f fe ren t ha l f par abo las

on e i ther s ide o f an in te r io r su ppo r t i f t he ad jacen t spans a re no t eq ua l , so tha t the

in tens i ti es o f the un i fo rmly d i s t r ibu ted reac t ions f rom the ha l f para bo la s d i ffe r

(F igure 6 .5 ). Th i s is o f no consequ ence as long as the cor rec t e qu iva len t loa ds a re

taken in the ca lcu la t ions .

I t i s impor tan t to apprec ia te tha t the concen t ra ted end shears due to the

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FLEXURE IN THE SERV ICEABIL ITY STATE 149

l l l l l

TT_TTTTT

Figure 6.5

Tendon p ro f i l e ov e r an in te rna l suppor t w i th u nequa l spans

e q u i v a l e n t l o a d s a r e i n t e r n a l t o t h e i n d e t e r m i n a t e s y s t e m a n d d o n o t c h a n g e t h e

o v e r a ll a p p l i e d l o a d . T h e s u m o f t h e r e d u n d a n t r e a c t i o n s a c t i n g o n a ll t h e

s u p p o r t s d u e t o t h e e q u i v a l e n t l o a d m u s t b e z e r o . R e a c t i o n s c a n o n l y b e

t r a n s f e r re d f r o m o n e s u p p o r t t o a n o t h e r w i t h o u t a f fe c ti n g t h e s u m t o t a l o f t h e

s e l f - w e i g h t a n d t h e a p p l i e d l o a d .

T h e p o i n t c a n p e r h a p s b e b e t t e r u n d e r s t o o d b y a n e x a m p l e . C o n s i d e r a s i m p ly

s u p p o r t e d b e a m c a r r y in g a lo a d w , a n d p o s t - t e n s io n e d w i th a p a r a b o l ic t e n d o n

w h o s e e q u i v a l e n t l o a d i s w e . A t t h e s u p p o r t , t h e s h e a r f o r c e i n t h e b e a m is

(w - we)L~2 w h e r e a s t h e s u p p o r t r e a c t i o n is w L / 2 . T h e d i f f e r e n c e , w e L l 2 , i s in f a c t

p r o v i d e d b y t h e s lo p e o f t h e t e n d o n a t t h e a n c h o r a g e a n d e q u a l s P t a n 0 i n v a l u e.

Example 6.1

Calcula te the required pres t ress for the s imply suppor ted span shown in Figure 6 .6 .

As s ume tha t the r a t io Pf/Pi = 0 .85 , tha t the t en don cen t r o id a t m ids pan i s 40 mm

above sof f i t and tha t the permiss ib le s t resses and des ign loads a re :

Pti

- - - -

1.8 N /m m 2 per = + 13.2

P c i - - +

12.5

P t f = - -

2.3

Im pos ed loads : dead 2 .0 , l ive 4 .0 kN /m 2

Solution

Area an d sec t ion centro id Yt f rom to p:

wid th x dept h = a rea (mm z) x y

1000 x 110 = 110.0 • 103 • 55

400 • 170 = 680 X 195

(60/2) • 170 = 5.1 • 167

= A y (mm 3)

= 6.05 x 10 6

= 13.26

= 0.85

A c = 183.1 x 10 3 Acy t

= 20.16

• 10 6

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150 POST-TENSIONEDCONCRETE FLOORS

Live 4.0 (kN/m2)

Dead 2.0 (kN/m2)

A

L= lO.Om

Yb

Figure 6.6 Example 6.1

F

- I

1000

, . i I I ,

O e 9 O 0

e 1_ 4 0 0 3

r "7

. . . I

1

Y t = A c Y t / A c = 1 1 0 m m

Yb --" D -- Yt - 17 0

e m " - Yb - - 40 = 130

M o m e n t o f i n e r t ia a n d s e c t i o n m o d u l i "

~ 4 0

A D Yt Y

1 1 0 . 0 x 1 03 x [ 1 1 0 2 / 1 2 + ( 1 1 0 - 5 5 ) 2 ] = 4 4 3 . 6 7 x 1 0 6 m m 4

6 8 . 0 x 1 03 x [ 1 7 0 2 / 1 2 + ( 1 1 0 - 1 9 5 ) 2 ] = 6 5 5 . 0 7

5 .1 x 1 03 x [ 1 7 0 2 / 1 8 + ( 1 1 0 - 1 6 7 ) 2 ] = 2 4 . 7 6

~

110

- - f

280

M o m e n t o f i n e r t i a I c = 1 1 2 3 . 4 0 x 1 0 6

Z t =

I r t

- " 1 0 . 2 1 3 X 1 0 6 m m 3

Z b - - l e / y b

= 6 . 608 x 106

S t =

1 - e m A r t

= - 1 .331

S b = 1 + e m A c / Z b

= + 4 . 6 0 2

S e c t i o n w e i g h t = 2 4 x 1 8 3 . 1 / 1 0 0 0 = 4 . 3 9 k N / m 2

M o m e n t s :

M o = 4 . 3 9 x 10 2 / 8 = 5 4. 9 k N . m / m s e l f - w e i g h t

M , = 6 . 0 0 x 1 0 2 / 8 = 7 5 . 0 o t h e r l o a d s

M o + M s = 1 2 9 . 9 t o t a l

P r e s t r e s s ( e q u a t i o n s 6 . 3 a a n d 6 . 3 b ) :

P f m i , = [ - 2 . 3 + 1 2 9 . 9 / 6 . 6 0 8 ] x 1 8 3 . 1 / 4 . 6 0 2 = 6 9 0 . 6 k N

P fm ax = 0 . 8 5 ( 1 2 . 5 + 5 4 . 9 / 6 . 6 0 8 ) X 1 8 3 . 1 / 4 . 6 0 2 = 7 0 3 . 7

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FLEXURE IN THE SERVICEABILITY STATE

151

Pfmin < Pfmax

U s e P f = 6 9 0. 6 k N

P i = P f / 0 . 8 5 = 8 1 2 k N

O K

A s s u m i n g 1 6 0 k N f in a l f o rc e p e r 1 5 .7 m m s t r a n d , t h e n u m b e r o f s t r a n d s p e r

r i b = 6 9 0 . 6 / 1 6 0 = 4 . 3 ~ 5

S t r e s s e s :

P J A r = 8 1 2 . 5 /1 8 3 . 1 = 4 . 4 4 N / m m 2

Opt =

S tP i / ae

- - - 5 . 9 1 ( f r o m 6 . 1 a )

0 p h i "--

S b e i / A e = 2 0 . 4 3 ( f r o m 6 . 1 b )

P f / A c = 6 9 0 . 6 /1 8 3 . 1 = 3 .7 7 N / m m 2

O'pt f "--

S tP f / A e

= - 5 . 0 2 ( f r o m 6 . 1 c )

O'pb f =

S b P f / A c - - 1 7 .3 6 ( f r o m 6 .1 d )

I n i t i a l s t r e s s e s ( E q u a t i o n s 6 . 2 a a n d 6 . 2 b )

oti : - 5 . 9 1 + 5 4 . 9 /1 0 . 2 1 3 = - 0 . 5 3 > P ti

O'bi

= - ~ - 2 0 . 4 3

- 53 .4/6 .60 8 = + 12.35 < Pci

F i n a l s t r e s s e s ( E q u a t i o n s 6 . 2 c a n d 6 . 2 d )

otf

= - - 5 . 0 2

+ 129 . 9 /1 0 . 213 = + 7 . 70 < Pcf

abf = + 17 . 36 - - 129 . 9 /6 . 60 8 = - - 2 . 23 > P t f

O K

O K

O K

O K

C o m m e n t :

A n o m i n a l a m o u n t o f r o d r e i n f o rc e m e n t m a y b e r e q u i re d t o c o m p l y w i t h th e

p a r t i c u l a r n a t i o n a l s t a n d a r d . T h e q u a n t i t y i s s u b j e c t t o u l t i m a t e s t r e n g t h c a l c u l a t io n s .

D e f l e c t i o n a n d s h e a r s t r e n g t h m a y b e c a l c u l a t e d a s d i s c u ss e d i n th e r e l e v a n t c h a p t e r s .

E x a m p l e 6 . 2

I n e x a m p l e 6 .1 n o r o d r e i n f o r c e m e n t i s r e q u i r e d t o c a r r y a n y f o r ce s a t t h e

s e r v i c e a b i l i t y s t a te , b e c a u s e t h e c o n c r e t e s e c t i o n i s s u f fi c i en t l y la r g e . T r y t h e s m a l l e r

s e c t io n s h o w n i n F i g u r e 6 . 7. A l l o w a b l e s t r es s e s a n d a p p l i e d l o a d s r e m a i n a s b e fo r e .

Solution

S e c t i o n p r o p e r t i e s "

Ac = 162 . 8 x 103 m m 2

Ic = 8 50 . 29 0 x 10 6 m m 4

~

110

t 4 o

Figure 6 .7

Example 6 .2

0 0 0

/ _ _ r

I_.

300

3 6 0

I

70

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152 POST-TENSIONEDCONCRETE FLOORS

Y t - - 9 8

m m

Yb - - 1 7 2 m m

e m - - 1 32 m m

Z t ~ - -

8 . 6 7 6 x 1 06 m m 3

Z b = 4 . 9 4 4 x 1 0 6 m m 3

S t = - 1 .47 7

S b = + 5 . 3 4 7

S e c t i o n w e i g h t = 2 4 x 1 6 2 . 8 / 1 0 0 0 = 3 .9 1 k N / m 2

M o m e n t s :

M o = 3 . 9 1 x 1 0 2 / 8 = 4 8 . 8 k N . m / m

M s = 6 .0 0 x 102 /8 = 75 .0

Mo + M s = 123 .8

P r e s t r e s s ( E q u a t i o n 6 . 3 a )

s e l f - w e i g h t

o t h e r l o a d s

t o t a l

P fm in = r - 2 . 3 + 1 2 3 . 8 / 4 . 9 4 4 ] x 1 6 2 . 8 / 5 . 3 4 7 = 6 9 2 . 4 k N

C a l c u l a t i o n s , n o t s h o w n h e r e, s h o w t h a t t h e p r e st r e s si n g f o r ce o f 6 9 2 .4 k N p r o d u c e s

a n i n i t i a l c o m p r e s s i v e s t r e s s a t t h e b o t t o m w h i c h e x c e e d s t h e p e r m i s s i b l e v a l u e .

T h e r e f o r e , t h e e c o n o m i c a l s o l u t i o n is t o c h o o s e a l o w e r v a l u e o f p r e s t r e s s i n g f o r c e,

s u c h t h a t t h e i n i ti a l s t re s s is n o t e x c e e d e d , a n d t o p r o v i d e s o m e r o d r e i n f o r c e m e n t a t

t h e b o t t o m o f t h e r ib .

F r o m E q u a t i o n ( 6 .2 b ),

( P i / A r M o / Z b = pr

P~ = (Pci +

M o / Z b ) A J S b

= ( 1 2 .5 + 4 8 . 8 / 4 . 9 4 4 ) x 1 6 2 . 8 / 5 . 3 4 7 = 6 8 1 . 1 k N

s a y P i = 6 5 0 k N

P f = 0 . 8 5 x e i = 5 53 k N

A s s u m i n g 1 6 0 k N f in a l f o rc e p e r 1 5 . 7 m m s t r a n d , n u m b e r o f s t r a n d s p e r

r i b = 5 5 3 / 1 6 0 = 3 . 5 ~ 4

S t r e s s e s :

P f / A c

= 5 5 3 / 1 6 2 .8 = 3 . 40 N / m m 2

O'pt

= S t /P f /Ac

= - 5 .02

apb e =

SbPf /Ar

= 18 .18

P.dAr

= 6 5 0 / 1 6 2 .8 = 3 .9 9 N / m m 2

O'pt i " -

StPi/ae

= - - 5 . 9 0

O'pb i = Sbei/ac = 2 1 . 3 3

F i n a l s t r e s se s ( E q u a t i o n s 6 . 2c a n d 6 . 2 d )

o'tf

- - - -

5 . 0 2 + 1 2 3 . 8 / 8 . 6 7 6 = + 9 . 2 5 < p cf

a b f = + 1 8 . 1 8 - - 1 2 3 . 8 / 4 . 9 4 4 = - - 6 . 8 6 < Ptf

I n i t i a l s t re s s e s ( E q u a t i o n s 6 . 2 a a n d 6 . 2 b )

o ti = - 5 . 9 0 + 4 8 . 8 / 8 . 6 7 5 = - 0 . 4 0 > P ti

a bi = + 2 1 . 3 3 - 4 8 . 8 / 4 . 9 4 4 = + 1 1 . 4 6 < p r

O K

A , t n e e d e d

O K

O K

T e n s i o n r e in f o r c em e n t "

h t - - h e i g h t o f t e n s i o n z o n e f r o m r i b b o t t o m

= 2 7 0 x 6 . 8 6 / ( 6 . 8 6 + 9 . 2 5 )

= l l 5 m m

b t = r i b w i d t h a t h e i g h t h t

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FLEXURE IN THE SERVICEABILITY STATE 153

= 3 0 0 + 6 0 x 1 1 5 / ( 2 7 0 - 1 10 ) = 3 43 m m

T = T o t a l t e n s i o n

= [115 x (300 + 34 3) /2 ] x 6 .86 / (2 x I00 0)= 126 .8 kN

A st = A r e a o f b o n d e d r e i n f o r c e m e n t p e r r i b

= 126 .8 x 1000/200 = 634 m m 2

= 4 x 16 m m dia . (804 m m 2)

T h e a m o u n t o f b o n d e d s t ee l ( 80 4 m m 2) is 2 6 % m o r e t h a n t h e r e q u i r e d 6 3 4 m m 2 a n d

t h e n u m b e r o f s t r a n d s p r o v i d e d ( 4 N o . ) i s 1 4 % m o r e t h a n t h e r e q u i r e d 3 .5 . A n

a d j u s t m e n t t o t h e s e c t i o n w o u l d b e r e q u i r e d t o a r r i v e a t a m o r e e f f i c i e n t s o l u t i o n .

A m o r e e c o n o m i c a l s o l u t i o n m a y b e o b t a i n e d i f t h e r i b s p a c i n g a n d w i d t h a r e

i n c r e a s e d b y 1 0 % . T h e s e c t i o n p r o p e r t i e s a n d t h e s e lf - w ei g h t p e r u n i t w i d t h r e m a i n

u n c h a n g e d a n d , t h e r e f o r e , al l o f t h e a b o v e c a l c u l a t i o n s r e m a i n v a l i d ; t h o u g h a

r e v i s io n to i n d i c a t e t h e c h a n g e i s re qu i r e d . T h e r e a d e r i s i n v i t e d t o t r y su c h a c h a n g e .

Exam p le 6.3

D e s i g n t h e p r e s t r e s s i n g fo r a r i b b e d p o s t - t e n s i o n e d f l o o r, c o n t i n u o u s o v e r t h r e e s p a n s

o f 1 0, 8 a n d 1 1 m l e n g th s , c a r ry in g im p o s e d d e a d a n d l i v e l o a d s o f 2 . 0 a n d 4 . 0 k N /m 2

re sp e c t iv e ly . Th e s e c t io n , sp a n s a n d lo a d s a r e sh o w n in F ig u re 6 . 8 ; a s su m e a n in i t i a l /

f i n a l p r e s t r e s s r a t i o o f 1 . 1 5 . Th e p e rm is s ib l e s t r e s s e s a r e :

Pti ---- -- 1.8 N / m m 2

Pr = + 12.5

P c f = d - 1 3 . 2

N / m m 2

P t f - - - - 2 . 3

S o l u t i o n

T h e s o l u t i o n p r e s e n t e d i n t h is e x a m p l e is b a s e d o n t h e u s e o f a p p r o x i m a t e v a l u e s o f

e q u i v a l e n t l o a d s . T h e m e t h o d i s s u i t a b l e f o r a p r e l i m i n a r y d e s i g n ; t h e f i n a l d e s i g n

s h o u l d b e b a s e d o n m o r e a c c u r a t e c a l c u l a t i o n s , w h e t h e r c a r r i e d o u t m a n u a l l y o r

u s i n g a c o m p u t e r p r o g r a m .

Th e t e n d o n p ro f i l e , c h o se n a f t e r a p r e v io u s t r i a l , i s a l so sh o w n in F ig u re 6 . 8 . Th e

d i a g r a m i n d i c a t e s t h e t r u e s a g in e a c h s p a n f o r th e p a r a b o l a ; h o w e v e r , t h e v a lu e t a k e n

i n t h e f o ll o w i n g c a l c u l a t i o n i s a n a p p r o x i m a t i o n , a s s u m i n g t h a t t h e l o w e s t p o i n t o f

t h e p a r a b o l a i s a t m i d s p a n , w h i c h i s n o t s t r i c t l y c o r r e c t b u t i s s i m p l e r .

Th e f in a l p r e s t r e s s in g fo r c e i s a s su m e d a c o n s t a n t 6 5 0 k N in a ll sp a n s . Th i s im p l i e s

a l t e r n a t e t e n d o n s b e i n g s t r e s s e d f r o m o p p o s i t e e n d s . T h e t e n d o n c e n t r o i d i n t h e

c e n t r e s p a n is a b o v e t h e s e c t i o n c e n t r o i d ; o b v i o u s l y , t h e t e n d o n f o rc e is t o o h i g h f o r

th i s sp a n .

S e c t i o n p r o p e r t i e s "

Ac = 179.6 x 103 m m 2

Yt - '- 106.5 m m

Yb = 163.5 m m

I c = 1011.5 x 106 m m 4

Z t = 9 .497 x 106 m m 3

Z b = 6 .186 x 106 m m 3

S e l f -w e ig h t = 2 4 x 0 . 1 7 9 6 = 4 .3 k N /m

E q u i v a l e n t l o a d s a n d t h e i r f i x e d - e n d m o m e n t s :

S p a n 1 - 2

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154

POST-TENSIONEDCONCRETE FLOORS

1

Live load 4.0 kN/m2

I - Dead load 2.0 + self weight

, o . o ; ~ 8 . o , i , , . o

2 3

' t 8 0 ,

d b

4

~

Sag

190 ,

t40'

2 3

Loading and tendon profile

1000

, ~ o , Z . / [ ' ~ o

4 o , 1 4 o o

,~

4 7 0

Section

~

64

t31.0 7 kN ~45.695 kN ~52.325 kN

62.14 kN 29 .25 kN 75.40 kN

Equivalent loads

t

7.70 kN

$

-0.408 kN -4.08 kNm ~ -4.625 kN

3 . 0 9 0 5 k N + 1 7 . 3 8 k N m 1 . 5 8 k N

S e c o n d a r y f o r ce s a n d m o m e n t s

Figure 6.8

Example 6.3

s ~ ( 2 3 5 + 1 6 4 ) / 2 - 8 0 = 1 1 9 . 5 m m

w e = - 8 x 6 5 0 • 0 . 1 1 9 5 / 1 0 2 = - 6 . 2 1 k N / m

FEM = 6 . 2 1 x 1 0 2/ /8 = 7 7 . 6 3 k N m

S p a n 2 - 3

s = 2 3 5 - 1 9 0

w e = - 8 x 6 5 0 • 0 . 0 4 5 / 8 2

FEM = 3 . 6 6 x 8 2 / / 1 2

= 4 5 . 0 m m

= - 3 . 6 6 k N / m

= 1 9 . 50 k N m

S p a n 3 - 4

s ~ ( 2 3 5 + 1 6 4 ) / 2 - 4 0 = 1 5 9 . 5 m m

w e = - 8 x 6 5 0 x 0 . 1 5 9 5 / 1 1 2 = 6 . 8 5 k N / m

FEM = 6 . 8 5 x 1 1 2/ /8 = 1 0 3 . 6 8 k N m

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FLEXURE IN THE SERVICEABILITY STATE 155

A n a l y s i s r e s u l t s "

M o m e n t s r e s u l ti n g f r o m t h e a n a l y s i s f o r d e a d , e q u i v a l e n t a n d t h r e e a l t e r n a t iv e l iv e

l o a d s a r e g i v e n i n t h e t a b l e b e l o w .

Su pport 1 2 3 4

M o m e n t s k N m

1 E q . l o a d 0 + 4 2 . 0 7 + 6 3 . 5 3 0

2 S e l f - w e i g h t 0 - 3 5 . 21 - 4 4 . 7 2 0

3 A p p l i e d d e a d 0 - 1 6 .3 8 - 2 0 . 8 0 0

4 L i v e, 1 - 4 0 - 3 2 . 7 6 - 4 1 . 6 0 0

5 L i v e , 1 - 2 , 3 - 4 0 - 2 1 . 1 1 - 3 0 . 6 2 0

6 L i v e , 2 - 3 0 - 1 1 . 7 5 - 1 0 . 9 8 0

F i n a l f l e x u r a l s t r e s s e s "

S p a n 1 - 2

L i v e l o a d C a s e 5 is c r i t i c a l

M 1 = 0

M 2 = + 4 2 . 0 7 - 3 5 .2 1 - 1 6 .3 8 - 2 1 .1 1 = - 3 0 . 6 3 k N m

M s = S p a n m o m e n t

( M 1 + M 2 ) / 2 +

(Ew)L2 /8

- 3 0 . 6 3 / 2 + ( 4 . 3 + 2 . 0 + 4 . 0 - 6 . 2 1 ) x 1 0 2 /8 = 3 5 . 7 6 k N m

a te = 6 5 0 / 1 7 9 . 6 + 3 5 . 7 6 / 9 . 4 9 7 = + 7 . 3 8 N / m m 2

O 'bf = 6 5 0 / 1 7 9 . 6 - - 3 5 . 7 6 / 6 . 1 8 6 = - - 2 . 1 6

O K

O K

S u p p o r t 2

L i v e l o a d C a s e 4 is c r i t i c a l

M 2 = + 4 2 . 0 7 - 3 5 .2 1 - 1 6 . 3 8 - 3 2 . 76

O ' t f = 6 5 0 / 1 7 9. 6 - 4 2 . 2 8 /9 . 4 9 7 = - 0 . 8 3 N / m m 2

a be = 6 5 0 / 1 7 9 . 6 + 4 2 . 2 8 / 6 . 1 8 6 = + 10 . 45

= - 4 2 . 2 8 k N m

O K

O K

T h e s e c t io n w o u l d n o r m a l l y b e s o l id o v e r t h e s u p p o r t , s o t h e s e c t io n p r o p e r t i e s f o r t h e

s o l id s e c t i o n w o u l d b e m o r e r e a l is t ic t h a n t h o s e o f t h e T - s e c t i o n u s e d .

T h e f i n a l s t r e s s e s a t o t h e r c r i t i c a l p o i n t s c a n b e c h e c k e d i n t h e s a m e m a n n e r .

I n i t i a l f l e x u r a l s t r e s s e s :

I n i t i a l p r e s t r e s s i n g f o r c e = 1 .1 5 x 6 5 0 = 7 4 7 . 5 k N

A v e r a g e s t r e ss = 7 4 7 .5 / 1 7 9. 6 = + 4 . 1 6 N / m m 2

S p a n 1 - 2

MI=0

M 2 = + 1 . 1 5 x 4 2 . 0 7 - 3 5 . 2 1 = + 1 3 . 1 7 k N m

M s = s p a n m o m e n t

1 3 . 1 7/ 2 + ( 4 . 3 - 1 .1 5 x 6 . 2 1) • 1 0 2 /8 = - 2 8 . 9 3 k N m

O 'tf--- + 4 . 1 6 - 2 8 . 9 3 / 9 . 4 9 7

a be = + 4 . 1 6 + 2 8 . 9 3 / 6 . 1 8 6

= + 1 .1 1 N / m m 2

= + 8 . 8 4

O K

O K

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156 POST-TENSIONED CONCRETE FLOORS

I n i t i a l s t r e s s e s a t o t h e r c r i t i c a l p o i n t s c a n b e c h e c k e d i n t h e s a m e m a n n e r .

S e c o n d a r y m o m e n t s :

M sx = M x -

P e

F o r M ~ s e e C a s e 1 o f A n a l y s i s R e s u l t s

A t 2 , M s , = + 4 2 . 0 7 - 6 5 0 x (2 3 5 - 1 6 4 ) /1 0 0 0 = - 4 . 0 8 k N m

A t 3 , M a , = + 6 3 . 5 3 - 6 5 0 x (2 3 5 - 1 6 4 ) / 1 0 0 0 = + 1 7 .3 8

Example 6.4

D e s i g n t h e p r e s t r e s s in g f o r a r ib b e d p o s t - t e n s i o n e d f l oo r , c o n t i n u o u s o v e r t h r e e s p a n s

o f 1 0, 8 a n d 11 m l e n g t h , c a r r y i n g i m p o s e d d e a d a n d l iv e l o a d s o f 2 .0 a n d 4 . 0 k N / m 2

r e s p e c t iv e l y . T h e f l o o r s e c t i o n i s s h o w n i n F i g u r e 6 . 9 ; a s s u m e a n i n it ia l / f in a l p r e s t r e s s

ra t i o o f 1 .15 .

T h e p r o b l e m i s t h e s a m e a s t h a t i n E x e r c i s e 6 . 3 , e x c e p t t h a t t h i s t i m e t h e

c a l c u la t io n s a r e t o b e b a s e d o n t h e a c t u al t e n d o n p r o f il e s r a th e r t h a n t h e a p p r o x i m a t i o n s ,

a n d t h e t e n d o n s a r e t o b e c u r ta i l e d t o s a v e o n t h e n u m b e r o f s t r a n d s .

Solution

F i g u r e 6 . 9 s h o w s t h e p r o p o s e d t e n d o n p r o f i l e a n d t h e f i n a l p r e s t r e s s i n g f o r c e s i n t h e

t h r e e s p a n s . T h e f u ll p r e s t r e s s i n g f o r c e o f t h e u n c u r t a i l e d t e n d o n s i s a v a i l a b l e a t e a c h

o f t h e p e n u l t i m a t e s u p p o r t s . T h e i n f l u e n c e o f t h e s h o r t l e n g t h s o f t h e c u r t a i l e d

t e n d o n s i n th e m i d d l e s p a n is n o r m a l l y i g n o r e d i n t h e c a l c u l a ti o n o f e q u i v a l e n t l o a d

a n d s e c o n d a r y f o r c es . T h e c o n t r a f l e x u r e p o i n t s o f th e p r o f il e a r e p l a c e d a t 0 . 1 L i n

e a c h s p a n . T h e p r o f i le a n d t h e p r e s t r e s s i n g f o r c e s a r e th e r e s u l t o f a n i n i t ia l a t t e m p t ,

n o t s h o w n h e r e .

S e c t i o n p r o p e r t i e s "

Ac = 179 .6

x 10 3 mm 2

Yt = 106 .5 m m

Yb

=

1 6 3. 5 m m

I c = 1011.5 x 10 6 m m 4

Z t - - 9 .49 7 x 10 6 m m 3

Z b = 6 .186 x 10 6 m m 3

S e l f -w e i g h t = 2 4 x 0 . 1 7 9 6 = 4 .3 k N / m

E q u i v a l e n t l o a d s : [ S e e T a b l e 5 .1 a n d E q u a t i o n s ( 5 .1 1 ) a n d ( 5 .1 9 ) ] .

S p a n 1 - 2 P f = 5 60 k N

T h e t e n d o n i s h i g h e r a t t h e f i g h t - h a n d s i d e ; t h e r e f o r e , w o r k f r o m s u p p o r t 2 .

a = 0 . 1 • 1 0 = l . 0 m

Y3

=

1 6 5 - 4 0 = 1 2 5 m m

Yl = 2 35 - 40 = 195 Y3/Y l= 0 .641

I n t e r p o l a t i n g f r o m T a b l e 5 . 1 ,

b / L

= 0 . 5 4 9

b = 5 .4 9 m m e a s u r e d f r o m s u p p o r t 2

w e = - - 2 P f y l / ( b 2 - a b )

= - 2 x 5 60 x 0 . 1 9 5 / ( 5 . 4 9 2 - 1 .0 0 x 5 . 4 9 ) = - 8 . 8 6 k N / m

W 1 2 - - s h e a r a t s u p p o r t 1 =

w e . ( L

- b - a ) = 3 1 . 1 0 k N

W 2 1 - s h e a r a t s u p p o r t 2 = w e . ( b - a ) = 3 9 . 7 8 k N

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FLEXURE IN THE SERVICEABILITY STATE

157

Live load 4.0 kN/m 2

I !-D ea d load 2.0 + self weig ht

i

' 1 t

0.0- 8.0 11.0

- r : ; .

I 2 3

Loading

40 235 40 235 40

4 .5 4 j_ 5 .46 _ 4 .0 L 4 .0 6 .0 _=_ 5 .0

_

r

' . . . . . ' . . . . . '

t

1 2 3 4

Tendon prof i le

1.0 8.0 1.0 0.8 6.4 0.8 1.1 8.8 1.1

I_ _1 _L_l_ .J ,L I_ _1

'-

-1T I-

- IT

I- - I

, I l i t

t , t

{l::} {l:}

1 co 2 ~.

, J '

f

i =

o O

3 Qo 4

Equivalent load

1000

I~ ;.[

o ~ T - - + - ~

1 1 § / 2 2 7 0

' 4 0 ' 4 6 0 ~1

4 I!-~

47o

65

F i g u r e 6 . 9 Example 6 .4

W 1 2 a n d W 2 1 a r e u n i f o r m l y d i s t r i b u t e d o v e r l e n g t h a = 1 . 0 m .

S p a n 2 - 3 P f = 1 4 0 k N

a = 0 . 1 x 8 = 0 . 8 m , a n d b y s y m m e t r y , b = 4 . 0 m

t o t a l s a g s = ( 2 3 5 - 4 0 ) / 1 0 0 0 = 0 . 1 9 5 m

w e = - 2 x 1 40 x 0 . 1 9 5 / ( 4 . 0 0 2 - 0 . 8 0 x 4 . 0 0 ) = - 4 . 2 7 k N / m

I4 /2 3 = I 4 /3 2 = 4 . 2 7 x ( 4 - 0 . 8 ) = 1 3 . 6 5 k N o v e r 0 . 8 m

S p a n 3 - 4 P f = 68 0 k N

a = 0 . 1 x l l = l . l m

y l = 2 3 5 - 4 0 = 1 9 5 m m

Y3 = 165 - 40 = 1 25

Y 3 / Y l

= 0 . 6 4 1

I n t e r p o l a t i n g f r o m T a b l e 4 . 1 ,

b / L

= 0 . 5 4 9

b = 6 .0 4 m m e a s u r e d f r o m s u p p o r t 3

W e = - 2 P f Y 3 / ( b 2 - - a b )

= - 2 x 6 8 0 x 0 . 1 9 5 / (6 . 0 4 2 - 1 .1 0 x 6 . 0 4 ) = - 8 . 8 9 k N / m

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158 POST-TENSIONEDCONCRETE FLOORS

W34 = shear at support 3 = w s ( b - a) = 43.91 kN ove r 1.1 m

W4a = shear at support 4 =

w ~ ( L - b -

a) = 34.30 kN

Analysis results:

Results of the analysis for dead, equival ent and three alt ernativ e live loads are given in

the table below.

S u p p o r t 1 2 3 4

Moments kNm

1 Eq. load

2 Self-weight

3 Applied dead

4 Live 1-4

5 Live 1-2, 3-4

6 Live 2-3

Shear forces kN

1 Eq. load

2 Self-weight

3 Applied dead

4 Live 1-4

5 Live 1-2, 3-4

6 Live 2-3

0 + 42.05 + 56.88 0

0 -35.2 1 -44. 72 0

0 - 16.38 - 20.80 0

0 -32 .7 6 -41 .6 0 0

0 -21.11 -30. 62 0

0 -11. 75 -10. 98 0

+ 0. 30 - 0.30

+ 17.98 + 25.02

+ 8.37 + 11.63

+ 16.72 + 23.27

+ 17.89 + 22.11

- 1 . 1 8 +

1.18

+ 1. 85 - 1.85

+ 16.01 + 18.39

+ 7.45 + 8.55

+ 14.89 + 17.11

- 1 . 1 9 +

1.19

+ 15.90 + 16.10

-

0.85 + 0.85

+ 27.72 + 19.58

+1 2. 89 + 9.11

+ 25.78 + 18.22

+ 24.78 + 19.22

+ 1. 00 - 1.00

Final flexural stresses"

Span 1-2:

Live load Case 5 is critical

V12 - " +0 .3 0 + 17.98 + 8. 37 + 17.89 = +44 .54 kN

V21 = - 0 . 3 0 + 25.02 + 11.63 + 22.11 = + 58.46 kN

w = net spa n U D L = + 6.30 - 8.86 + 4.0 = + 1.44 kN /m

Che ck to tal lo ad = (6.3 + 4.0) x 10.0 = 103.00 kN

Reactions = 44.54 + 58.46 = 103.00

X = Zero shear distance from 1

= ( V 1 2 - W 1 2 + a w e ) / w

= ( 44 .5 4- 31 .1 0- 1.0 x 8.860)/1.44 = 3.18 m

Msl = Moment in span 1

- - V 1 2 X - -

(W12 -

a W e ) . ( X - a / 2 ) - w X 2 / 2 + M t

= 44.54 x 3 . 1 8 - (31.1 + 1 .0 x 8.860) x 2 . 68 - 1.44 x 3.182/2

= 27.26 kN m

Prestress = 560 kN

trtf = 560/179.6 + 27.26 /9.497 = + 6.00 N/ mm 2

0"bf - - 560/179.6 - 27.26 /6.186 = - 1.29

Support 2:

At the support the section will be solid

A c = 270 x 10 -a mm 2 Z b = Z t = 12.15 x 10 -6 mm 3

O K

O K

O K

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FL E X U R E I N TH E S E R V I C E A B I L ITY S TA TE 159

M 2 = + 4 2 . 0 5 - 5 1 . 5 9 - 3 2.7 6 = - 4 2 . 3 0 k N m

tr tf = 5 6 0 / 2 7 0 - 4 2 . 3 8 / 1 2 . 1 5 = - 1 . 41 N / m m 2

t rb f = 560 /27 0 + 42 .38 /12 .15 = + 5 .56

O K

O K

F i n a l s t r e s s es in o t h e r s p a n s a n d a t s u p p o r t 3 , a n d i n i ti a l s tr e s s e s , c a n b e

c a l c u l a t e d i n a s i m i l a r m a n n e r .

T e n d o n r a d i u s o v e r s u p p o r t 2 "

F r o m E q u a t i o n 4 .1 8 , A 1 =

- y l / ab

= - 0 . 1 9 5 / ( 1 x 5 .4 9 ) = 0 . 0 3 5 5 / m

R a d i u s =

1/2A

= 1 4.1 m O K

I n t h is e x a m p l e , t h e n e g a t i v e m o m e n t s t r e ss is c h e c k e d a t t h e s u p p o r t c e n t r e l i n e ; t h e

c h e c k m a y b e m a d e a t t h e s u p p o r t f a ce w h e r e t h e m o m e n t w o u l d b e s o m e w h a t s m a l le r .

T h e f o l l o w i n g f u r t h e r c a l c u l a t i o n s m u s t b e c a r r i e d o u t t o c o m p l e t e t h e d e s i g n .

9 C h e c k o n d e f l e c ti o n s .

9 P r e s t r e s s l o s s e s a n d , i f n e c e s s a r y , c a l c u l a t i o n o f f l e x u ra l s t r e ss e s c o r r e s p o n d i n g t o

t h e n e w p r e s t r e s s i n g f o r c e s .

9 U l t i m a t e f l e x u ra l a n d s h e a r s t r e n g t h c a l c u l a t io n s . B o n d e d s te e l r e q u i r e m e n t a n d

t h e p o i n t w h e r e t h e s e c t i o n i s t o c h a n g e f r o m r i b b e d t o s o l i d .

9 A n c h o r a g e z o n e r e i n f o r c e m e n t

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7 P R E S T R E S S L O S S E S

Som e of the app l ied pres t ress ing force is los t during and af ter the s t ress ing

opera t ion , a nd , therefo re , the mag ni tude o f comp ress ive fo rce induce d in the

concre te i s som ewh at l es s than the fo rce app l i ed by the j ack ing equ ipm ent . In the

pas t , t he p rac t i ce was to as sum e a cer t a in a rb i t ra ry va lue fo r the overa l l loss. Th i s

is now cons idered to be too inac cura te a nd such assumed losses a re now accep ted

to have been under -es t imated .

Fo r in i t ia l des ign and app rox im ate quan t i t i es , an average f inal fo rce o f 0 .60fp,

to 0 .65fp, i s of ten assumed in the current pract ice. Assuming that the s t rands are

ini tia l ly s t ressed to 0 .75fp, , these values represen t a long-term loss of ab ou t 2 0%

to 15% respect ively . For the f inal des ign the losses should be calculated.

In th i s cha p ter the causes o f the loss in p res t ress a re cons idered in rough ly the

order in which they occur .

7 . 1 G e n e r a l

There a re severa l fac to rs which con t r ibu te to the loss o f p res tress . So me of these

are imme dia te , as they occur du r ing the s t ress ing opera t ion , whi le the o thers a re

t ime re la t ed and t ake p lace over a long per iod . Therefo re , the losses can

conven ien t ly be cons idered in two g roups"

a . Im m edia te , which occur dur ing the s t res s ing opera t ion . These a re o f in te res t

in assess ing the pres t ress ing force at the initial stage, i .e . immediately af ter

s t res s ing , befo re any o f the impo sed load i s app l i ed . Th i s g ro up com pr i ses :

9 f r ic t ion loss due to t end on curv a tu re and wob ble

9 e l ast ic shor ten ing o f concre te u nde r the induce d com press ion

9 an d an ch o rag e d raw - i n .

b . Lon g t e rm, which occur a t a g radu a l ly decreas ing ra te over the li fe o f the

member . These a re cons idered on ly a t the f inal s tage w h en i mp o s ed l o ad s

have been app l i ed . Th i s g roup cons i s t s o f :

9 sh r inkage o f concre te

9 creep of concrete

9 and relaxat ion of s teel .

In order to calculate the losses , values must be assumed for the in i t ia l s t resses at

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PRESTRESS LOSSES 161

c r it ic a l s e ct ions a nd the nu m be r o f s t r a nds m us t be gue s se d . T he in i ti a l

a ssu m pt io ns w i ll , a lm ost ce r ta in ly , need to be modif ied as a r e su l t o f ca lcu la t ion

of losses , and a r ev ised level o f s t re ss and a d i f fe ren t nu m be r of s t r ands wi l l ensue .

There fore , the loss ca lcu la t ions ca r r ied ou t have a lso to be rev ised . The losses

inc lude the e f fect o f the cab le prof ile whose a dequ acy , o r inad equa cy , wi l l no t be

judg ed unt i l a f te r mo s t o f the s t re ss ca lcu la t ions have been ca r r ied ou t . A fur the r

c om pl i c a t ion i s t ha t t he l o sse s due to t he va r ious c a use s a r e i n t e r de pe nde n t . T he

ca lcu la t io n process , the re fore , i s i t e ra t ive and cum bers om e, pa r t icu la r ly i f an

accura te force prof ile is to be ob ta ine d for the length of the m em ber .

Ca lcu la t ion of s t re sses r equi res f a ir ly accura te va lues of the p res t re ss ing force

a t t he c ri ti c al s e c tions in a m e mb e r . A n a c c u r a c y o f a bo u t 5% is c ons ide re d

a de q u a te ; e nde a vou r ing f o r a g r e a te r a c c u r a c y in c a l c u la t i ng los ses i s un l ike ly t o

resu l t in a grea te r degree of accuracy in the s t r e sses , because o f the u nce r ta in t ie s

a nd unp r e d ic t a b il i ti e s o f ma te r i a l c ha r a ct e r is t i cs a nd wo r km a nsh ip . F o r e xa m ple ,

t he mo du lu s o f e l a st ic i ty o f c onc r e te m a y va r y f r om the a s sum e d va lue , a c c u r a c y

o f i n s t a l l a t ion o f t e ndo ns i s unkno wn , a nd the a l lowa b le d im e ns iona l t o l e r a nc e s

permi t a va r ia t ion in sec t ion proper t ie s .

7.1. 1 Sim pl i fy ing assum pt ions

T he f o l lowing s imp l i fy ing a s sump t ions a r e m a de in t he c a l c u l a t i on o f l o sse s.

9 The angle be tween the tendon prof i le and the member ax is i s sma l l so tha t s in

0 ~ t a n 0 ~

0.

9 T he inc re a se i n t e ndon l e ng th be c a use o f i ts c u r va tu r e c a n be igno r e d a nd the

t e ndon l e ng th in a spa n e q ua l s t he spa n l e ng th .

F o r a n e x t e r na l spa n the t e ndon l e ng th is m e a su r e d f r om the a nc ho r a ge f a ce ,

a nd no t t he suppor t c e n t r e l i ne .

9 The losses va ry l inea r ly a lon g the ten do n leng th .

I n a g ive n spa n , t he f o r c e va r i e s f r om one suppor t t o midspa n to t he o the r

sup po r t , bu t i t i s cons id e red to be suf f ic ien t ly accura te to a ssess the ma gn i tud e of

the ne t p res t re ss ing force a t m ids pan and use these va lues for the whole of tha t

span . I f the losses a re judg ed to be c r i t ical the n i t i s necessa ry to ca lcu la te the

lo sse s a t t he su ppo r t s o f a c on t in uou s m e m be r a s wel l. T h is m a y be ne c e s sa ry i f

the ca lcu la ted s t r e sses a re ve ry c lose to the a l low able va lues , o r when te nd ons a re

c u r t a i l e d ne a r a suppor t .

7.1.2 High losses

T he lo s s i n p r e s t r e s s ing fo r ce i nc re a ses w i th t he num be r o f t e ndons . I f t he t o t a l

loss i s unac ceptab le then a f ter s t re ss ing a ll the ten don s , each o f them can be

re - s t r e ssed . Al te rna t ive ly , each tendon can be s t r e ssed to a d i f fe ren t load- - the

f ir st t e ndon to t he h ighe s t l oa d a n d the l a st t o t he l ow e s t - - so tha t a f te r s tr e ss ing

e a c h o f t he m wi ll ha ve a pp r ox im a te ly e q ua l f o rc e .

E i the r of these pro cedu res w i ll subs tan t ia l ly r educe the loss ; the form er wi ll be

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162 POST-TENSIONEDCONCRETE FLOORS

the m ore efficient. Ho we ver , th ey wi ll a lso ad d to the w ork lo ad of the s i te s taff

and pu t an u ndue ex t ra respons ib i li ty on them fo r wh at i s rea l ly a des ign p rob lem .

There fore, i t m ay be preferable to acc ept the high er losses if poss ible , even at the

expense o f a few add i t iona l t endons .

7.1.3 Long tendons

Accu ra te as sessment o f the loss o f p rest ress ing fo rce is se ldom car r i ed ou t because

the values of E c , /~, K , s hr ink age s t rain , creep coeff icient, e tc . , are al l based on

previous experience, and the values actual ly effect ive for the part icular tendon

being s t ressed may wel l be qui te d i fferent . Another d i ff icul ty in the accurate

de te rm ina t ion o f the loss st ems f rom the in te rdep endenc e o f the var ious fac to rs

contr ibut ing to the loss . For example, the elas t ic shortening loss at a sect ion

depends on the s t ress at the sect ion, which should be the value af ter e las t ic loss ;

creep loss should take in to account the elas t ic loss and the creep loss i t sel f .

In long t endons , ru nn ing over severa l con t inuous spans , the loss a t the fa r end

is over -es t ima ted i f ca lcu la t ions a re based on fo rces a t the j ack ing po in t . In such

tendons , be t t e r ac curacy resu lt s i f the overa l l mem ber l eng th i s d iv ided in to

severa l sec t ions and each sec t ion i s cons idered in success ion . For immedia te

losses in the second sect ion, the jacking force is taken as the force af ter f r ic t ion

an d elas t ic losses at the end of the f i rst sect ion, and the force for long-te rm losses

in the second section is tak en as the force at the en d of the first section, after al l losses.

For a long t endon s t res sed f rom bo th ends , the maximum loss occurs near i t s

m id-leng th . I f the losses are calculated in sections , then the loss calcu lat ions m ust

a l so be car r i ed ou t f rom each end up to the mid- leng th .

W here the loss o f p res t ress i s unac cep tab ly h igh , the fo l lowing op t ions a re

avai lable .

9 Red uce fr ic t ion loss by m ak ing the te nd on profi le as f la t as poss ible ,

par t i cu la r ly in the in te r io r spans .

9 Prov ide a dd i t iona l shor t l eng ths o f t endon s in the fa r spans .

9 App ly a h igher in i ti a l j ack ing fo rce , and then reduce the jack in g fo rce to the

nomina l des ign va lue befo re lock ing the anchorage .

The p roced ure i s, how ever , unre l iab le and g ives on ly a m arg ina l ad van ta ge .

M o s t n a t i o n a l s t an d a rd s l imi t th e ma x i mu m j ack i n g fo rce to 8 0 % o f t h e

tendon s t reng th so tha t the p rocedure can on ly be used where the des ign

jacking force is lower than th is l imi t ing value.

9 Re-s tress the t endo ns a f t e r some of the sh r inkage and c reep losses have t a ken

place.

The p roce dure a l so reduces the e l as ti c shor ten ing loss , bu t requ i res access to

the anchorag es fo r a longer per iod , thereby s lowing dow n the cons t ruc t ion and

r i sk ing damage to the s t res sed t endons .

9 St ress from both ends .

Pres t ress ing fo rces in ou te r bays a re h igher than those ob ta inab le f rom

al ternate end s t ress ing.

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PRESTRESS LOSSES 163

9 S t ress a l te rna te tend on s f rom the tw o ends .

T h i s me thod r e su l t s i n a mor e un i f o r m p r e s t r e s s ing f o r c e ove r t he t e ndon

le ng th tha n bo th e nd s t re s s ing , be ca use a t e a ch e nd , a l t e r na t e t e nd ons ha ve the

m a x i m u m a n d t h e m i n i m u m f o r c e .

9 S t ress each bay as i t i s cas t , us ing in te rmedia te anchors .

T h i s p r oc e du r e a l l ows good c on t r o l ove r t he p r e s t r e s s ing f o r c e a nd

min imize s sh r inka ge c r a c k ing o f c onc r e te . H owe v e r , e x t r a ha r dw a r e is

requi red and each tendon requi res seve ra l s t r e ss ing v is i t s .

7 . 2 F r i c t i o n lo s s e s

A pr op o r t i on o f t he j a c k ing f o rc e is l o s t du r ing s t re s s ing th r oug h c on ta c t f r ic t ion

be twe e n the s t r a nd a nd i t s su r r ound ing duc t - - g r e a se d p l a s t i c e x t r u s ion f o r

u n b o n d e d t e n d o n s a n d p r e f o rm e d s h e a t h fo r b o n d e d t e n d o n s . T h e c o n t a c t m a y

be in t e n t iona l due to t e n don c u r va tu r e , a s whe n a t e ndon is d r a pe d in t he sha pe o f

a pa r a bo la , o r un in t e n t iona l , c a use d by min o r de v ia t ions o f t he t e ndon f r om it s

in tended prof i le . The loss due to the in ten t iona l tendon contac t i s r e fe r red to a s

c u r v a t u r e f r i c t i o n ,

a nd th a t due to t he un in t e n t iona l c on ta c t a s

w o b b l e

o r

paras i t i c

f r i c t i o n .

In the f ir s t case , the con tac t force , and hence the loss , be twe en the te nd on and

the duc t i s p r opo r t iona l t o t he a ng le t h r ough wh ic h the t e ndon tu r ns s t a r t i ng a t

the j ack ing end; where the prof i le cons is t s o f a se r ie s of curves the loss i s

p r o po r t ion a l t o t he sum o f a ll ang les . Loss a t a po in t i n the t e nd on l e ng th due to

un in t e n t iona l de v ia t ion i s s t a ti s ti c a lly shown to be p r op o r t io na l t o t he l e ng th o f

the tend on f rom the j ack in g end . In bo th cases , the loss inc reases f rom ze ro a t the

l ive a nc ho r a ge to a m a x i m um a t t he de a d a n c ho r a ge , o r a t t he m id - po in t o f t he

ten do n length i f i t i s s t r e ssed f rom bo th ends .

Som e loss a l so occu rs due to f r ic t ion in s t r e ss ing j acks and anch orag es ,

pa r t i c u l a r ly i n mu l t i s t r a nd t e ndons whe r e t he s t r a nds c ompr i s ing a t e ndon

spread ou t to a l low suf fic ien t space for the j ack to gr ip them . The loss f rom these

f a c to r s va r i e s f r om one sys t e m to a no the r a nd i s u sua l ly a c c oun te d f o r i n t he

c a l ib r a t ion o f t he e q u ipme n t .

The two losses, due to f r ic tion and wobb le , a re g roupe d in the fo l lowing equa t ion :

Px = Po e -0 ,o + Kx) (7.1)

w h e r e e = N a p e r i a n l o g a r i t h m b a s e

0 = A ng le o f c ha nge o f t e nd on d i r e c t ion in r a d i a ns

x = d i s t a nc e f r om a nc ho r a ge

N o te t ha t K ha s t he d ime ns ion 1 / l eng th a nd the r e fo r e , i ts va lue de pe nds o n the

uni t o f length be ing used .

The pe rcen tage loss of p res t re ss is g iven by the express ion 100(1 - e - (~0 + r~) ).

Fo r smal l va lues of ( /~0 + K x ) , say less th an 0.3, th e exp ressio n e-(U0 + K~)

approximates to (1 - - #0 - - K x ) , a nd , t he r e f o r e t he e xp r e s s ions f o r P xa nd lo s s

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164 POST-TENSIONED CONCRETE FLOORS

Table 7.1 Va lues of lu and K

BS 8110 A CI 318 Comm entary

D uct mater ial la K /m # K/ metre K / f t

R ig id g a l v ani zed 0 . 2 5 0 . 0 0 1 7 0 .1 5 - -0 .2 5 0 .0 0 0 5 -0 .0 0 2 0 0 .1 5 -0 .6 1

x 1 0 - 3

Greased p last ic 0 .12 0 . 0 02 5 0 .0 5 -0 .1 5 0 .0003-0 .002 0 0 .09-0 .61 x 10 -3

Non-rigid 0.0033

percentage can be s impl i f ied to :

P , = P o( 1 - # 0 -

K x )

Percentage loss = 100(#0 + K x )

(7.2)

Ta ble 7 .1 gives the values of # an d K as specif ied in BS 8110 and A CI 318.

Al th oug h the coefficient of f r ic t ion, # , is lowe r for a greased te ndo n in an ex t rud ed

plas t ic sheath , such a tendon is not as r ig id as a metal duct and, therefore, i t i s

l iable to have a larger dev iat ion of profi le . Reco gnizing th is , the w obb le factor , K ,

is h ighe r for the greased s t ra nd in p las t ic ext rus ion. The 0 .0033 value of K , g iven

for a non-r ig id duct i s meant for sheaths which are not suff icient ly r ig id to res is t

be ing d i sp laced dur ing concre t ing .

Drawn s t rand has a smoother su r face and , in g reased p las t i c ex t rus ion , i t s

coeff ic i en t o f f r ic t ion i s m uch lower than tha t fo r the norm al shaped s t rand . The

value o f # fo r d raw n s t rand can be as low as 0 .03 fo r th i s p roduc t .

Supp l i e rs o f pos t - t ens ion ing hard w are g ive va lues o f/z and K fo r the i r p rod uct s

which are general ly lower than those speci f ied in BS 8110--- typical ly , # = 0 .19

and K = 0 .0008 /m fo r r ig id meta l shea th , and /z = 0 .06 and K = 0 .0005 /m fo r

g reased s t rand . The supp l i e r 's va lues ma y be used i f know n. I f no t , i t ma y be

prudent to use the coeff icients speci f ied in the relevant s tandard.

7 .3 A n c h o r a g e d r a w - i n

In m os t o f the pos t - t ens ion ing sys tems p resen t ly ava i l ab le fo r use in f loors , s t ran d

is gr ipped by the conical wedges as they s l ide in to a tapered hole in the barrel or

the bear ing p la te . In the p rocess , the l eng th o f the s t ressed t endo n is reduced by

t h e am o u n t o f d r a w - i n needed to lock the ancho rage , which is usua l ly a min im um

of 6 m m (0 .25 in ) . Hard w are m anufa c tu rers p rov ide f igures fo r the d raw- in fo r

the i r sys tems bu t these do no t m ake any a l lowance fo r the s t rand s l ipp ing in the

wedges before the gr ip becom es effective. Th e p oss ibi l ity of s t ran d s l ippage is

m uch l ess wi th j acks w here the wedges a re d r iven hom e hydrau l i ca l ly . Ho we ver ,

an overa l l a l lowanc e o f 8 m m (0 .3 in) d raw- in is reasonab le a t the des ign s t age .

The s lip m ay be on ly a few mi l limet res bu t i t is unpred ic tab le . Be ing un kn ow n

an d variable , i t i s d i ff icul t to ca lculate a ccu rately the loss of pres t ress a t the de s ign

s tage. The no rm al p rac t i ce is to measu re the ac tua l to ta l d raw - in dur ing s t res s ing

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PRESTRESS LOSSES 165

and m ake a l lowance fo r any such s l ip a t tha t s t age by ad jus t ing the ex tens ions fo r

the t endon s s ti ll t o be s tres sed . I f the mea sured d raw- ins a re fou nd to be ou t s ide

the rang e specif ied by the des igner , then the tend on forces are calculated from the

draw- ins and the se rv iceab i li ty s t res ses a re checked . Som e of the t endon s m ay

have to be re-s t ressed i f the s t resses are foun d to be u nacc eptab le .

Loss o f p res tress due to anch orage d raw - in becom es increas ing ly s ign if ican t as

the t endon l eng th reduces ; in shor t t endons th i s loss may be very h igh . A shor t

l eng th o f t endo n needs a re l a tive ly smal l to t a l e longa t ion o f the s t rand fo r the

desi red pres t ress ing force an d a n 8 m m draw -in loss i s m ore s ignif icant in th is case

tha n i t would b e in the case of a long tend on wh ere the in i tia l e long at ion is h igher .

Fo r exam ple , a 5 m long t en don , w i th a j ack ing ex tens ion o f 30 mm , wo uld lose

26% of the fo rce i f the s lip was 8 mm ; a longer t e ndo n wou ld lose a

p ro p o r t i o n a t e l y s ma l le r am o u n t o f f or ce .

The h igh loss o f p res t ress due to d raw - in , coup led wi th the unce r t a in ty over the

am ou nt o f s t rand s lip, make s i t di ff icu lt to pos t - t ens ion shor t m em bers . A usefu l

p roce dure to fo llow in the case o f shor t t en dons is descr ibed in Ch ap te r 13.

W hen the j ack is pu l l ing the t endo n , f r i c tion be tween the t endo n and the duc t

opposes the jack in g fo rce, so tha t the t endo n fo rce reduces g radua l ly aw ay f rom

t h e j ack . D u r i n g t h e an ch o rag e l o cki n g o p e ra t i o n t h e t en d o n mo v es i n th e

o p p o s i t e d i r ec t i o n - - i n t o t h e d u c t - - an d f r i c t i o n n o w o p p o s es t h i s mo v emen t .

After locking, the net s t ress in the tendon, therefore, increases away from the

anch orag e fo r a d i s t ance and then i t reduces jus t as i t d id whe n jack ing . A t th i s

s tage, the loss i s confined to a def in i te length of the tendon. This i s shown

diagram m at ica l ly in F igure 7 .1 ; F igures (a ) and (b) show the fo rces imme dia te ly

before and af ter lock-in .

Ca lculat io n of the length Z , wi thin w hich th is loss takes place, i s necessary to

check i f any of the cr i t ical sect ions fall in th is zone, an d i f any does , th en the

tendon force is calculated at that point . In th is calculat ion the loss i s assumed to

be linear a long the l eng th o f the mem ber .

7 .3 .1 Un bon ded tendons

Unbonded t endons , no t be ing g rou ted , remain f ree to move in the i r g reased

p las t i c shea th ing , a l lowing re la t ive movement to occur be tween the s t rand and

the concre te . The mo vem ent m ay poss ib ly be in i t ia t ed by v ib ra t ion , t emp era tu re

changes , sh r inkage o r c reep . Such a move m ent t ends to sp read the d raw - in loss

beyo nd the in i ti a l d i st ance Z , and g radu a l ly to the whole l eng th o f the t endon .

Therefo re , the long- te rm pres t ress d i s t r ibu t ion in an unbonded t endon d i f fe rs

f rom tha t show n in F igure 7 .1 (b ). The d raw- in loss fo r unb ond ed t endon s can be

ca lcu la ted on the bas i s o f the s imple assum pt ion tha t the w hole t endo n l eng th is

affected.

The s l ip movement in fact s tar ts wi thin a few minutes af ter s t ress ing and the

tend on force ma y reac h a uniform level wi thin a few hou rs thereaf ter . I f the in i t ia l

loss in a ten do n is judg ed to be excess ive then i t m ay be re-s t ressed wi thin a few

hours of the f i rs t s t ress ing.

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166 POST-TENSIONEDCONCRETE FLOORS

Tendon

movement

Contact pressure

on tendon

T e n d o n f or ce

- - 1

(a) Jacking

n0on

movement ~ ~

L Friction

l

Tendon force

(b) After lock-in

Figure

7.1 Draw -in loss of prestress

Loss in force =

( 6 / L t ) A p E s

f o r u n b o n d e d t e n d o n

w h ere 6 = An ch o rag e d raw - i n

(7.3)

7 .3 .2 B o n d e d t e n d o n s

In the case o f bon ded t endons , the duc t i s g rou ted soon a f t e r s tres s ing and the

draw - in loss is , t herefo re , locked wi th in the d i s t ance Z. F igure 7 .2 (a) shows the

case o f a long bo nde d t endo n , where the l eng th Z a f fec ted by the d raw - in i s l es s

than the to ta l t en don l eng th Z t . L ine ABC represen t s the t endo n fo rce a t j ack ing ,

befo re the t endon i s locked in , and l ine CDE represen t s the fo rce a f t e r lock- in ,

when the loss due to d raw- in has t aken p lace . Po in t s B and D are a t the

mid- leng ths o f AC and CE respect ive ly . The loss o f p res t ressing fo rce a t any po in t

wi th in the l eng th Z i s rep resen ted by the ver t i ca l d i s t ance be tween l ines AB C and

C D E .

Let f l repre sent the s lope of l ine AB C, i t s uni ts be ing loss of force per uni t

l eng th . The va lue o f /3 can be de te rmined by ca lcu la t ing f r i c t ion loss a t a

co n v en i en t p o i n t aw ay ~o m t h e an ch o rag e , s u ch a s a t t h e n ex t s u p p o r t o r a t

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P R E S T R E S S L O S S E S 167

o ~ _

7

I.,

I -

Jack ing force before

A / lock- in

E \ 13

af ter lock- in

x

w ~

Figure 7 . 2 L o s s o f p r e s t r e s s i n a b o n d e d t e n d o n

m idspan , a nd then d iv id ing th is loss by the d i s tance . I t i s reason ab ly accura te to

assume the s lope o f l ine CD E as - f l , because the va lues o f # and K rem ain the

same b u t the d i rec t ion o f f r ic t ion has cha nged . Th i s i s no t s t ri c tly accura te

because the s lope o f l ine CD E shou ld co r respon d to the t endon fo rce a t C, which

i s lower tha n the o r ig ina l j ack ing fo rce a t A, bu t i t is a reasonab le as sum pt ion .

Fo r the va lue of f l, calculate the loss over the span length L. L et 0 repres ent the

ang u lar dev ia t ion o f the t endon over leng th L . Th en

PL = Forc e a t d i s t ance L

= Pje -O~o+

KL) ,~ P j( 1 - # 0 -

K L )

(7.4)

,8 = ( P j -

PL) /L ,~ P j (#O/L + K )

(7.5)

The loss of force betwee n A a nd C is f lZ. The loss of force at B, represe nted by the

l ine BD , is a lso f lZ. This is the avera ge loss over length Z an d the co rresp ond ing

change in t endon l eng th mus t equa l the d raw- in 6 . Therefo re ,

6 = ( f l Z /Ap E~ ) Z

and so

Z - ( (~ApEs/~)~

(7.6)

Kn ow ing P j , and f l and Z f rom Eq uat ion s (7 .4 ) and (7.6 ), the t end on fo rce P i and

the fo rce a t any d i s t ance x f rom the anchorage can now be ca lcu la ted .

P i = P j - 2 (P j - P z ) = P j - 2 fl Z

F o r x ~< Z , Px = Pi + fix (7.7)

Fo r x i> Z , P~ = P j -

f i x

The above d i scuss ion has as sumed tha t the bonded t endon i s longer than the

d i s t ance Z, so tha t the t endon l eng th near the dead end i s no t a f fec ted by the

draw - in loss . In a shor t t endo n th is m ay no t be the case and the w hole o f i ts leng th

wil l then suffer a loss . This case i s shown in Figure 7 .3 . Line AF represents the

dis t r ibu t ion of the force before lock-in , an d l ine G E af ter lock-in . L t i s the length

f rom the anch orag e A to the dead end a t F , po in t s M and N be ing a t the midd le o f

l eng ths AF and GE respec t ive ly .

The va lue o f f l can be ca lcu la ted f rom Eq uat io n (7.5) , subs t i tu ting L t fo r L . L ine

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168 P O S T - T E N S I O N E D C O N C R E T E F L O O R S

1 Jacking orce before

,,o I A / lock-in

P j - ~ M g

N -

I x

"after ock-in

T

l~ 1.~ _1

r "]

F i g u r e 7 . 3

D r a w - i n l o s s i n s h o r t b o n d e d t e n d o n s

M N represe nts the av erage loss of force ove r the tend on length Lt ; le t th is loss be

repre sente d b y PMN" Th en, the c hang e in ten do n len gth und er the inf luence o f PMN

must equa l the d raw- in 6 .

(PMN/ApEs)Lt = 6 , a n d so

PMN = r

P i = P j - / ~ L t - PMN

P , ,

= Pi + f ix

x is less than or equal to

L t.

(7.8)

7.4 E last ic shor ten ing

As a tendon is s t ressed, i t induces compress ion in the concrete , which causes a

s ligh t reduc t ion in the l eng th o f the mem ber . T he sh or ten ing in l eng th an d the

app l i ca t ion o f p res t ress be ing s imul taneous , ther e is no loss o f fo rce in the

part icular tendon being s t ressed and, af ter s t ress ing, the tens i le force in the

tendon i s equa l to the compress ion in concre te .

W he n the se cond tend on is s t ressed the n i t suffers no loss from i ts own s t ress ing

but the elas t ic shortening caused by i t s s t ress ing reduces the force in the f i rs t

tendon. Similar ly , when the th i rd tendon is s t ressed, the forces in the previously

s t ressed two t endons reduce .

Con s ider a num ber o f t endons in c lose p rox imi ty w i th each o ther , which a re

s t ressed sequent ial ly .

PL = Loss o f fo rce in a t endo n whe n a no th er t endo n is s t res sed

= pc(Es /Ec i )A1 (7.9)

w h e re

PL = Loss o f fo rce in a t end on when ano ther i s s t res sed

Pc = s t ress in concrete at te nd on level due to P x

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PRESTRESS LOSSES 169

A x = t h e a r ea o f o n e t en d o n

P ~ = f o r ce i n o n e t en d o n a f t e r l o ck - i n

A s ex p l a i n ed ab o v e , t h e r e i s n o l o s s w h en t h e f i r s t t en d o n i s s t r e s s ed . W h en t h e

seco nd t e nd on i s s t r essed , i t ca r r i es i t s fu ll fo rce P ~ bu t t he fo rce i n t he f i rs t t end on

i s r ed u ce d b y P L . T h e av e r ag e l o ss fo r t h e t w o t e n d o n s eq u a l s h a l f t h e t o t a l l o s s ,

i .e. 0 .5 P L.

W h en t h e t h i r d t en d o n i s s t r e s s ed t h en i t c a r r i e s it s f ul l f o rce , th e s ec o n d t en d o n

n o w l o se s P L an d t h e f i rs t t en d o n h a s n o w s u s t a i n ed a l o ss o f 2 P L. T h e t o t a l l o s s i n

t h e t h ree t en d o n s i s 3 P L . W i t h f o u r t en d o n s s t r e s s ed s eq u en t i a l l y t h e t o t a l l o ss o f

force i s 6P L.

F o r N t en d o n s t h e t o t a l l os s o f f o rce is 0 . 5 N ( N - 1 ) PL , an d t h e av e r ag e l o ss

per t end on i s 0 .5 (N - 1 )PL, wh ich i s equ a l t o ha l f t he l oss i n t he fi r s t t en do n d ue

t o t h e s tr e s s i n g o f t h e r em a i n i n g ( N - 1 ) t en d o n s .

U s i n g E q u a t i o n ( 7.9) , t h e l o s s f o r N t en d o n s , o f t o t a l a r ea A p , i s g i v en b y

PLN =

T o t a l l o s s i n N t en d o n s

= 0 . 5 ( E s / E c i ) [ (N - 1 ) p c ] . [ S A 1 ]

- 0 . 5 [ ( S - 1)pc] . (Es /Ec i )A p

(7.10)

W ha t e xac t ly i s pc? As s t a t ed abo ve , i t is t he s t r ess i n t he concre t e a t t he l eve l o f t he

t en d o n s . W h i l e th e s l ab i s s u p p o r t e d o n f o r mw o r k , t h e s t re s s i n d u c ed b y a t en d o n

i s a u n i f o r m co m p r es s i o n o n t h e co n c r e t e s ec t i o n an d P c = P 1 / A c 9 A s m o r e

t en don s a re s t r essed , t he s l ab l if ts of f t he fo rm w ork an d then Pc i s t he r esu l t o f t he

ax i a l f o rce , th e m o m en t p r o d u ce d b y t h e eccen t ri c i t y o f t h e t en d o n s an d t h e s elf-

w e i g h t m o m e n t , g i v e n b y

Pc = P ~ I A c + P l e 2 1 I , - M o e p / ( N I r

= P~ /Ar +

[ P i ep -

M o / N ] e p / I ~

(7.11)

E q u a t i o n ( 7.11 ) is n o t s t r ic t l y co r r ec t , b ecau s e o f t h e ch a n g e i n t h e v a l u e o f Pc a s

t h e m e m b e r b e c o m e s s e l f -s u p p o rt i n g. A l so , i n c o n t i n u o u s m e m b e r s t h e v a l u e o f

P c i s mo d i f i ed b y t h e r e d u n d an t f o rce s . H o w ev e r , t h e l o s s o f p r e s t r e s s d u e t o

e l a s ti c s h o r t en i n g i s o n l y o n e o f s ev e r a l o t h e r l o s s e s, n o n e o f w h i ch can b e

ca l cu l a t ed w i t h ex ac t a ccu r acy ; f u r t h e r i n accu r acy a r is e s b ecau s e o f t h e u n ce r t a i n t y

i n t h e v a l u e o f E c i. T h e r e f o r e , a n ex a c t c a l c u l a t i o n o f t h e e l a s t ic s h o r t en i n g l o ss i s

u n n ece s s a r y , an d E q u a t i o n s ( 7 .1 0 ) an d ( 7.11 ) a r e co n s i d e r ed v a l i d f o r u s e i n th e

d es i g n o f p o s t - t en s i o n e d f l o o rs u s in g b o n d ed t en d o n s .

Eq ua t io ns (7 .10 ) an d (7 .11 ) can be s impl i f i ed by assu m ing N ~ N - 1 , so t ha t

Pen = P i / A c + [ P i e p - M o ] ep /I ~

(7.12)

PLN -- 0 .5. Pc,"

( E f f E c i ) A p

(7.13)

w h e r e P c , = t h e s t r e s s i n t h e co n c r e t e a t t en d o n l ev e l d u e t o N t en d o n s an d

P i = t h e fo r ce i n N t e n d o n s a f t e r al l i m m ed i a t e l o ss e s , i n c l u d i n g t h a t d u e

t o e l a s t i c s h o r t en i n g .

T h e l o ss e s in t h e t e n d o n f or c e ar e n o r m a l l y c a l c u l a t e d a t m i d s p a n s a n d s u p p o r t s ,

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170

POST-TENSIONED CONCRETE FLOORS

where the s tresses a re h igh. Th i s is the cor rec t p roce dure fo r bon ded t endon s an d

Eq uat ion s (7.12) and (7.13) can be used as they s tand . F or u nbo nde d t endons , the

tendo n s t ra in i s the average over i ts leng th . Assum ing a parabo l i c t en don p ro f il e

and a parabo l i c shape o f the se l f-weigh t m om en t d ia gram , the va lues o f the l as t

two t e rms in Equa t ion (7 .12)average ou t to 0 .53 . Us ing 0 .5 , fo r un bon ded t endo ns

p c , = P i / A c + 0 . 5 [ e i e p - M o ] e p / I r (7.14)

PLN = 0 .5pr (7.13)

The t e rm [Piep - Mo] in Equ at ions (7.12) and (7.14) represen t s the ne t m om en t

a t th e s ec ti o n u n d e r co n s i d e ra t i o n . Fo r co n t i n u o u s mem b er s t h e t erms Piep a n d

Mo are rep laced wi th the moments ob ta ined f rom the ana lys i s fo r the t endon

equ iva len t loads and the permanen t loads respec t ive ly .

In m os t o f the pos t - t ens ioned f loors , the se lf -weigh t is ba lanced by the t e ndo n

equ iva len t load , so tha t the t e rm [Piep- Mo] in Equat ions (7 .12) and (7 .14)

becom es smal l . Neg lec t ing th is t e rm, the two se ts o f equa t ions a re iden t i ca l ; the

loss due to e l as t i c shor ten ing can be ca lcu la ted fo r bo th bonded and unbonded

tendons f rom the much s impler Equat ion (7 .15) .

PL = the to ta l loss of force for al l tendo ns

= o . s E P i / a o j ( e j e o i ) a p

Pi is the to tal pres t ress ing force in al l ten don s af ter a l l im m edia te losses, including

tha t d ue to the e l as t ic shor ten ing . I t i s norm al ly assum ed as 90% of the j ack ing

force Pj .

7 .5 S h r i n k a g e o f c o n c r e t e

Shr inka ge o f concre te has been d i scussed in Cha p ter 2 and the re l a t ionsh ip

be tween re la t ive humid i ty , sec t ion dep th and sh r inkage shown in F igure 2 .3 . In

the UK c l imate , fo r p la in concre te a s t ra in o f 300

x 1 0 - 6

i s t aken fo r indoor

exposure an d 100 x 10 -6 fo r ou tdo or exp osure . M ore accura te va lues may be

ob ta ine d f rom Figure 2 .3 and m odi f i ed fo r wate r con te n t o f the concre te mix , i f

known, as d i scussed in Chap ter 2 . In the UK, the ambien t re l a t ive humid i ty i s

t ak en a s 4 5 % fo r i n d o o r ex p o s u re an d 8 5 % fo r o u t d o o r .

The bonded rod re in fo rcement normal ly p rov ided in beams and s l abs res i s t s

any change in concre te l eng th . Shr inkage s t ra in i s , t herefo re , reduced by the

bonded re in fo rcement . A s l ab normal ly has re in fo rcement on on ly one face and

the quan t i ty is smal l e r than tha t in a beam, which has a re in fo rcem ent cage wi th

s tee l on bo th faces . Consequen t ly , the sh r inkage i s smal l e r in a beam than in a

s l ab . I t would , how ever , be imprac t i ca l to as sum e d i f fe ren t sh r inkage ra tes fo r a

s l ab and i ts beams , an d the no rm al p rac t i ce i s e i ther to ignore the re in fo rcem ent

con ten t , o r to as sume a cons tan t va lue fo r the whole f loor . BS 8110 g ives the

fo l lowing re la t ionsh ip be tween re in fo rcement con ten t and sh r inkage s t ra in ,

a s s u mi n g a s y m met r i ca l p lacemen t o f b o n d ed r e i nfo rcemen t.

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PRESTRESS LOSSES

171

e s - e d ( 1 +

P K s )

( 7 . 1 6 )

w h er e ep = t h e u l t i m a t e s h r i n k ag e s t r a i n i n p la i n co n c r e t e

p = p r o p o r t i o n o f s te e l a r ea t o co n c r e t e

K s = a co n s t a n t , 2 5 f o r i n d o o r e x p o s u r e a n d 15 f o r o u t d o o r ex p o s u r e

S o me s h r i n k ag e w i l l a l r e ad y h av e t ak en p l ace w h en t h e co n c r e t e i s s t r e s s ed an d ,

t h e r e f o re , o n l y t h e r e s i d u a l s h r i n k ag e n eed b e co n s i d e r ed i n ca l cu l a t i o n o f l o ss e s.

P o s t - t en s i o n ed s l ab s a r e n o r m a l l y s t r e s s ed w i t h i n a f o r t n i g h t o f c a s t i n g , an d

s o me t i mes w i t h i n t h r ee d ay s . I t w o u l d b e s u f f i c i en t l y accu r a t e t o a s s u me t h a t

2 0 % o f s h r i n k a g e w i l l h av e t ak e n p l ace i n t w o w eek s an d t h a t w i t h i n t h i s p e r i o d

t h e r a t e o f s h r i n k ag e i s l i n ea r .

Lo ss of pre s t re ss in g force = esEsAp (7 .17)

F o r a r e s i d u a l s h r i n k ag e s t r a i n o f 2 4 0 x 1 0 - 6 a n d a 0 .2 % r o d r e i n f o r cem en t

co n t en t , t h e l o s s i n p r e s t r e s s i n a f l o o r i n t h e U K i n d o o r en v i r o n men t w o u l d b e

0 .0 4 5 k N / m m 2 .

7 .6 C r e ep o f conc r e te

C r e e p o f c o n c r e t e h a s b e e n d i sc u s s ed i n C h a p t e r 2 a n d t h e r e l a t io n s h i p b e t w e e n

r e l a t i v e h u mi d i t y , s ec t i o n d ep t h an d c r eep co e f f i c i en t i s s h o w n i n F i g u r e 2 .4 f o r

l o ad i n g a t d i f fe r en t ag es . T h e ap p r o p r i a t e v a l u e o f t h e c r eep co e f fi c ien t, C o , c an b e

r e a d f r o m t h e s e c u r v e s a n d t h e n

Loss o f p res t r es s ing fo rce = (r p

= ( o ' C f f E ~ ) E s A p

= r e (7.18)

wh ere c r = l oca l s t r ess a t t e nd on l eve l

C~ = creep coeff icient

e~ = cre ep a f ter s t re ss ing

O n s t r e s s i n g , t h e memb er b eco mes s e l f - s u p p o r t i n g , an d c r eep i s a l o n g - t e r m

p h en o men o n . T h e r e f o r e , t h e l o ca l s t r e s s a i n E q u a t i o n ( 7 .1 8 ) s h o u l d i n c l u d e f o r

p r e s tr e s s a n d t h e m o m e n t s d u e t o se lf -w e i gh t o f t h e m e m b e r a n d o t h e r p e r m a n e n t

d ead l o ad s ; l i v e l o ad i s g en e r a l l y i g n o r ed . T h e r e f o r e , f o r b o n d ed t en d o n s ,

r = ( P i / A ~ ) + (P iep - M o - M d ) . e / I ~ ( 7.1 9 )

T h e c h a n g e i n th e s t r a i n a t a n y p o i n t a l o n g a b o n d e d t e n d o n is t h e s a m e a s t h a t i n

t h e co n c r e t e a t t h e t en d o n l ev e l. T h e r e f o r e , t h e c r eep l o ss v a r i e s a l o n g t h e s p an i n

acco r d an c e w i t h t h e v a r i a t i o n o f co n c r e t e s t re s s.

I n a n u n b o n d e d m e m b e r , t e n d o n s c a n m o v e r e l a t iv e t o th e a d j a c e n t c o n c re t e .

The refo re , t he l o ss o f p res t r ess i s due t o t he ave rage va lue o f ~r fo r t he t en do n

l e n g th a n d n o t t h e l o ca l v a lu e . A s s u m i n g p a r a b o l i c m o m e n t s a n d t e n d o n p ro fi le ,

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POST-TENSIONEDCONCRETE FLOORS

20

15

t ~

0

m

,--

10

0

X

t r

5

OI I I

5 0 % 6 0 % 7 0 %

Susta ined load/st rength

, i J

8 0 %

F i g u r e 7 .4 Long-term relaxation of low relaxation strand

the va lue of the second te rm in E qu a t io n (7 .19) ave rages o u t to 0 .53 . Us ing 0 .5 ,

t he a ve r a ge va lue o f a t o be u se d f o r un bo nd e d t e ndon s i s

t r = ( P t / A c ) + 0.5(Pie p - M o - M d ) e p / I ~ (7.20)

I n m o s t p o s t - t e n s io n e d f l o o rs t h e p e r m a n e n t l o a d s a r e a p p r o x i m a t e l y b a l a n c e d

by the e q u iva l e n t t e nd on loa d , so t ha t t he s t re s s on the c onc r e t e s e c t ion is ne a r ly

uni form; in such a case , the second te rm in Equa t ions (7 .19) and (7 .20) can be

neglec ted . The exp ress ion for tr then reduces to

P i / A ~

a nd the l o s s o f p r e s t r e ss ing

f o rc e c a n be c a lc u l a t ed f r om E q u a t ion ( 7.21 ), wh ic h is a pp l i c a b le t o bo th bo nd e d

a n d u n b o n d e d t e n d o n s .

Loss of p res t re ss ing force =

C e ( P i / A , z ) A p ( E d E r

where E c i s the m od ulu s of e la s t ic ity of concre te a t the t ime of load in g .

(7.21)

7 .7 R e l a x a t i o n of t e n d o n s

Re la xa t ion o f s t r a nd ha s be e n d i s c us se d in C h a p te r 2 i n some de t a il . U se o f l ow

r e l a xa t ion s t r a nd i s now c ommon in pos t - t e ns ione d f loo r s ; howe ve r , l oc a l

c i r c ums ta nc e s ma y r e q u i r e no r ma l s t r a nd to be u se d .

F igu r e 7 .4 shows the l ong - t e r m r e l a xa t ion f o r l ow r e l a xa t ion s t r a nds a t

d i f fe ren t ambient tempera tures and va r ious in i t ia l s t r e ss leve ls . In the absence of

de f in i te da t a f r om the ma n uf a c tu r e r o f s t r a nd , t he r e l a xa t ion pe r c e n ta ge c a n be

read f rom these curves .

Loss o f p r e s t re s s ing fo r ce - - p% P i /1 00

whe r e p% = pe r c e n ta ge lo s s

(7.22)

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PRESTRESS LOSSES 173

The t end on fo rce to be t aken in the ca lcu la t ion o f re laxa t ion loss is the fo rce a f te r

the sh ort - t erm losses have occ urred , i .e . the jac king force less the losses due to

e last ic shor ten ing , f r ic t ion an d d raw- in . Th i s fo rce var ies a long the t end on l eng th ,

and an av erage va lue ma y be t aken , un less the des i red degree o f accurac y requ i res

the actual forces to be taken at each cr i t ical point .

7 .8 T e n d o n e lo n g a t io n

Du ring jack ing the t en don fo rce i s me asured wi th a hydrau l i c p ressure gauge . A

pressure ce ll ma y a l so be used as a conf i rm atory dev ice . In add i t ion to these d i rec t

measur ing dev ices , the normal p rac t i ce i s to use the e l as t i c e longa t ion o f the

tend on to mo ni to r the force . Elong at ion i s m easur ed a t fu ll j ack ing load befo re

lock ing the anch orag e ; the am ou nt o f d raw - in is a l so recorded .

The e xpec ted e longat ion , and the ma rg in o f accep tab le var i a t ion , i s spec if ied

by the des igner fo r each t endon . I f the me asure d e longat ion fal ls ou t s ide the

spec if ied range th en im me dia te s t eps can be t a ken to rem edy the s i tua t ion i f i t is

cons idered necessary . The e longat ion o f a t end on s t res sed by a force P i s:

6 = P L t / ( A p E ~ ) (7.23)

where P = average fo rce in a t endon

P shou ld be tak en as the averag e value of the pres t ress ing force, i.e ., e i ther the

fo rce a t the midp o in t o f the t end on l eng th , o r the avera ge o f the j ack ing fo rce

(before lock ing) and the fo rce a t the dead end .

A tend on, par t ic ular ly a bon de d one, nearly alwa ys has a s lack in it before it is

s t ressed. Du rin g the s t ress ing of bo nd ed tend ons , in i t ia l ly the jack ing effor t goes

in to t ak ing the s l ack up , and then i t s tresses the tendo n . U nb on de d t endon s have

v i r tua l ly no s l ack . The normal p rocedure i s to par t i a l ly s t res s the t endon , to

abo u t 10% of the spec if ied j ack ing fo rce , and t ake th is as the re fe rence zero po in t

fo r measur ing e longat ion . Th i s p rocedure au tomat ica l ly a l lows fo r any t endon

s lack and any s l ip a t the dea d end . The e lon gat ion , m easu red f rom the re fe rence

po in t , mus t be mul t ip l i ed by a co r rec t ion fac to r so tha t i t can be compared wi th

the calculated 6 .

6 = cor rec ted va lue o f ac tua l e long a t ion

= me/(e - e , )

w h ere 6 m = meas u red e l o n g a t i o n

Pr = Tendon fo rce a t the re fe rence po in t

(7.24)

7 .9 T e n d o n f o r c e f r o m e l o n g a t i o n

Som et imes , a t endo n m ay n o t ach ieve the e longat ion speci fi ed by the des igner, o r

a t endon may be los t dur ing s t res s ing th rough fa i lu re a t the anchorage o r the

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174

POST-TENSIONED CONCRETE FLOOR S

b r e a k i n g o f t h e t e n d o n . I n s u c h c i r c u m s t a n c e s , i f t h e a d j a c e n t t e n d o n s a r e

p e r c e i v e d t o p o s s i b l y h a v e s u f f i c ie n t r e s e r v e t o m a k e u p f o r t h e lo s s o f f o r c e , t h e n

t h e f o rc e s in th e s tr e s se d t e n d o n s a r e c a l c u l a t e d f r o m t h e d a t a r e c o r d e d d u r i n g

s t r e s si n g . T h i s i s a l s o a g o o d d i s c i p l i n e f o r s h o r t t e n d o n s w h e r e t h e l o s s d u e t o t h e

t e n d o n d r a w - i n m a y b e si g n if ic a n t . T h e t e n d o n f o rc e is g i v e n b y :

P i ~ e j - (t~d/~rn )" ( P j - e r )

w h e r e P~ = f o r ce i n t e n d o n , a l l o w i n g f o r i m m e d i a t e l o ss e s

6 d = m e a s u r e d d r a w - i n

6 m = m e a s u r e d e l o n g a t i o n f r o m P r t o P j

P j = r e c o r d e d j a c k i n g f o r c e

P~ = r e c o r d e d t e n d o n f o r ce a t t h e r e fe r e n c e p o i n t.

(7 .25)

Example

7.1

Calcu la te the losses in span 1-2 of Exa mp le 6 .3 .

A c = 179.6 x 103 mm 2 Ic = 1011.5 x 106 mm 4

Yb = 163.5 m m

fr = 40 N/ m m 2 E~ = 28 .0 kN /m m 2

fr = 25 N /m m 2 Er = 21 .7 kN /m m 2

So/ut /on

T e n d o n s :

P f - - 650 kN, a s sum e P j = 830 kN and P i - - 740 kN

Try 4 supers t rands 15.7 mm dia ,

A p = 4 X 1 5 0 = 6 0 0 m m 2

The p res t r es s ing fo rce i s a s sum ed cons t an t fo r a ll spans ; h a l f o f t he s t r ands a re

s t ressed f rom each end.

1. Frict ion loss

T e n d o n s l o p e

at a - 2 • 0.084/4.66 = 0.0361

_ c _

[ oO T ~t" |

1 0 I 1 .0 ~ ,0 .8 j 1 .1 ~ /

. . - , , , - -

F ig u r e 7 . 5

Exa mp le 7. 1: Tendon pro f i le

64 mm

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PRESTRESS LOSSES

175

a t b = 2 x 0 . 1 5 5 / 5 . 3 4 = 0 . 0 5 8 1

a t c = 2 x 0 . 0 4 5 / 4 . 0 0 = 0 . 0 2 2 5

a t d = 2 x 0 .045 / / 4 .00 = 0 .022 5

a t e = 2 x 0 . 1 9 5 / 6 . 0 5 = 0 . 0 6 4 5

T a k e # = 0 .1 a n d K = O . O 01 /m f o r u n b o n d e d t e n d o n s .

P o i n t C u m u l a t i v e 0 x ( # 0 + K x )

A 2 x 0 . 0 3 6 1 = 0 . 0 7 2 2 x 4 . 6 6

2 0 A + 2 x 0 .058 1 = 0 .18 84 x 10 .00

B 02 + 2 x 0 .02 25 = 0 .23 34 x 14 .00

3 0 B + 2 x 0 .02 25 = 0 .27 84 x 18 .00

C 0 3 + 2 x 0 . 0 6 4 5 = 0 . 4 0 7 4 x 2 4 . 0 5

= 0 . 0 1 1 9

= 0 . 0 2 8 8

= 0 . 0 3 7 3

= 0 . 0 4 5 8

= 0 . 0 6 4 8

A s s u m e t h a t f o r t e n d o n s s t r e s se d f r o m e n d 4 , ( /~ 0 +

K x )

a t A is 0 . 0 6 4 8 , t h e s a m e a s a t

C f o r t e n d o n s s t r e s se d f r o m e n d 1.

T h e n , l o s s in f o r c e a t A = 7 4 0 x ( 0 . 0 1 19 + 0 . 0 6 4 8 ) / 2 = 2 8 k N

2 . A n c h o r a g e d r a w - i n

A l l o w 8 m m d r a w - i n .

T e n d o n l e n g th = 2 9 m

L o s s o f f o r c e = ( 8 / 2 9 0 0 0 ) • 6 0 0 x 1 9 5 = 3 2 k N

3 . E l a s t i c s h o r t e n i n g

U s e E q u a t i o n s ( 7 . 1 4) a n d ( 7 .1 3 )

S e e E x a m p l e 6 . 3 , t h e e q u i v a l e n t l o a d o f 6 5 0 k N is t a k e n i n th e t a b l es f o r m o m e n t s a n d

s h e a r s . I n t h e p r e s e n t e x a m p l e P i is a s s u m e d 7 4 0 k N .

We = - 6 . 2 1 x ( 7 4 0 / 6 5 0 ) = - 7 .0 7 k N / m

M 2 - - + 4 2 . 0 5 x ( 7 4 0 / 6 5 0 ) - 3 5.2 1 = + 1 2.6 6 k N m

M s = 1 2 . 6 6 / 2 + ( 4 .3 - 7 . 0 7 ) x 1 0 2 /8 = - 2 1 . 9 7 k N m

Pen = Pi /Ac + 0 - 5 [ P i e - - M o ] e / I c

= 7 4 0 / 1 7 9 . 6 + 0 .5 [ - - 2 1 . 9 7 ] x ( 1 6 4 - - 8 0 ) / 1 0 1 1 .4 5 0 = 3 . 2 1 N / m m 2

PLN = 0 .5Pcn(Es/Er

= 0 . 5 x 3 . 2 1 x ( 1 9 5 / 2 1 . 7 ) x 6 0 0 / /1 0 0 0 = 9 k N

S h o r t - t e r m l o s s e s "

F r i c t i o n 2 8 k N

D r a w - i n 3 2 t T o t a l 6 9 k N

E l a s t i c 9

F o r P j = 8 3 0 k N , P i = 8 3 0 - 6 9 = 7 6 1 k N

T h e a s s u m e d v a l u e o f 7 4 0 k N is c l o se e n o u g h n o t t o n e e d r e p e t i t i o n .

4 . S h r i n k a g e o f c o n c r e t e :

F r o m F i g u r e 2 . 03 , f o r a 1 8 0 m m d e e p s e c t i o n a n d a t 4 5 % r e l a ti v e h u m i d i t y , t h e

3 0 - y e a r s h r i n k a g e i s 4 0 0 x 1 0 - 6 . A s s u m e 8 0 % r e s i d u a l s h r i n k a g e , a n d a b o n d e d

r e i n f o r c e m e n t c o n t e n t o f 0 . 1 3 % . T h e n , f r o m E q u a t i o n s ( 7 .1 6 ) a n d ( 7 .1 7 ),

ep = 0 . 8 x 4 0 0 x 1 0 - 6 / ( 1 + 2 5 x 0 . 0 0 1 3 ) = 3 1 0 x 1 0 - 6

L o s s o f p r e s t r e s s = 3 1 0 x 1 0 - 6 • 1 9 5 x 6 0 0 = 3 6 k N

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POST-TENSIONED CONCRETE FLOORS

5 . C r e e p o f c o n c r e t e :

F r o m F i g u r e 2 .0 4 , f o r a 1 8 0 m m d e e p s e c t i o n , a t 4 5 % r e l a ti v e h u m i d i t y , a n d f i na l

s t r e s s i n g a t 1 4 d a y s a g e , t h e 3 0 - y e a r c r e e p c o e f f ic i e n t is 3. 3. T h e a v e r a g e s t r e s s i n

c o n c r e t e m a y b e t a k e n a s

Pi/Ac

= 7 6 1 / 17 9 . 6 = 4 .2 N / m m 2

A s s u m i n g t h a t 2 0 % c r e e p h a s t a k e n p l a c e b e f o r e t h e c o n c r e t e i s l o a d e d , f r o m

E q u a t i o n ( 7 . 2 1 ) ,

Lo ss o f fo rce = 0 .8 x 3 .3 x 4 .2 x 600 x 195 /21 .7 /1000 = 60 kN

6 . R e l a x a t i o n o f s t r a n d

Pi/Pu = 7 61 / (4 x 265 ) = 0 .72

F r o m F i g u r e 7 .4 , f o r a n a m b i e n t t e m p e r a t u r e o f 2 0 ~ r e l a x a t io n l os s is 2 % .

L o s s o f f o r c e = 2 % o f 7 6 1 k N = 1 5 k N

F i n a l p r e s t r e s s i n g f o r c e = 7 6 1 - 3 6 - 6 0 - 1 5 = 6 5 0 k N

N o a d j u s t m e n t i s r e q u i r e d i n t h e a s s u m e d j a c k i n g f o r c e o f 8 3 0 k N .

O K

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8 U L T I M A T E

S T R E N G T H

F L E X U R A L

This chap ter dea l s wi th the u l t ima te f lexura l s t reng th o f pos t - t ens ioned m em bers

of a f loor and the s t reng th requ i reme nts in the anc hora ge zone; shear s t reng th i s

discussed in Chapter 10.

The p urpos e o f check ing the u l t imate s t reng th o f a me m ber i s to ensure tha t , i f

loaded beyond the nominal des ign value, i t wi l l not fai l before the load exceeds

the des ign load mul t ip l i ed by a p rede te rmined fac to r . S ince a number o f

com bina t ion s o f d i ffe ren t loads ( se lf -weigh t, o ther dead loads , l ive load and

perh aps l a t e ra l loads ) a re cons idered , the obv ious bas ic requ i re me nt fo r fl exura l

s t reng th i s tha t , a t an y sec t ion , the design u l t ima te m om en t o f res i st ance M , be

no t l ess than the ma xim um requ i red m om ent o f res is t ance M~ fo r the com bina t ions .

8 .1 F a i l u r e m e c h a n i s m s

As the load on a member i s increased , the cor respond ing moments and s t res ses

increase in proport ion to the load, unt i l the tens i le s t ress in the s teel reaches i t s

yield point , or the compress ive s t ress in the concrete reaches the l imi t of

proport ional i ty at a sect ion. After th is , the rotat ion in the short y ielded length is

d i sp ropor t iona te ly l a rge fo r any fu r ther increase in the load . I t can be assumed

tha t the sec tion has reached i ts peak mo m en t o f res i st ance and any a t t em pt to

increase the load wi ll only increase the ro tat io n; the sect ion, in fact , beha ves as a

h inge. Hinges , obv ious ly , fo rm a t the peak po in t s o f the bend ing m om ent d iagram .

Fo r a suspended m em ber to co l lapse , a nu m ber o f p las ti c h inges mu s t fo rm a t

the c r it ica l sec tions , t rans fo rm ing the w hole o r pa r t o f the f loor in to a m echan i sm .

An und ers tan d ing o f the fa ilu re m echan i sms is usefu l fo r the ca lcu la t ion o f the

u l t imate s t reng th o f pos t - t ens ioned f loors , par t i cu la r ly those wi th unbo nde d

tendons .

8.1.1 Modes of failure

Fai lu re a t a h inge m ay t ake p lace in one o f th ree poss ib le mod es , depen d ing on

the level of pres t ress .

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178

POST-TENSIONEDCONCRETE FLOORS

1 . Fo r l igh tly loaded m em bers , the mag ni tude o f the requ i red p res t ress ing fo rce

may be very low, so that in the serviceabi l i ty s tate the tens i le s t rength of the

concre te m ay p ro v ide m os t o f the requ i red f l exural s t reng th . I f over loa ded , the

concre te wi ll c rack , an d i ts con t r ibu t ion to the s t reng th o f the mem ber wil l be

los t . In such a case, it i s qui te poss ible for the ul t im ate s t reng th of the sect ion

based on the s tee l con te n t to be lower than i ts c rack ing s t reng th . Fa i lu re the n

occurs as soon as the sec t ion c racks , wi th no warn ing .

Th i s s t a t e i s to be avo ided . ACI 318 recommends a min imum average

pres t ress of 1 .0 N /m m 2 on the gro ss sect ion af ter a l l losses , an d requires

suff icient tendon area to be provided to develop a factored load at leas t 1 .2

t imes the c rack ing load .

2 . At a h igher level of pres t ress th an tha t of Case 1 , wi th increas ing load, the

tendo n reaches it s l imi t o f p ro por t iona te s t ra in . B eyond th i s po in t , t he

s t ress -s t ra in re l a t ionsh ip fo r the s t rand becomes increas ing ly non- l inear and

plas t ic h inges form at the cr i t ical sect ions . Not iceable ro tat ion takes place at

each h inge , the neu t ra l ax is r ises, and the dep th o f the com press ion b lock

reduces . The sec t ion even tua l ly fa i l s e i ther when the t endon b reaks o r when

the concrete crushes .

Th i s mo de o f fa ilu re is a lways acc om panied by a la rge def lec t ion a nd am ple

wa rning. M ost p ost - tens ioned f loors would fail in th is m od e i f load ed to failure .

3 . At a very high level of pres t ress , the concre te crushes before the tend on reache s

i ts l imi t of propo rt ional i ty . Fai lure in th is case is wi thout w arning, and explosive.

Such a h igh level of s t ress wo uld exis t if the conc rete sect ion w ere redu ced to

the m inimum . High levels of pres t ress are bes t avoided. Ho wev er , i f unav oidab le,

t h en ad equ a t e b i n d in g r e i n fo rcemen t s h o u l d b e p ro v id ed a ro u n d t h e co n c re t e

in the compress ion b lock .

8 . 1 .2 On e-w ay spans

Fo r m a t i o n o f t h e fir st h in g e in a o n e -w ay s p an n i n g co n t i n u o u s me mb er d o es n o t

l ead to i t s co l l apse . F igure 8 .1 shows the h inges requ i red in one-way spann ing

me m bers to t rans fo rm them in to mechan i sms . A can t i lever needs one h inge a t it s

roo t , as does a s imply suppo r ted span ; the end sp an o f a con t inuou s s t r ing needs

two h inges i f i t is f ree to ro ta te a t the ou te r supp or t ; and the inner span needs th ree

hinges , as does the o uter s pan i f i t is no t f ree to ro tate at the end s up por t .

The u l t imate loa d c ar ry ing capa c i ty o f a me m ber c an be ca lcu la ted i f i ts

m om ents o f res is t ance a re know n a t the c r it ica l po in t s fo r the load ing p a t t e rn .

Cons ider the inner span in the u l t imate s t a t e , ca r ry ing a un i fo rmly d i s t r ibu ted

load . For s t a t i c equ i l ib r ium,

wuLZ/8

= M , b - (M u a + M ur

( 8 . 1 )

where wu = u l t imate load per un i t l eng th

Mua , Mub and Muc are the values of the m om en ts of res is tance a t a , b and c;

normal ly , the suppor t moments Mua and Mac are expec ted nega t ive .

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UL TIMA TE FLEXURAL STRENGTH

179

j , ,Hinge

. . . . . . . • • . H i n g e s , , •

Hinges " ~

a b c

L

F i g u r e 8 . 1 Hinges in one-way spans

9 ' 9Cantilever

Hinges

~ Outer spans

a Inner spans

In the case o f an e x te rna l span , M r a is z e ro , an d fo r a si m p l y s u p p o r t ed s p an b o t h

Mra an d

Mrr are

zero .

8.1.3 Two-wayspans

In a f lo o r co n t a i n i n g a n u m b er o f t wo -w ay s p an n i n g p an e l s in each d i r ec t io n , i t is

ex t remely un l ike ly tha t a l l pane l s in a l ine wi l l ge t over loaded a t the same t ime,

and the poss ib i li ty o f co l lapse occu rr ing in the ma nn er show n in F igur e 8 .1 i s

v i r tua l ly n i l, espec ia l ly i f on ly ver t i ca l loads a re imp osed on the f loo r. M ore

pro bab le is the fa i lu re o f one pane l in i t ia l ly , wh ich m ay lead to the co l lapse o f

ad jac en t pane l s . A s ing le pane l ou t o f a two-w ay co n t inu ous f loo r can fa il on ly

wh en y ie ld l ines have deve lo ped so tha t the p ane l i s d iv ided in to severa l zones , see

Figure 8 .2 .

The y ie ld l ine mechan ism i s a fa i lu re pa t te rn and i t does no t re f lec t the

serv iceab i l i ty s t resses . Serv iceab i l i ty checks a re based on the p ropor t ion ing o f

loads in two o r th ogo na l d i rec t ions , so as to equa te the def lec t ion o f two s t r ips , a t

I

I Hogging j

. ~ . i ~ . . _

?

/ [Yield linesl

r_ 4_

/ ~-Beamsi

(a) Yield lines

Figure 8.2

Two-way slab panel

I I

J

L ]

I I

i I

(b) Critical sections in serviceab ility state

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180 POST-TENSIONED CONCRETE FLOORS

r igh t ang les , a t t he i r po in t o f in t e r sec t ion . The se rv iceab i l i ty s ta t e c r i t ica l sec t ions

a r e l oc a t e d a l ong t he su pp o r t l i n e s a nd i n the m i d spa n r e g i on , i n e a c h d i r e c t i on .

Ca l c u l a t i ons ba se d on y i e l d l ine s c a n be l e ng t hy , a nd se ve ra l tr i al s m a y be ne e d e d

t o a r r i ve a t t he m i n i m um l oa d i n t e ns i t y w h i c h w ou l d c a use c o l l a p se . F o r t h i s

r e a son , t he y i e l d l i ne m e t hod i s u se d on l y i n e xc e p t i ona l c i r c um s t a nc e s ; f o r

no r m a l f l oo r s l a b s c a r r y i ng un i f o r m l y d i s t r i bu t e d l oa d i ng , t he u l t i m a t e s t r e ng t h

i s c he c ke d a t t he s a m e se c t ions w h i c h a r e c r i ti c a l in t he s e r v i ce a b i li t y s t a t e - - a l o ng

t he suppo r t s a nd ne a r t he m i dspa n s . The c he c ks a r e c a rr i e d ou t i n e a c h d i r e c t i on .

Th i s l e a ds t o a m or e c onse r va t i ve de s i gn t ha n t he y i e l d l i ne a pp r oa c h .

I n t he c a se o f a pa ne l su pp o r t e d by be a m s , t he s la b a nd t he be a m e l e m e n t s a r e

c he c ke d se pa r a t e l y . Th i s a vo i d s t he d if f ic u lt y o f p r o po r t i on i ng t he l oa d be t w e e n

t he c o l um n a n d m i dd l e s t r ip s i f t h e w ho l e pa ne l w i d t h w e r e t o be c he c ke d a s one

un i t . The sum o f t he loa ds c a r r i e d by t he s la b a nd t he be a m s i n e a c h d i r e c t i on

m us t e q ua l t he t o t a l l oa d .

8 .2 Le v e l o f p re s t re s s

Us ing the t ens i le s t ren g th of concre te , i t i s t heore t i ca l ly pos s ib le to sa t is fy the

se r v i ce a b i li t y r e q u i r e m e n t s o f a f loo r spa n n i ng a sho r t d i s t a nc e w i t h ze r o

p r e s t r e s s~ t he c onc r e t e t e n s i l e s t r e ng t h be i ng su f f i c i e n t t o t a ke t he f l e xu r a l

t e n s i on . A t t he o t he r e x t r e m e , a ve r y t h i n s e c t i on c a n be p r e s t r e s se d t o suc h a n

e x t e n t t ha t , w he n o ve r l oa d e d , f a i lu r e oc c u r s by c r u sh i ng o f t he c onc r e t e be f o r e

the t endons fa i l .

Ne i the r o f the two i s des i rab le in a p rac t i ca l des ign . The f i r s t case does no t me e t

t he u l t i m a t e s t r e ng t h r e q u i r e m e n t , a nd f o r bo t h c a se s f a i l u r e w i l l b e sudde n a nd

w i t h o u t w a r n i n g .

A C I 3 1 8 l im i ts th e m a x i m u m a r e a o f s t e e l - - t e n d o n s p l us r o d r e i n f o r c e m e n t - - s o

t h a t

0.36fll > cop

> O ~ p + ( d / d p ) ( C o - o 9 ' )

where f l l = d c / d n

= 0 . 8 5 - o . o 0 8 ( L ' - 3 0 )

= 0.85 - 0 .05(fc ' - 400 0)/10 00

Ra t ios o f s t ee l to conc re te s t reng ths"

B on de d in t ens io n zone" 09 = A s f y / ( f ~ ' b d )

B o n d e d i n c o m p r e s s i o n z o ne " 0 9 ' =

A s ' fy / ( f ~ ' b d )

Pres t re s s ing t endon s" cop = A p f p J ( f c ' b d )

b = w i d t h o f c onc r e t e i n c om p r e s s i on ,

= w e b w i d t h i n the c a se o f a T - be a m

fps = t endon s t re ss a t des ign load

f o r f~ ' in N / m m 2

for fc ' in psi

(8.2)

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UL TIMA TE FLEXURAL STRENG TH 181

8 .2 . 1 C r a c k i n g lo a d

I n t he a bse nc e o f s e lf -we ight a nd a pp l i e d loa ds , a pos t - t e ns ion e d m e m be r c a r r ie s

only the pres t re ss in g s t r e sses and the wh ole of the sec t ion is in com press io n .

W he n a sma l l l oa d i s a pp l i e d , t he c ompr e s s ion r e duc e s on one f a c e . U nde r a n

inc r e as ing loa d , t he p r e - c om pr e s s ion r e duce s t o ze r o on the t e ns ion f a ce , a nd the n

the s t r e ss beco m es tens i le . As long as the tens i le s tr e ss is le ss tha n the m od ulu s of

r up tu r e o f t he c onc r e t e, t he s e c t ion r e m a ins un c r a c ke d a nd the de f le c t ion

c o r r e spon ds to t he m om e n t o f i ne r ti a o f t he g r os s s e ct ion .

W hen the tens i le s t re ss reaches the m od ulu s of rup tu re , the sec t ion c racks , and

the tens i le force , which was up to now ca r r ied by the uncracked concre te , i s

sudde n ly t r a ns f e rr e d to t he te ndo n . T he a c t ion is a c c om pa n ie d by a n imm e d ia t e

inc rease of de f lec tion . Th e ra te o f inc rease of de f lec t ion i s now h ighe r because of

the sm a l le r m om e n t o f i ne r t ia o f t he c r a c ke d se c tion .

The moment a t which the f lexura l tens i le s t r e ss on the sec t ion equa ls the

m odu lus o f r up tu r e is c al le d the crack ing mo ment . In ca lcu la t ing the c rack ing

moment , the f ina l p res t re ss ing force i s used , a f te r a l lowing for a l l losses .

Mcr =ft Z + P /A~(Z + epA~)

(8.3)

w h e re M , = c r ac k in g m o m e n t

f t = m o d u l u s o f r u p t u r e

Z = s e c t ion m odu lus o f t he e x t re me t e ns ion f ib r e

In BS 8110, the c rack ing moment and the pr inc ipa l tens i le s t r e ss a re used to

de t e r mine i f t he c r a c ke d o r t he unc r a c k e d se c tion shou ld be u se d in t he

ca lcu la t ion of shea r s t r ength . The l imi t ing va lue of the pr inc ip a l tens i le st r ess

reco m m end ed for use in th is case i s 0 .24(fcu) ~ in N-m m uni t s . This top ic i s dea l t

wi th in de ta i l in Chapte r 10 .

8 .3 Ap p l i ed loads

T he m in im um va lue o f t he r e q u i r e d m om e n t o f r e s is t a nce M r a t a s e c t ion i s

c a l c u l at e d by imp os ing se rv i c e ab i li t y loa ds m u l t i p l ie d b y the l oa d f a c to r s . W h e r e

a member i s r equi red to re s i s t a la te ra l force , the la te ra l force mul t ip l ied by the

a pp r op r i a t e l o a d f a c to r is i nc lude d in t he c a l c u l a ti on o f t he r e q u i re d m om e n t o f

resis tance .

A num be r o f l oa d c om bina t ion s a r e no r ma l ly c ons ide r e d in o r de r t o a rr ive at

t he mo s t a dve r se c ond i t i on a t e a c h c r it ic a l pos i t i on . T he va lue o f a l oa d f a c to r

de pe nds o n the imp or t a nc e o f t he loa d c o m bina t ion be ing c ons ide r e d , a nd , fo r a

pa r t i c u l a r c o m bina t ion , t he l oa d f a c to r ma y be d if fe re n t fo r e ac h loa d type . T he

de a d loa d a nd the s e c onda r y m om e n t s a r e, o f c ou r se p r e se n t i n a ll ca ses ;

howe ve r , t he c o r r e sp ond ing lo a d f a c to r va ri e s be twe e n 1 .0 a nd 1 .4 f o r de a d loa ds

de pe nd ing on the l oa d c om bina t ion a n d r e m a ins c ons t a n t a t 1 .0 f o r t he

s e c o n d a r y m o m e n t s .

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182 POST-TENSIONED CONCRETE FLOORS

The load ing pa t t e rns fo r s t reng th ca lcu la t ion a re the same as fo r the

serviceabil i ty state, viz:

Dead load on a l l spans wi th l ive load on two ad jacen t spans , and

Dead load on a l l spans wi th l ive load on a l t e rna te spans .

BS 8110 requ i res the fo l lowing load combina t ions to be cons idered ; the

num erical values of the load factors are g iven in the express ions .

(a) 1.4G + 1.6Q

(b) 1 .4G + 1 .4W

(c) 1 .2G + 1 .2Q + 1 . 2w

(d) 1.056 + 0.35Q + 0.35 W

(e) 1.05G + 1.05Q + 0.35 W

w h ere G = d ead l o ad

Q = nomina l l ive load , inc lud ing dynamic load

W = w i n d l o ad

(8.4)

Cases (d ) and (e) above represen t a bno rm al load ing cases and a re used i f i t is

nece ssary to conside r the pro bab le effects of excessive loads ca used by misuse o r

acc iden t , o r w hen cons ider ing the con t inue d s t ab il i ty o f a s truc tu re a f t e r it has

sus ta ined loca li zed dam age . On ly those loads l ike ly to be ac t ing s imul taneo us ly ,

or those l ikely to occur before measures are taken to repai r or offset the effect of

the damage need be inc luded . For bu i ld ings used p redominan t ly fo r s to rage o r

indus t r i a l purposes o r where the impo sed loads a re perm anen t , the whole o f the

impo sed load shou ld be take n a nd exp ress ion (e ) app l i ed ; fo r o ther cases

ex p re s s i o n (d ) s h o u l d b e u s ed .

Th e l o ad co m b i n a t i o n s r equ i r ed b y AC I 3 1 8 an d t h e co r r e s p o n d i n g v a lu es fo r

the load factors are:

(8.5)

( a ) 1 . 4 G + 1 .7Q

(b) 0 .75(1 .4G + 1 .7Q + 1 .7w )

(c) 0 .9G + 1 .3W

(d) 0 .75(1.4G + 1 .7Q + 1 .87E)

(e) 0.9G + 1.43E

(f) 0 .75(1.4G + 1 .7Q + 1 .4T)

(g) 1.4G + 1.4T

where E = load e f fects o f ear thqu ake

T = e ffec t o f t em pera tu re , se t t leme nt , sh r inkag e o r c reep

Express ions (d ) and (e) o f the AC I 318 requ i rem ents a re app l i cab le on ly i f the

effect of ea r th qu ak e is to be cons idered, an d (f) and (g) are to be used wh ere effects

o f d i ffe ren t ia l se t t lement , c reep , sh r inkage , o r t e m pera tu re change a re s ign i fi can t

in design.

8.4 Proc ed ure for ca lcu la t ing s t rength

The p roced ure fo r ca lcu la t ing the fl exura l s t reng th o f a pos t - t ens ioned m em ber is

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U L T IMA TE FL E X U R A L S TR E N G TH 183

s imi lar to that for a reinforced concrete member, the only di fference being the

ca lcu la t ion o f s tres s in the t endon s a t u l t im ate load . The bas ic as sum pt ion in bo th

cases i s that p lane sect ions remain plane.

The appl ied loads are increased by speci f ied

load actors

an d the s t resses in the

mater i a l s a re reduced by app ly ing

partial safety factors

t o the i r s t reng ths . The

load fac to rs t ake acco un t o f the poss ib le inaccurac ies in the assessments o f loads ,

and the to le rances . The par t i a l sa fe ty fac to rs fo r mater i a l s a l low fo r the

d i f fe rences be tween l abora to ry and ac tua l s t reng ths , loca l weaknesses and

inaccurac ies o f the p rope r t i es o f mater i a l s . The load fac to rs and the par t i a l sa fe ty

fac to rs a re spec i f i ed by the var ious na t iona l s t andards ; BS 8110 and ACI 318

requ i rements a re g iven in th i s chap ter .

The requ i red u l t imate m om ent is ca lcu la ted , a t each o f the c r it ica l sec tions, by

combin ing the moments fo r the fac to red loads in the mos t adverse manner .

Al though the t endon fo rce i s h igher a t the u l t imate s t a t e than a t se rv iceab i l i ty

s t a te , the e ffec t o f th is increase on the secon dary m om ents is ignored and the

s eco n d a ry mo m en t s a r e ad d ed u n fac t o red t o each o f t h e l o ad co m b i n a t io n s .

The level of pres t ress in a pos t - tens ion ed f loor is usual ly too low for an

explos ive fai lure; fai lure i s l ikely to be in i t ia ted by the s t ress in the s te el - - te nd on s

and bon ded re in fo rcem ent - - r eac h ing the limi t o f l inear ity o f s t res s and s t ra in .

Th i s , o f course , does no t app ly to h igh ly s t res sed mem bers , such as t rans fer beams .

The des ign u l t imate m om ent capac i ty o f a sec t ion i s ca lcu la ted by cons ider ing

s ta ti c equ i l ib r ium , w i th the t endon in the non - l inear s t a te . The to ta l t ens il e fo rce

in a l l s t ee l s , t endons and rod re in fo rcement , mus t equa l the to ta l compress ive

force in the concrete . T he fact that som e of the force in the ten don s is appl ied as

pres t ress i s i r relevant .

Co ncr ete i s assu m ed to ha ve reac hed i t s fai lure s t rain at the ext reme f ibre. Tests

ind ica te tha t in the u l t imate s t a te the shape o f the com press ion b lock i s near ly

parabo l i c fo r pos t - t ens ioned concre te , as fo r re in fo rced concre te. T he co mpress ive

fo rce and i t s cen t ro id can be ob ta ined f rom the s t res s -s tra in curve fo r the concre te

be ing used , and the par t i a l sa fe ty fac to rs app l i ed . The p rocedure , however , i s

complex , and i t does n o t necessar ily lead to a m ore accura te as sessment o f the

u l t imate s t reng th capac i ty o f a mem ber , because o f the possib le va r i a t ions in the

ma ter i a l p roper t i es . Th e shape o f the com press ion s tress b lock i s , t herefo re ,

normal ly s impl i f i ed to a rec tang le . In o rder fo r the rec tangu lar b lock to be

equ iva len t to tha t ob ta ine d f rom tes ts , o r the s t res s -s t ra in d iagram , i t is necessary

to app ly cer t a in fac to rs to the in tens i ty o f the s t res s and to the de p th o f the

com press ion b lock . The fac to rs defin ing the equ iva len t rec tangu lar com press ion

block di ffer in BS 8110 and ACI 318.

The t e ndo n s tres s in the u l timate s t a t e is usua l ly be tween the y ie ld po in t a nd

the b reak ing s tres s, and can be de te rm ined b y s t ra in co mp at ib i l ity a t the sec t ion

be ing cons idered . S t ra in com pat ib i l ity ca lcu la t ions a re , however , ra the r com pl i -

ca t ed , p a r t i cu l a r l y fo r u n b o n d ed t en d o n s , an d t h e v a r i o u s n a t i o n a l s t an d a rd s

g ive s imple ru les fo r de te rm in ing the s t res s in bonded and unb ond ed t endons . BS

8110 and ACI 318 recommendat ions a re g iven in th i s chap ter .

F igure 8 .3 shows a typ ica l s t res s d iagram fo r a pos t - t ens ioned member .

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184 POST-TENSIONED CONCRETE FLOORS

fcu or f6 Pcu

L.. L _1

i- ~1 I~ -1

~ _

Figure 8.3 Ult imate state st ress d iagrams

r

~r

, . ._ -

J _

C = Co mp res s i o n = p c u b d , ,

(8.6a)

T = Tens ion

= f p b A p + f s b A s = T p + T r

(8.6b)

wh ere PCu = com pre ssive stress in co nc rete at fai lure, as given in Sec tion 8.5.1

Tp

= f p b A p =

t ens ion in p res t ress ing t endons

T r = f ~ b A , = t en s i o n i n b o n d ed ro d r e i n fo rcemen t

f~b - - s t re s s in bond ed re in fo rceme nt

Tak ing m om ents o f the t ensi le forces abo u t the cen t re o f com press ion C g ives the

u l t imate c apac i ty o f the sec t ion M u, which , o f course , shou ld no t be l es s than the

requ i r ed u l t ima t e m o m en t o f r e s is t an ce M r .

M u = f p b A p ( d p - d c ) + f sbAs(dr - de)

(8.6c)

where d c = the dep th f rom c omp ress ion face to the cen t re o f the compress ive fo rce .

The above equa t ions assume a rec tangu lar sec t ion ; the equa t ions a re va l id fo r a

f l anged sec t ion p rov ided tha t the neu t ra l ax i s rem ains w i th in the dep th o f the

f lange. This i s normal ly the case in post - tens ioned concrete . A neutral axis

loca ted be low the f lange ma y be ind ica t ive o f a po ten t i a l ly b r i t t l e fa ilu re an d

shou ld be avo ide d by increas ing the dep th o f the sec t ion o r o f the f l ange .

If the tens ile force in the ten don s alone is no t suff icient to prov ide the req uired

s t r en g t h , t h en ro d r e i n fo rcemen t may b e ad d ed ; t h e s eco n d t e rm i n Equ a t i o n s

(8.6) rep resen t s rod re in fo rcement . In th is case the ca lcu la t ion p rocess beco me s

i t e ra t ive , because the dep th o f the comp ress ion b lock , and hence the cen t ro id o f

the compress ive fo rce, changes wi th the am ou nt o f bon ded rod re in fo rcem ent ;

howe ver , the ca lcu la t ion i s no t very sens it ive . The p r oced ure fo r the u l t imate s t a t e

check general ly consis ts of the fol lowing s teps .

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UL TIMA TE FLEXURAL STRENG TH 185

1 . Calcu la te the requ i red m om ent o f res is t ance M r

2. C alc ula te T r an d Tp, init ial ly Tr m ay b e a s s u med ze ro

3. Ca lculate d Xand d c

4 . Ca l cu l a t e M,

5 . I f M , < M r then ad d bo nde d re in fo rcement and repea t f rom 2 .

8 . 5 U l t i m a t e s t r e s s e s

The s tresses in the u l t imate s t a t e a re as sumed to equa l the s t reng th o f the mater i a l

modif ied by the part ia l safety factor , as d iscussed above. The factors d i ffer for

concre te , bonded rod re in fo rcement and p res t ress ing t endons .

8 .5 . 1 C o n c r e t e s t r e s s e s

The compress ive s t res s b lock in a f l exure member approach ing fa i lu re i s

parabo l i c , bu t a rec tangu lar compress ion b lock i s normal ly assumed , as

discussed above.

I f any c om press ion re in fo rcement i s p rov ided , then i t s st res s can be ca lcu la ted

f rom the s t ra in d iagram a t the sec t ion be ing cons idered . The compress ion

re in fo rcement w ould requ i re l inks to p reven t buck l ing . Use o f com press ion

reinforcement , however , i s very rare in post - tens ioned f loors , and i t i s not

discussed in any detai l in th is book.

BS 8110 requires :

9 the 28-day c oncre te s t ren gth to be divided by a pa r t ia l safety factor of 1 .5,

9 the concre te s t ra in a t the com press ion face to be assum ed as 0 .0035 ,

9 the rec tang ular s t ress b lock to have an avera ge s t ress Pc, of 0 .45fc, ,

9 the dep th o f the rec tang u lar c om press ion b lock (dx) to be 0 .9 t imes the dep th o f

the neutral axis (d ,) ,

9 tens ile s t reng th of conc rete to be ignored .

The above cond i t ions g ive the to ta l compress ive fo rce as 0 . 4 0 5 f c . b d . , ac t ing

0 .45d . f rom the co mp ress ion face o f the sec t ion .

AC I 318 takes a d i ffe ren t appro ach . Rath er th an spec ify ing par t i a l fac to rs fo r

the ind iv idua l mater i a l s , i t requ i res tha t the des ign s t reng th p rov ided by a

member be t aken as the nomina l ca lcu la ted s t reng th mul t ip l i ed by a s t reng th

reduc t ion fac to r o f 0 .9.

For the rec tangu lar compress ion b lock , ACI 318 spec i f i es tha t :

9 the concre te s t ra in a t the com press ion face be assume d as 0 .003 .

9 the rec tang ular s t ress b lock be assum ed to hav e an avera ge s t ress Pc, of 0 .85fc '.

9 the dep th o f the rec tang u lar com press ion b lock be t aken as /31 t imes the dep th

of the n eu tral axis. Tab le 8.1 gives the v alues of/3~ for different cylinder stren gths.

9 tens ile s t ren gth of concrete i s ignored .

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1 8 6 POST-TENSIONEDCONCRETE FLOORS

Table 8.1a

Com pression block depth ratio, AC I 318 (Metric units)

f 'c 30 35 40 45 50 55 N/m m 2

fll 0.85 0.81 0.77 0.73 0.69 0.65

Table 8.1b Com pression block depth ratio, AC I 318 (Imperial units)

f 'c 3000 4000 5000 6000 70 00 8 00 0psi

/31 0.85 0.85 0.80 0.75 0.70 0.65

8.5.2 Stress in rod reinforcement

f s b

The s t re ss in the rod re inforcem ent can be ca lcu la ted f rom the s t r e ss - s t r a in curve

f o r the s t ee l a nd the s t r a in d i a g r a m a t t he s e c t ion be ing c ons ide r e d . H o we v e r , i n

most cases i t reaches i ts yie ld s tress .

BS 8110 spec if ie s an up per l imi t for the s t re ss in the rod re inforcem ent equ a l to

i ts yie ld s tress divided by a par t ia l safe ty fac tor of 1 .15. The upper l imit specif ied

by the AC I 318 is 550 N /m m 2. The s t r eng th re du c t ion fac tor of 0 .9 impl ie s a l imi t

of 495 N /ra m 2.

8.5.3 Stress in bonded and unbonded tendons

A la r ge r o t a t i o n t a ke s p l a c e ove r t he sho r t l e ng th o f ea c h p l a s t ic h inge in t he

me c ha n i sm, so t ha t t he c onc r e t e s t r a in s a t t he h inge s a r e muc h l a r ge r t ha n

e lsewhere , and the s t r a ins in the lengths be tween the h inges can be ignored .

In a bonded tendon, the inc rease in s t r a in i s the same as the loca l s t r a in in

concre te a t the tendon leve l . The tendon s t r e ss , the re fore , inc reases a t a h igher

r a t e l oc al ly a t a h inge tha n e l se whe r e in it s l e ng th . T he be ha v io u r o f a n un bo nd e d

ten do n i s qu i te d i f fe ren t. I t s l ips r e lat ive to the concre te a t the h inge , an d the loca l

e longa t ion i s d i s t r i bu t e d e ve n ly ove r the w ho le l e ng th o f t he t e ndo n ; c o nse q ue n t ly

the s t r a in i s v i r t ua lly c ons t a n t ov e r t he t e ndo n l e ng th a nd is m uc h sma l l e r a t t he

h inge tha n tha t f o r a bond e d t e ndon . T he inc r e a se i n s tr e ss is al so c o r r e spond in g ly

smal le r . There fore , a bonded tendon deve lops a h igher f lexura l s t r ength a t the

h inge , bu t a n u nb on de d t e nd on a l lows a m uc h l a r ge r h inge r o t a t i o n to t a ke p l a c e

before i t r eaches i t s b reaking load .

T he a ve r a ge inc r e a se i n s t r a in , a nd s t r e s s , i n a n unbonde d t e ndon i s t he sum

to ta l o f e longa t ions a t a ll the h inge s d iv ide d by the t e nd on l e ng th . F o r

convenien ce , the ef fect ive length of an u nb on de d t end on i s take n as the ac tua l

t e nd on l e ng th d iv ide d by the num be r o f h inge s ne c e s sar y to c ha nge the spa n

unde r c ons ide r a t ion in to a me c ha n i sm. T he r e f o r e ,

Lte =

L t / N

(8.7)

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UL TIMA TE FLEXURAL STRENGTH 1 8 7

w h e r e Lt e = e ff ec ti ve l e n g th o f a n u n b o n d e d t e n d o n

N = n u m b e r o f h i n g e s n e ce s sa r y t o c h a n g e t h e s p a n i n t o a m e c h a n i s m

I n a s t r i ng o f

cont inuous spans , i t i s theore t ica l ly poss ib le for a l l spans to be

o v e r l o a d e d s i m u l t a n e o u s l y , b u t o n l y o n e s p a n is c o n s i d e r e d t o a p p r o a c h t h e

u l t i m a t e s t a te a t a t im e . C o n s i d e r i n g t h e o u t e r s p a n , o r th e p e n u l t i m a t e s u p p o r t

o f a s t ri n g o f s p a n s w i t h p i n n e d e n d s u p p o r t s , t h e in c r e a s e i n t e n d o n s t r a i n is d u e

t o th e r o t a t i o n a t tw o h i n g e s d iv i d e d b y t h e t e n d o n l e n g t h o f a ll sp a n s t h r o u g h

w h i c h i t p a s s e s .

8.5.4 Stress in bonded tendons f p b

F o r b o n d e d t e n d o n s , B S 8 1 1 0 g i v e s t h e v a l u e s i n T a b l e 8 . 2 f o r t h e t e n d o n s t r e s s

a n d t h e d e p t h o f t h e n e u t r a l a x i s i n t h e u l t i m a t e s t a t e .

F o r a g i v e n s e c t i o n , v a l u e s o f R p a n d (fpe/fm,) a r e c a l c u l a t e d a n d t h e

c o r r e s p o n d i n g v a l u e o f (fpb/O.87fr, u ) is o b t a i n e d f r o m T a b l e 8 .2 , f r o m w h i c h fp b

c a n b e d e d u c e d .

T h e A C I 3 1 8 r e q u i r e m e n t s f o r s t r e s s i n b o n d e d t e n d o n s a r e "

fpb = (1 --

Kpyp / f l l ) f pu

(8 .8)

K p = ( r j p J f ~ ' ) + d / d p ) ( C o - c o')

w h e r e 7p = f a c t o r f o r t h e t y p e o f t e n d o n

= 0 .55 fo r

fpy / fp ,

n o t l e s s t h a n 0 . 8 0

= 0 . 4 0 f o r

fpy / fp ,

n o t l e s s t h a n 0 . 8 5

= 0 .28 fo r fpy / fp , n o t l e s s t h a n 0 . 9 0

Table 8 .2

BS 8110 S tress in bond ed tendo ns and depth o f neutral axis.

Rp

fr

dn/d

for fpJfp~ = for fpJfp~ =

0.60 0.50 0.40 0.60 0.50 0.40

0.05 1.00 1.00 1.00 0.11 0.11 0.11

0.10 1.00 1.00 1.00 0.22 0.22 0.22

0.15 0.99 0.97 0.95 0.32 0.32 0.31

0.20 0.92 0.90 0.88 0.40 0.39 0.38

0.25 0.88 0.86 0.84 0.48 0.47 0.46

0.30 0.85 0.83 0.80 0.55 0.54 0.52

0.35 0.83 0.80 0.76 0.63 0.60 0.58

0.40 0.81 0.77 0.72 0.70 0.67 0.62

0.45 0.79 0.74 0.68 0.77 0.72 0.66

0.50 0.77 0.71 0.64 0.83 0.77 0.69

Rp = A~pu/(f~ubd)

d = dep th to steel centroid fro m compression face

0.87 is the inverse of partial fac tor 1.15

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188 POST-TENSIONEDCONCRETE FLOORS

f l l = d e / d n

o 9 = A ~ / ~ ' b d )

c o ' = A s ' f y / ~ ' b d )

r p - A p / b d p

= ra t io o f t endo n area

l as defined

in Sect ion 8 .2

I f any com press ion re in fo rcement i s t aken in to acco un t then the t e rm Kp sha ll be

taken as no t l es s than 0 .17 , and d ' (d i s t ance f rom ex t reme compress ion f ib re to

cen t ro id o f com press ion re in fo rcement ) sha ll be no g rea te r than 0 .15 dp.

In the absence o f any bo nde d rod re in fo rcement , the express ion (8.8) reduces to :

fpb = fpu - ( Y p / f l x ) E A p f p u / b d p f c ' ] f p u (8.9)

Values

o f f p J f p u

for a range of

( A r f p u / b d r f c ' ) ,

ca lcu la ted f rom Equat ion (8 .9 ) , a re

given in Table 8 .3 .

8.5.5 Stress in unbonded tendons

For unbonded t endons , BS 8110 g ives"

fpb = f p , , + ( 7 0 0 0 d / L t , , ) ( 1 - - 1.7Rp)

= fpe + 6p (8.10a)

d r , = 2 . 4 7 d R p ( f p b / f p u ) (8.10b)

where Rp =

A p f p u / ( f e u b d )

Ltr = effective tendon length, as defined in 8.5.3.

The value offpb should not be taken as greater than 0 .7fpu.

Eq uat io n (8 .10a) has been der ived by t ak ing the l eng th o f the zone o f

ine las t i c i ty wi th in the concre te as 10d . . The second t e rm, cor respond ing to 6p ,

represen ts the increase in the ten do n s t ress . Table 8 .4 gives the increase of ten do n

s t ress in N /m m 2, based on Eq uat io n (8 .10a), fo r a range o f Rp a n d L t e / d values .

Eq ua t io n (8 .10b) , def in ing the dep th of the neu tral axis , can be wri t te n in a

s impler form as"

a u / d - 2.47Rp ( f p b / f p u ) (8.11)

ACI 318 requirement for span-depth rat ios of 35 or less i s"

fpb =fpe + 70 + f c ' b d p / ( l O O A p ) N. m m 2 u n it s

--fpe +

10000 + f c ' b d p / ( l O O A p ) psi units (8.12a)

The value Offpb m ust no t exceedfpy, norfpe + 400 in N /m m 2 uni ts ( fpe + 600 00 in

psi units).

For span-depth rat ios h igher than 35, ACI 318 speci f ies

fpb =fpe + 70 + f r N. m m 2 u n i ts

- fpr + 10000 + f c ' b d p / ( 3 O O A p ) psi units (8.12b)

The value Offpb m ust no t exceedfpy, norfpr + 200 in N/ m m 2 uni ts ( fpe + 300 00 in

psi units).

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UL TIMA TE FLEXURAL STRENG TH

189

Table 8.3 Stress rat io

fpb/fp,

f or bonded t endons , A CI 318

Apf~u

f ' c N / m m 2 (psi )

30 40 50 60 70

7p bd pf ' c (4350) (5800) (7250) (8700) (101 50)

0.55

0.05 0.968 0.964 0.960 0.955 0.948

0.10 0.935 0.929 0.920 0.910 0.896

0.15 0.903 0.893 0.880 0.865 0.844

0.20 0.871 0.857 0.841 0.820 0.792

0.25 0.838 0.821 0.801 0.775 0.741

0.30 0.806 0.786 0.761 0.730 0.689

0.35 0.774 0.750 0.721 0.684 0.637

0.40 0.741 0.714 0.681 0.639 0.585

0.45 0.709 0.679 0.641 0.594 0.533

0.50 0.676 0.643 0.601 0.549 0.481

0.40

0.05 0.976 0.974 0.971 0.967 0.962

0.10 0.953 0.948 0.942 0.934 0.925

0.15 0.929 0.922 0.913 0.902 0.887

0.20 0.905 0.896 0.884 0.869 0.849

0.25 0.882 0.870 0.855 0.836 0.811

0.30 0.859 0.844 0.826 0.803 0.774

0.35 0.835 0.818 0.797 0.770 0.736

0.40 0.812 0.792 0.768 0.738 0.698

0.45 0.788 0.766 0.739 0.705 0.660

0.50 0.765 0.740 0.710 0.672 0.623

0.28

0.05 0.984 0.982 0.980 0.977 0.974

0.10 0.967 0.964 0.959 0.954 0.947

0.15 0.951 0.945 0.939 0.931 0.921

0.20 0.934 0.927 0.919 0.908 0.894

0.25 0.918 0.909 0.899 0.885 0.868

0.30 0.901 0.891 0.878 0.862 0.842

0.35 0.885 0.873 0.858 0.839 0.815

0.40 0.868 0.855 0.838 0.816 0.789

0.45 0.852 0.836 0.817 0.793 0.762

0.50 0.835 0.818 0.797 0.770 0.736

8.6 Stra in com pat ib i l i ty

F i g u r e 8 . 4 s h o w s t h e s t r a i n h i s t o r y a t a s e c t i o n a s t h e l o a d i s i n c r e a s e d f r o m n i l

( L i n e 1 ) t o c r a c k i n g l o a d ( L i n e 2 ), a n d f in a l ly t o i t s u l t i m a t e c a p a c i t y ( L i n e 3 ). T h e

i n i t i a l s t r a i n i n t h e t e n d o n c o r r e s p o n d s t o t h e p r e s t r e s s a l o n e a n d e q u a l s

e 1 = P / ( A p E s)

(8 .13a )

W h e n t h e m e m b e r is l o a d e d , u p t o t h e p o i n t w h e r e th e s e c ti o n re m a i n s

u n c r a c k e d , t h e s t r a i n l in e o n t h e d i a g r a m s w i n g s a b o u t t h e s e c ti o n c e n t r o i d ; t h e

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190 POST-TENSIONED CONCRETE FLOORS

Table 8.4 I n c re a se i n s t r e s s i n u n b o n d e d t e n d o n s 6p, N / m m 2 (B a s ed o n B S 8 1 10 )

Rp = fp.Ap/fr

Lte / d 0 .05 0 .10 0 .15 0 .20 0 .25 0 .30 0 .35 0 .40 0 .45 0 .50

10 641 581 522 462 403 343

20 320 291 261 231 201 172

30 214 194 174 154 134 114

40 160 145 130 116 101 86

50 128 116 104 92 81 69

60 107 97 87 77 67 57

70 92 83 75 66 58 49

80 80 73 65 58 50 43

90 71 65 58 51 45 38

100 64 58 52 46 40 34

110 58 53 47 42 37 31

120 53 48 43 39 31

130 49 45 40 36

140 46 42 37

150 43 39

283 224

142 112

94 75

71 56

57 45

47 37

40 32

35 28

31 25

28

164

82

55

41

33

27

105

52

35

26

Reminder: BS 8110 limits the value of (fpe + &p) to a maximum of 0.7fp..

Concrete in tension 4 - - ~ Concrete in compression

E C U = 1

1 - e A j z t ) ~ ,-' L ,_ m / A c E c

A ~ E c

Y t - - ' - - - ~ tion centroid

p - _ .

Tendon centroid

~ , / ' 1 / I I I , I x

I

Concrete rupture 1

_ E 3 _ 1 _ . . s _ _

E p u

Figure 8.4 Strain history

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U L T IMA TE FL E X U R A L S TR E N G TH 191

cha nge in strain in the stra nd 13 equ als the strain in the concre te at the ten do n level.

e 2 = {P/ (A~Er + epZAJI~) (8.13b)

W hen the sec tion c racks and the mem ber i s over loaded to fa ilu re , the neu t ra l ax is

r ises to d n be low the ex t reme c om press ion f ib re , and the m axim um concre te s t ra in

reaches i t s l im i t ing value e~u, which is 0 .0035 ac cord ing to B S 8110 or 0 .003 as

given by ACI 318. The increase in tendon s t rain i s

e3 = ecu(dp - d . ) / d . (8.13c)

The to tal s t rain in the tendo n at fai lure (ep.) i s the sum of the three s t rains an d eq uals

e p u - - e I -~" e 2 -~- e 3 (8.14)

K n o w i n g t h e s t ra i n an d t h e t en d o n stressfp b, t he dep th o f the compress ion b lock

d~ can be ca lcu la ted by cons ider ing the s t a ti c equ i l ib r ium . Fo r s t a ti c equ i l ib r ium ,

the to ta l compress ion in concre te equa l s the to ta l t ens ion in the t endon .

pcubd~ = A p f p b (8.15)

The n the fl exura l s t reng th ( M .) is found by t ak ing the m om ent o f the t endon fo rce

abo u t the l ine o f ac t ion o f the com press ive fo rce in the concre te .

m u = A p f p b ( d p - d,,/2) (8.16)

8 .7 A n c h o r a g e z o n e

Pres t ress i s app l i ed to the concre te th rough an anchorage assembly , which

usual ly consis ts of a s teel cas t ing, tho ugh a fla t s teel p late ma y som et ime s be used.

The bea r ing a rea o f the ancho rage i s on ly a smal l p ro por t ion o f the concre te a rea

assoc ia ted wi th an anchorage . The p res t ress ing fo rce , concen t ra ted on the

bear ing a rea , sp reads th rough the concre te , and can be cons idered to be

un i fo rmly d i s t r ibu ted on the concre te sec t ion a t a su i t ab le d i s t ance f rom the

an ch o rag e .

Exper iment s to de te rmine the s t res ses genera ted in the anchorage zone have

been car r i ed ou t by app ly ing the fo rce th rough a th ick p la te bear ing on the

sur face o f a concre te b lock ; the w id th o f the p la te w as the sam e as tha t o f the

concre te b lock . In p rac t i ce , an anchorage i s smal l e r than the concre te sec t ion ,

cove ring only a pro po rt io n of i t. In assess ing the burs t ing s t resses , the force from

the ancho rage is as sum ed to sp read wi th in a p ri sm o f concre te hav ing the l a rges t

ava i l ab le a rea o f concre te p laced symm et r i ca l ly a rou nd the anchora ge , see F igure

8.6 . Th e co m bine d effect of a grou p of anc hor age s i s assessed by replacing the

grou p wi th an equ iva len t s ing le ancho rage , o f the same overa l l a rea and ca r ry ing

the same to ta l fo rce as the g roup , and cons ider ing a co r respond ing symmet r i ca l

p r i sm of concre te .

F igure 8 .5 shows the sp read o f the compress ive fo rce , app l i ed to a sm al l a rea on

the surface of a co ncrete b lock. Along the axis of the force, the s t ress is

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192

POST-TENSIONED CONCRETE FLOORS

• • • ; : .

E x p e r i m e n t s

/ / /

~,,~,~ Assumed

D i s t a n c e

~ 0 . 2 5 Oe ._1

1 2 - '

V " ~ [

(a) T e n s i l e s t r e s s d i s t r i b u t i o n on x - x

f I

f

S p a l l i n g f ~ ~ - - - - -

O a , / / 7 I - - ~ " " - - - ' - - - - "

I / / /

/ ~ 1 \ \ \ ~ ~ - - - - - - - "

p ' ." 2 - : _ : -

- - -

L.. De ._l

I - V l

(b) S p r e a d o f c o n c e n t r a te d l o a d in a c u b e

x

m

D~

i

F i g u r e 8 . 5

A n c h o r a g e z o n e s t r e s s e s

c o m p r e s s i v e , g r a d u a l l y r e d u c i n g i n i n t e n s i t y a w a y f r o m t h e l o a d e d f a c e . A t a

d i s t a n c e e q u a l t o t h e d e p t h o f t h e s y m m e t r i c a l c o n c r e t e p r i s m t h e s t r e s s c a n b e

a s s u m e d u n i f o r m o v e r t h e s e c t i o n , t h e i n t e n s i t y b e i n g P/Ar The s tress a t r ig h t

a n g l e s t o t h e c o m p r e s s i o n , o n a p l a n e t h r o u g h t h e a n c h o r a g e a x i s, i s t e n s il e ; th e

t e n s i o n s t a r t s a t a d i s t a n c e o f a b o u t o n e - t e n t h o f t h e d e p t h o f t h e s y m m e t r i c a l

b l o c k f r o m t h e s u r fa c e a n d p e a k s a t a b o u t o n e - q u a r t e r o f th e d e p t h o f th e b l o c k .

Th i s t ens i l e fo rce i s ca l l ed the bursting force. The d i s t r ibut io n o f t ens i l e s t ress

a lo n g the b lo ck a x i s is u sua l ly s im p l if i ed to a t r ia ng u la r fo rm a s sho w n in F ig ur e

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ULTIMATE FLEXURAL STRENGTH

193

8 .5 (a) . Tens ion a l so deve lops near the su r face su r round ing the p la te a rea ; th i s

force tends to cause spa l l ing of the conc rete surface in heavi ly s tressed m em bers .

However , in normal post - tens ioned f loors th is effect i s negl ig ible .

These experimental s tudies d i ffer f rom the t rue s i tuat ion in several ways .

F i r s t ly , an anch orag e cas t ing i s approx im ate ly con ica l in shape , wi th a s ti ff f lange

near one end ; i t t rans fers the p res t ress ing fo rce to the concre te th rough d i rec t

bear ing over the f lange a rea and th ro ugh the cone ac t ion . The p rop or t io n o f load

t rans fer red th rough the two ac t ions depends , among o ther fac to rs , on the

geom et ry o f the anch orag e cas t ing . The wedge ac t ion o f the con ica l shape usua l ly

increases the burs t ing force, and affects the pe ak p osi t ion of the bu rs t ing force. A

fur ther dev ia t ion f rom the t es t cond i t ions i s tha t ancho rages a re never p laced on

the su r face o f concre te ; ins t ead , they a re loca ted in ind iv idua l o r c om m on

pocket s , rang ing in dep th f rom 75 mm up to 150 mm (3 to 6 inches ) .

Addi t iona l ly , the exper imenta l s tud ies have been l a rge ly conf ined to two-

d imen s iona l m odel s , whereas the p rob lem is th ree d imen s iona l . In sp it e o f these

l imi ta tions , usefu l da ta ha ve been ob ta ine d and m os t o f the cur ren t des ign

m ethod s a re based on the resu lt s o f such s tud ies .

The burs t ing fo rce i s normal ly con ta ined by rod re in fo rcement p rov ided as a

ser ies of m ats or as a ser ies of c losed l inks beh ind the an cho rag e; the la t ter i s the

m ore e f fi cient. T he spa l ling t ens ion m ay be t ake n c are o f by a re in fo rcement m at

nea r the surface, but this stress, being sm all in post-ten sioned floors, is often ignored .

Anc horages a re expec ted to func t ion sa t i s fac to r ily fo r t endon fo rces o f a ro un d

95 % of the t end on s t reng th , tho ugh mo s t o f the s t and ards l imi t the j ack ing fo rce

to abou t 8 0% of the t endon s t reng th . I t m ay be permiss ib le to t ake the des ign

force fo r anc horag e zone re in fo rcement as equa l to the spec if ied j ack ing fo rce , bu t

i t i s preferable to des ign for a force of 90 to 100% of the ten do n s t reng th as th is

prov ides a useful reserve for a del iberate o r inadv erte nt o vers t ress ing of a tendon .

Th e s t ress in the reinforce m ent is l imi ted to 200 N /m m 2 (30000 ps i) .

In des igning the ancho rage z one, ver tical and horizo ntal sect ions are considered,

f i r s t t h ro u g h i n d i v i d u a l an ch o rag es an d t h en t h ro u g h an ch o rag e g ro u p s . Th e

block of conc rete , asso ciated wi th each sect ion, consis ts of the larges t pr ism

concen t r i c wi th the cen t ro id o f the ancho rage o r g roup o f ancho rages . In a

conc rete f loor , howev er , a ver t ical crack d ue to the bu rs t ing force is very unl ikely

to occur (except possib ly fo r anchorag es near a co rner ) , because the w id th o f the

f loor acts as a deep res t raint . Design is , therefore, carr ied out for horizontal

c rack ing a t a l l anchorages and ver t i ca l c rack ing near the s l ab corners .

F igure 8 .6 ind ica tes some of the a r rang em ents o f ancho rages used in

pos t - t ens ioned f loors and the concre te p r i sm assoc ia ted wi th each . The jagged

l ines on the diagram indicate the posi t ions where tens i le s t resses should be

checked fo r each a r ra ngem ent . In the case o f a s lab anc hora ge n ear a co rner ,

burs t ing s t res ses shou ld be checked on bo th hor izon ta l and ver t i ca l p lanes

pass ing th roug h the cen t re o f the bear ing p la te , whi le fo r an in te r io r anc horag e o f

a s lab , i t i s only necessary to check the horizontal p lane for burs t ing s t ress . The

ancho rages in the bea m are shown p os i t ioned in two s t aggered l ayers ; two checks

are necessary in th i s case : an ind iv idua l anchorage fo r burs t ing on a hor izon ta l

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194 POST-TENSIONEDCONCRETE FLOORS

Slab edge Beam

Figure

8.6

Typical anchorage arrangements

Table 8 .5

Burs t in9 t ens il e forc e

Da/Dc 0.20 0.30 0.40 0.50 0.60 0.70

F ~ P a

0.23 0.23 0.20 0.17 0.14 0.11

Da = Depth of anchorage

F b =

Bursting force

Dc = Dep th of concrete prism Pa = A ncho rage force

p l a n e t h r o u g h t h e c e n tr e o f t h e b e a r i n g p l a te , a n d t h e g r o u p b u r s t i n g o n a

h o r i z o n t a l p l a n e h al fw a y b e t w e e n t h e u p p e r a n d l o w e r a n c h o r a g e s . F o r t h e

l a t te r , th e u p p e r a n d t h e l o w e r a n c h o r a g e s a r e r e p l a c e d b y a n h y p o t h e t i c a l s i n g le

a n c h o r a g e c o v e r i n g th e c o m b i n e d a r e a o f t h e tw o a n c h o r a g e s a n d t h e g a p

b e t w e e n t h e m . F o r c i r c u l a r a n c h o r a g e s , t h e l o a d e d s u r fa c e c a n b e a s s u m e d t o b e a

s q u a r e o f a n e q u a l a r e a .

T a b l e 8 .5 , r e p r o d u c e d f r o m B S 8 1 1 0 , g iv e s a s im p l e a p p r o a c h t o a s s e s s in g t h e

b u r s t i n g f o r c e b a s e d o n a l i n e a r r e l a t i o n s h i p w i t h a n u p p e r l i m i t o f 0 . 2 3. A s af e

r u l e o f t h u m b f o r b u r s t i n g s te e l a r o u n d a s i n gl e te n d o n is to p r o v i d e 1 .2 m m 2 o f

l in k a r e a f o r e a c h k N o f a n c h o r a g e f o rc e .

Exam pl e 8 .1

C heck the s t r eng th o f t he f loo r in E xam ple 6 .4 in span 1 -2 ; i ts geom et ry is show n in

F igure 6 .9 . O the r da t a a r e :

Sect ion:

S e c ti o n d e p t h = 2 7 0 m m

F l a n g e w i d th = 1 0 0 0 m m

d e p t h = 1 10 m m

S erv iceab il i ty l oads , and shea r s a t su ppo r t 1"

D ead load = 2 .0 + se l f -weight 4 .3 = 6 .3 kN /m 2,S he ar = 26.35 kN

L ive load = 4 .0 S hea r = 17 .89

Shea r force f rom seco nd ary ef fec ts = 0 .30

T e n d o n s

F ina l fo rce = 560 kN (4 supe r s t r ands 15 .7 m m )

M i n i m u m h e ig h t = 4 0 m m

Heigh t a t 1 = 235 m m

Dista nce a = 1 .0 m

D is t ance b = 5 .49 m ( low est po in t )

S t r and a rea = 150 mm 2

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U L T IMA TE FL E X U R A L S TR E N G TH 195

B r e a k i n g l o a d = 2 65 k N

N o t e t h a t f o u r s u p e r s t r a n d s a r e u s e d i n p r e f e r en c e t o t h e s t a n d a r d s t r a n d b e c a u s e o f

t h e i r s u p e r i o r r e l a x a t i o n c h a r a c t e r i s t i c s .

B e i n g t h e e n d s p a n , s u p p o r t m o m e n t s a r e n o t r e qu i r ed .

( i ) S o lu t i on : I t S 8 1 1 0 , Unbond e d t e nd ons

R e q u i red mom e n t o f r es i s ta nce M r

S p a n l o a d = 1 . 4 x 6 . 3 + 1 .6 x 4 = 1 5 . 2 2 k N / m

S h e a r a t 1 = 1 . 4 x 2 6 . 3 5 + 1 .6 x 1 7 .8 9 + 0 . 3 = 6 5 . 8 1 k N

S p a n m o m e n t = 6 5. 81 2 (2 x 1 5 .2 2 ) = 1 4 2 . 3 0 k N m

M o m e n t is m a x i m u m a t 6 5. 81 / 15 .2 2 = 4 . 32 m f r o m s u p p o r t 1

Tendon he igh t

F r o m E q u a t i o n ( 5 . 1 9 ) ,

A 2 = r l / ( b 2 - a b ) =

( 2 3 5 - 4 0 ) / [ 1 0 0 0 x ( 5 . 4 9 2 - 1 .0 x 5 . 49 ) ] = 0 .0 0 7 9 1 / m

F r o m E q u a t i o n ( 4 . 1 7 b ) ,

y = 0 .0 0 7 9 x (5 .4 9 - 4 .3 2 ) 2 x 1 0 0 0 = 1 1 m m

T e n d o n h e i g h t a t 4 . 3 2 m f r o m 1 = 4 0 + 11 = 5 1 m m

Tens i le forc e

T e n d o n a r e a = 4 x 1 50 = 6 0 0 m m 2, F i n a l f o r c e = 5 6 0 k N

fp e = 5 6 0 0 0 0 / 6 0 0 = 9 3 3 N / m m 2

T h r e e s t r a n d s a r e s t o p p e d a t s u p p o r t 1, o n e c o n t i n u e s f o r t h e w h o l e 2 9 m l e n g t h

o f t h e s l a b .

E q u i v a l e n t t e n d o n l e n g t h = (3 x 1 0 + 1 x 2 9 ) / 4 = 1 4 .7 5 m

T w o p l a s t i c h i n g e s a re r e q u i r e d fo r s p a n 1 - 2 t o b e c o m e a m e c h a n i s m , a t s u p p o r t

a n d i n s p a n

d = 2 7 0 - 51 = 2 1 9 m m ,

s a y 2 2 0 m m t o a l l o w f o r r e i n f o r c e m e n t b a r s a t a l a r g e r l e v e r a r m

Lte /d

= ( 1 4 . 7 5 / 2 ) / 0 . 2 2 = 3 3 .5

Rp =fpuAp/ f~ubd = 2 6 5 x 1 0 0 0 x 4 / ( 4 0 x 1 0 0 0 x 2 2 0 ) = 0 . 1 2 0

F r o m E q u a t i o n ( 8 . 4 ) ,

f pb = 9 3 3 + ( 7 0 0 0 / 3 3 .5 ) ( 1 - - 1 .7 X 0 . 1 2 ) = 1 0 9 9 N / m m 2

0 .8 7 fp u = 0 . 8 7 x 2 6 5 x 1 0 0 0 / 1 5 0 = 1 5 3 7 > 1 0 9 9 O K

T e n d o n f o r ce = 1 09 9 x 6 0 0 / 1 0 0 0 = 6 5 9 .4 k N

P r o v i d e 2 N o . 1 0 m m d i a m e t e r b a r s a t b o t t o m , f r = 4 6 0 N / m m 2

A r e a = 1 57 m m 2 F o r c e = ( 15 7 x 4 6 0 / 1 0 0 0 ) / 1 .1 5 = 6 2 .8 k N m a x i m u m

T o t a l t e n s i o n = 6 2 .8 + 6 5 9 . 4 = 7 2 2 . 2 k N

N o t e t h a t i n c a l c u l a t i n g R p , th e s e c t i o n is t a k e n a s r e c t a n g u l a r o f w i d t h e q u a l t o t h e

f l a n g e w i d t h . T h i s i s v a l i d i f t h e n e u t r a l a x i s is w i t h i n t h e f l a n g e d e p t h .

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196 POST-TENSIONEDCONCRETE FLOORS

Section s trength

C o n c r e t e s t re s s i n s t re s s b l o c k = 0 . 45 x 4 0 = 1 8 N / m m 2

D e p t h o f c o m p r e s s i o n b l o c k = 6 9 9 .4 x 1 0 0 0 / ( 1 00 0 x 1 8 ) = 3 8 .9 m m

L e v e r a r m = 2 7 0 - 5 1 - 3 8 . 9 / 2 = 1 9 9 .6 m m f o r t e n d o n s

a n d 2 7 0 - 4 0 - 3 8 .9 / 2 = 2 1 0 .6 m m f o r r e i n f o r c e m e n t

S t r e n g t h = (6 5 9 . 4 x 1 9 9 .6 + 6 2 .8 x 2 1 0 . 6 ) / 1 0 0 0 = 1 4 4 .8 k N m > 1 4 2 . 3 0 O K

C h e c k s tr e ss i n r e i n f o r c e m e n t b a r s

d , = 3 8 . 9 / 0 . 9 = 4 3 . 2 m m

d i s t a n c e b e t w e e n b a r s a n d n e u t r a l ax i s = 2 7 0 - 4 0 - 4 3 .2 = 1 8 6 .8 m m

s t r a i n i n b a r s = 0 . 0 0 3 5 x 1 8 6 . 8 / 4 3 . 2 = 0 . 0 1 5 1

E s x s t r a i n = 2 0 0 x 1 0 0 0 x 0 . 0 1 5 1 = 3 0 0 0 N / m m 2 > 0 . 8 7 • 4 6 0

(i i ) Solution: ItS 8110, Bonded tendons

A s s u m e a fl a t d u c t a n d a l l o w 2 m m r is e o f t e n d o n i n d u c t

R e q u i r e d m o m e n t o f r e s i s ta n c e = 1 4 2.3 k N m a s f o r (i) a b o v e

T e n d o n h e i gh t = 51 m m

F i n a l t e n d o n s t re s s f pe - 9 3 3 N / m m 2

Tensile forc e

P r o v i d e 2 N o . 8 m m b a r s , f y

R p = 0 . 1 2 a s a b o v e , fpe/fpu

F r o m T a b l e 8 . 1 , fpb/O.87fpu

T e n d o n f o r c e

F o r c e i n 2 b a r s 8 m m d i a .

T o t a l t e n s i l e f o r c e

C o n c r e t e s t r e s s i n s t re s s b l o c k = 0 . 4 5 x 4 0

D e p t h o f c o m p r e s s i o n b l o c k = 9 5 3 x 1 0 0 0 / (1 0 0 0 x 1 8)

L e v e r a r m f o r t e n d o n s = 2 7 0 - 51 - 5 3 / 2

a n d f o r r o d r e i n f o r c e m e n t = 2 7 0 - 4 0 - 5 3 / 2

S t r e n g t h = ( 9 1 3 x 1 9 2 . 5 4 - 4 0 x 2 0 3 . 5 ) / 1 0 0 0

= 4 6 0 N / m m 2

= 9 3 3 / ( 2 6 5 x 1 0 0 0 / 1 5 0 ) = 0 . 5 2 8

= 0 . 9 9

= 4 x 0 . 9 9 x 2 6 5 x 0 . 8 7 = 9 1 3 k N

= 2 x 5 0 x 0 .8 7 x 4 6 0 / 1 0 0 0 = 4 0

= 9 1 3 4 - 4 0 = 9 5 3 k N

= 1 8 N / m m 2

= 5 3 m m

= 1 9 2 .5 m m

- 2 0 3 . 5 m m

= 1 8 3 .9 k N m .

> 1 4 2 . 3 0

S t r e s s i n r e i n f o r c e m e n t b a r s c a n b e c h e c k e d a s in ( i).

O~i) Solution: ACI 318, Unbonded tendons

f e' = 4 0 N / m m 2

Required mom ent o f res is tance

S p a n l o a d = 1 .4 x 6 . 3 4 - 1 .7 x 4 = 1 5 . 6 2 k N / m

S h e a r a t 1 = 1 .4 x 2 6 . 3 5 + 1 .7 x 17 . 8 9 4 - 0 . 3 = 6 7 . 6 0 k N

S p a n m o m e n t = 6 7 . 6 0 2 /( 2 x 1 5 .6 2 ) = 1 4 6 .2 8 k N m

M o m e n t is m a x i m u m a t 6 7 . 6 0 /1 5 . 6 2 = 4. 33 m f r o m s u p p o r t 1

O K

O K

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UL TIMA TE FLEXURAL STRENGTH 197

Tendon height

= 5 1 m m a t 4 . 33 m f r o m s u p p o r t 1

Tensile Force

fp e = 5 6 0 0 0 0 / 6 0 0 = 9 3 3 N / m m 2

dp = 2 7 0 - 5 1 = 2 1 9 m m

f pb = 9 3 3 + 7 0 + 4 0 X 1 0 0 0 X 2 1 9 / ( 1 0 0 X 4 X 1 5 0 ) = 1 1 4 9 N / m m 2

C h e c k : s p a n / d e p t h = 1 0 / 0 .2 7 = 3 7 > 3 5

L i m i t i n g s t re s s = 9 3 3 + 2 0 0 = 1 1 3 3 , U s e 1 1 33 N / m m 2

T e n d o n f o r c e = 1 1 33 x 4 x 1 5 0 / 1 0 0 0 = 6 7 9 . 8 k N

F o r c e i n b a r s = 1 5 7 x 4 6 0 / 1 0 0 0 = 7 2 .2

T o t a l t e n s i o n = 6 7 9 . 8 + 7 2 . 2 = 7 5 2 . 0

Section strength

C o n c r e t e s t re s s i n s t re s s b l o c k - 0 . 85 x 4 0 = 3 4 N / m m 2

D e p t h o f c o m p r e s s i o n b l o c k = 7 52 x 1 0 0 0 /( 3 4 x 1 00 0 ) = 2 2 m m

L e v e r a r m = 2 7 0 - 5 1 - 2 2 / 2 - 2 0 8 m m f o r t e n d o n s

a n d 2 7 0 - 4 0 - 2 2 /2 = 2 19 m m f o r r e i n f o rc e m e n t

S t r e n g t h = 0 . 9 x ( 6 79 . 8 x 0 . 2 0 8 + 7 2 .2 x 0 . 2 1 9 ) = 1 5 7 .2 k N m

> 1 4 6 . 2 8

C h e c k s t r e s s i n r o d r e i n f o r c e m e n t

f l l = 0 . 85 - 0 . 00 8 x ( 40 - 30 ) = 0 . 77

d . = 2 2 / 0 . 7 7 = 2 8 .6 m m

s t r a i n i n b a r s - 0 . 0 0 3 x ( 2 7 0 - 4 0 - 2 8 . 6 ) / 2 8 . 6 = 0 . 0 2 1

E s x s t r a i n - 2 0 0 x 1 0 0 0 x 0 . 0 2 1 = 4 2 0 0 N / m m 2 > 4 6 0

O K

O K

Examp le 8 .2

D e s i g n t h e a n c h o r a g e z o n e r e i n f o r c e m e n t f o r a 2 0 0 m m d e e p s l ab w i t h 1 25 m m x 7 5

m m a n c h o r a g e s ( 75 m m is t h e v er t ic a l d i m e n s i o n ) s p a c e d a t 5 0 0 m m c e n tr e s . T h e

j a c k i n g f o rc e in e a c h t e n d o n is 20 0 k N .

Solut ion

P a = 2 0 0 k N D a = 7 5 m m D c = 2 0 0 m m

D J D c = 0 . 3 7 5

I n t e r p o l a t i n g f r o m T a b l e 8 . 5 ,

Ft/P, = 0 . 2 0 7 5 F t = 0 . 2 0 7 5 x 2 0 0 = 4 1 . 5 k N

A st = 4 1 . 5 • 1 0 0 0 / 2 0 0 = 2 1 0 m m z

P r o v i d e 2 l i n k s (4 l eg s ) 8 m m d i a, a t 75 m m a n d 1 25 m m , b e h i n d t h e a n c h o r a g e f l an g e .

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9 D E F L E C T IO N A N D

V I B R A T I O N

Excessive f loor def lec t ion o r v ib ra t ion can cause a la rm to bu i ld ing o ccup an t s and

may resu l t in damage to non-s t ruc tu ra l e l ements such as par t i t ions o r f in i shes .

Sag o f roo f s labs can a l so cause pond ing o f ra inwa ter . The def lec t ion pe r fo rm ance

of pos t - t ens ioned f loors i s genera l ly be t t e r than tha t o f re in fo rced con cre te f loors ,

s ince the ac t ion o f the p res t ress ing fo rce causes mem bers to hav e a n in i t ia l

upwards curva tu re , o r camber , which coun terac t s the sag due to the se l f -weigh t

and l ive loads . H ow ever , the g rea te r s l enderness o f pos t - t ens ioned f loors m akes

them more suscep t ib le to v ib ra t ion p rob lems than re in fo rced f loors . Bo th

def lec t ion an d v ib ra t ion a re se rv iceab i li ty p rob lem s w hich shou ld , therefo re , be

checked us ing unfac to red loads .

9 .1 De f lec t ions

Ca lcu la t ion o f def lec t ions fo r a pos t - t ens ioned f loor is simpler th an fo r a

re in fo rced concre te member , s ince a pos t - t ens ioned sec t ion can usua l ly be

assum ed to be uncra cked , so tha t the p roper t i es o f the g ross concre te sec t ion can

be used in the calculat ions . (St r ic t ly speaking, for s labs wi th bonded tendons the

t rans fo rmed sec t ion shou ld be used , whi l e fo r those wi th unbonded t endons the

ne t sec t ion shou ld be used . ) Other as sumpt ions commonly made in def l ec t ion

analys is are that :

9 the force in a tend on is con stan t a lon g i t s length

9 the s lope o f the t endo ns is smal l , so tha t the hor izo n ta l com pon en t o f the

pres t ress i s constant

9 any chan ge in pres tress in the tend ons caused by the deflect ions m ay be neglected

Def lec t ion ca lcu la t ions fo r beams and one-way spann ing s l abs a re based on

e last ic beam theory an d a re re l a t ive ly s t ra igh t fo rwa rd , i f som ew hat l abo r ious .

Fo r tw o-w ay spann ing s labs , since rea l is t ic mode l l ing o f the s t ruc tu ra l b eha v iou r

can bec om e ex t remely com plex , it is usua l to ado p t one o f a num ber o f s impl i fi ed

appro ache s . The resu l ting loss o f accurac y is cons idered accep tab le b ecause o f the

h igh degree o f unce r t a in ty inh eren t in such ca lcu la tions . The des igner is un l ike ly

to have an a ccura te know ledge o f the mo dulus o f e l ast ic i ty and c reep p roper t i es

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Table 9.1 D ef lec t ion l imi t s for f loors

DEFLECTION AND VlBR AT ION

199

Member type

Limi t

Deflection tO consider

ACI318 Roofs not support ing any

non-structural elements

Floors not suppor t ing any

non-structural elements

Floors or roofs support ing

bri t t le non-structural elements

Floors or roofs support ing

non-bri t t le non-structural

elements

L/180

L/360

L/480

L/240

Immedia t e

under l ive load

Deflect ion occurring

after instal lat ion

of non-structural

elements

BS811 0 Any visible structural mem ber

Floors or roofs support ing

bri t t le non-structural elements

Floors or roofs support ing

non-bri t t le non-structural

elements

L/250

Smaller of

L/500 or 20

m m

Smaller of

L/350 or 20

m m

Total deflection

Deflect ion oc curring

after instal lat ion of

non-structural elements

o f t h e co n c r e t e , o r o f t h e ex ac t l o ad s t h e s t r u c t u r e i s l i k el y to ca r r y o v e r a p e r i o d o f

s ev e r a l y ea r s . F i e l d mea s u r e m en t s o f f l o o r d e f l ec ti o n s s h o w ex t r eme l y w i d e

s ca t t e r , ev en i n ap p a r en t l y i d en t i c a l f l o o r s w i t h i n t h e s ame b u i l d i n g ( A C I

C o mmi t t ee 4 3 5 , 1 9 9 1 ) .

D e f l e c ti o n l im i t s r e c o m m e n d e d i n t h e B r it i sh a n d A m e r i c a n c o n c r e t e c o d e s a r e

s h o w n i n T a b l e 9 . 1 , i n w h i c h L i s t h e s h o r t e r s p a n l e n g t h . T h e r e c o m m e n d e d

l i m i t s f o r r o o f s a r e n o t i n t en d ed t o s a f eg u a r d ag a i n s t p o n d i n g . C h eck i n g ag a i n s t

t h e s e c r it e r i a is l i ke l y t o r eq u i r e t h e ca l c u l a t i o n o f b o t h s h o r t an d l o n g - t e r m

d e f l ec t i o n s, t h o u g h i n m o s t i n s t an ces t h e l o n g - t e r m b eh av i o u r i s t h e m o r e c r i t i c a l,

s i nce it a ff ec t s t h e b e h av i o u r o f f in i sh e s an d p a r t i t i o n s . I n o r d e r t o c a l cu l a t e

r ea l i s t i c l o n g - t e r m d e f l ec t i o n s , i t i s e s s en t i a l t h a t t h e l o ad i n g h i s t o r y an d t h e

m ag n i t u d e o f t h e s u s t a i n ed l o ad a r e ca r e fu l l y co n s i d e r ed .

9.1 .1 In f luence o f load ing h is tory

D ef l ec t io n s o f p o s t - t en s i o n ed s l abs h av e t w o co m p o n en t s ; a d o w n w ar d s d e f lec t io n

cau s ed b y t h e s e l f - w e i g h t an d an y i mp o s ed l o ad s o n t h e s l ab , an d an u p w ar d s

cam b er d u e t o t h e p r e st r e s s . U n d e r s h o r t - t e r m l o ad i n g t h e s e ef fec ts a r e c a l cu l a t ed

e l a s t i c a l l y . F o r l o n g - t e r m l o ad i n g , t h e ad d i t i o n a l d e f l ec t i o n d u e t o c r eep i s

c a l c u l a t e d b y m u l t i p l y i n g t h e s h o r t - t e r m d e f l e c t i o n b y a n a p p r o p r i a t e c r e e p

co e ff ic i en t C c. T h e cam b er m u s t a l s o b e mo d i f i ed t o t ak e acc o u n t o f an y p r e s t r e s s

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200

POST-TENSIONEDCONCRETE FLOORS

losses . The to tal def lect ion is then the su m of the sh ort - te rm an d creep deflect ions .

Creep coeff icients have been discussed in detai l in Chapter 2 . In addi t ion to

env i ronmenta l fac to rs , the c reep behav iour i s dependen t on the average load

in tens i ty and the age a t which loads were app l i ed . The load ing h i s to ry i s ,

therefo re , impor tan t .

The s ignif icance of the loadin g his tory ca n bes t be i llus t rated by conside ring a

typical f loor . Im m edia tely a f ter s t ress ing the deflect ion is l ikely to be ne gat ive ( i.e.

upwards ) because the upwards -ac t ing ba lanc ing fo rce due to the curved t endons

is grea ter th an the sel f-weight of the f loor . I f no con st ruc t ion loads or l ive loads

are appl ied for some m onth s , then the creep deflect ion wi l l a lso be negat ive. I t i s

thus poss ible tha t , even af ter the app l icat ion of l ive loads , the re sul tan t def lect ion

wil l be negat ive. I f , however , const ruct ion loads are appl ied short ly af ter

s t ress ing, and are quickly fol lowed by l ive loads , then the long-term deflect ion

will be posit ive.

As can be seen from the deflect ion l imi ts in Table 9 .1 , i t may be necessary to

calculate def lect ions at several s tages w i thin the l i fe of a s t ructu re. I f detai led

in fo rm at ion o n the cons t ruc t ion t imetab le , inc lud ing de ta i ls o f bac k-p rop p ing

an d t ime of app l icat ion of l ive loads , i s avai lable , th en th is can be u sed in the

def l ec t ion ca lcu la t ions . Otherwise , i t i s reasonab le to approx imate the load ing

his tory by the fol lowing three-s tage sequence:

1 . The f loor is ba ck -pr op pe d, an d so effect ively carr ies no load, for betw een on e

and four weeks . The exac t du ra t ion o f th is phase wi ll dep end on the

cons t ruc t ion p rogramme, bu t two weeks i s a reasonab le average .

2 . Thereaf t e r , t he f loor car r ies it s own weigh t and a cons t ruc t ion load w hich is

typical ly 25 % of i ts sel f-weight , tho ug h th is can be subject to large va riat ions .

Af te r approx imate ly th ree months , f in i shes a re added , se rv ices a re ins ta l l ed

and the cons t ruc t ion load i s remov ed . Ho we ver , s ince the weigh t o f f in i shes

e tc . is l ike ly to be o f the same o rde r as the con s t ruc t ion load , i t may be assum ed

tha t the load in tens i ty remains unchanged .

3 . Once the bui ld ing is f in ished, i t i s occupied and addi t ional l ive loads are

app l i ed . F or a l a rge bu i ld ing , th i s m ay t ake over a year , bu t s ix m onth s i s m ore

typical .

9.1 .2 L ive load in tens i ty

As well as the t ime at which th ey are app l ied, the m agn i tude of the loads i s

impor tan t . Whi le the se l f -weigh t and o ther permanen t loads can usua l ly be

accura te ly assessed , the va lue o f live load w hich m ay be regarded as a sus ta ined

long- te rm loa d is m uch ha rder to de te rmine . Calcu la t ing the long- te rm def lec t ion

on the assu m pt ion tha t the m axim um l ive load is app l i ed a t a l l t imes y ie lds va lues

of def lec t ion co ns iderab ly l a rger tha n the ac tua l , m easured va lues, and is as

unrea l i s t i c as as suming no l ive load a t a l l . BS 8110 : Par t 2 recommends tha t

expected values shou ld be used to g ive a bes t es t imate of the l ikely be ha vio ur of

the s t ructure whi le , for calculat ions to sat i s fy a par t icular l imi t s ta te , upper or

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DEFLECTION AND VIBRATION 201

l ower bound va lues shou ld be used , depend ing on whether o r no t the e f fec t i s

beneficial .

L i v e l o ad can b e co n s i d e red a s co mp r i s i n g a p e rman en t o r n ea r -p e rman en t

com pon en t wp due to i tems such as o ff ice fu rn i tu re , and a t rans ien t com pon en t w

due to p ersonnel , veh ic les et c . In es t imat ing long- te rm behav iour , i t is reasonab le

to t rans fo rm these loads in to an equ iva len t perm ane n t ly sus ta ined loa d w~ on the

bas is o f the du ra t io n fo r which the t rans ien t load is p resen t:

W s = W p - t - w tR t ( 9 . 1 )

whe re R t i s the fract ion of the to ta l t im e for which the t ran s ient load is present .

N o t e t h a t w v and w are the ac tua l es t imated loads , no t the recommended va lues

for f lexure des ign. Their sum wil l not usual ly equal the nominal l ive loading

specified for a floor.

App lying th is formula to publ ic car parks , a typical vehicle weighs approx im ately

1300 kg an d occup ies a f loor a rea o f a ro und 13 m 2 , g iv ing an average t rans ien t

load in tensi ty w of 1 .0 kN /m 2 (20 psf) . Assum ing n o f in ish o n the f loor , wp is zero .

In shopp ing a reas a car pa rk is usua l ly occup ied for a rou nd t en hours per day , s ix

day s per week ( i.e. 35% of the t ime), bu t i s unl ikely to be ful ly loaded for the

who le of th is t ime. Eve n i f ful l occ upa ncy durin g these h our s i s assum ed, th is g ives

R t - 0 . 3 5 and hence an equ iva len t sus ta ined l ive load o f 0 .35 kN /m 2 (7 ps f) is

appropriate for the parking bays . The sus tained load in the ais les i s negl ig ible; a

nom ina l va lue o f, say, 0.1 kN /m 2 (2 ps f ) ma y be a ssume d i f des ired . C ar parks in

long- te rm use , such as a t a i rpor t s , may be occup ied fo r twen ty hours per day ,

seven da ys per week (R t = 0 .85), g iving an equiv alent sus taine d load of

0 .85 kN /m 2 (17 psf) in the pa rkin g bays and , say, 0 .25 kN /m 2 (5 psf) in the ais les .

Clear ly , these values fal l wel l below the recommended load in tensi ty of

2 .5 kN /m 2 (50 ps f ) fo r car parks in the U ni ted King dom . A sus ta ined load o f 30%

of the recom m ende d des ign load , a pp l i ed over the en t i re f loor a rea inc lud ing the

aisles, wo uld, therefore, be am ple for es t ima t ing long -term deflections in car park s .

The reco m m ende d l ive load fo r o f f i ce f loors , exc lud ing s to rage a reas , i s

2 .5 kN /m 2 (50 ps f ), b u t the cur ren t t rend is toward s 4 .0 kN /m 2 (80 ps f) . There is

no ev idence tha t a load as l a rge as 4 .0 kN /m 2 is ever app l i ed to a no rm al o f fi ce

f loor , wi th surveys suggest ing that a value of 1 .25 k N /m 2 (25 psf) , excluding

par t i t ions and serv ices , i s appropr ia t e (Mi tche l l and Woodgate , 1971 ; Choi ,

1992). Since the t ran s ient loads du e to perso nne l in offices are smal l com pa red to

the w eight of furnishings , i t i s reas ona ble to ta ke R t = 1 .0 in spi te of the fact tha t

m ost offices are fu lly occu pied only ab ou t 25 % of the t ime. There fore, a sus tain ed

l ive load of 1 .25 k N /m 2 (25 psf) i s suitable for lon g-ter m deflect ion calculat ions .

Publ ic bui ld ings such as ar t gal ler ies , theat res and cinemas are l ikely to be

occu pied for 25 % of the t ime o r less. In th is case, the t ran s ient load due to c row ds

of peop le i s the l a rges t com pon en t o f the live load . A sus ta ined load ing o f 30% of

the des ign l ive load i s a reasonab le upper bound .

W areho uses and s to rage a reas a re expec ted to be fu lly loaded m os t o f the t ime

and the deflect ion in such bu i ld ings sho uld be calculated for the full des ign load.

Excep t ing wa rehouses , i t can be a rgue d tha t 30% of the des ign live load

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202 POST-TENSIONEDCONCRETE FL OORS

represen t s a sens ib le u pper bo un d o f load in tens ity fo r use in long- te rm def l ec t ion

ca lcu la tions . I t shou ld be reme m bered tha t , s ince upw ards as well as dow nw ards

def l ec t ions can cause p rob lems , too h igh a load ing i s as e r roneous as too low.

9.1.3 One-way spanning slabs

Sh ort - te rm deflect ions of one -w ay s labs can be eas ily calculated us ing elas t ic

beam theory . The no rm al p roc edure is to cons ider the loads ac t ing on a one met re

wide s tr ip o f s lab . The to ta l do wn wa rds load is the sum of the se lf -weigh t and any

app l i ed loads p resen t , whi l e the equ iva len t load cor respond ing to the p res t ress

norm al ly ac t s upwa rds . A de ta i led d i scuss ion o f equ iva len t loads due to var ious

tendo n p ro fi les is g iven in Cha p ter 5 . In genera l , parabo l i c p ro f il es co r respo nd to

un i fo rmly d i s t r ibu ted loads and harped p ro f i l es co r respond to concen t ra ted

loads at the poin t of the reversal in s lope. Fo r def lect ion calculat ions i t i s

accep tab le to ignore the smal l reverse parabo las near the suppor t s as these wi l l

mo s t ly cause compress ion o f the suppor t s ra the r tha n ben d ing o f the f loor span .

Simi la r ly , the smal l parabo las under the load po in t s in harped p ro f i l es can be

neglected.

Knowing the loads , and t rea t ing co lumns o r wal l s as s imple suppor t s , t he

reac t ions can be ca lcu la ted and a bend ing moment express ion se t up in t e rms o f

the d i s tance x f rom one end o f the s lab . The cu rva tu re o f the me m ber i s re l a t ed to

t h e b en d i n g mo men t b y

d2y /dx 2 = M(x ) / (Er162

(9.2)

and the displacement y i s found by in tegrat ing (9 .2) twice, wi th the resul t ing

co n s t an t s o f i n t eg ra ti o n d e t e rmi n ed f ro m t h e b o u n d a ry co n d i t io n s . U n d e r

uniform load ing the pea k de flect ion o f a s lab panel , 6 , i s the value o fy a t i ts centre .

I t may be necessary to ca lcu la te def l ec t ions in bo th ou ter and inner pane l s in

order to de te rmine the wors t case .

As an al ternat ive to Equat ion (9 .2) , i t may be poss ible to obtain a suff icient ly

accura te es t imate o f the defl ec tion us ing wel l -known s tand ard resu l ts such as

those t abu la ted in the S tee l Des igne rs ' M an ual (SCI , 1992) . Som e par t i cu la r ly

useful results are:

s imp l y s u p p o r t ed b eam: 6 = 5wL4/(384EI)

f ixed-end beam" & =

wL4 / (384EI )

cant i lever: 6 = wL4 / (8EI )

( 9 . 3 a )

(9.3b)

(9.3c)

The p eak def lec t ion o f an in te r io r pane l in a m ul t i span s l ab can be c losely

approx imated by us ing the resu l t fo r a f ixed-ended beam.

Having ca lcu la ted the def l ec t ions due to the app l i ed loads and due to the

pres t ress , the net short - te rm deflect ion is s imply the sum o f the two values . Th e

long- te rm def lec t ion i s found by mod i fy ing the def l ec tion due to the app l i ed loa ds

to t ake acc oun t o f c reep , and tha t due to the p rest ress to accoun t fo r bo th c reep

and loss of pres t ress .

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DEFLECTION AND VIBR ATION 203

9.1.4 Tw o-way span ning slabs

Acc ura te ca lcu la t ion o f def lec tions in two-wa y span n ing f loors i s fa r more

difficult to achieve than for one-wa y spanning sys tems, due to the three-dimen sional

na tu re o f the p rob lem and the d if fe rence in p res tress ing a r ran gem ents in the two

span d i rec t ions . A thoro ugh review of the num erou s semi -ana ly t i ca l and

ap p ro x i ma t e ca l cu l a t i o n me t h o d s w h i ch h av e b een p ro p o s ed i s g i v en b y ACI

Commi t t ee 435 (1974) . Only the mos t wide ly used methods a re d i scussed here .

Where s l abs a re suppor ted on beams , the beam def lec t ion o f t en makes a

s ignificant co ntr ib ut io n to the to tal s lab deflect ion. The refore, for the purp oses of

def lec t ion ca lcu la t ions , tw o-w ay s l abs shou ld be t ak en to inc lude s l abs suppo r ted

on beams , even though these a re normal ly t rea ted as one-way spann ing in o ther

par ts o f the des ign process .

C lassica l p late theory

Fo r s qu a re o r r ec t an g u l a r p an e ls w h o s e ed ges can r ea s o n ab l y b e ap p ro x i ma t ed

as f ixed or p inned, formulae based on class ical e las t ic p late theory can be used.

The c en t ra l def lec t ion o f a rec tang u lar p la t e o f p lan d imens ions a x b (where

b > a) unde r a un i fo rm load w per un i t a rea is g iven by

t~ = ~ l w a 4 / H (9.4)

wh ere ct~ = a factor depen ding on the aspec t rat io

H = the plate bending s t i f fness

= ( E 0 3 ) / [ 1 2 ( 1

- v2)-I

D = the plate th ickness .

For a central point load HI, the deflect ion is

6 = ~1 W a 3 / H (9.5)

Tim oshe nko and W oinow sky-K r ieger (1959) g ive t abu la ted va lues o f ~1 fo r

num erou s com bina t ion s o f edge cond i t ions , o f which a few s imple cases a re

reproduced in Tab le 9 .2 . Unfor tuna te ly , p la t e so lu t ions a re ra re ly adequate fo r

f loor s l abs as the boundary cond i t ions a re usua l ly cons iderab ly more complex

tha n s imple pins or f ix i ty . How eve r , these solut ions are useful as par t of the

f rame-and-s lab method , d i scussed l a t e r .

Timoshenko and Woinowsky-Kr ieger a l so g ive an ana lys i s fo r an in te r io r

pane l wi th in a f i a t s l ab . Assuming the co lumns p rov ide po in t suppor t on ly , the

cen t ra l def l ec t ion under a un i fo rm load i s g iven by

t~ = o~2 wa 4/H

(9.6)

where t~2 is tak en from Tab le 9 .2 .

This form ula i s sui table for in ter ior p anels o f large, uni form ly loade d f loors ,

bu t canno t be app l i ed to o ther cases (ou te r pane l s , o r non-un i fo rm loads ) which

are of ten more cr i t ical .

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2 0 4

POST-TENSIONED CONCRETE FLOORS

Table 9.2 Values of ~1 and a 2 or rectangular panels

b/a

O~ 1 O~ 1 O~1 O~2

F ix ed e d g e s P i n n e d d g e s P i n n e d d g e s In ter io r pan el

un ifo rm l o a d u n i f o r m o a d c e n t r a l oint u n i f o r m oad

load

1.0 0.00126 0.00406 0.01160 0.00581

1.1 0.00150 0.00485 0.01265 0.00713

1.2 0.00172 0.00564 0.01353 0.00888

1.3 0.00191 0.00638 0.01422 0.01105

1.4 0.00207 0.00705 0.01484 0.01375

1.5 0.00220 0.00772 0.01530 0.01706

1.6 0.00230 0.00830 0.01570 0.02150

1.7 0.00238 0.00883 0.01600 0.02664

1.8 0.00245 0.00931 0.01620 0.03254

1.9 0.00249 0.00974 0.01636 0.03623

2.0 0.00254 0.01013 0.01651 0.04672

C r o ss in g b e a m m e t h o d s

There a re seve ra l m e th od s ava i lab le based on the idea l iza t ion of a s lab as two

pe r pe nd ic u la r s e ts o f be a m s , o f wh ic h the S imp le s t a nd m os t w ide ly u se d i s

desc r ibed he re (Rice and K ulk a , 1960). S t r ips of the s lab in each d i rec t ion a re

t rea ted as beams , and the cen t ra l de f lec t ion is then g iven by the sum of the

m idsp a n de f l e ct ions o f a c o lum n s t r i p spa nn ing in one d i r e c t ion a n d a m idd le

s t r i p spa nn ing pe r pe nd ic u la r t o i t , F igu r e 9 .1 . B e a ms r unn ing a long the c o lumn

l ines can be eas ily inc luded in the ca lcu la t ion of the equiva len t b eam prop er t ie s

for the co lu m n s t r ips . The de f lec t ions of the equiva len t beam s can be ca lcu la ted

us ing e la s t ic me thods or s imple formulae , a s d iscussed in Sec t ion 9 .1 .3 .

T he c o r r e c t p r op o r t io n in g o f l oa d be tw e e n the e q u iva le n t be a m s t r ip s c a n be

d i ff ic u lt ; u sua l ly t he c o lum n s t ri p s o f a s l a b c a r ry r a the r l a r ge r m om e n t s t ha n the

midd le s t ri p s . If p r io r a na ly s is ha s a l r e a dy y i e lde d va lue s o f t he m om e n t s on

c o lum n a n d midd le s t r ip s , t he n the se ma y be u se d d i r e ct ly . I f on ly t he a ve r a ge

loa d on the f l oo r i s known , t he n th i s mus t be d i s t r i bu t e d a pp r op r i a t e ly . F o r

r e in f o r c e d c onc r e t e f l a t s l a bs , A C I 318 r e c omme nds tha t c o lumn s t r i p s be

a s sume d to c a r r y 1 .2 t ime s the a ve r a ge loa d a nd midd le s t r i p s 0 .8 t ime s the

a ve r a ge . If no m or e a c c u r a t e i n f o r m a t ion i s a va il a b l e , t he n th i s r ough gu ide c a n

a lso be used for pos t - ten s ion ed s labs . I t is suf fic ien t ly accura te to a ssu m e th a t th e

loa d i s un i f o r m a long the l e ng th o f e a c h s t r i p , neg le c ting a ny r e duc t ions due to

sha r ing o f t he l oa d whe r e t he tw o s t ri p s me e t .

T h i s s imp l i f i e d me thod obv ious ly l a c ks r i gou r , bu t ha s be e n f ound to g ive

reaso nab ly re l iab le e s t ima tes o f s lab de f lec tions .

F r a m e - a n d - s l a b m e t h o d

T hi s me thod i s q u i t e s imi l a r t o t he c r o s s ing be a m me thod d i sc us se d a bove , bu t

t a ke s r a the r be t t e r a c c ou n t o f t he two- w a y spa nn in g be h a v iou r o f t he c e n t r a l

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DEFLECTION AND VIBRATION

2 0 5

.- Co lum n l ine

J

~ A 1

s

~ B 2

(~A2

Midspan deflection 5 = 8A1

+ (~A2 =

~B1

+ ~B2

Figure 9.1 Crossing beam method

por t ion of a s lab panel . Com par i so ns wi th measure d values sugges t tha t the

method s l igh t ly underes t imates ac tual def lec t ions (Ni l son and Wal ters , 1975) .

As with the crossing beam method, the s lab is f i rs t spl i t into s t r ips along the

colum n l ines , F igure 9 .2(a). Eac h c o lum n s t r ip i s assum ed to have a w id th of 0 .4

t imes the re levant span d imen s ion , so that the edges of the s t r ips corre spond

roughly to l ines of contraflexure in the s lab. (Of course, the exact l ines of

cont raf lexure are hard to determine, and are a lmos t cer ta in ly not s t ra ight . ) The

rectang ular panel b ou nde d by the co lumn s t rips is t rea ted as an e las ti c p la te wi th

s imply suppor ted edges . Each co lumn s t r ip i s assumed to car ry uni formly

dis t r ibu ted dead and imposed loads , together wi th l ine loads a long i t s edges

corresp onding to the supp or t react ions of the in ter ior panel . The to ta l cent ra l

def lec t ion of the s lab panel is then fou nd as the sum of three com pon ents , F igure

9.2(b):

9 the centrel ine deflect ion of a colum n str ip sp ann ing b etwe en co lum ns, 61,

calcu la ted us ing e las t i c beam theory;

9 the deflect ion of the edge of the column str ip relat ive to i ts centrel ine, 62

9 the cent ra l def lect ion of the re c tangu lar panel between colum n s t rips , 63,

calculated using the Equat ions (9.4) and (9.5) .

F i n i t e e l e m e n t m e t h o d

Numerous sophis t i ca ted f in i te e lement packages are now avai lab le , making i t a

re lat ively s imple m at ter to genera te and analyse a s imple p la te or gr i l lage mod el

of a concre te f loor . The d i ff icu lty of choos ing a ppro pr ia te bo un da ry condi t ions a t

the edges of ind iv idual panels m eans that i t wo uld be necessary to create a model

of an ent i re f loor, or at least of a substa nt ial a rea, in order to hav e confidence in

the accuracy of the resu l ts . The f in ite e lement m odel requi red wo uld thu s be qui te

large and the cost of the analysis w ould no t be negl igible.

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206

POST-TENSIONED CONCR ETE FLOORS

(b)

\

\

\

d \

o \

(a)

b

a

~ ) 1 . L ~ ) ~ 3

. . . .

Figure 9 .2

Frame-and-slab method: (a) Division of slab into column str ips an d ce ntral

panels (b) Section through slab at midspan, showing component deflect ions

C h o i c e o f m e t h o d

The cho ice o f which o f the above m ethod s to use in a par t i cu la r ap p l i ca t ion

depends on the s t ruc tu ra l de ta i l s and , to an ex ten t , on persona l p re ference . The

class ical p late so lut ion is only l ikely to be useful in qui te a sma l l num be r of cases .

Fo r mo s t s t r u c t u re s b o t h t h e c ro s s i n g b eam ap p ro ach an d t h e f r ame-an d - s l ab

m etho d p rov ide re asonab le es t imates o f def lec t ions , wi th the fo rmer usua l ly

preferred on the g rou nds of i ts s impl ici ty . Given the level of un cer tainty

sur ro und ing the ca lcu la tions , the soph i s t i ca t ion o f the fini te e l ement m etho d is

only l ikely to be jus t i fied for s t ructure s w i th ir reg ular o r unu sual geom etr ies . A

com para t ive exa mple o f def lec t ion ca lcu la t ions us ing these meth ods i s g iven a t

the end of th is chapter .

9 . 2 V i b r a t i o n

Concern over v ib ra t ion p rob lems has t rad i t iona l ly been focused on l igh tweigh t

f loors such as those o f t imber o r com pos i t e s t ee l -concre te cons t ruc t ion . In recen t

years , however , i t has become increas ing ly c lear tha t v ib ra t ions may a l so be an

im po r tan t des ign c r i t e rion in concre te f loors , par t i cu la r ly those wi th long sp ans

an d h i g h s l en d e rn es s r a t i o s . Th e mo s t co mmo n p ro b l em i s an n o y an ce o r

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D E FL E C T I O N A N D V I B R A T I O N 207

di scomfor t to bu i ld ing o ccupa n t s caused b y v ib ra t ions se t up by hu m an foo tfa ll .

I t shou ld be em phas ized th a t the overwhelm ing major i ty o f pos t - t ens ioned

f loors do no t g ive r i se to any v ib ra t ion p rob lems and tha t , i n genera l , t he

v ib ra t iona l p er fo rm ance o f such f loors is fa r be t t e r than tha t o f o ther cons t ruc t ion

types . Ne vertheless , as post - ten s ione d f loors of increas ing s lenderne ss are

cons t ruc ted the impo r tance o f v ib ra t ion increases . I t is , t herefo re , recom m ende d

tha t some s imple o rder -o f -magni tude checks a re per fo rmed a t the des ign s t age ,

wi th a m ore d etai led an alys is carr ied ou t only i f the s imple checks suggest that

t h e r e may b e a p ro b l em.

In assessing the accep tab i l i ty o f a f loor v ib ra t ion , bo th the f requency an d

am pl i tude o f the mot ion a re im por ta n t . Usua l ly , am pl i tude i s expressed in t e rms

of the peak acce le ra t ion , thou gh occas iona l ly o the r m easures such as ve loc ity o r

d i sp lacement a re used . In genera l , humans a re more respons ive to qu i t e low

frequency v ib ra t ions , so tha t g rea te r acce le ra t ions a re accep tab le a t h igh

frequencies than at lower ones .

A com m on way o f as sess ing f loor v ib ra t ions in the m ajor i ty o f bu i ld ings i s the

use of percep t ib i l ity scales . These scales are sui table for m ost c onv ent io nal forms

of bui ld ing occ up anc y (off ices, apa rtm ents , e tc .) . How eve r , for a very sensi tive

occupancy such as a microe lec t ron ics workshop , a lower v ib ra t ion th resho ld

m ay be app ropr ia t e . Such cases a re l ike ly to requ i re spec ial is t adv ice , and a re no t

discussed fur ther here.

One of the most useful of the percept ib i l i ty scales i s the one given in the

C ana d ian s t ee lwork code , CA N3 -S 16 .1 -M89, show n in F igure 9 .3 . Th i s cons is t s

of a series of i so-perc ept ib i l i ty l ines cor resp ond ing to d i f fe rent types o f exc i ta t ion

and leve ls o f dam ping . Curves fo r o ther d am ping levels can be found by

in te rpo la t ion . Abov e 8 Hz, the l ines slope upward s , re f l ec t ing the g rea te r h um an

toleranc e of h igher frequ ency vib rat ions . F or a g iven f loor , a poin t on the scale i s

found c or respo nd ing to the na tu ra l f requency o f the f loor and the peak

acce lerat ion ca used by the act iv i ty . I f the po int l ies abo ve the re levant

iso-percept ib i l i ty l ine then the vibrat ion is excess ive, o therwise i t i s acceptable .

Th e con t inuous v ibra tion l ine is in tend ed for assessme nt o f v ibrat ion s las t ing for a t

least ten cycles, while the walking v ibrat ion l ines are used to assess a floor 's

accep tab i l i ty fo r walk ing v ib ra t ions based on the peak acce le ra t ion caused by a

heel dro p tes t . The tes t is perfo rm ed b y a person of avera ge weight r i s ing to the

bal ls of thei r feet, then dro pp ing onto thei r heels a t the centre of the f loor .

The p ercep t ib i l i ty l ines show n in Figu re 9 .3 are for relat ively quiet occ upanc ies

such as off ices , where the occupants are l ikely to not ice qui te smal l v ibrat ions .

Fo r ac t ive occupancies such as car parks a nd sho pp ing cen t res , Pern ica a nd Al len

(1982) recommend that these accelerat ion l imi ts be increased by a factor of 3 .

In order to assess the f loor us ing such a scale , i t i s necessary to determine the

f requency and dam ping charac te r i st i cs o f the f loor and i ts response to wa lk ing o r

hee l d rop exc i t a t ion . Whi le a l l t hese parameters a re qu i t e easy to de te rmine by

tes t ing, they are d i ff icul t to predict accurately at the des ign s tage. The bes t

ava i l ab le methods and su i t ab le numer ica l va lues a re d i scussed in the fo l lowing

sections.

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POST-TENSIONED CONCRETE FLOORS

100

3 0

1 0

3

t -

O

(1)

U

0

-~ 0.3

13.

0.1

0 . 3 -

0.01

1

, Walking vibration

_ Walking vibration

Walking vib rat ion ~ '~''kee

Continuous vibration

I I I i I I I

2 3 5 10 20 30 50

Frequency (Hz)

100

Figure 9.3

The CSA Vibration Perceptib i l i ty Scale

The re la t ionsh ip be tween f loor na tu ra l f requency and v ib ra t ion ampl i tude i s a

com plex one . F lo ors wi th a h igh ra t io o f mass to s ti ffness have low n a tu r a l

f requencies , so tha t resonan ce wi th wa lk ing f requencies i s l ike ly . Ho we ver , such

f loors a lso have h igh inert i a , mean ing tha t a g iven ma gni tud e o f load ing wi ll

produce a smal ler response than in a s t i f fer , l ighter f loor . Thus , whi le an es t imate

of the na tu ra l f requency i s an essen ti a l par t o f any assessment , i t is no t a lone

suff icient . Wyat t (1989), for example, c lass i f ies f loors in to two categories ,

depe nd ing o n w hether the i r lowes t na tu ra l f requency is above o r be low the th i rd

har m onic o f the walk ing f requency (which is typ ica lly a rou nd 2 Hz) . F or f loors

below this cut -off , resonance wi th walking exci tat ions i s l ikely . For f loors above

the limi t ing va lue , p rob lem s m ay s ti ll occur , bu t the v ib ra t ions a re caused by the

impuls ive load ing o f a s ingle foo t fa l l ra ther than by resonance .

9.2.1 Prediction of natural frequency

N um ero us s impl if ied meth ods ex is t fo r the p red ic t ion o f na tu ra l f requencies o f

f loors, bu t com par i son wi th measured va lues shows tha t they e r r qu it e cons iderab ly

on the low s ide. This error wi l l general ly be conservat ive, so an acceptable

procedure would be to use a s impl i f i ed method fo r a f i r s t es t imate and on ly

per form a mo re so phis t icated analys is (e.g. us ing fin ite e lem ents) i f the s imp le

method ind ica ted tha t v ib ra t ion p rob lems were l ike ly .

As wi th defl ec tions, the be hav iour o f s l abs sup por te d on b eam s i s un l ike ly to

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D E FL E C T I O N A N D V I B R A T IO N 209

fo l low s imple one-way spann ing assumpt ions . S ince the beams used wi th

pos t - t ens ioned f loors a re usua l ly qu i t e wide and sha l low, they do no t p rov ide

suff icient s t i f fness to act as supports . Therefore, two al ternat ive modes of

v ib ra t ion a re poss ib le : loca l bend ing o f the s l ab in the shor te r span d i rec t ion , a t

r igh t ang les to the beam s; an d overa l l bend ing o f the s lab and b eam s a long the

length of the beam s, i .e . in the longe r span di rect ion (Pavic e t a l . , 1994). Without

calculat ion , i t i s not usual ly poss ible to ascer tain w hich of these two m ode s

predomina tes in a par t i cu la r f loor . Beam-and-s lab f loors shou ld , therefo re , be

app roa che d by ca lcu la t ing the na tu ra l f requencies o f bo th mo des , as sum ing

one-way spann ing behav iour in each case , i . e . by cons ider ing :

9 a one-me t re wide s tr ip o f s l ab span n ing be tw een the beams , as sum ing no b eam

m o v e m e n t , a n d

9 a T-sec t ion cons i s t ing o f the beam toge ther wi th a t r ib u ta ry wid th o f s l ab ,

spann ing be twe en supp or t ing co lumns . T he cho ice o f t r ibu ta ry wid th o f s l ab to

use in ca lcu la t ing the T-beam proper t i es shou ld fo l low code gu ide l ines ; fo r

examp le , BS 8110 recom me nds us ing a wid th o f 0 .2 t imes the d i s t ance be twee n

the po in t s o f con t ra f l exure o f the s l ab under un i fo rm load ing .

The fundamenta l mode i s then the one hav ing the lower na tu ra l f requency .

E q u iv a le n t b e a m m e t h o d

This app roa ch is wide ly used fo r com pos i t e f loors , and can be app l i ed to concre te

f loors hav ing a p redo m inan t ly one-w ay ac t ion . A sec t ion o f the f loor is t rea ted as

a simple , one-w ay span n ing be am of l eng th L , and supp or t ing a weigh t per un i t

l eng th w. I ts f requency fo i s then de te rm ined us ing the fo rm ula

f o = ( n / 2 ) E ( E r 1 6 2 ~ = 0 . 1 8 ( 9 / 6 ) 0 .5

(9.7)

whe re 0 = gra vi tat io nal acce lerat ion (9 .81 m /s 2 , or 32.2 f t / s 2).

6 = def l ec t ion due to dead and imposed loads (no t p res t ress )

Fo r f loors cons is t ing o f several con t inuou s spans , Eq uat io n (9.7) tends to

unde res t ima te the na tu ra l f requency due to the assum pt ion o f s imple suppo r t s

ra the r than con t inu i ty a t the mem ber ends . To acc oun t fo r th i s, i t i s recom m ende d

tha t the span l eng th L fo r a con t inuou s f loor is reduced by a fac to r o f 0 .9 befo re

app ly ing Eq ua t ion (9.7) . Th i s is equ iva len t to increas ing the na tu ra l f requency by

24% . In do ing so , i t shou ld be no ted tha t the ef fec t o f end con t inu i ty on the

d y n am i c b eh av i o u r can v a ry co n s i d e rab ly b e t w een f l o or s (Cav e r s o n e t a l . , 1994),

so tha t even th i s reduced va lue o f span l eng th m ay som et imes g ive ra the r low

values o f na tu ra l f requency .

Two.way spanning f loors

The U .K. Con cre te Soc ie ty (1994) has p roposed a v ib ra t ion assessment p rocedure

which assumes tha t a f loor v ib ra tes in two ind epen den t se ts o f mo des in the two

perpen d icu la r spa n d i rec t ions. Th e na tu ra l f requency o f the lowes t mod e in each

d i rec t ion i s ca lcu la ted us ing a beam-type fo rmula , bu t wi th modi f i ca t ions to

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210 POST-TENSIONEDCONCRETE FLOORS

a c c o un t f o r t he t w o- w a y na t u r e o f t he f l oo r a nd t he c on t i nu i t y a t t he e dge s o f a

pa ne l . I t shou l d be no t e d t ha t e ve n f l oo r s w h i c h a r e t r e a t e d a s one - w a y spa nn i n g

i n o t he r p a r t s o f t he de s i gn p r oc e s s fr e q ue n t l y e xh i b i t tw o - w a y v i b r a t i ona l

be ha v i ou r . F o r t h i s r e a son , e q ua t i ons a r e g i ve n he r e f o r r i bbe d f l oo r s , e ve n

t h o u g h s u c h f lo o r s a r e n o r m a l l y r e g a r d e d a s o n e - w a y m e m b e r s . S u c h f lo o r s m a y

r e a s o n a b l y b e as s es s ed u s i n g e it h e r th e e q u i v a l e n t b e a m m e t h o d o r t h e p r o c e d u r e

ou t l i ne d be l ow .

F o r m u l a e a r e g i ve n f o r a t w o- w a y c on t i nuous f l oo r . Eq ua t i ons ( 9 . 8 ) t o ( 9 . 11 )

and (9 .14) re l a t e to v ibra t ion of the s l ab in the x-d i rec t io n ; the cha ra c te r i s t i c s o f

t he y - d i r e c t i on m ode a r e de t e r m i ne d by i n t e r c ha ng i ng t he x a nd y subsc r i p t s i n

t he se e q ua t i ons .

Th e e f fec tive a spec t ra t io o f a s l ab pa ne l i s de f ined as

2 x = ( n x L , , / L y ) [ I y / I x ] ~ (9.8)

I , a nd I y a r e m o m e n t s o f i ne r t ia i n x a nd y d i re c t i ons

L~ and

Ly are

t h e spa n l e ng t h s i n t he t w o d i r e c t i ons

nx and ny a re the n um be rs o f bay s in the two d i rec t ions .

Thi s , i n tu rn , i s used to ca lcu la t e a modi f i ca t ion fac tor k~ .

F o r so lid or waffle slabs" k x = 1 + (1/2x 2) (9.9)

F o r a r ib be d slab" k x = [1 + (1/2x4)] ~ (9.10)

F o r s l a b s w i t h be a m s a l ong t he c o l um n l i ne s , t he na t u r a l f r e q ue nc y i s t he n

f x ' = ( k x z r / 2 ) [ ( E r ~

= 0.18kx(O/6y) ~ (9.11)

w he r e w = t he l oa d pe r un i t a r e a.

(~y - " s t a t i c de f lec t ion of a 1 m w ide s t r ip spa nn ing in the y-d i rec t ion .

F o r s l a b s w i t hou t pe r i m e t e r be a m s t he f r e q ue nc y g i ve n by ( 9 . 11 ) i s m od i f i e d by

the ca lc u la t ion of a f requenc y fb- Fo r so l id o r waf fl e sl abs"

0r/2)[(Er 4 ) 3 0 . 5

[1 + ( l x L y 4 ) / ( I y L x 4 ) 3 ~

0 .18 (g / f x)~

- (1 + (~y/6x) 0"5 (9.12)

wh ere 6x = s t a t i c de f l ec tion of a 1 m w ide s t r ip span nin g in the x-d i rec t ion .

F o r r i bbe d s l a b s "

(Tz/2)[(Er 4 ) 3 ~

fb : {1 + [ ( I x L y 4 ) / ( I y L x 4 ) - ] 2 / 3 } 3 / 4

O . 1 8 ( g / 6 x ) ~

[-1 + ( 6 y / 6 x ) 2 / 3 ] 0 ' 7 5

(9.13)

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DEFLECTION AND VIBRA T/ON 211

T he na tu r a l f r e q ue nc y i s t he n

f x = f x ' - ( f x ' - f b ) [ 1 / n , , + 1 / n y ] / 2

(9.14)

C o m p a r i s o n w i t h m e a s u r e d v a l u e s s u g g e s t s t h a t t h i s m e t h o d u n d e r e s t i m a t e s

na tu r a l f r e q ue nc ie s , some t ime s by a s muc h a s 50% ( W i l l i a ms a nd W a ld r on ,

1994) . Never the less , i t r emains use fu l a s a qu ick , o rde r -of -magni tude check . As

wi th de f lec tions , a mo re accu ra te a ssessm ent i s l ikely to r equ i re the c re a t ion of a

de ta i led f in i te e lement mode l of the en t i r e f loor .

9.2.2 Damping

D a m pin g i s a c o ll e ct ive t e r m use d to c ove r the num e r ou s m e c ha n i sm s o f e ne r gy

d i s s ipa t ion w i th in a v ib r a t ing s t r uc tu re . A c e r t a in a m ou n t o f da m ping w i ll be

inheren t in the ma te r ia l f rom which the s t ruc ture i s bu i l t , bu t cons ide rab le

a dd i t i ona l da mping c a n a r i s e f r om non- s t r uc tu r a l sou r c e s suc h a s f i n i she s ,

se rv ices , fu rn ish ings and even the bu i ld ing occupants themse lves . I t i s thus no t

poss ib le to pred ic t ana ly t ica l ly the damping which wi l l be presen t in a f loor .

I n s t e a d , r e c om m e nde d va lue s a r e a ve r age s o f num e r ou s me a su r e d va lue s .

Da m pin g i s u sua l ly e xp r e s se d a s a pe r c e n ta ge o f a no t ion a l c r i ti c al va lue wh ic h

wo uld cause v ib ra t ion s to cease com ple te ly in a s ing le ha l f cyc le . The fo l lowing

pe r c e n ta ge s a r e r e c omme nde d f o r pos t - t e ns ione d f loo r s :

9 ba r e c onc r e t e : 2 - 3 %

9 w ith fa lse f loors , ce i lings, services, furn i ture : 4 -6 %

9 wi th non -s t ru c tu ra l p a r t i t ion s ex ten ding for the fu ll he igh t o f the f loor : 5 -8 %.

These va lues a re typ ica l for most f loors . Whi le i t i s poss ib le tha t lower damping

va lues may occas ion a l ly occur , des ign ing to such low va lues w ould be excess ive ly

conse rv a t ive for the vas t m a jor i ty of f loors .

9.2.3 Prediction of floor response

P r e d ic t ing the pe a k a c c e l e ra t i on o f a f l oo r is e ve n mor e p r ob le m a t i c t h a n the

e s t ima t ion o f na tu r a l f r eq ue nc y . F o r one - wa y s l abs i t is pos sib l e t o a pp ly the

a p p r o a c h r e c o m m e n d e d b y t he C a n a d i a n S t a n d a r d s A s s o c ia t io n ( C S A ) f o r

c om pos i t e f l oo rs , in wh ic h the pe a k a c c e l e ra t i on f o r a he el d r op is e s t ima te d a nd

use d in c on j unc t ion w i th t he pe r c e p t ib i l i t y s c a l e shown in F igu r e 9 .3 . F o r

two - wa y f loo r s , the c a l c u l a t i on o f a c c e l e ra t i ons i s t oo e r r o r - p r on e , so t he

m a gn i tude o f t he r e sponse i s e s t ima te d u s ing a n e m p i r ic a l r e sponse f a c to r.

T h e C S A m e t h o d

T hi s a pp r ox im a te m e tho d o f c a l c u l a ti ng the r e sponse to a he el d r o p loa d f o r

one - wa y spa nn ing f loo r s a s sume s tha t t he he e l d r op p r ov ide s a sudde n impu l se

H, which resu l t s in a pe ak acce le ra t ion a o g iven by

a o = 2 n f o H g / W (9.15)

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POST-TENSIONEDCONCRETE FLOORS

where W is the w eigh t o f a s imple osc i ll a to r hav ing the sam e dy nam ic

charac te r i st i cs as the f loor. Th i s i s depe nden t on the wid th o f f loor which

par t i c ipa tes in the response an d o n the mo de shape . Typ ica l ly , i t is as sum ed th a t

the wid th par t i c ipa t ing i s a ro un d 40 t imes the f loor th ickness and tha t the mo de

shape i s s inuso ida l ; i t can then be show n tha t the weigh t o f the equ iva len t

osci l la tor i s 0 .4 t imes the to tal d is t r ibuted load on the f loor . Using these values ,

Equat ion (9 .15) becomes

a o = ( ~ r f o H g ) / ( 8 w L D ) (9.16)

whe re D is the th ickness o f the f loor . F or r ibbe d o r waff le s labs , D shou ld be ta ke n

as the th ickness o f a so lid s l ab hav ing the sam e m om ent o f iner t i a as the r ibbed

sect ion. The heel drop impulse H is general ly taken as 70 Ns (15.7 lb .s ) .

Kn ow ing the peak acce le ra t ion , f requency and da m ping , the accep tab i l i ty o f a

f loor can be assessed f rom Figure 9 .3 . Howe ver , because o f the num erou s

app rox im at ions ma de , i t shou ld be reme m bered tha t the es t imate o f a o is p ron e to

qui te large errors . Therefore, values close to a borderl ine on the f igure should be

t rea ted wi th cau t ion .

Concrete Society method

Fo r tw o-w ay f loors , the Co ncre te Soc ie ty (1994) g ives a me thod o f ca lcu la t ing a

response fac to r R which fo llows on f rom the na tu ra l f requency equa t ions , (9.8) to

(9 .14). F i rs t , tw o dime nsionless response coeff icients , N~ an d C x are calcula ted.

For sol id or waff le s labs :

N~ = 1 + (0.5 + 0.1 loge ()2~

and fo r r ibbed s l abs :

(9.17)

N~ = 1 + (0.65 + 0.1 log e ()2x

wh ere ( i s the fract ion of cr i t ical d am pin g, as d iscussed in sect ion 9 .2 .2 .

F or fx < 3 Hz : C~ = 244.8/(f~2()

fx be tween 3 and 4 Hz: C x = 27 .2 / (

fx be tween 4 and 5 Hz: C~ = ( 8 3 .2 - 14f~)/(

fx b e tw e e n 5 a n d 2 0 H z : C x = 0 . 8 8 ( 2 0 - f x ) / ~ + 2 ( f ~ - 5 )

f~ grea ter tha n 20 Hz : C~ = 30

(9.18)

(9.19)

Eq ua t ion (9.19) i s i l lus t rated g raphic al ly in Fig ure 9 .4 for the range s of f reque ncy

and damping va lues l ike ly to be encoun tered in p rac t i ce . The response fac to r in

the x-di rect ion is then

R x = l O 0 0 C x N x g / ( w n , , n y L , , L y )

(9.20)

Af te r repea t ing the f requency an d respo nse ca lcu la t ions fo r the s lab spa nn ing in

the y-di rect ion, the overal l response factor i s

R = R x

+ Ry

( 9 . 2 1 )

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DEFLECTION AND VIBRATION 213

1600

1400

1200 ~ ~

1000 ~ i

I l l I I

~ 800 ~ .

I ~ ' - - - ~ L

6 o o - t

oo

400 i

o -_ _

0 5 10

I

fx (Hz)

15 20 25

Figure 9.4 Var ia t ion o f respons e co ef f ic ient C x wi th natu ra l f requen cy (E quat ion 9 .19)

F o r t h e v i b r a t io n b e h a v i o u r t o b e a c c e p ta b l e , R m u s t n o t e x c e e d 8 i n a n o r m a l

o f fi c e o r 1 2 i n a b u s y o f f ic e w h e r e t h e r e a r e f r e q u e n t v i s u a l a n d a u d i b l e

d i s t r a c t i o n s . I n a n e n v i r o n m e n t w h e r e t e c h n i c a l t a s k s r e q u i r i n g p r o l o n g e d

c o n c e n t r a t i o n a r e p e r f o r m e d , R s h o u l d b e l im i t e d t o 4.

C o m p a r i s o n s w i t h fi el d m e a s u r e m e n t s o f h e el d r o p t e st s i n d i c a te t h a t t h e

r e s p o n s e f a c t o r i s a r e a s o n a b l e g u i d e t o f l o o r a c c e p t a b i l i t y , e v e n t h o u g h t h e

c a l c u l a ti o n m e t h o d is b a s e d o n r a t h e r c o n s e r v a t i v e p r e d i c t io n s o f n a t u r a l

f r e q u e n c y ( W i l l i a m s a n d W a l d r o n , 1 9 9 4 ) .

E x a m p l e 9 . 1 D e f l e c t i o n c a l c u l a t i o n f o r a o n e - w a y s l a b .

Calcula te the def lec tion of the two-sp an c ar park s lab shown in F igu re 9 .5(a) . The

appl ied loads on the s lab are"

Co nst ruc t ion load = 1 .5 kN /m 2 (appl ied after 1 m onth )

F inish & services = 1 .5 kN /m 2 (appl ied after 3 m onth s)

Design l ive load = 2 .5 kN /m 2 (appl ied af ter 6 mo nths)

Susta ined l ive load = 0 .3 x 2 .5 = 0 .75 kN /m 2

Th e slab is prestressed by tend ons fol lowing the profile show n in Figur e 9.4(a) , giving

a pres t ress ing force af ter imm edia te losses of 600 kN. Fu r the r losses after 3 m onth s

a re 20% , an d a f te r s evera l yea r s a r e 30%. Cyl inde r s tr eng th f c '= 40 N /m m 2.

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2 1 4

POST-TENSIONEDCONCRETE FLOORS

112.5 mm 75 mm

J,

9 .0 m =I ~ 9 .0 m .j_~

(a) Elevation showing tendon profi le

W

t , _ t

" v l

3/8 wL 5 /4 wL 3/8 wL

(b ) Suppor t react ions for a un i formly loaded two-span beam

Figure 9.5

Example 9.1

S o / u t / o n

S e c t io n d e p t h = 2 0 0 m m

S e l f - w e i g h t = 0 . 2 x 2 4 = 4 . 8 k N / m 2 ( a p p l i e d a f t e r 1 m o n t h )

M o m e n t o f i n e r t i a I r 1 0 0 0 x 2 0 0 3 /1 2 = 6 . 6 6 7 x 10 8 m m 4 .

U s i n g t h e B S I r e c o m m e n d a t i o n s g i v e n i n C h a p t e r 2 ,

E r = 2 8 k N / m m 2 , r i si n g t o 3 2 .2 k N / m m 2 a f t er 1 y e a r .

C c = 1 .8 f o r l o a d s a p p l i e d a f t e r 1 m o n t h

C r = 1 .2 f o r l o a d s a p p l i e d a f t e r 6 m o n t h s

E q u i v a l e n t u n i f o r m l o a d

w e = 8 P s / L 2

= 8 x 6 0 0 x 0 . 1 1 2 5 /9 . 0 2 = 6 . 6 7 k N / m

F o r a t w o - s p a n b e a m c a r r y i n g a u n i f o r m l y d i s t r i b u t e d l o a d w p e r u n i t l e n g t h , th e

r e a c t i o n s a r e a s s h o w n i n F i g u r e 9 . 5 ( b ) a n d t h e b e n d i n g m o m e n t e x p r e s s i o n f o r t h e

l e f t - h a n d s p a n i s "

M = 0 . 3 7 5 w L x - 0 .Swx 2 = E e l e [ d 2 y / d x 2]

I n t e g r a t i n g t w i c e g i v e s a n e x p r e s s i o n f o r t h e d e f l e c t i o n y a t a n y p o i n t i n t h e s p a n "

E r = w L x 3 / 1 6 - w x 4 / 2 4 + A x + B

S u b s t i t u t i n g y = 0 a t x = 0 a n d a t x = L g i v e s

A = - w L 4 / 4 8 ,

a n d B = 0

5 = O . O 0 5 2 1 w L 4 / E r 1 6 2 a t m i d s p a n

A s l ig h t l y l a r g e r d e f l e c ti o n o c c u r s j u s t a w a y f r o m m i d s p a n , b u t t h e d i f fe r e nc e i s

s u f f i c i e n t l y s m a l l t o b e n e g l e c t e d .

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D E FL E C T I O N A N D V I B R A T I O N 215

S i m i l a rl y , i n a t w o - s p a n b e a m w i t h a u n i f o r m l o a d w o n o n e s p a n o n l y :

= O.O0915wL4/Eclc a t m i d s p a n

U s i n g t h e s e r e s u l t s , t h e d e f l e c t i o n s a t v a r i o u s s t a g e s d u r i n g t h e l if e o f t h e s t r u c t u r e c a n

b e c a lc u l a t e d . U s e u n i t s o f N a n d m m , a n d t a k e d o w n w a r d s d e f le c t io n s a s p o s i ti v e .

(a) Initial elastic deflections"

A f t e r 1 m o n t h , t h e p r o p s a r e r e m o v e d a n d t h e s l a b c a r r ie s i ts s e l f -w e i g h t , a

c o n s t r u c t i o n l o a d a n d t h e p r e s tr e s s .

S e l f- w e i g ht a n d c o n s t r u c t i o n l o a d"

0 . 00521 x (4 . 8 + 1 . 5 ) x 90004

3 =

28 x 103 x 6 .6 67 x 10 a

= 1 1 .5 m m

P r e s t r e s s e q u i v a l e n t l o a d :

= 1 1 .5 x ( - 6 . 6 7 / 6 . 3 ) = - 1 2 .2 m m

N e t d ef le c ti o n = - 0 . 7 m m

(b) Afte r 3 mo nths (just prior to installation o f services and finishes)"

T h e d e f l e c t io n s c a l c u l a t e d a b o v e m u s t b e m o d i f i e d t o a l l o w f o r e a r l y c r e e p a n d

p r e s t r e ss l os s. A s s u m e t h a t 5 0 % o f t h e l o n g - t e r m c r e e p h a s t a k e n p l a c e.

S e l f - w e i g h t a n d c o n s t r u c t i o n l o a d ~ = 1 1.5 x (1 + 1 .8 x 0 . 5) = 2 1 . 9 m m

P r e s t r e s s e q u i v a l e n t l o a d ~ = - 1 2 . 2 x (1 + 1 .8 x 0 .5 ) x 0 .8 = - 1 8 . 5 m m

N e t d e f l e c ti o n = 3 .4 m m

(c) After several years (no transient loads present)"

F u l l c r e e p a n d p r e s t r e s s l o ss h a v e n o w o c c u r r e d . N o t e t h a t o n l y t h e s u s t a i n e d p a r t o f

t h e l i v e l o a d i s u s e d i n t h e c r e e p c a l c u l a t i o n s .

Se l f -w eigh t , s e rv ic es , f in i shes ~ = 11 . 5 x (1 + 1 . 8 ) = 32 . 3 m m

P r e s t r e s s e q u i v a l e n t l o a d & = - 1 2 .2 x (1 + 1 .8 ) x 0 .7 = - 2 3 . 9 m m

S u s t a i n e d l i v e l o a d 3 = (1 + 1 .2 ) x 1 1 .5 x ( 0 . 7 5 / 6 . 3 ) = 1 3.1 m m

N e t d e f l e c t i o n = 1 1 .5 m m

(d) Short- term deflect ions:

I n a d d i t i o n t o t h e s u s t a i n e d l o a d s c o n s i d e r e d a b o v e , t h e f u ll d e s i gn l o a d m a y b e a c t i v e

f o r s h o r t p e r i o d s . T h e l a r g e s t d e f le c t i o n w il l o c c u r w h e n o n l y o n e o f th e t w o s p a n s is

l o a d e d "

A d d i t i o n a l l o n g - t e r m d e f l e c t i o n

0 . 0 0 9 1 5 x ( 2 . 5 - 0 . 7 5 ) x 9 0 0 0 4

= = 4 . 9 m m

32.2 x 103 x 6 .667 x 108

C o m p a r i n g w i t h t h e B S 8 1 1 0 r e q u i r e m e n t s ( T a b l e 9 . 1 ) "

M a x i m u m t o t a l d e f l e c t io n = 1 1 .5 + 4 . 9 = 1 6 . 4 m m < L / 2 5 0 a c c e p t a b l e .

M a x i m u m t o t a l d e f l e c t io n o c c u r r i n g a f t e r th e i n s t a l l a t io n o f f in i sh e s

= 1 6 . 4 - 3 .4 = 1 3.0 m m < L / 5 0 0 a c c e p t a b l e .

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2 1 6

POST-TENSIONED C ONCRETE FLOORS

kx

I

F

9 9 9 9

9 9 9 9 9

9 9 9 9 9

m 9 9 9

8.4 8.4 8.4

i - . ._1~ ~ I_ .. , _1 _

~i~ ,. .-i ~,, ,r p

L,~ Plan

~ X

8.4 ._,. 8.4

m

r

m J

i ' M

i

2 2 5

m m

0 7 0 - 6 0 7 0 - 6 0 7 0 - 6 0 7 0 - 6 0 7 0 0 . ,- -T e n d o n

~ j ~ ~ j ~ _~ . ~ ~ eccentricity

I I I i l II II I

Sing le tendons @ 600 mm cen t res

Sect ion X - X

0 7 0 - 4 0 70 - 4 0 7 0 0

I 1 1 I I I

8 tendons along interior column lines

4 tend ons along exterior column lines

Section Y - Y

F i g u r e 9 . 6

Example 9.2: Two-way post-tensioned f lat slab

E x a m p l e 9 . 2 D e f l e c t i o n c a l c u l a t i o n f o r a t w o - w a y s l a b

The v a r ious ap proach es a re i l lu s t r a ted fo r pane l B 3 o f t he f la t s lab show n in F igure

9.6. Deflection is calculated for self-weight only.

Assu me E c:

Px:

Py"

2 8 k N / m m 2

400 kN /m pres t re s s ing fo rce in x -d i r ec t ion

1600 kN pres t ress ing force a long in ter ior colu mn l ines in

y-di rec t ion

800 kN pres t ress ing force a lon g end colu mn l ines in y-di rec t ion.

Sel f-weight = 24 x 0 .225 = 5 .4 kN /m 2

S o l u t i o n

Eq uiva len t l oads p roduced by the p res t r e s s "

Along X-X direction, tendo ns are evenly spaced, giving a uniformly distr ibuted load"

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D E FL E C T I O N A N D V I B R A T I O N

217

Sp an s 1 ,5 9 w e = 8 P s / L 2 = 8 x 4 0 0 x 0 . 1 0 / 8 . 4 2 = 4 . 5 4 k N / m 2

S p a n s 2 , 3 , 4 : w e = 8 x 4 0 0 x 0 . 1 3 / 8 . 4 2 = 5 . 9 0 k N / m 2

I n t h e Y - Y d i r e c t i o n , t h e b u n c h e d c a b l es g i ve a l in e l o a d a l o n g t h e c o l u m n l in e s :

S p a n s A , C : w e = 8 x 1 6 0 0 x 0 . 0 9/ 7 .2 2 = 2 2 . 22 k N / m

S p a n B 9 w e = 8 x 1 6 0 0 x 0 . 1 1 / 7 . 2 2 = 2 7 . 1 6 k N / m

(a) Classical plate theory:

A u n i f o r m l o a d o v e r t h e p a n e l i s r e q u i r e d i n o r d e r t o c o m p u t e t h e d e f l e c t i o n u s i n g

E q u a t i o n ( 9 .6 ) . T h e r e f o r e , d i s t r i b u t e t h e e f fe c t o f t h e t e n d o n s i n th e y - d i r e c t i o n a c r o s s

t h e fu l l w i d t h o f t h e p a n e l m t h i s i s c r u d e , b u t b e t t e r t h a n n e g l e c t i n g t h e i r e f fe c t e n t i r e l y .

N e t d i s tr i b u t ed l o a d = 5 . 4 - 5 .9 - ( 2 7 . 1 6 / 8 . 4 ) = - 3 . 7 3 k N / m 2

I n t e r p o l a t i n g f r o m T a b l e 9 . 2 f o r

b/a

= 8 . 4 /7 . 2 = 1 . 17 , 52

" - "

0 . 0 0 8 3 5

P l a t e b e n d i n g p a r a m e t e r H = 2 8 x 1 03 x 2 25 3 /[ 12 (1 - 0 . 2 2 ) ] = 2 .7 6 85 x 1 0x ~ N m m

U s i n g E q u a t i o n ( 9.6 ),

0 . 0 0 8 3 5 x ( - 3 . 7 3 ) x 1 0 - 3 x 7 2 0 0 4

6 = o~2wa4/H = 2 . 7 6 8 5 x 1 0 l ~ = - 3 . 0 m m

(b) Crossing beam method:

C o n s i d e r a 4 . 2 m w i d e s l a b s t r i p a l o n g t h e c o l u m n s i n t h e y - d i r e c t i o n .

I c = 420 0 x 2253 /12 = 3 . 987 x 109 m m 4 .

U s i n g a l o a d d i s t r i b u t i o n f a c t o r o f 1 .2 f o r a c o l u m n s t r i p , t h e n e t lo a d s p e r u n i t l e n g t h a r e "

S p a n s A , C : 5 .4 x 4 . 2 x 1 . 2 - 2 2 . 22 = 5 .0 k N / m

S p a n B 9 5 . 4 x 4 . 2 x 1 . 2 - 2 7 . 1 6 ~ 0

T h e v e r t ic a l r e a c t i o n s d u e t o t h e s e lo a d s a r e s h o w n i n F i g u r e 9 . 7 (a ) . T h e n , s e t t i n g u p

a n d i n t e g r a t i n g t h e m o m e n t - c u r v a t u r e e x p r e s s i o n g i ve s t h e de f lec t ion 6 x a t t h e c e n t r e

o f s p a n B

61 = - 0 . 7 5 m m .

N o w c o n s i d e r a 3 . 6 m w i d e m i d d l e s t ri p i n th e x - d i r e c t i o n .

I c = 3600 x 225 3 /12 = 3 . 417 x 109 m m 4 .

U s i n g a l o a d d i s t r i b u t i o n f a c t o r o f 0 .8 f o r a m i d d l e s t r i p , t h e n e t l o a d s a r e "

S p a n s 2 ,3 ,4 " ( 5 . 4 - 5 .9 ) x 3 .6 x 0 .8 = - 1 . 4 4 k N / m .

T h e d e f l e ct i o n s o f a n i n t e r i o r s p a n o f a m u l t i s p a n b e a m c l o s el y r e se m b l e t h o s e o f a

f i x e d - e n d e d b e a m , s o a t t h e c e n t r e o f s p a n 3"

6 2 - - -wL g/384Er162= - 1 . 4 4 x 8 4 0 0 4 / ( 3 8 4 x 28 x 1 0 3 x 3 .1 4 7 x 1 0 9)

= - 0 . 2 0 m m

T h e r e f o r e , t h e t o t a l d i s p l a c e m e n t i s 6 - 6~ + 6 2 - - - - 0 . 9 5 m m .

(c) Frame and s lab method

W i d t h o f c o l u m n s t r i p i n y - d i r e c t i o n = 0 . 4 x 8 . 4 = 3 .3 6 m

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218 POST-TENSIONEDCONCRETE FLOORS

(a)

(b)

5 kN/m 5 kN/m

T l

6.2 kN ] 9 .8 kN 19.8 kN 16.2 kN

5.4 kN/m

0.45 kN/m 0.45 kN/m

l 1 1 l

.5 kN 23.2 kN 23.2 kN 0.5 kN

I

I

I

I

I

I

I

i Column

L. strip

I

I

I

I

I

I

I

( c ) I

I

I

I

I

I

I / '

V

v

I

I -

I

I

Load t ransferred

to column strip

/ ,,%

/

/

/

/

/ "

~ . 4 5 ~

5.04 m

r a m - - I

I

I

I

I E

I

I

1 _ '

(d)

0 .4 kN/m

~ t t t t t t

1 .68 m

0.86 kN

T

Figure 9 .7

Exam ple 9.2 (a) Sup port reactions for crossing beam exam ple (b) Sup port

reactions for frame-and-slab exam ple (c) Subdivision of slab load for frame-an d-slab

example (d) Canti lever loads for frame-and-slab example

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DEFLECTION AND VIBRA I ON 219

I c = 3 .189 x 109 m m 4

Lo a d s p e r u n i t l e n g t h "

Sp a n s A , C : 5 .4 x 3 .3 6 x 1.2 - 2 2 . 22 = - 0 . 4 5 k N / m

S p a n B 9 5 . 4 x 3 .36 • 1 . 2 - 2 7 . 1 6 = - 5 . 4 k N / m

T h e v e r t i c a l r e a c t i o n s u n d e r t h e s e l o a d s a r e s h o wn i n F i g u r e 9 . 7 ( b ) . A g a i n ,

i n t e g r a t i n g t h e r e s u l t i n g mo me n t - c u r v a t u r e e x p r e s s i o n g i v e s :

d e f l ec t io n a t t h e c en t r e li n e o f t h e s la b s t ri p in s p a n B , 61 = - 1 . 0 2 mm .

N o w c o n s i d e r t h e i n t e r io r p a n e l b e t we e n s l a b s t ri p s . T h is h a s t h e d i me n s i o n s s h o w n

in Figure 9 .7 (c ) , and ca r r i e s a load o f :

( 5 . 4 - 5 .9 ) x 0 .8 = - 0 . 4 k N / m 2.

I n t e r p o l a t i n g f r o m T a b l e 9 . 2 , f o r

b/a

= 5 .04/4 .32 = 1 .17, ~1 = 0 .00 540

(~3--~lwa4/D

= 0 . 0 0 5 4 x ( - 0 . 4 ) x 1 0 - 3 • 4 3 2 0 4 / (2 . 7 6 8 5 x 1 0 1 ~

= - 0 . 0 3 m m

T h e p r o p o r t i o n o f t h e p a n e l l o a d wh i c h i s t r a n s f e r r e d i n t o t h e a d j o i n i n g s l ab s t r ip i s

g i v e n b y th e s h a d e d a r e a i n F i g u r e 9 . 7( c) . A s s u m i n g t h e l o a d i s u n i f o r ml y d i s t r i b u t e d

a l o n g t h e p a n e l e d g e gi v es a l in e l o a d o f - 0 . 8 6 k N / m . N o w a n a l y s e a 1 m wi d e

segm en t o f the s l ab s t r ip a s a can t i l eve r wi th i t s roo t a t the co lum n l ine , F igu re 9 .7 (d ) .

T ip de f l ec t ion

( ~ 2 ~ - -

wL g/(8Eclc) + WLa/(3Eclc)

= [ ( - 0 . 4 x 1 6 8 0 4 )/ 8 + ( - 8 6 0 x 1 6 8 0 3 ) / 3 ] /( 2 8 x 1 03 x 9 . 4 9 2 x 1 0 a )

= - 0 . 0 7 m m

Tota l de f l ec t ion

-- ~1 "~- ~2 -~- ~3 ~" --

1.02 - 0.03 - 0.07 ~ - 1.1 m m .

C o m m e n t

Clea r ly the th ree m e th od s u sed he re y ie ld qu i t e d i f f e ren t r e su l ts . S ince they a l l con ta in

s i g n i f i c a n t a p p r o x i ma t i o n s , i t i s h a r d t o s a y w i t h c o n f i d e n c e wh i c h i s t h e mo s t

accura te . As s t a t ed in Sec t ion 9 .1 .4 , the s impl ic i ty and wide app l i cab i l i ty o f the

c r o s s i n g b e a m a p p r o a c h m a k e i t p r o b a b l y t h e m o s t s u i t a b l e m e t h o d f o r e v e r y d a y

d e s i g n u s e . I t s h o u l d b e n o t e d t h a t t h e c e n t r a l p a n e l e x a mi n e d h e r e ma y n o t b e t h e

mo s t c r it ic a l l o c a t i o n f o r t h i s s t ru c t u r e , r e q u i r i n g a n u m b e r o f d e f l ec t io n c h e c k s t o b e

pe r fo rmed . Las t ly , i t wi l l be necessa ry to ca lcu la te long- te rm de f l ec t ions us ing a

s i mi l a r a p p r o a c h t o t h a t i l l u s t r a t e d i n E x a mp l e 9 . 1 f o r o n e - wa y s p a n n i n g s l a b s .

E x a m p le 9 .3 V ibr a tion a s s e ss m e n t o f s l a b s

The two f loo r s whose de f l ec t ions have been ca lcu la ted in Examples 9 .1 and 9 .2 wi l l

n o w b e a s s e s s e d f o r v i b r a t i o n s .

(a) One-w ay spanning slab

Co n s i d e r t h e f l o o r s l a b o f E x a mp l e 9 . 1 . A s s u m e t h a t o n l y t h e s u s t a i n e d p a r t o f t h e li ve

l o a d i s p r e s e n t . S i n c e t h e s l a b p a n e l i s c o n t i n u o u s o n l y a t o n e e n d , n o r e d u c t i o n i n

s p a n l e n g t h i s ma d e t o a c c o u n t f o r t h e c o n t i n u i t y . T h e n a t u r a l f r e q u e n c y i s f o u n d

f r o m E q u a t i o n ( 9 . 7 ) , a n d t h e a c c e l e r a t i o n r e s p o n s e f r o m E q u a t i o n ( 9 . 1 6 ) .

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220 POST-TENSIONED CONCRETE FLOORS

Fo r a 1 m w ide s t r ip , I c = 6 .667 x 10 -4

m 4

w = (4 .8 + 1 .5 + 0 .75) = 7 .05 k N /m z

[ l o ,

8 x 109 x 6 .667 x 10 4 x 9 .81

fo = (n /2 ) x = 3 .1 Hz

7.05 x 10000 • 94

n x 3 . 1 x 7 0 x 9 .8 1

a o = = 0 . 6 7 % 9

8 • 1 0 0 0 x 9 •

A s s u m i n g 3 % d a m p i n g , t h e l i m i t i n g a c c e l e r a t i o n g i v e n b y t h e C S A s c a le , F i g u r e 9 . 3 ,

i s 1 . 0 5 % g , s o t h e c a l c u l a t e d a c c e l e r a t i o n i s a c c e p t a b l e .

(b) Tw o-way spannin9 slab

T h e f la t sl a b o f E x a m p l e 9 . 2 is a s s e ss e d u s i n g t h e C o n c r e t e S o c i e t y a p p r o a c h .

F o r t h e x - d i r e c t i o n ,

D = 2 2 5 m m ,

EcI,,= Ecly

= 2 . 6 5 7 8 x 1 07 N m 2 / m

2 x = 5 x 8 .4 / 7 .2 = 5 .833

kx = 1 + 1/5.8332 = 1.029

w = 5 . 4 + 2 5 % f i n i s h e s = 6 . 7 5 k N / m 2

A f ir st e s t i m a t e o f th e n a t u r a l f r e q u e n c y i s g i v e n b y E q u a t i o n ( 9 .1 1 ):

1 . 0 2 9 ~ F 2 . 6 5 7 8 x 1 07 x 9 . 8 1 ] ~

f ' x -

2 [ ~ x 1 0 00 x 7-.2 -g J =

6 . 1 3 H z

A s th i s is a fi a t s l a b , t h is v a l u e m u s t b e m o d i f i e d b y t h e c a l c u l a t i o n o f a s e c o n d

f r e q u e n c y , E q u a t i o n ( 9. 12 ) , a n d t h e s la b f r e q u e n c y is t h e n f o u n d f r o m E q u a t i o n ( 9. 14 ) .

_~F2.6578 x 107 x 9 .8 1] 0 .5

Iooo•

fb = = 3 .53 Hz

(1 + 7.24/8.44)~

f~ = 6.13 - - (6 .13 - - 3 .53)(~ + 3!) /2 = 5 .44 H z

W i t h 4 % d a m p i n g , t h e c o e ff ic i e nt s r e q u i r e d f o r t h e c a l c u l a t i o n o f r e s p o n s e f a c t o r a r e

N , = 2 . 0 4 a n d C , = 3 2 1 .2 . T h e x - d i r e c t i o n r e s p o n s e f a c t o r i s g i v e n b y E qu a t i o n ( 9 .2 0 ).

1000 x 2 .04 x 321.2 x 9 .81

R~ = = 1.05

6 .7 5 • 1 0 0 0 x 5 x 3 x 8 . 4 x 7 . 2

R e p e a t i n g t h e c a l c u l a t i o n s f o r t h e y - d i r e c t i o n g i v e s f y = 4 . 6 3 H z a n d R y = 1 .0 8

T h e o v e r a l l r e s p o n s e f a c t o r i s R x

+ Ry =

2.13.

T h i s i s w e ll b e l o w t h e m o s t s t ri c t o f t h e s p e c i fi e d l im i t s , m a k i n g t h e f l o o r a c c e p t a b l e

f o r al l k i n d s o f u s e. T h i s e x a m p l e s h o w s t h a t e v e n f l o o rs o f v e r y lo w n a t u r a l f r e q u e n c y

c a n g i v e a c c e p t a b l e v i b r a t i o n p e r f o r m a n c e , a s t h e ir h i g h m a s s m e a n s t h a t o n l y l o w

a c c e l e r a t i o n s r e s u l t f r o m t y p i c a l e v e r y d a y l o a d i n g s .

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1 0 S H E A R

In th is chap ter , the me chan i sm s o f shear fa i lu re a re d i scussed , meth ods fo r

ca lcu la t ing shear s t reng th a re p resen ted and recommendat ions a re g iven fo r

re in fo rc ing mem bers to res is t shear load ing . The chap ter f i rs t dea l s wi th shear in

b eams an d o n e -w ay s l ab s , t h en w i t h p u n ch i n g s h ea r i n t w o -w ay s p an n i n g

mem bers . M etho ds o f ana lys is a re g iven which com ply wi th bo th Br i ti sh and

Amer ican codes o f p rac t i ce .

Fai lu re of f loors in shea r is an ul t ima te s t ren gth cr i ter ion, w hich is usual ly

checked a f t e r the f l exura l des ign i s comple te . In beams , and in one-way s l abs

spann in g be tw een b eam s o r w all s, excess ive shear s t res ses resu l t in the fo rm at ion

of d iagon al t ens ion c racks . In f l a t s labs , punch ing shear fa ilu re a ro un d a co lum n

or under a very l a rge concen t ra ted load i s the p r inc ipa l concern . In p rac t i ce ,

punch ing shear i s the more impor tan t c r i t e r ion ; one-way s l ab shear i s ra re ly

cri t ical in des ign. For ins tance, beam s t r ips , a l though in other respects des igned

as one-way spann ing , shou ld be checked fo r punc h ing shear . Never the less , since

an u nder s t and in g o f one-w ay shear i s an essen t ia l p re requ i s i te to the assessment

of punc h ing shear , the fo rm er top ic i s dea l t wi th in some de ta i l here.

A major d i f fe rence be tween p res t ressed and re in fo rced concre te i s tha t the

vert ical component of the pres t ress ing force wi l l , in nearly al l cases , oppose the

shear due to the app l i ed loads , thus reduc ing the she ar fo rce which the concre te

sec t ion is requ i red to wi ths tand . Fo r exam ple , F igure 10 .1 shows the fo rces ac t ing

on a sec t ion o f a s imply suppo r ted b eam und er un i fo rm load ing . The resu l t an t

shear on the r igh t -hand face i s

V = - ( w L / 2 ) + w x + P sin

The m agn i tude o f th is fo rce is smal l e r than the va lue in the absence o f the

pres t ress so long as the ten do n is s loping d ow nw ard s ( i.e. ~ i s betw een 0 ~ an d 90 ~

In a f loor , however , b o th the average p res t ress and the inc l ina t ion o f the t endons

are qu i t e low, so tha t the con t r ibu t ion o f the p res tress to the shear s t reng th i s

smal l . Addi t iona l ly , tes ts have sho wn that th is beneficial effect i s nor m al ly only

effect ive at locat ions where the concrete sect ion is uncracked. At cracked

sect ions , therefore, the effect of incl ined ten don s is assu m ed to oc cur only i f i t

increases the effect ive shear force on a sect ion.

The n orm al p roce dure fo r shear des ign is to com pare the capac i ty o f the

concre te sec tion , inc lud ing rod re in fo rcement an d p res t ress ing t endons , wi th the

m axim um app l i ed shear fo rce , inc lus ive o f u l t imate loa d fac to rs . Values fo r load

fac to rs reco m m end ed by BS 8110 and AC I 318 a re g iven in Ch ap te r 8 . I f the

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2 2 2 POST- TENSIONED CO NCRET E FLOO RS

w /u n i t le n g t h

I V

P

x ( < L /2 )

Figure 1 0 . 1 Ef fec t o f i nc l i ned tendon on e f fec t ive she a r fo rce

capac i ty i s ade quate then n o fu r the r ac t ion is requ i red . I f the app l i ed shear fo rce is

excess ive then som e add i t ion a l capac i ty m us t be p rov ided , e i ther by inc lus ion o f

shear re in fo rcement o r by increas ing the c ross -sec t ion . S hear re in fo rcem ent m ay

take the fo rm of conv en t iona l s t i r rups , p ro pr ie t a ry re in fo rc ing cages (shearhoops)

or s t ruc tu ra l s t eel shearheads . These a l t e rna t ives a re d i scussed fu r ther in Sec t ion

10.4.

Fo r r ibbed o r waf fl e s l abs , increases in c ross-sec t ion a re ach ieved by c onve r t ing

to a so l id sec t ion c lose to the suppor t s . For so l id s l abs , d rop pane l s o r co lumn

heads can be p rov ided , bu t these requ i re spec ia l fo rmwork and so can cause

s ignif icant increases in const ruct ion t ime and cost . This solut ion is , therefore,

normal ly used only when i t i s not poss ible to provide suff icient shear s t rength

us ing re in fo rcement .

10 .1 Sh e a r s t ren g t h o f c o n c re t e

Th e b eh av i o u r o f co n c re te i n s h ea r is co mp l ex an d r ema i n s p o o r l y u n d e r s t o o d .

Pla in concre te deve lops shear s t reng th p r imar i ly by the mec han i sm of aggreg a te

inter lock and fr ic t ion between the const i tuent par t ic les . Long i tudinal reinforc eme nt

increases the shear s t reng th by dowel ac t ion , an d by ac t ing as a t ie across shear

cracks , p reven t ing them f rom open ing . Addi t iona l ly , the shear behav iour o f

conc rete elem ents i s c losely related to the tens i le s t rength of concre te , s ince a

ver t ica l shear fo rce usua l ly causes fa ilu re by the deve lopm ent o f d iago nal t ens ion

cracks .

Code s o f p rac t i ce give gu idance on the appro pr ia t e m ater i a l s t reng ths to be

used in shear ca lcu la t ions . For uncracked sec t ions , BS 8110 re la t es the shear

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Table 10 .1

Values o f conc re te shear s t reng th , N/ m m 2 ( B S 8110)

SHEAR 223

IOOAs E f fec t ive dep th (mm)

bvd 125 150 175 200 225 250 300 >_ 400

< 0.15 0.53 0.50 0.48 0.47 0.46 0.44 0.42 0.40

0.25 0.62 0.60 0.57 0.55 0.54 0.53 0.50 0.47

0.50 0.78 0.75 0.73 0.70 0.68 0.65 0.63 0.58

0.75 0.90 0.85 0.83 0.80 0.77 0.76 0.73 0.67

1.00 0.98 0.95 0.91 0.88 0.85 0.83 0.80 0.74

1.50 1.13 1.08 1.04 1.01 0.97 0.95 0.91 0.84

2.00 1.24 1.19 1.15 1.11 1.08 1.04 1.01 0.94

> 3.0 0 1.43 1.36 1.31 1.26 1.23 1.19 1.15 1.06

Notes:

*In calculating the reinforcement ratio, BS 8110 recommends hat the areas of rod reinforcement and

prestressing steel sho uld simply be summ ed. How ever, this fails to tak e adequate a cco unt of the

greater strength of the tendons. S om e engineers, therefore, mu ltiply the area of prestressing tendons

by the ratio

fpu/fy

prior to com bining it w ith the rod reinforcement.

*The values show n are for a concrete strength fcu = 40 N/mm 2 or greater. For lower strengths the

values in the tab le shou ld be m ultiplied by (fcJ4 0) 1/3.

*Fo r m embers prestressed by un bonded tendons, the value obtained should be m ultiplied by 0.9.

s t r e n g t h t o t h e t e n s i l e s t r e n g t h o f t h e c o n c r e t e a s o u t l i n e d i n S e c t i o n 1 0 .2 . 2. T h e

t e n s i l e s t r e n g t h i s , i n t u r n , r e l a t e d t o t h e c u b e s t r e n g t h b y

f t = 0 .24fc~ 5 N /m m 2 un i t s o r

ft = 2.89f~ "5 psi un its .

F o r s e c t i o n s c r a c k e d i n f l e x u r e , a s f o r r e i n f o r c e d c o n c r e t e , t h e s h e a r s t r e n g t h i s

f o u n d u s i n g a n e m p i r i c a l f o r m u l a t i o n i n t e r m s o f t h e s e c ti o n d e p t h , t h e

r e i n fo r c e m e n t r a t i o a n d t h e c u b e r o o t o f t h e c o m p r e s s i v e s t re n g t h . T a b u l a t e d

v a lu e s o f s h e a r s t r e n g t h f o r c o n c r e te h a v i n g c o m p r e s s i v e s t r e n g th 4 0 N / m m 2

(5800 ps i ) a re g iven in Tab le 10 .1 .

A C I 3 1 8 d e f in e s t h e s h e a r s t r e n g t h a s a l in e a r f u n c t i o n o f t h e s q u a r e r o o t o f t h e

c o m p r e s s i v e s t r e n g t h f e ' , t h e v a l u e o f t h e c o e f f ic i e n t, tip , v a r y i n g w i t h t h e n a t u r e o f

t h e s h e a r l o a d i n g . G u i d a n c e o n t h e c h o i c e is g iv e n in t h e r e l e v a n t s e c t i o n s l a t e r in

t h i s c h a p t e r .

W h e n l i g h t w e i g h t c o n c r e t e i s u s e d , t h e s h e a r s t r e n g t h a s s u m e d i n t h e

c a l c u l a t i o n s s h o u l d b e m o d i f i e d a p p r o p r i a t e l y . B S 8 1 1 0 r e c o m m e n d s t h a t t h e

d e s i g n s h e a r s t r e s s vc s h o u l d b e t a k e n a s 0. 8 ti m e s t h e v a l u e o b t a i n e d f r o m T a b l e

1 0 .1 , w i t h t h e e x c e p t i o n o f g r a d e 2 0 li g h t w e i g h t c o n c r e t e , f o r w h i c h t h e v a l u e s

s h o u l d b e t a k e n f r o m T a b l e 1 0.2 . N o r e c o m m e n d a t i o n is g i v e n f o r t h e te n s il e

s t r e n g t h , b u t i t i s r e a s o n a b l e t o a s s u m e t h e s a m e r e d u c t i o n f a c t o r a s f o r s h e a r

s t r e n g t h .

A C I 3 1 8 r e c o m m e n d s t h a t a l l v a l u e s o f x /fe ' a f f e c ti n g t h e c a l c u l a t i o n o f s h e a r

c a p a c i t y o r c r a c k i n g m o m e n t s h o u l d b e m u l t i p l i e d b y 0 . 7 5 f o r c o n c r e t e i n w h i c h

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224

POST-TENSIONED CONCRETE FLOORS

Table 10.2

V alues o f v c f o r G r ade 20 l i gh twe igh t c onc r e t e

lOOAs/bvd Values o f v c

N / m m 2 p si

0.15 0.25 36

0.25 0.30 44

0.50 0.37 54

0.75 0.43 62

1.00 0.47 68

1.50 0.53 77

2.00 0.59 86

> 3.00 0.68 99

a l l the aggrega tes a re l igh tweigh t , and by 0 .85 fo r sand- l igh tweigh t concre te .

La s t ly , the shear s t re ng th o f a p res t ressed sec t ion is to an ex ten t in f luenced by

w h e t h e r t h e t e n d o n s u s e d a re b o n d e d o r u n b o n d e d . B o n d e d t e n d o n s a c t i n m u c h

t h e sam e way a s n o rm a l r e i n fo rcem en t , p ro v i d i n g d o we l ac t i o n an d l o ad t r an s fe r

across c racks , bu t unbonded tendons a re l es s e f fec t ive in th i s respec t . The

C o n c re t e So c i e t y (1 9 9 4 ) r eco m m en d s t h a t t h e s h ea r s t r en g t h s fo u n d f ro m T ab l e

1 0.1 s h o u l d b e r ed u ced b y a f ac to r o f 0 .9 fo r m em b er s p re s t r e s sed b y u n b o n d e d

t en d o n s .

1 0 .2 B e a m s a n d o n e - w a y s l a b s

T h e n a t u re o f t h e s h ea r f ai lu re in a b eam o r o n e -w ay s l ab is d ep e n d en t o n t h e

i n t e r ac t i o n b e t ween t h e b en d i n g m o m en t an d t h e s h ea r fo rce i n t h e m em b er . I f

t h e b en d i n g s tr e s s a t a g i v en l o ca t i o n i s b e l o w t h e m o d u l u s o f ru p t u re , t h en t h e

co n c re t e s ec t i o n r em a i n s u n c rack ed . T h e s t r e s s e s o n a s m a l l e l em en t n ea r t h e

n eu t r a l ax is a r e a s s h o wn i n F i g u re 1 0 .2 (a ). F r o m a s i m p le M o h r ' s c ir c le an a l y s is ,

F igure 10 .2 (b ) , the p r inc ipa l s t resses a re found to be as shown in F igure 10 .2 (c) ,

s o t h a t f a i lu re o ccu r s b y t h e d ev e l o p m e n t o f d i ag o n a l t en s i o n c rack s . I f, o n t h e

o ther hand , the bend ing s t ress i s su f f ic ien t to cause some f lexura l c rack ing , the

shea r s tresses wi ll cause these c racks to g row an d to be com e increas ing ly inc l ined

tow ard s the neu t ra l ax i s o f the sec t ion . The she ar beh av i our in th i s case is s imi la r

t o t h a t i n a n o n -p re s t r e s s ed m em b er .

F o r s i m p l y s u p p o r t e d s p a n s u n d e r p r e d o m i n a n t l y u n i f o r m l o a d s , t h e p e a k

b e n d i n g m o m e n t o c c u r s a t m i d s p a n a n d t h e p e a k s h e a r f o r c e a t t h e s u p p o r t s ,

m a k i n g t h e d ev e l o p m en t o f co m b i n ed f l exu re /s h ea r c r ack s u n li ke ly . F o r co n t i n u o u s

s p an s , h o wev e r , t h e s i t u a t i o n i s r a t h e r m o re co m p l ex , s i n ce b o t h t h e b en d i n g

m o m en t a n d t h e s h ea r fo rce a r e m ax i m u m a t th e s u p p o r t s . F i g u re 1 0.3 i l lu s t r a t e s

t h e m o s t l i k e ly fa i lu re m o d es i n v a r i o u s p a r t s o f an i n t e r i o r s p an o f a co n t i n u o u s

b eam . U s u a l l y , b o t h m o d es o f f a il u re m u s t b e co n s i d e red , w i t h th e s h ea r s t r en g t h

t ak en a s t h e l o wer o f t h e t wo v a l u es t h u s ca l cu l a t ed .

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f cp

v l

v

I 1 '

v

(a)

v

~ p

SHEAR

225

S h e a r A ~ . / f. 2 v 2 )O .5

s t re u s = I - - ~ +

/ i C e n t r e ~ 1 / D i r e ~ t

stress

[

~ ~ ~ ~ ( fc p' V )

( b )

\ ,oo

T e n s i o n - ~. . ~ ~ , , ~ 4 - - + - f f

c r a c k 7 V N ~ + ~)o.5_

fcp fcp

(c)

Figure 10.2 Mo hr 's c i r c le ana lys is fo r a concre te e lemen t near the neu t ra l ax is

Very o f ten , the c ri ti ca l loca t ion fo r shear in con t inuo us beam s i s appro x im ate ly

one- fi fth o f the way a long the span , where the m om ent is very low, o r even

negat ive , so tha t combined f l exure / shear c rack ing domina tes .

1 0.2 . 1 C a l c u l a t i o n o f a p p l i e d s h e a r f o r c e

The shear fo rces to which a one-way spann ing member i s sub jec ted can be

ca lcu la ted by any s imple e l as t i c method . For s imply suppor ted spans the

ca lcu la t ion i s t r iv ia l . Con t inuous spans a re normal ly ana lysed in un i t wid ths ,

wi th beams a nd w al ls t rea ted as s imple suppor t s . T he m axim um shear is l ike ly to

occur when the ful l l ive load is present on adjacent spans . Since excess ive shear

can cause s t ruc tu ra l co l lapse , the loads mus t be mul t ip li ed by ap pro pr ia t e sa fe ty

factors , as d iscussed in Chapter 8 .

For uncracked sec t ions , the usua l p rac t i ce in the UK i s to modi fy the shear

fo rce thus ca lcu la ted by inc lus ion o f the ver t ica l com pon en t o f the t end on fo rce .

In nea rly al l cases th is wi ll lead to a re duc t ion in the shear w hich the sect ion mu st

car ry , though s i tua t ions may occas iona l ly a r i se in which the shear fo rce i s

increased . F or c racke d sec t ions, the ver t ica l com pon en t o f the p res t ress is

inc luded on ly i f i t i ncreases the app l i ed shear fo rce . The app roa ch ado p ted in the

US A is s l ightly d ifferent , in that the vert ical co m po ne nt of the ten do n force is

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2 2 6 POST-TENSIONED CONCRETE FLOORS

I I I

I I 1

I I I

I i I

I I I

I I I

I I I

C o m b i n e d

I f lex ure an d I I

I I I

I she ar I I

I I ) , J , ] I I

i ~ / l l l

I I I / t i l l

I I i I

'

" , ,

I Pure Pure

I ~ she ar I f lex ure

F i g u r e 1 0 . 3 L ike l y f a i lu re m odes i n var i ous par t s o f a con t inuous span

i n c l ude d du r i n g t he c a lc u l a t i on o f she a r r e s i st a nc e , r a t he r t h a n be i ng c om b i ne d

w i t h t he a pp l i e d l oa d - - se e S e c t i on 10 . 2 . 3 . No t e t ha t , w he r e a s t he l oa ds on t he

s t r uc t u r e a r e m u l t i p l i e d by l oa d f a c t o r s f o r de s i gn pu r pose s , t he ve r t i c a l

c o m po ne n t o f t he p r e s t re s s i s a l w a ys a s s i gne d a l oa d f a c t o r o f 1 .0 .

10.2 .2 BS 8110 she ar s t rength ca lcu la t ion

M o s t n a t i o n a l c o d e s a d o p t a s im i l a r a p p r o a c h f o r t h e c a lc u l a t io n o f s h e a r

s t r e ng t h . F i r s t , t h e sm a l le s t m o m e n t r e q u i r e d t o c a use c r a c k i ng i s c a l c u l a t e d a n d

c o m p a r e d w i th t he m a x i m u m a p p li e d m o m e n t i n o r d er t o d e t er m i n e w h e t h e r t h e

sec t ion u nd er c on s id e ra t io n i s l i ke ly to be c rack ed in f lexure . I f t he sec t ion i s

unc r a c ke d , t he n t he she a r s t r e ng t h c a n be c a l c u l a t e d u s i ng e l a s t i c t he o r y . F o r

c om bi ne d f l e xu r e / she a r c r a c k i ng no t he o r e t i c a l f o r m u l a t i on i s a va i l a b l e , so a n

e m p i r i c a l f o r m u l a i s u se d . The sh e a r s t r e ng t h o f a c r a c ke d se c t ion i s t he n t a ke n a s

t he sm a l l e r o f t he c om b i ne d f l e xu r e / she a r a nd t he unc r a c ke d va l ue s .

BS 8110 s t ipu la t e s up per l imi ts on the av e rage s hea r s t re ss ove r the c ross- sec t ion ,

r e ga r d l e s s o f a ny r e i n f o r c e m e n t p r e se n t . F o r no r m a l w e i gh t c onc r e t e t he she a r

s t re ss ma y no t exceed the sm al l e r o f 5 N /m m 2 (725 ps i ) and 0 .8~/fc u (un i t s o f N

a nd m m ) o r 9 .6~/fc u ( un i ts o f p s i) . F o r l i gh t w e i gh t c on c r e t e t he c o r r e sp on d i ng

va lues a re 4 N /m m 2 (580 ps i ) and 0 .63~/f r (un i t s o f N and m m ) o r 7 .6~/f~ u (un i t s

o f p s i) . It is w o r t h c he c k i ng t h i s r e q u i r e m e n t be f o r e p r oc e e d i ng w i t h a m o r e

de ta i l ed s t reng th check , a s , i f t he av e rage shea r s t re ss is excess ive , the re i s no

op t i on bu t t o i nc r e a se t he c r o s s - se c t i on .

I n BS 8110 the m om e n t M o r e q u i r e d t o p r o duc e ze r o s t re s s in t he e x t r e m e f i b re

of the s ec t ion i s ca l cu la t ed as :

M o = 0 . 8 ( P Z t ~ r , / A ~

+

M p )

(10.1)

w he r e P = p r e s t r e s s i ng f o r c e , i nc l ud i ng a n a l l ow a nc e f o r l o s se s

Z te = e l a s ti c m od u l u s o f f ib r e w h i c h w o u l d no r m a l l y be i n t e n s i on un de r

a pp l i e d l oa ds

M p = t o t a l m o m e n t d u e t o p r e s tr e s s ( i.e . s u m o f p r i m a r y a n d s e c o n d a r y

m o m e n t s )

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SHEAR

227

U nde r no r ma l c i r c ums ta nc e s , t ha t i s , w i th t he t e ndon e c c e n t r i c i t y t owa r ds t he

tens io n face , bo th of the te rms wi th in the brack e ts w i ll be pos i t ive . The m ul t ip l ie r

of 0.8 is a safe ty fac tor .

I f t he a pp l i e d m om e n t M is le ss t ha n M o the n i t c a n s a fe ly be a s sum e d tha t no

f le xu ra l c ra c k ing ha s oc c u r r e d . T he a pp l i e d she a r fo r ce Vm a y the n be a d j u s t e d to

t a ke a c c oun t o f t he ve r t ic a l c om pon e n t o f t he p r e s tr e s s. A s d i s c us se d a bov e , t h i s

will usual ly lead to a red uctio n in the effec tive shear force that the sect ion m us t carry.

The shea r s t r en gth of an un crack ed sec t ion , Vco, can be found f ro m the e la s t ic

ana lys is a l r eady in t roduced in F igure 10 .2 . The maximum tens i le s t r e ss in the

sec t ion i s shown in F igure 10 .2(c ) , ac t ing a t r igh t angles to the tens ion c rack .

Eq ua t ing th is s t re ss to the tens i le s t r ength of the concre te , f t , and aga in

in t r odu c ing a s a fe ty f a c to r o f 0 .8 on the p r e s tr e s s , the m a x im um a l lowa b le she a r

s t re ss on the sec t ion can be expressed as

/)all--" (it 2 "~ 0"82Pavft)~

NOW for a r ec tang ula r sec t ion i t can be sho wn us ing c la ssica l ela s tic theory tha t

the ave rage sh ea r s t r e ss is 2 /3 of th i s peak v a lue . Hence the shea r force tha t the

se c tion c a n c a r r y c a n be f ound by t a k ing the a ve r a ge she a r s tr es s a nd mu l t ip ly ing

by the shea r a rea to g ive

Vco = 0 .67b vD ( f t z + 0.8Pavft) ~ (10.2)

C a lc u la t i on o f t he unc r a c k e d she a r s t r e ng th c a n be s imp l if ie d by u s ing the va lues

of the a ve rage s t r ess Vr given in Table 10 .1 . These have been ca lcu la ted

us ing Eq ua t io n (10.2) . The co ncre te s t r eng ths an d p res t re ss leve ls g iven cover the

ranges l ike ly to be encounte red in pos t - tens ioned f loors . Va lues ou ts ide these

ranges can be ca lcu la ted d i rec t ly f rom Equa t ion (10 .2) .

F o r t he c a se o f c om bine d f l e xu ra l / she a r c r a c k ing , a n e m p i r ic a l f o r mu la t ion i s

used"

Vr = [ 1 - 0.5 5fpe/fpu]l~cbvd + M o V I M (10.3)

where d - - dep th to cen t ro id of a l l s tee l, t end on s a nd rod re inforcem ent

vr = concre te s t r ength

I n th i s e q ua t ion , t he a pp l i e d she a r f o rc e V a nd be nd ing m om e n t M shou ld be

t a ke n a s pos it i ve , a s shou ld the c r a c k ing m om e n t M o. I n pos t - t e ns ione d f loo r s

the ra t io fpe/fpu is no r m a l ly be twe e n 60 a nd 70% , so tha t E q ua t io n ( 10 .3 ) c a n be

simplif ied to

Vc~ = 0 . 65 v r + M o V / M (10.4)

< O . l b v d f ~ 5

N - m m u n i t s

o r

1.2bvdfr 5

lb - in un i t s

The sh ea r capac i ty o f the sec t ion , Vr i s take n as Vco for an u ncrac ked sec t ion , and

as the lo w er of V~o an d VCr for a cra ck ed sectio n.

The co ncre te s t r eng th re , g iven in Table 10 .3 , i s r e la ted to the rod re inforcemen t

r a t i o a nd the s e c t ion de p th . I n c a l c u l a t i ng the r e in f o r c e me n t r a t i o , bo th r od

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228

POST-TENSIONEDCONCRETE FLOORS

Table 10.3 Un cra ck ed shear strength (Vco/b~d), N /m m 2, calculated using Equation (10.2)

Average

prestress

Pav (N /m m 2) 25

Conc rete strength f~u (N /mm 2)

30 35 40 50 60

1.0 1.04 1.12 1.19 1.26 1.38 1.49

2.0 1.23 1.31 1.39 1.46 1.58 1.70

3.0 1.39 1.48 1.56 1.63 1.77 1.89

4.0 1.54 1.63 1.72 1.79 1.93 2.05

5.0 1.67 1.77 1.86 1.94 2.08 2.21

6.0 1.80 1.90 1.99 2.07 2.22 2.36

8.0 2.02 2.13 2.23 2.32 2.48 2.63

10.0 2.23 2.34 2.45 2.55 2.72 2.87

r e i n f o r cemen t an d p r e s t r e s s i n g s t e e l s h o u l d b e i n c l u d ed , w i t h t h e a r ea o f

p res t r e ss ing s t ee l m u l t i p l i ed b y t he f ac to r fpu / fy , t o acc ou n t fo r it s g re a t e r s t r en g th .

10 .2 .3 AC I 318 she ar s t reng th ca l cu la t ion

T h e ap p r o ach g i v en i n A C I 3 1 8 i s b a s ed o n s i m i l a r p r i n c i p l e s t o t h e B S 8 1 1 0

me t h o d , b u t t h e ex ac t f o r mu l ae d i f f e r s l i g h t l y . O n e s i g n i f i c an t d i f f e r en ce i s t h a t

m a t e r i a l s a f e ty f ac t o r s a r e n o t i m p l i c i tl y i n c l u d ed i n t h e A C I f o r mu l ae . I n s t ead , a

f ac t o r o f 0 .8 5 is i n t r o d u ce d a f t e r t h e ca l cu l a t i o n o f s h ea r s t r e n g t h - - s e e S ec t i o n

1 0 .2 .4 . T h e s h ea r s t r en g t h i n t h e ab s en ce o f f l ex u ra l c r ack i n g i s d e f i n ed a s

Veo = [flp(fc') ~ + 0.3pav]bvd + Vp (10.5)

w h e r e , f o r a o n e - w a y s p a n n i n g m e m b e r ,

tip = 0 .3 N - m m u n i t s

or 3 .5 lb- in uni t s .

T h e t e r m i n s q u a r e b r a c k e t s i n E q u a t i o n ( 10 .5 ) is a n a p p r o x i m a t i o n t o t he s q u a r e

r o o t ex p r e s s i o n i n E q u a t i o n ( 10 .2 ), w i t h t h e t e r m ~ / fc ' r ep r e s en t i n g t h e t en s i le

s t r en g t h . N o t e t h a t t h e v e r t ic a l co m p o n e n t o f t h e p r e s tr e s s , V p, i s i n c l u d ed

ex p l i c i t l y i n t h e eq u a t i o n f o r t h e s h ea r s t r en g t h , r a t h e r t h an b e i n g t r e a t ed a s p a r t

o f t h e ap p l i ed s h ea r f o rce .

T h e e x p r e s s i o n f o r c o m b i n e d f l e x u r a l / s h e a r c r a c k i n g i s

Vet = 0.0 5(fj)~ d +

M e t V / M

i n N - m m u n i t s ( 1 0 .6 a )

or = 0.60(f~') ~ + Met V I M i n l b - in un i t s (10 .6b )

w h e r e M e t = l iv e l o a d m o m e n t r e q u i r e d t o c a u s e c r a c k i n g

= Z t [0 .5 ( f~ ' ) ~ + f~ t] N -m m un i t s

or Zt [6 .0( fc ' ) ~ + f~t ] lb- in uni t s

f e t - ~ - s t r e s s a t ex t r eme t en s i o n f i b r e d u e t o d ead l o ad s .

(10 .7a)

(10.7b)

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SHEAR 2 2 9

N ote that the inclus ion o f the square ro ot term s in Eq uat io ns (10.5) to (10.7)

me ans tha t they a re no t d im ens iona l ly cons i s ten t , and so requ i re the use o f app ro-

pr i a t e un i t s. C are m us t , t herefo re , be t aken to use the appro pr ia t e fo rm o f each

equ at ion , and to use cons i s t en t un i ts . The shea r s t reng th o f the sec t ion shou ld be

take n as the sma l ler of the two v alues calculated us ing Eq uat io ns (10.5) and (10.6) .

10.2.4 Shear reinfo rcement

I f the app l i ed shear fo rce exceeds the ca lcu la ted shear s t reng th , then some

remedia l ac t ion mus t be t aken . In beams , the mos t usua l method i s to p rov ide

reinforce m ent in the form of closed vert ical shear l inks . I t is poss ible to p rovid e

shear l inks in one-w ay s l abs, bu t in p rac ti ce th i s is no t no rm al ly don e . I f the shear

is excessive then usual ly the s lab depth is increased an d the calcu lat ion repea ted.

The shear re in fo rcement in a beam mus t be su f f i c i en t to car ry the d iagonal

t ens ion induced by the shear load ing , and so p reven t c rack ing . Th i s l eads to a

s imple fo rmu la re l a t ing the num ber and s ize o f l inks to the i r s t reng th a nd to the

shea r forces act ing. Th e B S 8110 vers ion of the form ula i s"

Asv /s v > (V - Vr ) (10.8)

whe re Asv = to tal cross-sect ional area of the tw o vert ical legs of a l ink

Sv = long i tudina l spacing of l inks

fyv = yield stren gth of she ar steel

= 460 N/mm (66.7 ks i ) for h igh tens i le s teel

= 250 N/mm (36.3 ks i ) for mi ld s teel

The m ul t ip l ier of 0 .87 is a m ater ial par t ia l safety factor for the sh ear reinforce-

ment .

In ca lcu la t ing shea r re in fo rcement us ing th i s app roac h , a t t en t ion m us t be pa id

to the min imum shear re in fo rcement requ i rements g iven in BS 8110 . For

pres t ressed b eam s , no shear re in fo rcement is requ i red i f the app l i ed she ar fo rce V

is less tha n 0.5 Vr If V exceed s 0.5 Vr bu t is less tha n Vc + 0.4bvd, the n a no m ina l

am ou nt o f shear re in fo rcement is chosen by se t ting the t e rm (V - Vr Eq uat io n

(10.8) equ al to 0.4bvd; th is is don e as us ing the actual value of ( V - Vr wo uld

resu l t in an unaccep tab ly l a rge l ink spac ing . For l a rger va lues o f V , Equat ion

(10.8) is appl ied as i t s tands . The spacing of l inks along the len gth o f a m em ber

shou ld no t exceed 0 .75d unde r norm al c i rcum stances , and shou ld no t exceed 0 .5d

when V is g rea te r tha n 1 .8 Vr There a re no min imu m shear s tee l requ i rem ents fo r

slabs.

Becau se of i ts s l ightly d ifferent ap pr oa ch to the inclus ion of safety factors the

ACI 318 formula i s"

A ,,/Sv >_ ( V - 0.85Vr

(10.9)

Again , there i s no min imum shear s t ee l requ i rement fo r s l abs o r fo r beams in

which V is l es s than 0 .5 Vr Otherw ise , the m in im um shear l ink a rea to spac ing

ra t io may be t aken as the smal l e r o f

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230

P O S T-TE N S I O N E D C O N C R E TE FL O O R S

I

v - -

Figure 10.4

Punch ing cone fo rma t i on a t a co l umn connec t i on

or

A~v/S, > bv/(3fy v) ( lO.lOa)

A~v/S ~ >_ [(Apfpu)/ (8Ofy ~d )](d /b~) ~ ( lO.lOb)

1 0 . 3 T w o - w a y s la b s

In two -wa y spann ing f loors the c r it ica l des ign case is punc h ing shear a rou nd the

co lum ns o r und er very la rge conc en t ra ted loads . The exac t fo rm o f the fa ilu re

mechan i sm var i es wi th the re in fo rcement de ta i l s , bu t in genera l i t cons i s t s o f

c rack ing th rou gh the s l ab , usua l ly a t an an g le cons iderab ly f l a t te r th an 45 ~ so

tha t an inver t ed , t runc a ted cone o f ma ter i a l is d i s lodged , as show n in F igure 10.4.

In des ign, rath er th an t ry ing to assess the cone an gle, it i s usual to c alculate the

shear carr ied by a ver t ical fai lure zone whose edges are a f ixed dis tance from the

face o f the co lumn. The zone thus def ined is know n as the cr i t ica l per imeter . Since

the p res t ress ing a r rang em ents in the two perpen d icu la r span d i rec t ions a re ra re ly

the sam e, the she ar capaci t ies o f each pai r of paral le l edges are l ikely to be

d i f fe ren t and so mus t be ca lcu la ted separa te ly .

10.3.1 Ap plied punching shear force

Obv ious ly , the max im um punc h ing shear fo rce in a s lab occurs on a per im eter a t

the face o f the co lum n, a nd equa l s the to ta l ver t ica l load be ing t rans fer red f rom

the f loor in to the co lumn. Fo r a c r it ica l per imeter a t some d i s t ance f rom the face

of the co lumn, th is m ax im um shear fo rce is reduced by sub t rac t ing the ver t ica l

load ac t ing on the a rea o f s lab ins ide the per imeter .

For uncracked sec t ions , the punch ing shear can be fu r ther reduced by the

vert ical co m po ne nt of the pres t ress ing force, Vp, wi thin the cr i tical per ime ter . F or

exam ple , F igure 10 .5 shows a typ ica l p res t ress ing t end on pass ing over an in te r io r

co lumn and the resu l t ing equ iva len t ver t i ca l loads . Note tha t the en t i re reverse

curv a tu re o f the t endo n lies wi th in the c r i ti ca l per ime ter . The con t r ibu t ion o f th is

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S H E A R 231

Point of Edge of cr i tical

cont raf lexure,

,~ zone

' _ _ ;

I I

Equivalent

loads:

Wl

w2 w2

I_.. a .- I

i--. Vl

Figure

10.5 Tendon over a suppor t

t endon i s w l a - w 2 ( b - a ) and the to ta l va lue o f Vp is found by sum m ing the

con t r ibu t ions o f a ll tendo ns ( in bo th span d i rec t ions ) which pass wi th in the

cr it ica l per imeter . I t shou ld be remem bered tha t p lac ing t endon s so as to ach ieve

a high reverse cu rva ture over a colum n is d i ff icul t, and th at smal l var iat ion s in the

cable profi le can c ause qu i te large change s in l ip . I t is , therefore, p rud en t to

eva lua te th i s t e rm con serva t ive ly , an d i t is norm al to t ake Vp as zero fo r tendon s

whose points inf lexion l ie outs ide the cr i t ical zone.

As wi th one-way shear , appropr ia t e load fac to rs as s t ipu la ted in the na t iona l

codes shou ld be app l i ed to the ex te rna l loads caus ing the punc h ing shear , whi le

the reduc t ion t e rm du e to the ver ti ca l com pon en t o f the p res tress shou ld a lways

be ass igned a load factor of 1 .0 .

1 0. 3.2 M o m e n t t r a n s f e r

I f the to ta l shear fo rce ac t ing on a per ime ter a ro un d a co lum n i s V and the leng th

of the perim eter i s u , then the avera ge sh ear s t ress v act ing on the assu m ed fai lure

plane is s imply given by V/ud . Ho w ev e r , i n man y f l o o r s t h e s l ab -co l u mn

conn ec t ion i s requ i red to t ransm i t a m om ent M t f rom the f loor s l ab in to the

co lumn . The va lue o f th is mo m ent c an be de te rmined f rom the s t ruc tu ra l ana lys i s

carr ied o ut for the f lexure des ign. The t ransfe r of m om en t acro ss the cr it ical

per ime ter occurs by a co m bina t ion o f f lexure and eccen t r ic i ty o f shear , resu l ting

in a n o n -u n i fo rm s h ea r d is t r ib u t i o n a ro u n d t h e p e r ime t e r , w i t h a m ax i m u m s h ea r

s t res s which may be cons iderab ly l a rger than the average va lue . F igure 10 .6

shows the com m only assum ed fo rms o f shear s tres s d i s t r ibu t ions a ro un d bo th

in te rna l and edge co lumn connec t ions where moment t rans fer occurs . The

various nat ional codes of pract ice give appro xim ate form ulae, based on experimental

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232 POST-TENSIONED CONCRETE FLOORS

I I

(a)

M ,

/

(b)

Figure

10.6 Effect o f mom ent t ransfer on shea r s tress d is t r ibut ions (a) In ternal co lum n

(b) Edge co lumn, w i th mom ent t ransfer par a l le l to the edge

t e s t ing , f rom which the e ffec t o f m om en t t r ansfe r on the shea r s t r e ss d is t r ibu t ion

can be ca lcu la ted . The de ta i l s o f these app roa che s a re d iscussed fur th e r in

Sections 10.3.3 and 10.3.4.

1 0 .3 .3 B S 8 1 1 0 s t r e n g t h c a l c u l a t i o n

Punching shea r ca lcu la t ions a re dea l t wi th in s l igh t ly d i f fe ren t ways in Br i ta in

and the USA . The us ua l Br i t i sh prac t ice i s to sum the shea r capac i t ie s of the four

e dge s o f t he c ri ti c al pe r ime te r a nd c om pa r e t he va lue thus ob ta in e d w i th t he t o t a l

shea r load on the pe r ime te r . The shea r s t r ength in a g iven d i rec t ion i s ca lcu la ted

by us ing the f o r mu la e f o r one - wa y spa nn ing me mbe r s g ive n in s e c t ion 10 .2 .

The a rea a ro un d a co lu m n is d iv ided in to a se r ie s of c r i tica l pe r ime te r s , the f i r st

1 .5d f rom the face of the co lum n, wi th su bseq uen t pe r im e te r s spaced a t in te rva ls

of 0 .75d , a s show n in F igu re 10 .7 . The shea r f a i lu re a ssoc ia ted wi th a g iven

pe r ime te r i s a s sume d to oc c u r w i th in a n a r e a e x t e nd ing 1 .5d inwa r ds f r om the

pe r ime te r t ow a r ds t he c o lumn . F o r e xa mple , t he no t iona l fa i lu r e zone a s soc i a t e d

wi th the f i r st pe r ime te r i s sho wn shad ed in F igu re 10 .7 . I f f ree edges or o pen ings in

the s lab ex is t in the v ic in i ty of the c o lum n, then these wi ll s ign i fican t ly a l te r the

shea r d is t r ib u t io n a ro un d the pe r im e te r , s ince the shea r s t r ess a t a fr ee edge mu st

be ze ro . This is take n in to acco un t by reduc in g the length of the c r it ica l pe r ime te r

as shown in F igure 10 .8 .

The sh ea r s t r eng th i s f ir s t a ssessed on the inne rm ost pe r ime te r . I f the s t r en gth i s

ade qua te , then n o fur the r ac t ion i s requi red . I f the shea r s t r e sses a re excess ive,

the n she a r r e in f o r c e me n t o r a n inc re a se in s l a b th ic kne s s mu s t be p r ov ide d . T he

process i s r epea ted for success ive pe r ime te r s , un t i l a pe r ime te r i s r eached where

the s t r e ng th i s a de q ua te .

Ra the r tha n assess ing in de ta i l the va r ia t io ns in shea r s tr e ss a ro un d the c r it ica l

pe r ime te r c a use d by m om e n t t r ans f e r, B S 8110 s imp ly r e c om m e nds the u se o f a n

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SHEAR 233

- r

I

I r

i I

I I

F

I I

~ I / / / /

, I / /

I I V / /

, ,

I I

I I Z

I I

I L . . . . . . .

I

L . . . . . . . . .

I

, I

I I 1st critical perimeter

/- -- / - ] ~j ~ length u = 2 (x1 + x2)

/ / j ~ i i

/ / z r ~ I

/ / / / I ! I Notional ailure zone

/ / / ! ~ ~' f o r 1st cri tical

/ ~ / ~ I perimeter

/ / i ,

I i

i I

1.5d 0.75d 0.75d

I I

Jr,

I

9/ ~ I I i

Failure corresponding to

1st critical perimeter

F i g u r e

10.7 BS 8110 definition of critical perimeters

effective shea r force,

V e f f .

I f p r io r a na ly s i s ha s y i e lde d the va lue o f the m om e n t

M t

t r a ns fe r r e d to t he c o lum n , t he n

Vet f

may be ca lcu la ted as , fo r in te rna l co lumns"

Vef = 1,'[1 +

1.5Mt/(Vx2) ]

(10.11)

o r , fo r e dge c o lum ns w he r e be nd ing is a bou t a n a x i s pe r pe nd ic u la r t o t he fr ee e dge

Vef

=

V[1.25 + 1 .5Mt/ (Vx2) ]

(10.12)

where x 2 = l ength of the edge of the c r i tica l sec t ion pa ra l le l to the m om en t ax is -

see Figure 10.6.

In the absence of ca lcu la ted va lues of Mt,

V e f f

m a y be t a k e n a s 1 .15V f o r i n t e r io r

c o lum ns , 1 . 4V f o r e dge c o lumn s whe r e m om e n t t r a ns fe r is a bou t a n a x i s

pe r pe nd ic u la r t o t he f re e e dge , a nd 1 .25V f o r e dge c o lum ns whe r e m om e n t

t ransfe r i s about an ax is pa ra l le l to the f ree edge .

The va lue of V e t f t hus c a l c u l a t e d mus t be c ompa r e d w i th t he de s ign she a r

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234

POST-TENSIONED CONCRETE FLOORS

Free edge

/

Loaded

area

Normal critical

perimeter

/

v

Alternative

critical perimeter

(use the shorter

one)

\ ~ Opening

\ /

\ /

(a) (b)

Effective

perimeter

Figure

10.8

Effect of (a) free edges and (b) openings on assumed length of cr i t ical

p e r ime te r

s t reng th g iven by Equat ions (10 .1 ) to (10 .4 ) , and wi l l depend on whether the

sec t ion i s c racked o r uncracked in f l exure . When app ly ing Equat ions (10 .3 ) o r

(10.4) , the value of the con crete shea r s t reng th vc m us t be foun d f rom Tab le 10.1 ,

as a func t ion o f the sec t ion dep th a nd the re in fo rcement ra t io Adbvd. I n m a n y

s labs , t endons a re p laced in bands a long the co lumn l ines , g iv ing a very h igh

re in fo rcement ra t io immedia te ly ad jacen t to the co lumn, bu t a much lower va lue

in the nearby s lab . The C oncrete Society (1994) recom me nds tha t the reinforcemen t

ra t io sh ou ld be avera ged over a w id th o f twice the s ide o f the c r i ti ca l per imeter , in

orde r to avo id exagg era t ing the e ffec t o f ba nde d p res t ressing t endon s .

Near ly a l l s l abs wi l l have d i f fe ren t p res t ress ing and rod re in fo rcement

conf igura t ions in the two p erpen d icu la r span d i rec t ions . I t is , t herefo re , necessary

to ca lcu la te the she ar cap ac i ty o f each pa i r o f para l l e l edges o f the c r i t ica l

perimeter separately . I t i s then suff icient ly accurate to sum the shear res is tances

a ro u n d t h e p e ri me t e r an d co m p are t h e v a l u e th u s o b t a i n ed w i t h t h e to t a l ap p l i ed

shear force,

Veff.

If the to tal sh ear res is tance ar ou nd a per im eter exceeds Wef then the s lab is

ade qua te. I f , on the oth er ha nd , i t i s less tha n

Wef som e

s t reng then ing is requ i red .

Fo r s l abs g rea te r than 200 mm (8 in ) deep , s t reng th i s norm al ly p rov ided by the

inclus ion of l inks. L inks in th in ner s labs are ext rem ely di ff icul t to f ix an d are l ikely

t o b e to o s h o r t t o a ll o w th e d ev e l o p me n t o f ad equ a t e an ch o rag e . L i n k s mu s t b e

prov ided a round a t l eas t two per imeters wi th in the no t iona l fa i lu re zone

cor respond ing to the c r i t i ca l per imeter be ing checked . The innermos t per imeter

o f re in fo rcem ent shou ld be appro x im ate ly 0 .5d f rom the face o f the co lum n, w i th

subsequen t per imeters no t more than 0 .75d apar t . The l ink spac ing a long a

per im eter shou ld no t exceed 1 .5d. The to ta l a m ou nt o f shear re in fo rcem ent

prov ided m us t co mp ensa te fo r the d i ffe rence be tween the app l i ed shear fo rce and

the concrete shear s t rength , i .e . :

XAsv ~ (V e ff - VcF(O.87fyv)

(10.13)

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SHEAR

2 3 5

l - " " -

l -

J _

- - t

0 . 7 5 d 0 . 7 5 d

L. _1_., _1

I ~ " - 1 ~ " - 1

" - - ~ ,~ . C r it ic a l p e r im e t e r

_ I . L _

- - I . C o lu m n

- r

I

,,. 0 0 0 0 0 0 , ~ 0 'Omtbeer ~iO ~O rl~e i ~ n t

F i g u r e

1 0 . 9

Typica l arran gem ent o f l inks and lac ing b ars to res is t punc h ing she ar in

accordance wi th BS 8110

Obvious ly the fa i lu re zones cor respond ing to success ive c r i t i ca l per imeters

over lap . In as sess ing the re in fo rcement requ i rements a t a par t i cu la r per imeter ,

BS 8110 a l lows ac coun t to be t ake n o f any shear re in fo rceme nt ly ing wi th in the

re levan t zone , even i f i t was p rov ided to re in fo rce o the r zones . F or exam ple ,

F igure 10 .9 shows a su i tab le a r ra nge m ent o f shear l inks cor respo nd ing to the

inn erm ost cr i t ical per im eter . As well as reinforcing th is zone, the o uter set of l inks

also l ies wi thin the zone c orre spo ndin g to the nex t cri t ical per imete r . The refore, if

i t i s subsequent ly found that shear i s excess ive on th is second perimeter , then

l inks would be requ i red a round on ly one add i t iona l per imeter wi th in the second

no t iona l fa i lu re zone .

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236

POST-TENSIONED CONCRETE FLOORS

10.3.4 ACI 318 punching strength calculation

The p ro cedu re ado p ted by AC I 318 fo llows the sam e bas ic p r inc ip les , bu t d i f fe rs

from the Bri t i sh pract ice in some s ignif icant respects . Fi rs t ly , the in i t ia l shear

check i s per fo rmed on a rec tangu lar per imeter on ly 0 .5d f rom the co lumn face ,

com pare d wi th 1 .5d in the Br i t ish code . Seco nd ly , the check is based on the shear

s tress on the m os t heav i ly loaded edge o f the per imeter , which i s ra the r m ore

conserva t ive than the Br i ti sh p rac t ice o f check ing the to ta l shear a ro un d the

per imeter . In AC I 318 , the recom m ende d re in fo rc ing a r ra nge m ent cons i st s o f a

crosshead o f re in fo rc ing bars suppor t ing conve n t iona l re c tangu lar l inks , as

show n in F igure 10.10 . The d i s t ance th is c rosshead mu s t ex tend f rom the face o f

the co lumn i s then de te rmined by check ing shear on a new, oc tagonal c r i t i ca l

perim eter . As can be seen from Fig ure 10.10, th is consis ts of four edges para l le l

and equa l in l eng th to the s ides o f the co lum n, a t a d i s t ance

d/2

f rom the end o f the

main bars , connec ted by s t ra igh t l ines .

F or s labs of uniform thickness , sh ear i s check ed alon g a s ingle cr it ical

per im eter loca ted a t a d i s t ance 0 .5d f rom the co lum n face . For s l abs wi th vary ing

thickness , such as where drop panels are used, i t i s a lso necessary to consider a

per im eter a t the edge o f the d rop pane l . Adjus tm ent s to the c ri ti ca l per im eter to

accoun t fo r open ings o r f ree edges a re made in the same way as ou t l ined in

Sect ion 10.3 .3 and i l lus t rated in Figure 10.8 .

The e ffec t o f m om ent t rans fer on the d i s t r ibu t ion o f shear s t ress i s dea l t wi th b y

the c alc ulatio n of a fac tor Yv defined as:

Y v - 1 - 1 /[1 +

0.67(x1/x 2) ~

(10.14)

wh ere Xl and x 2 are the s ides of the cr i t ical per im eter respect ively paral le l a nd

perpe nd icu la r to the d i rec tion o f m om ent t rans fer , as shown in F igure 10 .6 . I t is

as sumed tha t the non-un i fo rm shear d i s t r ibu t ion shown in F igure 10 .6 accoun t s

fo r the t rans fer o f a m om ent ~ /vMt, the rem ainde r be ing t rans fer red by f l exure o f

the s l ab -co lumn connec t ion . Refer r ing to F igure 10 .6 , the resu l ting express ions

fo r the shear s tresses on edges AB and C D of the c r i ti ca l per imeter a re :

VAB = (V/ud) -F ()~vMtx l / 2 J e )

( lO.15a)

VCD = (V /ud ) - (YvMtXl /2Jr

( lO.15b)

wh ere Jr i s a geom etr ic pro pe rty of the cr i t ical sect ion, def ined for the in ternal

co lumn shown in F igure 10 .6 (a) as

Je = dx13/6 + xld3/6 + dXEXl2/2

and fo r the edge co lumn, F igure 10 .6 (b ) as

Jc = dx13/12 + xld3/12 + dx2x12/2

(10.16a)

N ot e th at i t is nec essa ry to c alcu late b oth VAB and/)CD, as i t is the sh ea r stress o f

gre ates t m ag nit ud e tha t is cri t ical . If M t is very la rge, the n i t is possible fo r VCD to

be more cri t ical than VAB.

(lO.16b)

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SHEAR

237

r a n d

/ N Cr i t ic a lsection

/ \ N

/ ] \

/

/ ] \

/ rH I \

I I

\ "11 I ' /

r /

\ 11 I I /

\ '11 II' /

Plan \ ] i I I ' /

N /

\ /

, ,4 ,

Elevation

4 ,

Figure 10.10

Typ ical a r rangem ent o f l inks and lac ing bars to res is t punch ing s hear in

accordance w i th AC 318

T h e m a x i m u m s h e a r s t r e s s t h u s c a l c u l a t e d m u s t b e c o m p a r e d t o t h e s h e a r

s t r e ng t h o f t he s l a b . AC I 318 a l low s t he pun c h i ng she a r c a pa c i t y t o be c a l c u la t e d

on t he a s sum p t i on t ha t t he s e c t i on i s unc r a c ke d so l ong a s c e r t a i n r e q u i r e m e n t s

f o r m i n i m u m a m o u n t s o f b o n d e d r e in f o r c e m e n t a r e s a ti sf ie d . T h e s h e a r s t r en g t h

Vc c a n , t he r e fo r e , be f ound f r om t he e xp r e s s i on f o r a n unc r a c ke d m e m be r ,

E qu a t i on (10 .5) , w i th the Vp t e rm d e te rm ined as ou t l ined in Sec t ion 10 .3 .1 . S ince

t he e dges AB a n d C D g i ve t he w or s t sh e a r s t re s ses , t he she a r w i d t h bv i n Eq ua t i on

(10 .5) shou ld b e t ak en as the s ide x 2 o f the c r i t ica l pe r im e te r . The pa ram ete r tip i s

t a ke n t o be t he sm a l l e r o f

tip = 0.3 or tip = (~d/12u + 0 .125)

tip = 3.5 or tip = (o~d/u + 1.5)

N - m m u n i t s

lb - in un i t s

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238 POST-TENSIONEDCONCRETE FLOORS

where ~ = 40 fo r in te rna l co lumn s

= 30 fo r edge co lumns , and

= 20 fo r co rner co lumns

Incorpora t ing a s t reng th reduc t ion fac to r in to the ca lcu la ted s t reng th , the

al lowable shear s t ress around the cr i t ical per imeter i s

1)al I : 0. 85 V c / x 2

.d.

I f the magn i tude o f the max im um shear s t res s ca lcu la ted by Equ at ion (10.15)

exceeds th i s va lue then e i ther the s l ab th ickness mus t be increased o r shear

re in fo rcement mus t be p rov ided .

The requ i red a m ou nt o f l inks can be de te rm ined f rom Eq uat io n (10 .9 ). A

s y mm et r ica l a r r an g em en t o f l in k s a ro u n d t h e co l u mn a r ea a s i ll u s tr a t ed i n F i g u re

10 .10 i s reco m m ende d by AC I 318 . Here , the l ink size and spac ing a re chosen on

the bas i s o f the m os t h eav i ly loaded o f the four edges o f the c r i ti ca l per imeter , and

so incor pora te a degree o f conserva t i sm fo r the o ther d i rec tions . A CI 318

s t ipu la tes tha t the l ink spac ing shou ld n o t exceed 0 .75D or 600 mm (24 in ). The

requ i red d i s t ance to which the l inks mus t e x tend f rom the face o f the co lum n is

de te rm ined by def in ing a new, e igh t - s ided c r i ti ca l per imeter

d/2

beyond the l as t

l ink, as shown in Figure 10.10, and checking that the shear s t resses on th is new

per imeter a re accep tab le .

An accurate shear check on th is new perimeter i s ext remely di ff icul t , s ince

severa l o f it s edges a re ro ta ted a t 45 ~ to the p res t ress a nd m om en t t rans fer

d i rec t ions , so tha t the p reced ing equa t ions canno t eas i ly be app l i ed to these

edges . The fo l lowing s impl i f i ed p rocedure i s , t herefo re , recommended . F i r s t ly ,

the shear s t ress due to the vert ical shear force alone is calculated on the new

per imeter , as

V/ud.

I f an u n b a l an ced m o m en t is b e in g t r an s mi t t ed b y t h e

connect ion, then the resul t ing addi t ional shear s t ress i s calculated for the or ig inal

per imeter , us ing the second t e rm in Equat ion (10 .15a) . The maximum shear

s tress on the per ime ter is then t ak en as the sum of these two t e rms . Th i s m axim um

stress mu st n ot the n exceed the permiss ible s t ress of the concre te alone, neg lect ing

the prestress, that is, V,l = flp(fc')~ Th i s ra the r conserva t ive ap pro ach is

recom m ende d on the g rou nds o f i ts s impl ic ity .

1 0.3 .5 D e c o m p r e s s i o n l o a d m e th o d

An a l t e rna t ive to the approaches ou t l ined above i s the decompress ion load

method , deve loped by Regan (1985) . Th i s g ives a s imple and conven ien t way o f

re la t ing the pu nch ing s t reng th o f a p res t ressed s lab to tha t o f a re in fo rced

concre te f loor . The app roac h is simi la r to the ca lcu la t ion o f shear s t reng th fo r

pres t ressed beams cracked in f lexure, Equat ions (10.3) and (10.6) .

Th e

decompression load

i s def ined as the force requ ired to cancel o ut the in i t ia l

com press ive s t ress caused by the pres t ress a t the face of the s lab which is no rm al ly

in t ens ion under app l i ed loads . At loads above the decompress ion fo rce , the

concrete sect ion s tar ts to crack and so behaves s imi lar ly to a reinforced sect ion.

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S H E A R 239

x2

Critical perimeter

I / Edges 1

I I I

. . . . I[

_ _ .

Edges 2

Tendons 2

Tendons 1

Figure 10.11

Typical tendon arrangement through a cri t ical perimeter

The s t reng th o f the p res t ressed s l ab is taken as the sum of the decom press ion fo rce

and the shear s t reng th o f an equ iva len t re in fo rced me mb er . (The equ iva len t

reinforced concrete sect ion is assumed to be geometr ical ly ident ical to the s lab

und er cons idera t ion , and to have a re in fo rcement qua n t i ty equa l to the sum of the

rod re in fo rcement a rea and the p res t ress ing s t ee l a rea mul t ip l i ed by fpu / f y . )

N ow for a g iven set of pres t ress ing ten don s , the pres t ress can inf luence only

those edges o f the c r i ti ca l per imeter p erpend icu la r to the t end ons . I t is , t herefo re ,

necessary to con s ider the two span d i rec t ions separa te ly . F igure 10 .11 shows a

typ ical a r rang em ent o f p res t ressing t endons pass ing th rou gh a c ri t ica l per imeter ,

wi th the two t end on d i rec t ions and the cor respo nd ing perpe nd icu la r edges o f the

shear per imeter den o ted by the subscr ip ts 1 and 2 . The shear s t reng th V of the

equivalent reinforced concrete sect ion is s imply calculated as

v , = v , ~ + v , ~

= 2 v ~ l x l d 1

+ 2V e2x2d 2 (10.17)

whe re the values o f vr are foun d from Ta ble 10.1 .

Nex t , i t i s necessary to ca lcu la te the decompress ion loads fo r the two span

d i rec t ions . For a g iven s t ruc tu ra l geomet ry and p res t ress ing a r rangement , the

decom press ion load Vo can be re l a t ed to the mo m ent Me g iven by Equ at ion

(10 .1 ). For exam ple , fo r a concen t ra ted load a t the cen t re o f a span o f l eng th L

Vo = 4 M o / L

The p unch ing s t reng th o f the p res t ressed sec t ion i s then g iven by

vr = v , + Vo~(GllVr) + Vo~(V~dV~)

(10.18)

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240

POST-TENSIONEDCONCRETE FLOORS

Thus the overa ll decom press ion load fo r the s lab is t aken as the weigh ted ave rage

of the decom press ion load s cor respo nd ing to the two d i ffe ren t p res t ress ing

ar rangements . Th i s i s a conserva t ive approx imat ion to the exac t fo rmula t ion ,

which is d i ff icul t to solve. Eq ua t io n (10.18) represents the m ost g enera l case of the

decom press ion load ap proa ch . Fo r a s lab p res t ressed in one d i rec t ion on ly it can

be simplified to

vo = v, + Vo,( Vr, /V ,)

(10.19)

Fo r a s lab wi th the same pres t ress ing arrang em ent in each direct ion Vo ~ = V o 2 = V 0

and Equat ion (10 .18) reduces to

Vr = F r + Vo (10.20)

1 0 .4 A l t e r n a t i v e s t o c o n v e n t i o n a l s h e a r r e i n f o r c e m e n t

The p ro v i s ion o f conv en t iona l s t i r rups in s l abs can o f t en be a p rob le m; in o rder to

be ful ly effect ive, the l inks must be anchored in the compress ion zone of the s lab

and t ied in to the ma in tens io n s teel. Fo r th is reason , it i s no t norm al ly po ss ible to

prov ide l inks in s l abs th inner than abo u t 200 mm (8 in) . Even in th icker s l abs , the

de ta i ling an d f ix ing o f the l inks can be d i ff icu lt and t ime-con sum ing opera t ions .

Cons iderab le e f fo r t s have , therefo re , been expended in the deve lopment o f

a l t e rna t ive systems a imed a t reduc ing con s t ruc t ion t ime and cos t whil e imp rov ing

st ructural rel iabi l i ty .

10.4. 1 P rop r ie ta ry shearh oop s

On e a t t rac t ive a l t e rna t ive i s the use o f p refabr i ca ted cages o f shear re in fo rc em ent

which can be s imply p laced on s it e and t i ed in to the m ain ben d ing re in fo rcem ent .

The des ign o f such sys tems requ i res cons iderab le care ; s l abs re in fo rced by

prefabr ica ted cages which s imply s i t be tween the top and bo t tom re in fo rc ing

mats have been found to fa i l a t loads lower than p red ic ted by the BS 8110

approach , because the l inks a re no t e f fec t ive ly t i ed in to the main t ens ion

re in fo rcement .

A sys tem of shea rhoo ps com ply ing wi th the requ i reme nts o f BS 8110, which

prov ides fu ll anch orag e whi le main ta in in g ease o f as sembly , has been deve loped

by Ch an a (1993). The sys tem is com me rcial ly ava i lable in a wide range of s izes

an d rein forcem ent densi t ies . A typical she arh oo p consis ts of a ser ies of special ly

shaped l inks he ld toge ther by two hor iz on ta l hoo p bars , F igure 10.12 . The l inks

are des igned to cover the maximum poss ib le dep th wi thou t v io la t ing the cover

requ i rem ents , an d to be easy to t ie in to the m ain b end ing re in fo rcement , o f fe r ing

a bar r i e r to the fo rm at ion o f shear c racks b e tween the l inks and the main s tee l.

The hoop bars ho ld the cage toge ther , mak ing i t easy to hand le , p rov ide some

add i t iona l anchorage to the l inks and o f fe r some conf inement to the s l ab .

S t ruc tu ra l t es t ing o f the she arho op sys tem sugges ts tha t i t resu lt s in punch in g

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I

SHEAR

241

Figure 10.12

Prefabricated shearhoop

shear capac it ies a ro und 10% grea te r than those o f s labs re in fo rced by conven t iona l

l inks.

She arhoo ps a re pos i t ioned a f t e r the fixing o f the bo t to m s lab re in fo rcement .

They s i t on spacers , wi th the l inks given the same cover as the lowest layer of

bending s teel . The top s teel i s then f ixed around them, the inner layer being

loca ted un der a nd a t r igh t ang les to the p ro t rud in g hor izon ta l l egs o f the links ,

thus ensuring that the l inks are fu l ly t ied in to the tens ion reinforcement . Fixing

tr ia ls suggest that the t ime ta ke n to f ix she arh oo ps is less tha n h al f tha t req uired

for conven t iona l shear re in fo rcement .

10.4.2 Structural steel shearheads

AC I 318 perm its the use of shearhe ads m ade of s t ructural s teel I - or channel-sect ions

in p lace o f conv en t iona l shear re in fo rcement . She arhead s usua l ly cons i st o f

iden t i ca l sec t ions , pos i t ioned a t r igh t -ang les and connec ted by fu l l -pene t ra t ion

welds, see F igure 10.13. These can be p refabr i ca ted awa y f rom s it e and q u ick ly

posi t ioned in the s lab pr ior to cas t ing, leading to s ignif icant reduct ions in f ix ing

t imes on s i t e . Shearheads a l so make a s ign i f i can t con t r ibu t ion to the moment

capac i ty o f the s lab a t the co lumn c onnec t ion , resu l ting in reduced am oun ts o f

conven t iona l bend ing re in fo rcement .

Shea rheads m us t sa ti sfy two des ign c r i te r i a . F i r s t ly , they mus t ex tend f rom the

co lumn face to a per imeter a t which the shear in the s l ab can be car r i ed by the

concre te a lone . Second ly , they mus t have su f f i c i en t moment capac i ty to ensure

tha t the requ i red shear s t ren g th o f the s l ab is reached befo re the f l exural s t reng th

of the sh ear hea d arm s is exceeded. T he f i rs t of these cr i ter ia can be used to assess

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242

POST-TENSIONED CONCRETE FLOORS

Figure

10.13 Structural steel shearhead

t he requ i red l eng th o f the shearhead a rm s by a simple i t e ra tive p rocedu re . Fo r a

guessed a rm leng th , the c ri ti ca l shear per im eter i s deduced as ou t l ined be low. I f

the shear s t ress on th is per imeter proves excess ive, then the arm length is

increased and the ca lcu la t ion repea ted un t i l an accep tab le shear s t res s i s

ach ieved . Once the requ i red l eng th o f the shearhea d a rm s has been found , then

the mo m en t capac i ty c r i te r ion can be used to es t ab l ish the requ i red sec t ion s ize.

Con s ider f i r st ly the requ i red l eng th o f the shearhe ad a rms . Fo r a g iven leng th

of shearh ead , the c r it ica l per imeter i s as sum ed to pass th roug h each a rm of the

shearh ead a t th ree-qu ar te rs o f the d i s tance f rom the face o f the co lum n to the end

of the a rm. T he shea rhead m us t ex tend be yon d th i s c r it ica l loca t ion in o rde r to

en s u re ad equ a t e an ch o rag e . Th e r ema i n d e r o f t h e p e ri me t e r , b e tw een t h e

shearh ead a rms , i s m ade up o f s t ra igh t lines , wi th the p rov i so tha t the per ime ter

need never come c loser than d/2 to the face of the colum n. T he resu l t ing cr i t ical

per imeters fo r s l abs re in fo rced by shearhe ads o f var ious s izes a re show n in F igure

10.14.

The shear capac i ty o f the concre te sec t ion can then be checked us ing an

approx imate approach s imi la r to tha t descr ibed in sec t ion 10 .3 .4 fo r s l abs

reinforced by l inks . The sh ear s t ress due to the v ert ical shear force is calcu lated on

the c r i t i ca l per imeter , whi l e the shear s t res s due to any moment t rans fer i s

ca lcu la ted by t ak ing a rec tangu lar p er imeter

d/2

f rom the face o f the co lum n. The

to ta l shear s t res s i s then found by summing the two t e rms . Th i s va lue i s then

co m pa red to the al lowab le s t ress, ignoring the effect of the pres t ress , th at i s ,

flpx/fc'- I f the sh ear s t ress is excessive, then the len gth of the she arh ead arm s m ust

be increased accord ing ly . The necessary ad jus tment i s easy to ca lcu la te , s ince

only the f i rs t of the two shear s t ress terms is affected by the change in cr i t ical

per imeter .

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SHEAR 2 4 3

0 . 7 5 [ I v - c I -

S h e a r h e a d

2 / ] , - ~ , , : r m s C r it ic a l

/ Y "

I I

\,, / / d / 2

\ \ / / 1 9 1 ~

/

/

S h e a r h e a d

, ~ a r m s

/ ' 1 I \ C r i t i c a l

/ / ~ ~ \ < P : I i ~ e te r

I i

/

/

/

/

\ /

\ /

( a ) ( b )

F i g u r e

1 0 . 1 4 Crit ical

p e r i m e t e r s a r o u n d s h e a r h e a d s a t i n t e r n a l c o l u m n s

(a) Small

s h e a r h e a d ( b ) L a r g e s h e a r h e a d

c

~ v - c / 2

, , ,4

. ' ~ . o . ~ . ., z ~ .. ~ - .

- ~ . .

.-.O o , % % 9

" ' ~ " I h v

/ v _ 1

-.~

yl

(I - - ( Z v )

0 . 9 q q

V V

q

F i g u r e

1 0 . 1 5 Elevat ion

an d assu m e d sh ear fo rce d is t ribu tio n a lo n g a sh earh ead a rm

H a v i n g d e t e r m i n e d t h e le n g t h o f th e s h e a r h e a d a r m s , i t is th e n n e c e s s a r y t o

c h o o s e a s e c t i o n h a v i n g a n a d e q u a t e f l e x u r a l s t r e n g t h . F i g u r e 1 0 . 1 5 s h o w s t h e

e l e v a t i o n o f a t y p i c a l a r m o f a sh e a r h e a d , t o g e t h e r w i t h t h e a s s u m e d d i s t ri b u t i o n

o f s h e a r fo r c e a l o n g t h e a r m . H e re "

r / = t h e n u m b e r o f a r m s o f t h e s h e a r h e a d ( u s u a l ly t h r e e f or a n e d g e c o l u m n

o r 4 f o r a n i n t e r n a l c o l u m n ) , a n d

c~v = t h e r a t i o o f t h e f l e x u r a l s t i ff n e s s e s o f t h e s t e e l s e c t i o n a n d o f t h e

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POST-TENSIONEDCONCRETE FLOORS

s u r r o u n d i n g c o n c r e t e s l ab , ta k i n g a w i d t h o f s la b e q u a l to t h e c o l u m n

w i d t h p l u s t h e e f f e c t i v e d e p t h

o~ = E s l s / E b c + d ) d 3 / 1 2 ]

w h e r e b c = c o l u m n w i d t h

T h e m o m e n t e x e r t e d o n t h e s te e l s e c t i o n a t t h e fa c e o f t h e c o l u m n i s t h e i n t e g r a l o f

t h e s h e a r f o r c e d i s t r i b u t i o n i n F i g u r e 1 0 . 1 5 . F o r t h e s e c t i o n t o b e a d e q u a t e , i t s

f le x u r al c a p a c i t y m u s t n o t b e le ss t h a n t h is m o m e n t . A C I 3 1 8 r e c o m m e n d s t h a t

t h e c a l c u l a t i o n b e s i m p l i f i e d b y a s s u m i n g V c ,~ V / 2 , w h i c h r e s ul ts in a m i n i m u m

r e q u i r e d p l a s t i c m o m e n t c a p a c i t y g i v e n b y :

0 . 9 M p = ( V / 2 ~ l ) E h v + 0~v(/v - 0 . 5 C ) ] ( 1 0 . 2 1 )

w h e r e h v = d e p t h o f s h e a r h e a d

Iv = d i s ta n c e f r o m c e n t r e o f c o l u m n t o e n d o f s h e a r h e a d a r m

c - c o l u m n c r o s s -s e c t io n a l d i m e n s i o n i n t h e d i r e c ti o n o f t h e s h e a r h e a d a r m .

T h e m u l t i p l i e r o f 0 .9 i s t h e A C I 3 1 8 s a fe t y f a c t o r f o r b e n d i n g s t r e n g t h .

A n a d e q u a t e s te e l s e c ti o n c a n t h e n b e c h o s e n b y c a l c u l a ti n g t h e r e q u i r e d p l a s ti c

m o d u l u s S , g i ve n b y S

=

Mp/fyw h e r e f y i s t h e y i e l d s t r e n g t h o f t h e s t e e l ( i n c l u s iv e

o f a m a t e r i a l s a f e ty f a c t o r ) .

Exam p le 1 0.1 One-way shear calculation

Calcula te the on e-way shear capa ci ty of a solid s lab con t inuous over two equa l spans ,

suppor t ed on 250 mm th ick wa l ls . A ssume tha t t he t endo n p rof il e is conc ordan t a nd

that the tendon reverse curvature over the suppor t i s def ined by the parabola

e = 0.000375x 2 - 65 in m m units

relat ive to an origin at the intersect ion of the slab an d the inte rior wall centrel ine, w ith

a downward eccent r ic i ty taken as pos i t ive . Other data are :

Span 9 .0 m

Imp osed l ive load 2 .5

kN/m 2

S lab dep th 200 m m

Co ncrete s t rength f~u 40 N /m m 2 ( fc ' = 32 N/ m m 2)

Ro d re inforcement 0 .2 %

Steel y ie ld point fy 460 N /m m 2

Tendon s 15 .7 mm @ 300 mm cen tr e s

Tendo n u l t ima te s tr e ss 1770 N /m m 2

Ten don force , f ina l 170 kN each

Solutions

Two so lu ti ons a r e p rov ided , comp ly ing wi th BS 8110 and A CI 3 18 . The two

approaches resul t in qui te di f ferent values for the shear s t rength. The ACI method

takes a ra ther co nservat ive appro ach to the assessment of concre te shear s t rength,

wi th no accou nt take n of the re inforceme nt level , but i t inc ludes a m ore rea l is t ic

assessment of the cracking m om ent than B 8110, which ignores the tensi le s t rength of

the con cre te .

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SHEAR 245

T h e c r i t i c a l s e c t i o n i s a t t h e f ac e o f t h e c e n t r e w a l l , a t a d i s t a n c e o f 8. 8 75 m f r o m t h e

c e n t r e l in e o f t h e o u t e r s u p p o r t .

S e c t i o n a r e a

S e c t i o n m o d u l u s

E c c e n t r i c i t y

A c = 0 . 2 m 2 m

Z = 0 . 0 0 6 6 7 m 3 m

e = 0 . 0 0 0 3 7 5 x 1 25 2 - 6 5 = - 5 9 m m

Solution 1: U sing B S 8110.

Fa c t o r e d lo a d = 2 4 x 1 .4 x 0 .2 + 1 .6 x 2 .5 = 1 0 .7 2 k N /m 2

R e a c t i o n a t o u t e r s u p p o r t = 0 . 3 75 x 1 0 .7 2 x 9 . 0 = 3 6 .1 8 I N / m

M o m e n t a t c r it i ca l s e c t i o n = 3 6 .1 8 x ( 8 . 8 7 5 ) - 1 0. 72 x (8 .8 7 5) 2/ 2 = - 1 0 1 . 1 k N m / m

Sh e a r a t c r i t i c a l sec t io n = 0 .6 2 5 • 1 0 .7 2 x 9 .0 - 1 0 .7 2 x 0 .2 5 /2 = 5 8 .9 6 k N /m

C r a c k i n g m o m e n t ( E q u a t i o n 1 0.1 )

M o = 0 . 8 [ P Z t e n / A e + M p ] = 0 . 8 ( P ) [ Z t e n / A c + e ]

0 . 8 ( 1 7 0 / 0 . 3 ) x [ 0 . 0 0 0 6 6 7 / 0 . 2 + 0 . 0 5 9 ] = 4 1 .9 k N m / m

U l t i m a t e m o m e n t ( 1 01 . 1) e xc e e d s M o. T h e s e c t i o n is c r a c k e d , s o u s e E q u a t i o n ( 10 .4 ).

Ef fec t ive de pt h d = (0 .2 x 165 + 159) /0 .45 = 162 m m

To ta l s t ee l co n ten t = 0 .2 + 1 0 0 x ( 1 5 0 /0 .3 ) x (1 7 7 0 /4 6 0 ) / ( 1 0 0 0 x 1 6 2 )= 1 .3 9 %

F r o m T a b l e 1 0 .1 , vc = 1 .0 3 N / m m 2

F ro m Eq . ( 1 0 .4 ) , Vc r= 0 .6 5 X 1.03 X 1 62 + 4 1 .9 X 5 8 .9 6 /1 0 1 .1 = 1 3 2 .9 k N /m > 5 8 .9 6

N o s h e a r r e i n f o r c e m e n t i s r e q u i r e d .

Solution 2: U s ing AC I 318 .

F a c t o r e d l o a d = 2 4 x 1 .4 x 0 .2 + 1 .7 x 2 .5 = 1 0 .9 7 k N / m 2

R e a c t i o n a t o u t e r s u p p o r t = 0 . 37 5 • 1 0. 97 • 9 . 0 = 3 7 . 0 2 k N / m

M o m e n t a t c r i t i c a l s e c t i o n = 3 7 .0 2 x ( 8 . 8 7 5 ) - 1 0.9 7 • ( 8. 87 5) 22 = - 1 0 3 . 5 k N m / m

S h e a r a t c r i t i c a l s e c t i o n = 0 . 6 2 5 • 1 0 . 97 x 9 . 0 - 1 0 . 9 7 x 0 . 2 5 / 2 = 6 0 . 3 0 k N / m

T h e s t r e s s a t t h e e x t r e m e f i b r e s t r e s s d u e t o p r e s t r e s s = ( P / A c ) x (1 + e A J Z )

[ 1 7 0 1 0 0 0 3 1 [ , 9 1 7 0 1 0 0 3 1

= ~ 1 3 x i ~ - - 0 _] x 1 + 0 . 0 0 6 6 7 x 1 - ~ J = 7 . 9 0 N / r a m 2

F r o m E q ( 1 0 . 7) , M ~ t = 6 . 67 x 1 06 x ( 0 . 5 x /3 2 + 7 .9 0 ) = 7 1 .5 k N m / m

T h e a c t u a l m o m e n t e x c e e d s th i s v al u e . T h e s e c t i o n i s c r a c k e d , s o us e E q u a t i o n ( 10 .6 )

V ~ r - ( 0 . 0 5 x / 3 2 ) x 1 65 + 7 1 .5 x 6 0 . 3 / 1 0 3 .5 = 8 9 . 7 k N / m

I n c o r p o r a t i n g a s h e a r s t r e n g t h f a c t o r o f 0 .8 5 g iv e s t h e s h e a r s t r e n g t h o f 76 .3 k N / m ,

w h i c h e x c e e d s t h e a c t u a l s h e a r f o r ce o f 6 0 .3 k N / m . T h e r e f o r e , n o s h e a r r e i n f o r c e m e n t

i s n e e d e d .

Exam p le 1 0.2 P unching shear calculation

C h e c k t h e p u n c h i n g s h e a r c a p a c i ty o f t h e in t e r io r s l a b - c o l u m n c o n n e c t i o n s h o w n i n

F i g u r e 1 0 . 16 , g i v e n t h e f o l l o w i n g d a t a "

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2 4 6

POST-TENSIONEDCONCRETE FLOORS

Ions

i00 kN

Tendons @ 600 m m c / c

P = 400 kN/m

L

0 0

I '~ , 1 3 .. O

e -

._o

o

r

0

. m

o

I

0 t'Xl

cO

I I

P o , n , o ,

~

. ~ contra flexu re

300

Section in x-direction

Figure 10.16

Example 10.2

S l a b t h i c k n es s D 2 2 5 m m

C o n c r e t e s t r e n g t h fc u = 4 0 N / m m 2 , f c' = 3 2 N / m m 2

F a c t o r e d p u n c h i n g s h e a r V = 7 0 0 k N

M o m e n t s i n s l a b a t c o l u m n c e n t r el in e

M ( x )

= 6 0 k N m / m a n d

M ( y )

= 25 k N m / m

M o m e n t t r an s f e r r e d to c o l u m n M ( x ) = 2 5 k N m / m a n d M(y) = 0 k N m / m

T e n d o n p r o fi le s c an b e a s s u m e d c o n c o r d a n t . E c c e n tr i ci t i e s o f t h e re v e rs e p a r a b o l a s i n

t h e tw o d i r e c t i o n s , r e l a ti v e to a n o r i g i n a t t h e in t e r s e c t i o n o f t h e s l a b a n d c o l u m n

c e n t r e l i n e s , a r e g i v e n b y t h e f o l l o w i n g e q u a t i o n s .

F o r t h e x - d i r e c t i o n t e n d o n s , e x = 0 . 0 0 0 1 6 8 X 2 - - 7 2

F o r t h e y - d i r e c t i o n t e n d o n s , e y = 0 . 0 0 0 0 8 8 x 2 - 5 7

SO L U TI O N 1 : Us ing B S 8 11 0.

F o r d e t e r m i n i n g t h e c r i t i c a l p e r i m e t e r , t a k e d a s t h e t e n d o n d e p t h a t t h e c o l u m n

c e n t r e l i n e .

x - d i r e c t i o n t e n d o n s : d = 2 2 5 - 4 0 = 1 8 5 m m , x 2 = 3 0 0 + 3 x 1 8 5 = 8 8 5 m m

y - d i r e c t i o n t e n d o n s " d = 2 2 5 - 5 5 = 1 7 0 m m , x , = 3 0 0 + 3 x 1 7 0 = 8 1 0 m m

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SHEAR 247

U s i n g E q u a t i o n ( 10 .1 1 ),

V ef f - - V ] [ 1 + 1 . 5 M t / ( V X 2 ) ]

= 7 0 0 1 1 + 1 .5 x 2 5 / (7 0 0 x 0 .8 5 5 ) ] = 7 4 4 k N

T h e x 2 e d g es a r e a t a d i s t a n c e 4 05 m m f r o m t h e c o l u m n c e n t r e l in e , a n d t h e x l e d g e s

a r e 42 8 m m f r o m t h e c e n t r e l i n e . S o t h e t e n d o n e c c e n t r ic i t i es a r e :

x - d i r e c t i o n e = 0 . 0 0 0 1 6 8 x 4 0 5 2 - 7 2 = - 4 4 m m

y - d i r e c t i o n e = 0 . 00 0 0 88 x 4 2 8 2 - 5 7 = - 4 1 m m

F o r t h e x - d i r e c ti o n , w h e r e t h e t e n d o n s a r e c o n c e n t r a t e d i n a n a r r o w b a n d , a s s u m e t h e

p r e s t r e s s i s d i s t r i b u t e d o v e r a w i d t h e q u a l t o t w i c e t h e e d g e o f t h e c r it i c a l p e r i m e t e r , s o

t h a t t h e f o r c e p e r m e t r e w i d t h i s 1 6 0 0 /( 2 x 0 . 8 5 5 ) = 9 3 6 k N . F r o m E q u a t i o n ( 1 0. 1 ),

t h e c r a c k i n g m o m e n t s p e r m e t r e w i d t h a r e"

Mo(x )

= 0 . 8 (9 3 6 x 0 .0 0 8 4 4 / 0 . 2 2 5 + 9 3 6 x 0 . 0 4 4 ) = 6 1 . 0 k N / m

Mo(y ) = 0 . 8 ( 4 0 0 x 0 . 0 0 8 4 4 / 0 . 2 2 5 + 4 0 0 x 0 . 0 4 1 ) = 2 5 . 1 k N / m

T h e s e c t io n is u n c r a c k e d i n b o t h d i r e ct i o n s , a n d s h e a r c a p a c i t y c a n b e f o u n d f r o m

T a b l e 1 0 .3 . S in c e t h e t e n d o n p o i n t s o f i n f l e x io n l ie o u t s i d e t h e c r i ti c a l p e r i m e t e r , t h e

v e r t i c a l c o m p o n e n t o f t h e p r e s t r e s s is n o t i n c l u d e d i n th e c a l c u l a t i o n s .

E d g e s x 2 ( x - d i r e c t i o n t e n d o n s ) :

P ay - - 9 3 6 X 1 0 0 0 / ( 1 0 0 0 X 2 2 5 ) = 4 . 1 6 N / m m 2

Vr = 1 .8 1 x ( 2 x 8 5 5 x 2 2 5 ) /1 0 0 0 = 6 9 6 k N

E d g e s x ~ ( y - d i r e c t i o n t e n d o n s ) "

Pa~ = 4 0 0 x 1 0 0 0 / ( 1 0 0 0 x 2 2 5 ) = 1 .7 8 N / m m 2

Vr = 1 .4 2 x ( 2 x 8 1 0 x 2 2 5 ) /1 0 0 0 = 5 1 8 k N

T o t a l s h e a r r e s i s t a n c e = 6 9 6 + 5 1 8 = 1 2 14 k N

T h e s h e a r r e s i s t a n c e e x c e e d s t h e a c t u a l s h e a r , s o n o s h e a r r e i n f o r c e m e n t i s n e e d e d .

Solution 2: Using A C I 318.

U s i n g t h e s a m e v a l u e s o f d a s i n S o l u t i o n 1 , t h e e d g e s o f t h e c ri t ic a l p e r i m e t e r a r e"

x l = 3 0 0 + 1 70 = 4 7 0 m m

x 2 = 3 0 0 + 1 85 = 4 8 5 m m

M o m e n t t r a n s f e r is a c c o u n t e d f o r u s i n g t h e p a r a m e t e r

Yv = 1 - 1 / [1 + 0 .6 7 (4 7 0 /4 8 5 ) ~ - 0 .4 0

J c is f o u n d c o n s e r v a t i v e l y b y u s i n g t h e s m a l l e r o f t h e t w o d v a lu e s .

Jc = 1 7 0 x 4 7 0 3 /6 + 4 7 0 x 1 7 0 3 /6 + 1 7 0 x 4 8 5 x 4 7 0 2 /2 = 1 2 .4 3 x 1 0 9 m m 4

F r o m E q u a t i o n ( 10 .1 5 ),

700 x 103 0 .4 x 25 x 106 • 47 0

VA8 2(4 70 + 485 ) X 170 + 2 X 12 .43 X 109 = 2 .16 + 0 .19

= 2 .3 5 N / m m 2

VCD = 2 .1 6 - - 0 .1 9 - - 1 .9 7 N /m m 2

t ip i s t h e sma l l e r o f :

0 .3 an d (4 0 x 1 7 0 ) / [1 2 x 2 x ( 4 7 0 + 4 8 5 ) ] + 0 .1 2 5 = 0 .4 2 . Ta k e t ip = 0 .3

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POST-TENSIONEDCONCRETE FLOORS

Pay = 4 .16 N /m m 2 a s fo r S o lu t io n 1

U s i n g E q u a t i o n ( 1 0 . 5 ) ,

Vco = [flp(fc') ~ +

0.3Pav]bvd + Vp

Vco/(bva

= 0 .3x /32 + 0 .3 x 4 .16 = 2 .95 N /m m 2

M u l t i p l y i n g b y a s t r e n g t h r e d u c t i o n f a c t o r o f 0 . 85 g i v es a d e s i g n s h e a r s t r e n g t h o f

2 .5 N / m m 2 , wh i c h is g r e a t e r t h a n 2 .3 5 . So n o s h e a r r e i n f o r c e m e n t i s n e e d e d .

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11

S L A B S O N G R A D E

Grou nd-b ea r ing f loor s, com mo nly used in commerc ia l and indus t r i a l bu i ld ings ,

a re norma l ly requi red to car ry a v ar ie ty of load-p a t te rns , such as wheel loads

f rom fork t r ucks , hand l ing machine ry , concen t r a t ed loads f r om r ack ing and

stacking sys tems, and uni formly dis t r ibuted loads sep ara ted by u nload ed a is les.

Any f ixed machinery i s usua l ly founded on i t s own separa te base .

The design process is based on empir ical rules backed by elast ic analysis , and

the const ruc t ion techniques have la rge ly evolved through exper ience over a

pe r iod . Des ign gu ides and background da ta a r e pub l i shed by a number o f

organiza t ions in di f fe rent count r ies ; these a re based on exper ience , and on the

loadings re levant to the par t icula r count ry and should be re fe r red to when

designing a gro un d bear ing f loor. The pu bl ica t ions a lso conta in useful guidance

on the prop er t ies of the var ious soi ls , the loads an d load ing pa t te rns , con st ruc t ion

techniques an d de ta i l s of jo ints in use in the cou nt ry . Th e o rganiza t ion s inc lude :

T h e P o r t l a n d C e m e n t I n s t i t u t e - - U S A

T h e P o s t - te n s io n i n g I n s t i t u t e - - U S A

The Br i t i sh Cement Assoc ia t ion

T h e C o n c r e t e S o c i e t y - - U K

This chapter dea ls wi th the des ign aspec ts per t inent to pos t - tens ioning, though

there is considerable over lap wi th con vent ion a l f loors con st ruc ted in re inforced

or pla in concre te . I t a l so gives a compar ison be tween convent iona l re inforced

and post - tens ioned f loors , and da ta on the e las t ic s t resses produced under a

concent ra ted or l ine load, and uni form loading wi th unloaded a is les .

A grou nd s lab fundam enta l ly di ffer s f rom a suspende d s lab by vi r tue of it s

be ing cont inuou sly sup por ted by the ground. I t s func t ion is to dis t r ibute the load

con cent ra t ion s to the grou nd in a sa fe m ann er . A s lab car rying a uni form load

over the whole of i ts a rea se rves only as a separa t ion me dium be tween the g roun d

and the load; s t ruc tura l ly i t i s redu nd ant . The per form ance o f a grou nd s lab i s,

therefore , s t rongly l inked wi th the charac te r i s t ics of the sub-base and ground,

and, of course , the loading.

Reinforced gr ou nd f loors tend to develop a var iety of defects . The m ost

com m on p roblem is tha t of unco nt rol led c racking of the s lab pane ls . In t ime, the

concre te begins to spa l l a t the c racks . In order to minimize such c racking, the

f loor is provided wi th a gr id of jo in ts - -co nsis t ing of saw-cuts , con t rac t ion joints

or expan sion joints - - f i l led wi th a sea lant . Und er co nt inua l wheel traf fic, the

joints gradual ly de te r iora te . Another problem is tha t of curling of the edges at

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250 POST-TENSIONED CONCRETE FLOORS

jo in t s , caused by var i a t ion o f s t ra in in the dep th o f the s l ab , which ma y be due to

s h r i n k ag e o r t emp era t u re g r ad i en t .

Fr om the user ' s po in t o f v iew, c racks and jo in t s in a floor a re h igh ly

undes i rab le . They gene ra te dus t , cause dam age to the veh ic les , and s low dow n o r

imped e the w ork ing o f the s t ack ing sys tems .

Pos t - tens io ning al lows large a reas of the f loor , exceeding 100 x 100 m

(330 x 330 f t) , t o be cons t ruc ted wi tho u t any jo in t s . Th i s apprec iab ly redu ces the

prob lem s assoc ia ted wi th c racks a nd jo in t s, an d a l lows the f loor to be f in i shed to

the f l a tness accuracy requ i red by the var ious mechan ized s t ack ing sys tems . A

pos t - t ens ioned f loor is a l so m ore to le ran t o f occas iona l ove r load ing , bec ause

m inor c racks t end to c lose and hea l in t ime . By com par i son , a non-pos t - t ens ion ed

floor norm al ly rel ies on the tens ile s t rength o f concrete for d is t r ibut ing con cen trated

loads; once the concrete has cracked, i t s s t rength is los t for ever .

To a much smal le r ex ten t , pos t - t ens ioned f loors can su f fe r f rom the same

prob lems as the normal non-pres t ressed f loors . The p rob lems and the remedies

are equa l ly imp or ta n t in bo th types o f cons t ruc t ion , and i t is , t herefo re , usefu l to

d i scuss the per fo rma nce o f non-pres t resse d f loors f ir st .

1 1 . 1 T h e d e s i g n p r o c e s s

In the pas t , t he s t reng th r equ i rem ent o f a g ro und f loor was o ft en speci fi ed in

te rms o f a un i fo rmly d i s t r ibu ted load ing . T h i s is s truc tu ra l ly m ean ing less ,

because und er a un i fo rm load ing , app l i ed over the whole o f the f loor a rea , the

s lab has no s t ruc tu ra l func t ion ; i t ac t s on ly as a separa t ion medium be tween the

load and the subgrade . The cur ren t t rend i s to spec i fy the magni tude o f

conc en t ra ted loads , such as the wheel load o r the l eg load f rom a rack ing sys tem,

or to speci fy ais le widths b etwe en are as of b lock s tacking.

The long- te rm se t t leme nt beha v iour o f a f loor depends on the charac te r i s t i cs o f

the subgra de and the average in tens i ty o f the sus ta ined load ing . Excep t fo r loca l

effects , i t i s ind epe nde nt of the s t re ngth of the s lab . Advice on soil prope rt ies

shou ld be sough t f rom spec ia l i s t consu l t an t s .

The des ign p rocess i s mos t commonly concerned wi th the s t res ses deve loped

by the conc en t ra te d loads and the shor t t e rm s t reng th o f the s lab . The ca lcu la t ion

of s t resses i s base d on an elas t ic analys is of the f loor-s ubg rade sys tem, assu m ing

the f loor to be a th in e l as ti c p la t e con t inu ous ly supp or ted by an e las ti c me d ium .

The theo ry deve loped by W es terg aard in the 1920s fo rms the bas is o f the cur ren t

des ign meth ods . In the future, fin ite e lem ent analys is techniq ues are l ikely to hav e

an increas ing role .

The p roce dure fo r the des ign o f a g rou nd bear ing f loor us ing e las ti c theory

consis ts of the fol lowing s teps:

1. D eterm ine the charac te r i s t i cs o f the subgrad e

2 . D e t e rmi n e t h e l o ad i n g p a t t e rn an d m ag n i t u d e o f l o ad s

3 . D e t e rm i n e t h e ap p ro p r i a t e s a fe ty fac t o r fo r th e n u m b er o f l o ad mo v em en t s

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SLABS ON GRADE

251

4 . Assum e th icknesses o f the sub-base and the concre te s lab

5. Calculate the s lab s t rength

6 . Calcu la te the s t res ses p roduced by the loads

7. Revise s tep 3 an d repe at i f necessary

The s tress a t a po in t m ay be in f luenced by severa l d i scre te co ncen t ra ted loads in

the v ic in ity . M etho ds o f as sess ing th is a re g iven in C& CA (now BC A) Techn ica l

R ep o r t 5 50 (Ch an d l e r , 1 9 8 2) an d b y F a t em i -Ard ak a n i , Bu r ley an d W o o d (19 89 ).

Exper imenta l and theore t i ca l work in the Uni ted Sta tes and in the UK has l ed

to the der iva t ion o f semi -empi r i ca l ru les and no m ogr aph s fo r de te rmin ing the

th icknesses o f the var ious l ayers o f the f loor cons t ruc t ion wi tho u t the need fo r

i terat ion.

A groun d s l ab can be an a lysed on a com puter as a g rid o r us ing fini te e l ements ,

replacing the soi l wi th equiv alent spr ings . The ana lys is can be exten ded to predict

the long-te rm set t leme nt of the f loor . Specialized des ign software i s a lso avai lable

f rom var ious sources .

Th e W es t e rg aa rd equ a t i o n s , t h e mo me n t s p ro d u ce d b y p a r t i ti o n l o ad s an d t h e

uniform load ing o f a f loor wi th un load ed aisles are g iven later in th is cha pter .

11 .2 Fa ctors a f fect ing the des ig n

The factors which inf luence the s t resses produced in a f loor and determine i t s

s t reng th a re :

9 load ing , pa t t e rn and in tens ity

9 loaded a rea

9 s lab th ickness

9 modulus of e las t ic i ty of concrete

9 f lexural tens i le s t reng th of the conc rete

9 am ou nt o f p res t ress

9 f r ic t ion b e tween the s l ab and the sub-ba se /g rade

9 and e last ic cha rac te r i st i cs o f the su b-base /g rad e

Som e of the ab ove fa ctors , and the safety factor used in the des ign, are d iscussed

below.

11.2 . 1 Lo ad ing

The loading pat terns on a f loor can be class i f ied in four categories .

1 . W heel loads f rom fo rk li ft t rucks and s t ack ing m ach ine ry . The t rend tow ards

h igh s t ack ing has necess i ta t ed a min im um of e l ast ic sp r ing ing o f the ma ch ine

and sol id tyres . The loa d is appl ied over a smal l area o f the f loor; the average

con tac t p ressure has been me asured as 13 .9 N /m m 2 (2000 ps i ), and the

m axim um s ing le whee l st a ti c load reg i st e red was 5 .6 tonnes , (Fa tem i -Ardak an i ,

Burley and Wood, 1989). The s tat ic load is magnif ied by the i r regular i t ies in

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252 POST-TENSIONEDCONCRETE FLOORS

t he f loor su rface, par t i cu la r ly by o pen jo in t s . Th e dyn am ic load can be up to

twice the s tat ic value, depen ding on the qua l i ty of the f loor . Hig h s tac king

mo bi le equ ipm ent i s par t i cu la r ly d e t r ime n ta l to op en jo in t s in the f loor.

Typ ica l exam ples o f l aden ty re con ta c t p ressures a re :

3 tonne coun terba lan ce lif t t ruck 2 .4 N /m m 2 350 ps i

2 tonne reach t rucks 5 .6 N /m m 2 800 ps i

2 tonne pa l l e t t rucks 11.0 N / ram 2 1600 ps i

W heel loads in the reg ion o f 8 .5 tonnes a re no t unusua l .

2 . Leg loads f rom rack ing sys tems . Leg loads o f the o rder o f 20 tonnes a re

common. Each l eg i s p rov ided wi th a basep la te , which may vary f rom a

m in im um of abo u t 100 m m square (4 in ) up to 500 m m square (20 in ) o r mo re .

The basep la te o f ten cons is t s o f a s tee l p la t e wi thou t ade qua te r ibs o r

gussets to enable i t to d is t r ibute the load over the whole of i ts area. In o rde r to

be effect ive, the basepla te s hou ld be s t if fer than the conc rete s lab . A basep late

of s ti ffness equ al to tha t o f the s lab is only a bo ut 50% eff icient. Eve n for th is

eff ic iency, assuming an E s / E ~ rat io of 8, the re quired th ickness of a s teel p late

(wi thout gussets) i s hal f that of the concrete s lab as the fol lowing s imple

calculat ion shows.

E s t 3 = E e D 3

therefore,

t= (Ec/Es)I /3D = (1 / 8 ) ' / 3 0 = 0 / 2 .

3 . Areas o f f loor car ry ing un i fo rmly d i s t r ibu ted load ing , separa ted by a i sl es . The

m ax im um tensile s tress develops in the middle of the aisle at the top of the s lab .

4 . Line loads . Occa sional ly , a f loor s lab is requ ired to supp ort a par t i t io n w al l.

T h e loads a re usua l ly perm ane n t and a ffec t the long- te rm se t t lem ent o f the

f loor . The impl ica t ion o f supp or t ing heavy par t i t ions on the f loor shou ld be

s tud ied in con junc t ion wi th the se t tl ement charac te r is t ics o f the subgrade .

The wheel load and the l eg load f rom a rack ing sys tem are bo th concen t ra ted

loads ac t ing on a re la t ive ly smal l a rea o f the f loor . Rack load ing m ay vary in

intensi ty but the legs s tay in posi t ion for long durat ions; they di rect ly affect the

long- te rm se t tl ement . T he wheel load i s t rans ien t and i t s in fluence on the

long- te rm se t t l ement i s neg l ig ib le ; the repe t i t ious wheel load ing may , however ,

affect the s t ren gth of the f loor thr ou gh fat igue. The suggested load factors are

discussed in Section 11.2.4.

11 .2 .2 Modu lus o f rup tu re

The m odulu s o f rup tu r e is expressed in t e rms o f a pow er o f the concre te s t reng th .

The express ions a re o f the fo rm

f t = C ( f ~ ) n

(11.1)

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Table 11.1

Mo du l i o f r up tu re (N /mm 2) , U K prac t ic e .

S L A B S O N G R A D E 253

fcu 25 30 35 40 45 50

ft 28 3.36 3.79 4.21 4.60 4.97 5.33

ft 9o 3.70 4.17 4.63 5.06 5.47 5.87

where C = a coeff ic ient

n = a c on s t a n t

fc = 28-d ay c onc re te s t ren g th fcu or fc '

In the U K prac t i ce , C is usua l ly t ak en as 0 .393 an d n a s 2 . A pe r io d o f 90 day s is

e xpe c t e d t o l a p se be f o r e t he f l oo r i s l oa de d . A t 90 da ys , the m od u l u s o f r up t u r e is

a s sum e d t o ha ve i nc r e a se d by 10% ove r t he 28 - da y va l ue . The r e f o r e

ft = 0.393(feu ) 2/3 at 2 8 d ay s

= 0.432(f~u) 2/3 at 90 da ys (11.2)

Ta b l e 11 . 1 g i ves t he m o du l i o f r up t u r e f o r a r a nge o f 28 - da y c onc r e t e s t r e ng t h s .

The va l ue s sh ow n on t he s e c ond l i ne a r e at 28 da ys a n d t hose on t he t h i r d l i ne a t

90 days .

I n Am e r i c a n p r a c t i c e , t he m odu l u s o f r up t u r e is e xp r e s se d a s C( f c')~ psi .

R i ngo a nd Ande r son ( 1992 ) s t a t e t ha t t he c oe f f i c i e n t C c om m on l y va r i e s f r om 9

t o 11 f o r ba nk - r u n g r a ve l a nd c r u she d s t one a gg r e ga t e r e spe c ti ve ly . The P o r t l a n d

Ce m e n t A ssoc i a t i on r e c om m e nd s a c oe ff ic ie n t o f 9 . The d e s i gne r e it he r a s sum e s

on e of these coeff ic ients (7 .5, 9 or 11), or speci f ies tests to be ru n fro m the t r ia l

c o n c r e te m i x t o d e t e r m i n e a n a p p r o p r i a t e v a l u e f o r t h e m o d u l u s o f r u p t u r e . A

defau l t va lue of 7 .5 i s normal ly used for the coe f f i c i en t C .

The I m p e r i a l a nd m e t r i c e q u i va l e n t s o f t he c oe ff ic i en t C a r e"

Im per ia l 7 .5 9 .0 11 .0 ps i un i t s

Me t r i c 0 . 62 0 .75 0 . 91 N - m m un i t s

The I m p e r i a l va l ue s o f the m odu l i o f r up t u r e f o r va r i ous c y l i nde r s t r e ng t h s f c'

c o r r e sp ond i ng t o t he t h r e e va l ue s o f t he c oe ff ic ie n t C a r e sho w n i n Ta b l e 11.2 .

Table 11.2

M od ul i o f rup ture (ps i) , U S prac t i ce

C f ' c

3000 3500 4000 4500 5000 5500 6000

7.5 410 444 474 503 530 556 581

9.0 493 532 569 604 636 667 697

11.0 602 651 696 738 778 816 852

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254 POST-TENSIONEDCONCRETE FLOORS

lO0

"

90'

E

~ 8 0

Z

~ 70

i 6 0

t-

O 50

_

~ 4 0

L

~ 30

~ 2 0

o

o

~

10

J

f

f

j l /

11

/

J

/

3 4 5 6 7 8 9 1 0

Cal i fornia Bearing Rat io-%

f

15 20 30

F igu re 11 . 1 Relationship between subgrade modulus and CBR

11 .2 .3 Sub grade m odu lus k

T h e m o d u l u s o f s u b g r ad e r eac t i o n i s a meas u r e o f i ts e l a s ti c co mp r es s i b i l i t y . I t i s

d e f i n ed a s t h e s tr e s s o n t h e s u r f ace o f t h e s u b g r a d e w h i ch cau s e s i t t o d e f o r m b y

o n e u n i t o f l en g t h , an d i s ex p r e s s ed i n M N / m 3 o r p c i. It s v a l u e can b e d e t e r m i n ed

b y a p l a t e te s t o n t h e p r e p a r e d g r a d e , o r a n a p p r o x i m a t e v a l ue m a y b e o b t a i n e d

f r o m t h e C a l i f o r n i a B ea r i n g Ra t i o ( C B R) ; t h e r e l a t i o n s h i p , a s g i v en b y C h an d l e r

(1982) , i s shown in F igu re 11 .1 .

G e n e r a l g u i d a n c e o n t h e v a l ue o f k c a n b e o b t a i n e d f r o m T a b l e 1 1.3 . F o r m o r e

d e t a i l r e f e r t o t h e C o n c r e t e S o c i e t y T ech n i ca l Rep o r t 3 4 ( 1 9 8 8 ) .

T h e v a l u e o f k o n t h e s u r f ace o f t h e s u b - b as e d i ff er s f r o m t h a t o n t h e s u b g r a d e .

C h an d l e r an d N ea l ( 1 9 8 8 ) g i v e t h e en h an ced v a l u e s s h o w n i n T ab l e 1 1 .4 f o r

g r a n u l a r s u b - b a s e .

T h e s t re s s e s g en e r a t ed i n a s lab b y a l o ad a r e n o t v e r y s ens i ti v e t o t h e v a l u e o f k ,

a s c an b e j u d g e d f r o m S ec t i o n 1 1.6. L i t t l e ad v an t a g e i n th e s l ab t h i ck n es s i s

g a i n e d b y t a k i n g v a l u es h i g h e r t h a n 8 0 M N / m 3.

Table 11.3

Sub grade modul i

Descr ipt ion k M N /m 3 k pc i

Coarse grained gravel ly soi ls

Coarse grained sand and sandy soi ls

Fine grained soils-silts and clays

High plasticity clays

54-82 200-300

54-82 200-300

27-54 100-200

14-27 50-100

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Table 11 .4 Enhancement o f k (M N /m 3) wi th sub-base

SLABS ON G RA DE 255

k value for Sub-base thickness , mm

subgrade alone 150 200 250 300

13 18 22 26 30

20 26 30 34 38

27 34 38 44 49

40 49 55 61 66

54 61 66 73 82

60 66 72 81 90

11.2.4 Sa fety fac tor F

s

I n t h e U K , a s a f e t y f ac t o r o f 1 . 5 i s co n s i s t en t l y ap p l i ed t o t h e l o ad s w h e r e t h e

l o ad i n g p a t t e r n s a r e k n o w n . T h i s i s ap p r o x i ma t e l y i n l i n e w i t h a p a r t i a l s a f e t y

fac to r o f 1 .3 fo r con cre t e and 1 .2 fo r l oad in g . F o r r epe t i t i ve w hee l l oad ing , f a t i gue

m ay a f fec t t h e s t r en g t h o f t h e co n c r e t e s l ab . C h a n d l e r ( 19 8 2 ) h a s s u g g es t ed a n

increase i n t he f ac to r o f sa fe ty , sho w n in Tab le 11 .5 . One cyc l e o f l oa d in g cons i s t s

o f a v eh i c le t r av e l l i n g lad en an d r e t u r n i n g u n l ad en a l o n g t h e a i s le .

I n t h e U S A , a r an g e o f v a l u e s f o r t h e s a f e ty f ac t o r F s i s u s ed , d ep e n d i n g o n t h e

p a r t i c u l a r c o n d i t i o n s a s j u d g e d b y t he d e s ig n e r. A c c o r d i n g t o R i n g o a n d

A n d e r s o n ( 1 9 9 2 ) :

9 A co n s e r v a t i v e v a l u e o f 2 .0 is co m m o n l y u s ed . T h i s i s ap p r o p r i a t e w h e r e

l o a d i n g s a r e n o t a c c u r a t e l y k n o w n a t t h e t im e o f d e s ig n , o r w h e r e s u p p o r t

Table 11.5 Safety factors for fatigue loading

Cycles Sa fety factor

> 400,000 2.00

400,000 1.96

300,000 1.92

240,000 1.87

180,000 1.85

130,000 1.82

100,000 1 .79

75,000 1.75

57,000 1.72

42,000 1.70

32,000 1.67

24,000 1.64

18,000 1.61

14,000 1.59

11 ,000 1 .56

8,000 1.54

< 8,000 1.50

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256

POST-TENSIONEDCONCRETE FLOORS

condi t ions , o r any o th e r key i tems, a re e ithe r no t accura te ly kn ow n o r a re suspect .

9 A va lue of 1 .7 i s acceptab le cons is ten t w i th the lo ad fac tor s used in o th e r

c onc r e te de s ign a pp l i c a t ions . T h i s c a n be u se d w he r e l oa d ing i s f r e q ue n t a n d

inpu t va lue s ( de s ign pa r a me te r s ) a r e r e a sona b ly we l l known .

9 A va lue of 1.4 i s acceptab le for use un der ce r ta in con di t ions . F or exam ple ,

whe r e impa c t l oa d ings do no t e x i s t, o r whe r e t he loa d ing in t e ns it ie s a n d /o r t he

f requency of load app l ica t ion a re less , then va lues be tween 1 .4 and 1 .7 m ay be

a pp r op r i a t e .

11.2.5 Restraints to latera l mov eme nt

T he de s ign o f a g r ou nd s l ab a s sume s th a t , a p a r t f r om the fr i ct i on be twe e n the s l a b

a nd the subg r a de , i t i s f re e to u nde r go c ha nge s i n l e ng th a r i sing f r om the v a r i a t i on

in t e mp e r a tu r e , m o i s tu r e c on te n t o f t he c onc r e t e a nd hu mid i ty . I f t he r e s t r a in t

induces tens i le st r esses h igher tha n the s t r en gth of the s lab , then i t c racks .

Re s t r a in t t o l a t er a l mo ve m e n t m a y be c a use d by a num be r o f f a c to r s , inc lud ing

the fo l lowing .

9 F r i c t i on be twe e n the sub g r a de a nd the f l oo r sl ab . T he c onv e n t iona l r e in fo r c e d

concre te s lab i s normal ly provided wi th suf f ic ien t r e inforcement to avoid

c racks w hen sh r inkag e takes p lace a t the ea r ly age of the concre te . Th is is

d iscussed in more de ta i l in sec t ion 11 .3 .

C ha n ge s i n t e mp e r a tu r e o r hum id i ty m a y c a use h ighe r t ens il e s tr e sse s in a

f u lly l oa de d s l ab th a n in a n u n loa d e d one , be c a use o f t he a dd i t i on a l f r i c ti on

due to the load i t sel f. The tens ion in a loaded s lab is m uch m ore seve re than in

a n un loa de d s l ab . T he a dd i t i ona l f r i c ti on due to t he l oa d i s, howe v e r , no r m a l ly

ignored in the des ign process .

9 W here a f loor i s r equi red to ca r ry pe r m an en t l ine loads , such as a t a pa r t i t ion , i t

is of ten m ad e lo cal ly deep er to increase i ts st if fness . Since the soff it of the f lo or is

no longer leve l , the subgrade has to have a loca l t rough, whose ve r t ica l o r

s lop ing side s h inde r t he mov e m e n t o f t he f loo r . E i the r t he subg r a de m us t m ove

wi th the s lab or the s lab has to r ide the s lope i f i t i s no t to c rack . In the case of a

s ing le l ine l oa d , t he t r ough m a y f o r m a n a n c ho r po in t f o r a ll m ove m e n t . S om e

com press ib le ma te r ia l i s usua l ly pro vide d a lo ng the ve r t ical , o r s lop ing , faces of

the t roughs to r educe the tens i le forces which would o the rwise deve lop . The

com press ib i l i ty of the ma te r ia l sh ould be s tud ied w i th re fe rence to the expec ted

m ove m e n t t o a s se s s the m a gn i tude o f t he t e ns i le f or ce wh ic h w ou ld de ve lop .

An a l te rn a t ive i s to p rovid e sha l low s lopes , say of the ord e r of 5%.

9 The pen e t ra t io ns for the in te rna l co lum ns form a gr id of r ig id po in ts in a la rge

f loor .

A s imi la r r e s t ra in t m ay occur a lo ng the o u ts ide edges of a f loor i f the ex te rna l

c o lum ns p r o j e c t i n to t he bu i ld ing so tha t t he e dge o f t he s l ab ha s t o be no tc he d

a r o un d the c o lumns . T he no tc he s p r e ve n t t he sho r t e n ing o f t he s la b a long the

par t icu la r edge .

I n t he se l oc a t ions t oo , t he m a gn i tude o f t he t e nsi le s t re s s t ha t m a y de ve lop i s

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S L A B S O N G R A D E

2 5 7

con t ro l l ed by p rov id ing a compress ive l ayer be tween the f loor and the

co lumn s . The re m arks above , ab ou t the compress ib il i ty o f the mater i a l , a l so

apply to these cases .

The m ass o f a co lum n foun dat ion represen t s a hard spo t und er the s l ab . If the

subgra de has no t been p rop er ly com pacted , o r the co lumns a re founde d on s t if fe r

foun dat io ns such as p iles , the f loor gra dua l ly set tles bu t the smal l are a of the s lab

immedia te ly above a base does no t . Th i s causes f l exura l t ens ion a t the bo t tom

surface o f the s l ab a rou nd the co lum n base .

Al te rna t ive ly , if t he g rou nd p ressure und er the found at ions is h igher tha n the

general f loor loading, as i s of ten the case, then the foundat ions set t les , leaving a

gap be twee n the base an d the f loor . In th is case the s lab develops tens ion at i ts top

surface in flexure.

11 . 3 T r a d i t i ona l R C f l oo r s

Trad i t iona l ly , g rou nd f loors have cons i s ted o f an unre in fo rced , o r l igh t ly

re in fo rced , concre te s l ab l a id on a s l ip membrane on p repared g rade . The s l ip

m em bran e usua l ly consi s ts o f one o r two l ayers o f heavy po ly thene . A sub-base ,

cons i s ting o f com pacte d ba l l as t o r hardcore , and sand b l inded , m ay a l so be

prov ided be low the concre te . F igure 11 .2 shows the t rad i t iona l c ons t ruc t ion .

For i t s capac i ty to d i s t r ibu te load concen t ra t ions , a f loor re l i es upon the

f lexural tens ile s t rength of the concrete . The f loor is assume d to beh ave as a fu lly

elas t ic th in plate , sup po rted by a semi-inf in i te elas t ic grade. I f the f loor cracks , i ts

physical propert ies chang e dras t ical ly , gross ly reducing i t s load ca rrying capa ci ty .

Di rec t shear has no t been found to be c ri ti ca l in de te rmin ing the load capac i ty

of a floor ; punc h ing shear m ay need to be checked u nder heavy conc en t ra ted

loads . Shear i s t rans fer red across f ine c racks th rough aggrega te in te r lock .

I t is recogn ized tha t fo r a f loor to per fo rm in a sa ti s fac to ry m ann er i t m us t n o t

be al lowed to crack. Apart f rom overloading and deficiencies in the grade, a f loor

can c rack f rom a num ber o f causes, inc lud ing hea t gene ra t ion dur ing concre t ing ,

d ry ing sh r inkage and subseq uen t t em pera tu re changes . O f these , the m os t

important i s shr inkage. As a f loor i s a l lowed to dry up, poss ibly months af ter i t s

cons t ruc t ion , the loss o f m ois tu re i s acco m panied by a reduc t ion in the vo lume of

R e i n f o r c e m e n t

.~ , ... f . ~ ? o n c r e t e s l a b

.. .. .. . " ~ - ~ ~ " 4 " ~ .-~ - S l ip m e m b r a n e

. = . . . ' . o " , . , 'i . . . ~ , , , ..

- ~ ; Sand-blinded sub-base

S u b - g r a d e

Figure

1 1 . 2

A t rad i t i ona l conc re te f l oo r

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POST-TENSIONEDCONCRETE FLOORS

c o n c r e te . T h e h o r i z o n t a l m o v e m e n t a s s o c i a t e d w i t h t h e r e d u c t i o n i n t h e p l a n s iz e

o f t h e f l o o r i s r e s i st ed b y f ri c t i o n b e t w e en t h e co n c r e t e an d t h e g r ad e b e l o w , w h i ch

se t s up t ens i l e s t r esses over t he co ncre t e sec t i on . I f t he t ens i l e s t ress exceeds t he

s t r en g t h o f t h e co n c r e t e , t h en i t c r ack s .

T h e s l i p m e m b r a n e , b e i n g u s u a l l y w a t e r p r o o f , a c t s a s a v a p o u r b a r r i e r . T h e

m o i s t u r e f r o m t h e b o d y o f t h e c o n c re t e m i g r a t e s u p w a r d s a n d e v a p o r a t e s f r o m

t h e t o p s u r f ace . T h e s u r face i s, t h e r e f o r e , d r y e r t h a n t h e b o d y o f t h e co n c r e t e ; an d

co n s e q u en t l y , t h e s u r face z o n e h a s a h i g h e r s t r a i n a n d a h i g h e r t en si l e s t re s s . T h e

c r ack s , t h e r e f o r e , s t a r t f r o m t h e t o p s u r f ace an d p en e t r a t e i n t o t h e co n c r e t e .

S eas o n a l an d d a i l y t emp e r a t u r e v a r i a t i o n s s u p e r i mp o s e f u r t h e r t en s i l e s t r e s s e s ,

cau s i n g t h e c r ack s t o t r av e l d eep e r i n t o t h e co n c r e t e . A f r eez e - t h aw cy c l e i s

p a r t i c u l a r l y d e t r i m e n t a l .

T h e m e a s u r e s t a k e n t o c o n t r o l t h e c r a c k s m a y i n c lu d e o n e o r m o r e o f t h e

fo l l owing .

9 Red u ce t o t a l s h r i n k a g e s t r a i n . U s e p l a s t i c iz e r s t o r ed u ce t h e w a t e r co n t en t .

H i g h - s t r e n g t h c o n c r e te s u s u a l l y c o n t a i n a g r e a t e r q u a n t i t y o f c e m e n t , w h i c h

n eed s m o r e w a t e r . T h e r e f o r e , p r e f e r a l o w s t r en g t h c o n c r e t e i f p o s s i b l e .

9 R e d u c e t h e r a te o f s h r i n k a g e . M o s t o f t h e s p r a y e d c u r i n g m e m b r a n e s l o w e r ,

an d d e l ay , t h e r a t e o f l o s s o f mo i s t u r e . T h i s r ed u ce s t h e i n t en s i t y o f th e t en s il e

s t r esses , and t hey o ccu r l a t e r i n t he l if e o f t he co ncre t e , w hen i t i s s t ron ger .

9 P r o v i d e c r ack co n t r o l j o i n t s a t f r eq u en t i n t e r v a l s . J o i n t s , h o w ev e r , a r e w eak

s p o t s d e l i b e r a t e l y i n t r o d u c ed a t s t ra t eg i c l o ca t i o n s . M o v em en t o f a f l o o r i s

o f t en co n cen t r a t ed a t th e j o i n t s , w h i ch l e ad s t o a g r ad u a l d e t e r i o r a t i o n o f t h e

s u r face . T r av e r s i n g w h ee l s a cce l e r a t e t h e p r o ces s . A j o i n t m ay e v en t u a l l y f a il a s

t h e a r r i se s g e t w o r n o u t , t h e r e b y w i d en i n g i t , o r ev en cau s i n g a s t ep t o d ev e l o p

across i t .

9 U s e s te e l f ib r e t o en h an c e t h e t en s i le s t r en g t h o f co n c r e t e . F l o o r a r ea s o f

a r o u n d 1 00 0 m 2 h a v e b e e n c a s t w i t h o u t a n y jo i n t s u s i ng t h i s a p p r o a c h .

9 P r o v i d e a r e i n f o r cemen t me s h f ab r i c n ea r t h e t o p s u r face , w h e r e t h e t en s i le

s t r esses a re h ighes t . Th i s i s a f a r f rom idea l so lu t i on as t he mesh o f t en ge t s

d i s p l aced t o a lo w e r t h an i n t en d ed l eve l an d i t r e s t ri c t s t h e m o v em en t o f t h e

co n c r e t i n g c r ew .

T h e a m o u n t o f r o d r e i n f o rc e m e n t is n o r m a l l y b a s e d o n t h e drag theory. In a f l oo r

o f l en g t h L , f r i c ti o n a l r e s i s t an ce t o s h r i n k ag e w i ll b u i l d u p l i n ea r l y t o a ma x i m u m

a t t h e m i d - p o i n t o f t h e f lo o r l e ng t h . T h e r e i n f o r c e m e n t m u s t b e c a p a b l e o f

o v e r c o m i n g t h is f r ic t i o n , so t h a t i t c an d r ag t h e f l o o r t o w ar d s t h e mi d d l e o f i ts

l en g t h . T h e r e f o r e ,

Asfy/F d > #woOL~2 (11.3)

w h e r e F d = p a r t i a l s a f et y f ac t o r

A s = s t ee l a r ea m mZ m w i d t h

# = t h e co e ff ic i ent o f f r ic t i o n b e t w e en s l ab an d g r ad e

I n o r d e r t o en s u r e t h a t t h e r e i n f o r cemen t r ema i n s w i t h i n t h e e l a s t i c r an g e , t h e

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Table 11.6

Co e f f i c i e n t o f f r i c t i o n

M a t e r i a l p

Sand and gravel 2.0

Gra nu lar sub-base 1.7

Lay er of sand 1.0

Polythene 0.9

S L A B S O N G R A D E 2 5 9

va lue of F d i s usu a l ly t ak en be tw een 1 .5 an d 1 .15 . The coe f f ic i en t o f f r i c t ion

de pe n ds o n t he f ir m ne ss a nd r ough ne ss o f t he g r a de , a nd on t he na t u r e o f t he s li p

m em bra ne . T ypica l va lues o f the coe f fi c ien t o f f r i c t ion ag a ins t the in i ti a l

m ove m e n t a r e g i ve n i n Ta b l e 11 . 6 .

Ta k i ng F d = 1 .15 , f y = 460 N / m m 2 a nd # = 1 .5 , a nd m e a su r i n g D i n m i ll i-

m e t r e s a n d L i n m e t r e s , fo r c onc r e te o f no r m a l d e ns i t y ,

A s = 0 . 0 4 5 D L m m 2 / m w i d t h ( l l . 4 a )

I n i m pe r i a l un i t s , w i t h F d - 1 .15 ,f y = 600 00 p s i , a nd m e a su r i ng D in inc he s a nd

L in feet ,

A s = 0 .000 16D L in2 /f t w id th (11 .4b)

The p r ob l e m w i t h t he d r a g s te e l c onc e p t i s tha t , w o r k i n g a t a s t re s s in t he r a nge

f y /1 . 5 t Ofy /1 .15 , t he s t r a i n i s h i gh e nou gh t o c a use t he c onc r e t e t o c r a c k - - t h e ve r y

de f e c t i t i s suppose d t o gua r d a ga i n s t . Adm i t t e d l y , t he c r a c ks a r e ve r y f i ne a nd

w e l l d i s t r i bu t e d , ne ve r t he l e s s t he c onc r e t e c a nno t r e m a i n i n t a c t . W h i l e t he

p r e se nc e o f r e i n f o r c e m e n t doe s n o t s t o p t he c onc r e t e f r om c r a c k i ng , i t doe s ho l d

t he c onc r e te t i gh t l y t oge t he r , m a i n t a i n i n g g ood a gg r e ga t e i n t e r l oc k f o r tr a n s f e r o f

shea r .

H o w e v e r , t h e c o n c r e te i s a s s u m e d t o b e u n c r a c k e d a n d c a p a b l e o f w i t h s t a n d i n g

a m o m e n t o f

Mcr

= f t Z / F s

(11.5)

w he r e F s = F a c t o r o f s af e ty

11 .4 P o s t - t e n s io n e d g ro u n d f lo o rs

I n a pos t - t e n s i one d f l oo r t he p r a c t ic e i s t o p r ov i de a c onc e n t r i c p r e s t r e s s , w h i c h

e na b l e s t he f l oo r t o c ope w i t h s t re s s r e ve rsa l s. Th e a m ou n t o f p r e s t r es s i n a

g r o und - be a r i ng f l oo r i s de pe nde n t o n t he le ng t h o f t he fl oo r , t h e l oa d i ng pa t t e r n

a nd i n t e ns i ty , a nd t he q ua l i t y o f t he subg r a de . P r e s t r e s s i s a pp l i e d i n bo t h

d i r e c t i ons a nd t he a ve r a ge s t re s s is u sua l l y m uc h sm a l l e r t ha n t ha t i n a su spe nde d

f loor . An a ve rage f ina l s tre ss o f 1 .0 N /m m 2 (150 ps i ) a ft e r losses is qu i t e co m m on ,

t ho ug h i t m a y be a s l ow a s 0 .4 N / m m 2 ( 60 p s i) i n fl oo r s o f sho r t l e ng t h .

The a m ou n t o f p r e s t r es s i s c hose n t o s a ti sf y tw o r e q u i r e m e n t s :

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260 P O S T-TE N S IO N E D C O N C R E TE FL O O R S

1 . C o n c r e t e m u s t n o t c r a c k w i t h t h e d r a g

2 . C o n c r e t e m u s t n o t c r a c k u n d e r t h e l o a d i n g

The f i r s t r equ i remen t i s sa t i s f i ed fo r t he un loaded s l ab i f :

e > t~wcDL /2. (11.6)

U s i n g / ~ = 1 .5 , w c = 2 4 k N / m 3, an d m eas u r i n g D in m i l l i me t r e s an d L i n me t r e s ,

P = 0 . 0 1 8 D L k N / m w i d t h , a n d

Pay -- P / A c = 0 .0 1 8 L N / m m 2 ( l l . 7 a )

I n I m p e r i a l u n i t s , w i t h a co n c r e t e d e n s i t y o f 1 44 p c f, D mea s u r ed i n i n ch es an d L

i n f ee t , t h e eq u i v a l en t eq u a t i o n s a r e :

P = 9 D L l b / f t , and

P a y = P / A e

= 0 . 7 5 L p si ( l l . 7 b )

I f t h e l o ad ed s l ab i s co n s i d e r ed , t h en t h e t r a n s i en t w h ee l l o ad s m ay b e i g n o r ed

a n d t h e r a c k l o a d i n g t r a n s l a t e d i n t o a n e q u i v a l e n t u n i f o r m l y d i s t r ib u t e d l o a d i n g .

T h e n

P = (0 .018D + 0 .75w )L in m et r i c un i t s

P = (9D + 0 .75w)L in Im per i a l un i t s (11 .8 )

w h e r e w = t h e e q u iv a l e n t u n i f o r m l y d i s t r i b u t e d l o a d

A s r eg a r d s t h e s eco n d r eq u i r e me n t , t h a t o f t h e s t r en g t h , t h e co n c r e t e s ec t i o n d o es

n o t s u f f e r f r o m an y c r ack s w h en p o s t - t en s i o n ed t o s a t i s f y t h e f i r s t r eq u i r emen t .

T h e co n c r e t e b e i n g i n t ac t , t h e t o t a l t en s i l e s t r en g t h av a i l ab l e i s t h a t d u e t o t h e

m o d u l u s o f r u p t u r e an d t h e p r e s t r e s s , l es s t h e f r ic t i o n lo s s. T h e r e f o r e ,

M u = ( f t + P av

- -

#wex)Z/Fs (11.9)

where x = d i s t ance f rom the s l ab edge ~< L / 2

F~ - t he par t i a l sa fe ty f ac to r app l i e d t o s t r esses

N o t e t h a t t h e p re s t re s s i s m a x i m u m s o m e d i s t a n c e a w a y f r o m t h e e d g e ; i t is l o w e r

n e a r t h e a n c h o r a g e b e c a u s e o f t h e d r a w - i n a n d i t r e d u c e s t o a m i n i m u m a t t h e fa r

en d i n a s lab s t r e s s ed f ro m o n e en d . I n a s l ab s t r e s s ed f r o m b o t h en d s , o r w h e r e t h e

L

I.. d

Subgade friction

/

Prestress -~-

Figure 11.3 St resses i n an un load ed pos t - tens ione d f l oo r

Net available

prestress

Draw-in loss

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SL ABS ON G RADE

261

t e ndons a r e s t r e s se d a l t e r na t e ly f r om e a c h e nd , t he p r e s t r e s s i s min imum a t

m id- len gth of the s lab . The s t r en gth of the f loor va rie s cor respo ndin gly , see

Figure 11.3.

In a bo nd ed f loor th is s ta te of s t re sses i s locked in a f ter g rout in g . Fa c to r s

a ff ec ting the l e ng th o f a f l oo r, suc h a s t e mpe r a tu r e a nd hum id i ty c ha nge s , m a y

te nd to e ve n ou t t he p r e s t r e s s i n a f l oo r u s ing unbonde d t e ndons , bu t i t i s

cons ide red sa fe r to use the d is t r ibu t ion s how n in F igure 11 .3 in the des ign .

1 1 .5 E l as t i c an a l ys i s

This sec t ion dea ls wi th the e la s t ic ana lys is o f con cent ra ted , l ine and uni form ly

dis t r ib u ted load s . In des ign ing a f loor on the bas is g iven he re , a f loor cons t r uc t io n

is in i t ial ly a ssum ed, a nd the load , inc luding d yn am ic e ffec t i f any , i s m ul t ip l ied b y

the ap pro pr ia te f ac tor of sa fe ty . The resu l t ing tens i le s t re ss , o r mo m ent , i s

checked a ga ins t the pe rm iss ib le s tr e ss, o r the c rack ing m om en t . I f necessa ry , the

f loor cons t ruc t ion i s r ev ised .

T he f o r mu la s g ive n in t h i s s e c t ion a r e ba se d on W e s te r ga a r d ' s e q ua t ions ,

wh ic h a r e now be l ie ve d to ove r - e s t ima te t he s t re s se s p r oduc e d by c on c e n t r a t e d

loads . As s ta ted in Sec t ion 11 .1 , o the r s have devised methods based on e la s t ic

the o r y a nd e xpe r ime n ta l me a su r e me n t s , wh ic h y i e ld mor e e c onomic a l de s igns .

Some of these a re r e fe r red to in th is sec t ion but no t d iscussed in de ta i l .

1 1.5. 1 C o n c e n t r a t e d l o a d s

T he e l a s ti c a na ly s i s o f g r ou nd- b e a r ing f loo r s c a r r y ing i so l a t e d c on c e n t r a t e d

loa ds i s u sua l ly ba se d on the e q ua t ions o f W e s te r ga a r d ( 1948) a nd the i r

modif ica t ions , which assume a th in e la s t ic p la te bea r ing on an inf in i te or

semi- inf in i te e la s t ic medium.

T he ba s i c e q ua t ion s do n o t t a ke a c c o un t o f t he s e pa r a t ion be twe e n the s l a b a nd

the g r a de wh ic h ma y oc c u r be c a use o f c u r l ing a nd t e m pe r a tu r e g r a d ie n t a c r os s

the s lab th ickness . The s t r e sses g iven by these equa t ions a re immedia te and do

no t c on s ide r t he l ong - t e r m se t t l e me n t o f t he subg r a de . N o r e li a b le c o r r e l a t i on

exist s be tw een the e la s t ic p roper t ie s of a so il and i ts long- te rm d eform at ion

c ha r a c te r i st i cs . F o r p oo r so i ls , t he l ong - t e r m se t t le me n t m a y be up to 40 time s the

im m edia te de f lec t ion . Spec ial i st so il s enginee rs shou ld be cons ul ted for pred ic t ing

the long - t e r m be ha v iou r o f a f loo r .

The re la t ive s t if fnesses of the s lab and the gro un d a re represen ted by a ra d i u s o f

relat ive st i f fness r, m easu red in un i t s o f length . I t s va lue i s g iven by:

r = { E eD 3 / [1 2 (1 - / t Z ) k }

~

whe r e v = P o i s so n ' s r a t i o f o r c onc r e t e

k = m odu lus o f subg r a de r e a c t ion

Fo r v = 0 .2 , the express ion reduces to r = 0 . 5 4 3 { E cD 3 / k } ~

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262

POST-TENSIONED CONCRETE FLOORS

Table 11 .7a Values of radius o f relat ive s t if fness (m m)

k ( M N / m 3)

Slab thickness (mm)

150 175 200 225 250 275 300

13 887 996 1101 1203 1301 1398 1492

27 739 830 917 1002 1084 1164 1243

54 621 498 771 842 912 979 1045

82 560 628 695 759 821 882 942

Table l l .7b Values of radius of relative stif fness (in)

k (pci) Slab thickness ( inches)

6 7 8 9 10 11 12

50 34.8 39.1 43.2 47.2 51.1 54.9 58.6

100 29.3 32.9 36.4 39.7 43.0 46.2 49.3

200 26.8 27.7 30.6 33.4 36.1 38.8 41.4

300 22.3 25.0 27.6 30.2 32.7 35.1 37.4

W i t h Er i n k N / m m 2, D i n m m a n d k in M N / m 3,

r = 1 7 . 1 7 { E c . D 3 / k } ~ m m

( l l . l O a )

and , wi t h E c i n ks i , D in i nches an d k in pc i

r = 3 . 0 5 4 { E e D 3 / k } ~

i n ch es

(11 .10a)

Ta b les 11 .6a and l l .6 b g ive t he va lues o f t he r ad iu s o f r e l a t i ve s t i f fness fo r

d i ff e re n t c o m b i n a t i o n s o f s u b g r a d e m o d u l u s a n d s l ab t h i c k n e ss i n m e t r i c a n d

I m p e r i a l u n i t s r e s p ec ti v e l y . T h es e a r e b a s e d o n a co n c r e t e m o d u l u s o f e l a s t i c i t y o f

2 8 k N / m m 2 ( 40 0 0 k s i) .

T h e m ax i m u m v a l u e s o f s t re s s e s i n t h e s lab a r e g i v en b y t h e fo l l o w i n g

e x p r e s si o n s a t a n i n t e r i o r p o i n t , a t t h e e d g e a n d n e a r a c o r n e r . T h e i n t e r i o r a n d

e d g e s t r e s s e s a r e m a x i m u m d i r e c t l y b e l o w t h e l o a d a n d h a v e t e n s i o n a t t h e

b o t t o m o f t h e s la b w h e r e a s t h e s tr e ss d u e t o a c o r n e r l o a d i s m a x i m u m s o m e

d i s t a n c e a w a y f r o m t h e c o r n e r a n d h a s t e n s i o n o n t h e t o p s u r fa c e . T h e e d g e a n d

c o r n e r l o a d s a r e a s s u m e d t o b e a p p l i e d s o t h a t t h e s l a b e d ge s a r e t a n g e n t i a l t o t h e

c i r cu l a r l o ad ed a r ea . T h e l o g a r i t h ms a r e t o b a s e 1 0 .

(7

=

l O 0 0 ( P / D 2 ) [ 1 . 2 6 4 l o g ( r / a ) +

0 .3 3 7 9 ] ( l l . l l a )

(7 e = l O 0 0 ( P / D 2 ) [ 3 . 2 0 8 l o g ( r / a ) + 0 . 5 5 1 8 a / r -

0 .0 2 6 ]

( l l . l l b )

ar = l O 0 0 ( P / D 2 )3 [ 1 - ( a ~ / 2 / r ) ~

( l l . l l c )

w h e r e P = l o a d i n k N

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S L A B S O N G R A D E 263

a = r a d i u s i n l o a d e d a r e a m m

o i - - - s tr e ss i n N / m m 2 u n d e r a n i n t e ri o r l o a d , a w a y f r o m t h e e d g e s

r = s t re s s in N / r a m 2 u n d e r a l o ad a t an ed g e N / m m 2

a~ = s tr e ss i n N / m m 2 d u e t o a l o a d a t a c o r n e r N / m m 2

T h e emp i r i c a l o b s e r v a t i o n s o f T e l l e r an d S u t h e r l an d ( 19 4 3 ) s h o w t h a t t h e s t re s s

d u e t o a co r n e r l o ad i s g i v en mo r e accu r a t e l y b y

tz e = lO 0 0 (P / D 2)3[ 1 - (a x / 2 / r ) ~ '2] (11.1 ld )

E q u a t i o n s ( l l . 1 2 a ) t o ( l l . 1 2 d ) a r e t h e Im p e r i a l v e r s io n s o f t h e a b o v e f o u r

e q u a t i o n s , ( l l . 1 2 d ) b e i n g t h e T e l l e r a n d S u t h e r l a n d e q u a t i o n .

6 i = ( p / D 2 ) [ 1 . 2 6 4 l o g ( r / a ) +

0 .3379 ] (11 .12a)

a~ = ( p / D 2 ) [ 3 . 2 0 8 l o g ( r / a ) + 0 .5 5 1 8 a / r - 0 .026 ] (11 .12b)

t r r ( p / D 2 ) 3 [ 1 - ( a x / 2 / r ) ~

(11 .12c)

trr = ( p / D 2 ) 3 [ 1 - ( a x / 2 / r ) L 2 ] (11 .120)

E q u a t i o n s ( 1 1 .1 1 a ) , ( 1 1 .1 1 b ) an d ( 1 1 .1 1 d ) can b e ex p r e s s ed i n t h e co n v en i en t

fo rms o f Eq ua t io ns (11 .13 ) and F ig ure 11 .4 g ives t he va lues o f t he coef f ic i en t s fo r

a r an g e o f a / r values .

t7 = I O 0 0 ( P / D 2 ) K i

a~ - I O 0 0 ( P / D 2 ) K e (11.13)

rye = I O 0 0 (P / D 2)K r

6

4

\

~ 2 ~ ,

o

-1

- 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 11.4

V a lu es o f K i , K e an d K r

1 . 0

a/r

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2 6 4 POST-TENSIONEDCONCRETE FLOORS

Table 11.8 Increase in con centrated load near join ts and edges

Join t descr ip tion Increase

Tied joints

Dow elled contract ion joints, opening 1 to 6 mm

Dowel led cont ract ion jo in ts opening m ore than 6 mm,

induced cont ract ion jo in ts opening m ore than 1 mm ,

free edges and isolation joints

Nil

33%

85%

T h e ab o v e eq u a t i o n s g i v e t h e s t r e s s e s f o r i s o l a t ed l o ad s o n l y , an y ad j acen t l o ad s

a r e i g n o r ed . F r o m a p r ac t i c a l p o i n t o f v i ew , t h e s tr e s se s p r o d u c ed b y an y l o ad

l o ca t ed b ey o n d a d i s t an ce o f 3 r f r o m t h e p o i n t u n d e r co n s i d e r a t i o n a r e n eg li g i b le

an d i t c an b e i g n o r ed . W h er e a f l o o r i s co n s i d e r ed c o n t i n u o u s , a j o i n t w i t h i n a

d i s t an ce o f 1. 5r o f t h e p o i n t u n d e r co n s i d e r a t i o n i n t e r r u p t s t h e co n t i n u i t y .

C h an d l e r ( 19 8 2 ) h a s g i v en me t h o d s f o r c a l cu l a t i n g , a t a g i v en p o i n t , t h e e f fec t

o f a n ea r b y l o ad .

M u c h o f t h e d e t a i l ed d e s i g n o f t h e s l ab is co n c e r n ed w i t h s t r e ss e s g o v e r n ed b y

l o ad s c l o s e t o t h e ed g es an d co r n e r s . T R3 4 ( 1 9 8 8 ) s u g g es t s t h a t , w h e r e t h e l o ad

cen t r e i s w i t h i n 3 0 0 m m ( 12 i n ) o f an ed g e o r j o i n t , t h e c o n c en t r a t ed l o ad s h o u l d

be i ncreas ed b y t he f ac to r s g iven i n Tab le 11 .8 . See sec t i on 11 .6 fo r a def in i t i on o f

jo in t s .

11.5.2 Line loads

Li n e l o ad s , s u ch a s t h o s e f r o m p a r t i t i o n s , a r e s o me t i mes ap p l i ed a t a s l ab ed g e .

F o r an i n t e r i o r l i n e l o ad , t h e b e a m o n e l as t ic f o u n d a t i o n t h e o r y eq u a t i o n s a r e g i v en

b y t h e f o l l o w i n g ex p r e s s i o n s ; an g l e

f i x

i s meas u r ed i n r ad i an s .

m = (0.25w//3)e-aX(sin f ix - cos f ix)

pg = (0.5w/3 )e-aX(c os fix + sin/3x )

(11.14)

w h e r e w = l i n e l o ad p e r u n i t l en g t h

x = d i s t an ce f r o m t h e l o ad

r = rad ius of relat i ve s t i ffness

/3 = 1 / (x /2r ) per uni t l en gth

m = m o m e n t p e r u n it w i d t h

pg = g r o u n d p r e s s u r e

T h e f o l l o w i n g r e s u l t s c an b e d e r i v ed f r o m t h e ab o v e eq u a t i o n s :

9 T h e m a x i m u m m o m e n t o c c u r s d ir e c tl y b e lo w t h e l o a d a n d t a k e s t h e v a l u e

0 .3536wr .

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~

w p e r u n i t

l eng th

9~.~,

S L A B S O N G R A D E

2 6 5

b : . " " " " , " ~ , . .

0 .4

0.3

0 .2

0.1

- 0 . 1

- 0 . 2

--0.3

--0.4

lwr

2 r

m/wr

3r

4r 5 r

Figure

11.5

Mo m e n t a n d g r o u n d p r e s s u r e u n d e r a li n e l oa d

I f r i s m e a s u r e d i n m i l l im e t r e s a n d w i n k N / m u n i ts t h e n t h e m o m e n t i n

k N m/ m i s g i v en b y 0 .3 5 3 6 w r / 1 0 0 0 .

T h e mo men t d r o p s t o z e r o a t a d i s t an ce o f 1 . 1 r f r o m t h e l o ad , an d t h en

r ev e r se s it s si g n , p eak i n g a t a d i s t an ce o f 2 .2 r f ro m t h e l o ad w h e r e i ts mag n i t u d e

i s 2 1 % o f t h a t u n d e r t h e l o ad .

9 T h e g r o u n d p r e s s u re i s m a x i m u m u n d e r t h e lo a d , i ts m a g n i t u d e is 0.3536w/r.

I f w is m e a s u r e d i n k N / m a n d r in m i l li m e t r e s t h e n t h e g r o u n d p r e s su r e i n

k N / m 2 is g i v e n b y 353.6w/r.

T h e g r o u n d p r e s s u r e i s z e r o a t a d i s t an ce o f 3 . 3 3r fr o m t h e l o ad , b e co m i n g

n e g a t i ve b e y o n d t h a t p o i n t , p e a k i n g a t a d i s ta n c e o f 4 . 4r f ro m t h e l o a d w h e r e

i ts v a l u e 4 .3 % o f t h a t b e l o w t h e l o ad .

F i g u r e 1 1.5 s h o w s t h e d i s tr i b u t i o n o f t h e m o m e n t a n d t h e g r o u n d p r e s s u re f o r a

u n i f o r m l i ne l oa d . T h e m o m e n t d i a g r a m is d r a w n o n t h e t e n s io n f a ce o f t h e

c o n c r e t e . N o t e t h a t t h e p a r a m e t e r m/wr is n o n - d i m e n s i o n a l , a n d pg/wr h as t h e

u n i t o f p e r u n i t a r ea .

T h e ab o v e v a l u e s a r e b a s ed o n t h e l o ad b e i n g ap p l i ed o v e r an i n f i n i t e l y s ma l l

w i d t h , w h i c h is n o t t h e c a se w i t h p a r t i ti o n s . T h e p e a k v a l u es o f t h e m o m e n t a n d

g r o u n d p r e s s u r e , d i r ec t l y b e l o w t h e l o ad , a r e r ed u ced t o t h e v a l u e s s h o w n i n

T ab l e 1 1.9 i f t h e l o ad i s a s s u m ed t o b e ap p l i ed o v e r a n o t i o n a l w i d t h o f 6 0 0 m m.

S u c h a w i d t h w o u l d r e s u l t f r o m a 2 00 m m p a r t i t i o n s u p p o r t e d o n a 2 00 m m s la b ,

a s s u m i n g a s p r ead o f l o ad a t a n an g l e o f 4 5 ~

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266

POST-TENSIONEDCONCRETE FLOORS

Table 11.9

M o m e n t a n d 9 r o u n d p r e s s u re u n d e r a p a r t it i o n

r ( m m ) M o m e n t G r o u n d p r e ss u r e

( k N m / m ) ( k N / m 2)

500 0.102w 0.658w

750 0.185w 0.456w

1000 0.271w 0.347w

1250 0.358w 0.279w

1500 0.445 w 0.234w

1750 0.532w 0.201w

2000 0.619w 0.177w

11 .5 .3 UDL an d a is les

I n som e s t o r a ge bu i l d i ngs , t he f l oo r m a y be r e q u i r e d t o c a r r y a un i f o r m l y

d i s t r i bu t e d l oa d ove r l a r ge a r e a s w i t h r e l a t i ve l y na r r ow a i s l e s , w h i c h r e m a i n

u n l o a d e d . T h e l o a d i n g i s e q u i v a l e n t t o a f l o o r c a r r y i n g a n u p w a r d u n i f o r m l y

d i s t r i bu t e d l oa d o n t he a i sl e w i d t h s on l y , w i th t he r e s t o f t he f l oo r un l oa d e d . Th e

m a x i m u m m o m e n t i n a f l o o r t h u s l o a d e d d e v e l o p s in t h e m i d d l e o f t h e ai sle , a n d

ha s t e n s i on on t he t op o f t he s l ab . F o r a n i n f i n it e ly l ong t h i n e l as t ic p l a t e on a n

e l a s t i c m e d i u m , l o a d e d a s s h o w n i n F i g u r e 1 1 . 6 , t h e m o m e n t s a n d t h e g r o u n d

p r e s su r e a t po i n t O a r e g i ve n by t he f o l l ow i ng e xp r e s s i ons :

F o r x < a ,

m = - ( w / 4 f l 2 ) . [ e - p (" + b)

sin{f l (a + b)} - e -aa s in(f la) ]

pg - - - ( w / 2 ) E e -p (a + b)

cos{ f l (a + b)} - e -pa cos(f la) ] (11.15)

M o m e n t s a n d g r o u n d p r e s s u r e s w i th i n t h e w i d t h b c an b e c a lc u l a te d i f t h e

d i s t a nc e a is m a de ze r o a nd t he l oa d on t he t w o s i de s is c ons i de r e d i n t w o se pa r a t e

c a l c u l a t i ons . V a l ue s i n the m i dd l e o f l e ng t h b a r e g i ve n by :

m =

- ( w / 2 f l 2 ) e -p b/ 2

s i n ( f i b ~ 2 )

pg = w[1 - e - pb/2 c o s ( f i b ~ 2 ) ] (11.16)

F ig ure 11 .7 sho ws the re l a t ion sh ip be tw een the rad ius o f re l a t ive st if fness r an d

t he m o m e n t f o r ai sl e w i d t h s o f 1.2 a nd 1 .5 m e t r e s ; a l so sho w n f o r c om pa r i so n i s

t h e r e l a t i o n s h i p b e t w e e n r a n d m o m e n t / w f o r p a r t i t i o n l o a d s a p p l i e d o v e r a

600 m m w i d t h . The f i gu re is ba se d o n Eq ua t i on ( 11. 16 ) . The un i t s a re kN m / m f o r

1

o T

L..

F

w/unit length

a ~l_.. b _j

~1~ v l

x

L..

I ~ v r

F i g u r e 11.6

Part ial uni formly distr ibuted load

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S L A B S O N G R A D E 2 6 7

t -

" O

. ~

E

E

O

O

t,D

t -

O

t -

O

. ~

t ~

f t . .

t ~

m

t ~

" O

t ~

O

m

t -

+ 0 . 8 -

+ 0 . 6 -

+ 0 . 4

-

+0.2 -

0 .0

- 0 . 2 -

- 0 . 4 -

- 0 . 6 -

- - 0 . 8 -

I I I I I 1 I

5 0 0 7 5 0 1 0 0 0 1 2 5 0 1 5 0 0 1 7 5 0 2 0 0 0

r ( m m )

F i g u r e 1 1 . 7

M o m e n t a n d g r o u n d p r e s s u r e d u e t o

a pa r t i t i on , and an

u n l o a d e d a i s l e

m o m e n t ( m ), k N / m 2 f or u n if o rm l o a d i n g ( w ) , k N / m f or p a r t it io n l o a d ( w ) , a n d

mi l l imetres for r .

W i t h re f erence t o F i gure 1 1 .7 , no t e t hat t he par t i t i ons produce t ens i l e s t re s s a t

t h e b o t t o m o f t h e sl a b w h e r e a s t h e u n l o a d e d a i sl es h a v e t e n s i o n a t t h e to p . T h e

g r o u n d p r e s s u r e s h o w n f o r t h e u n l o a d e d a i s l e s r e p r e s e n t s a r e d u c t i o n f r o m t h a t

u n d e r a u n i f o r m l y l o a d e d s l a b .

1 1 . 6 C o n s t r uc t io n

U n b o n d e d t e n d o n s a r e o f te n p r e fe rr ed i n p o s t - te n s i o n e d g r o u n d f lo o r s b e c a u s e

o f t h e e a s e o f c o n s t r u c t io n , t h o u g h b o n d e d t e n d o n s h a v e b e e n u s ed . T h e a m o u n t

o f p r e s t r es s r eq u i r e d b e i n g m u c h s m a l l er t h a n t h a t in a s u s p e n d e d f lo o r , 1 3 m m

strand ( 89 n) i s of ten used .

T he f l oors are usua l l y pres t res sed i n bot h d i rec t i ons . In t he l ong d i rec t i on t he

p res t ress i s ap p l i ed w i t h i n 2 4 to 48 hou rs o f cas t i ng t he concre t e . In t he transverse

d i r e c ti o n , t h e t i m e o f a p p l i c a ti o n o f t h e p r e s tr e s s d e p e n d s o n t h e m e t h o d o f

cons t ruc t i on , w i t h ear l y pres t res s i ng pre f erred .

T h e c u r r e n t ly f a v o u r e d m e t h o d s o f c o n s t r u c t i o n a r e b r ie fl y d e s c r ib e d b e l o w .

T h e y a r e s im i l a r to t h e m e t h o d s u s e d f o r r e in f o r ce d c o n c r e t e f l o o rs . T h e c h o i c e o f

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268 POST-TENSIONEDCONCRETE FLOORS

t he meth od depend s on the s ize o f the f loor and the ava i l ab il i ty o f equ ip m ent and

l ab o u r :

9 The long s t r ip method has become es tab l i shed as the conven t iona l method o f

con st ruc t ing f loors o n grade . The f loor is cas t in a ser ies of s t r ips up to 6 m

(20 f t ) wide. The s t r ips may be la id consecut ively or a l ternately , though the

lat ter requires t ransverse pres t ress ing to be delayed.

9 The wide s t r ip const ruct ion, in which the f loor i s cas t in bays of up to 20 m

(65 f t ) in width . Special ized machinery and ear ly s t ress ing in both di rect ions

are required for such wide bays .

Th e m a i n ad v a n t ag es a r e t h o s e o f t h e r ed u ced n u m b er o f co n s t ru c t i o n jo i n t s

an d t h e h ig h e r p ro d u c t iv i t y co m p ared w i t h th e lo n g s tr ip m e t h o d . Ho w ev e r ,

the l a rger floor a rea to be f in i shed in one opera t ion requ i res mo re sk il led and

at tent ive f in ishers .

9 The l a rge bay m ethod , in which severa l thou san d square m et res o f a f loor a re

cas t in one opera t ion . High s lump concre te i s d i scharged d i rec t ly on the

p rep a red g rad e (o r memb ran e ) , an d s p read an d s c r eed ed man u a l l y , f o l l o w ed

by surface f loat ing wh en the concrete h as suff icient ly harden ed. Superplas t ic izers

are used to ach ieve the h igh s lump ; v ib ra to rs o r o ther mec han ica l co m pac t ing

devices are not used.

The a dvan tages o f the sys tem are the ease and speed o f cons t ru c t ion , because

no co m pact io n is requ i red . V ery la rge a reas o f f loor can be cas t in a day . T he

d i sadvan tage i s tha t the l a rge a rea requ i res much more sk i l l ed and a t t en t ive

f in ishers . Also, in the in teres t of speed of con st ruc t ion, an d for conven ience, the

fabr ic re in fo rcement i s somet imes p laced ne ar the bo t to m of the s l ab , where i t

may be less effective.

The c ur ren t t re nd i s towa rds the use o f h igh s lump concre te co n ta in ing

p las ti c izers and o ther adm ix tu res to g ive the se t t ing and harde n ing t imes to su i t

the con t rac to rs . The h igh s lump makes i t much eas ie r to ach ieve good

com pact ion , o f t en the concre te is descr ibed as

self-compacting.

G ra d e 4 0 co n c re t e

(fc ' - 3000 ps i ) i s m ost c om m on ly used. A highe r s t reng th m ay be used i f i t i s m ore

economica l .

The s lab is usual ly cas t af ter the external wal ls have been erected. Access to

anch orage s w ould , therefo re , be very res t r ic t ed i f they were pos i t ioned a long the

edge o f the s lab . The usua l p roced ure is to con s t ruc t the pos t - t ens ioned f loor

abo u t two m et res ( say six feet ) shor te r in l eng th tha n the c l ear d i s t ance be tw een

the extern al wal ls , leaving a m etre a t each e nd. T he gap is f il led at a la te s tage to

a l low t ime fo r as much sh r inkage and c reep to t ake p lace as the p rogramme

al lows . The gap s t r ip i s cons t ruc ted in re in fo rced concre te ; re in fo rcement

pro jec t ing f rom the pos t - t ens ioned f loor i s bonded in to the gap s t r ip .

A pos t - tens ioned f loor has a muc h sm al le r nu m ber o f jo in t s tha n a re in fo rced

concre te f loor , and a p roper ly des igned and cons t ruc ted pos t - t ens ioned f loor

shou ld be mo re s t ab le and re la tive ly c rack f ree . Jo in t s be ing unde s i rab le f rom the

user ' s po int of v iew, the opin ion is of ten expressed tha t , in floors where h ygiene

an d a d u s t -f r ee a t mo s p h e re a r e n o t p a r t icu l a r l y i mp o r t an t , an u n ex p ec t ed c r ack

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SL ABS ON G RADE 269

Standard mesh

Bonded reinforcement

Standard mesh

, \ Induced crack

. L _ . ,_ _ _ _ _ _ ~ . \ ~ . . t

l Standard mesh

Crack inducer

Dowel-bars

Crack inducer

Dowel cap, end filled

with comoressible material

Compressible Dowel-bars

filler

Tied joints Dowelled contraction and

expansion joints

Figure 11.8

M o v e m e n t j o i n ts

is m o r e s t a b l e a n d c a u s e s fe w e r p r o b l e m s t h a n a p u r p o s e - m a d e j o i n t. C r a c k i n g is ,

o f c o u r s e , u n s i g h t l y .

T h e j o i n t s , i f r e q u i r e d i n a p o s t - t e n s i o n e d f l o o r , a re s i m i l a r t o t h o s e i n

r e i n f o r c e d c o n c r e t e . T h e o n l y e x c e p t i o n is t h e j o i n t b e t w e e n t h e w a ll s a n d t h e

e d g e o f t h e f l o o r , w h i c h s h o u l d b e d e s i g n e d t o c a t e r f o r t h e r e l a ti v e l y l a r g e r

c o n t r a c t i o n o f t h e f lo o r d u e t o s h r i n k a g e a n d c r e ep .

T y p i c a l t i e d a n d d o w e l l e d jo i n t s , a s u s e d i n r e i n f o r c e d c o n c r e t e f l o o r s , a re

s h o w n i n F i g u r e 1 1 . 8 .

E x a m p l e 1 1 . 1

De term ine the pres t ress in a 100 m long 200 mm thick gro un d s lab for the fol lowing

loads , which include the safe ty fac tors :

C on cen t r a t ed load o f 150 kN app l i ed ove r a 300 x 300 m m a rea

Un iform loadin g of 30 kN /m 2 wi th 1 .25 m wide a is les

P a r t i t i on load ing o f 40 kN /m app l i ed ove r an e ff ec tive f loo r w id th o f 600 m m

Assum e f~u = 40 N/ m m 2 and k = 50 M N /m 3

S o l u t i o n

The W esterg aard equa t ion i s used for ca lcula t ing flexura l s t ress und er the conce nt ra ted

load.

From Table 11.1 ,

F rom Tab le 2 .3 ,

F r o m E q u a t i o n ( l l . 1 0 a ) ,

f t90 = 5 .06 N /m m 2

E c = 30.3 kN /m 2

r = 1 7 . 0 8 7 { E c D 3 / k } ~

= 17.087{30.3 x 2003/50}0"25= 800mm

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270 POST-TENSIONED CONCRETE FLOORS

N o t e " I n t e r p o l a t i o n f r o m T a b l e 1 1 .7 a gi v es 79 3 m m , a s l i g h tl y s m a l l e r b u t a c c e p t a b l e

v a l u e .

Concent rated load

P = 1 5 0 k N

D = 2 0 0 m m

a = r a d i u s o f c o n t a c t a r e a

= ( 3 0 0 x 3 0 0 / r 0 ~ = 1 7 0 m m

U s i n g E q u a t i o n ( l l . l l a ) ,

f f i - -

IO00 (P/D 2) . [ 1 .264 log(r /a) +

0 . 3 3 7 9 ]

= 1 0 0 0 ( 1 5 0 / 2 0 0 2 ) [ 1 . 2 6 4 l o g ( 8 0 0 / 1 7 0 ) + 0 . 3 3 7 9 ] = _+ 4 . 4 6 N / m m 2

Part i t i on load

w = 4 0 k N / m

I n t e r p o l a t i n g f r o m T a b l e 1 1.9 ,

m = 0 . 2 0 2 2 w = 8 .1 k N m / m

tr = 8 . 1 x 1 0 0 0 / ( 2 0 0 2 / 6 ) = _+ 1 .2 2 N / m m 2

P g = 0 . 4 3 4 w = 1 7 .3 6 k N / m 2

U DL w i t h a i s l e

w = 3 0 k N / m 2

F r o m F i g u r e 1 1 .7 ,

m = 0 .2 x w = 6 .0 k N m / m

a = 6 . 0 x 1 0 0 0 / ( 2 0 0 2 / 6 ) = __ 0 . 9 0 N / m m 2

P g = 0 . 6 w = 1 8 k N / m 2

Prest ress

O f t h e t h r e e l o a d i n g c a s e s , t h e c o n c e n t r a t e d l o a d is t h e c r it i ca l o n e ,

s t r e s s = _+ 4 . 4 6 N / m m 2

U s i n g # = 1 .5 , f r o m E q u a t i o n ( 1 1 .7 a ) , Pav t h e a v e r a g e s t r e ss r e q u i r e d t o o v e r c o m e

d r a g f r i c t i o n ,

P ,v = 0 . 0 1 8 L = 1 .8 N / m m 2

T h e r e f o r e , t o t a l r e q u i r e d p r e s t r e s s a f t e r a l l o w i n g f o r f t

= 4 . 4 6 + 1 .8 - 5 . 0 6 = 1 . 2 N / m m 2

P = r e q u i r e d f i n a l p r e s t r e s s i n g f o r c e

- 1 .2 x 2 0 0 - 2 4 0 k N / m w i d t h

A s s u m i n g a f i na l f o r ce o f 17 0 k N p e r 1 5 .2 m m s u p e r s t r a n d ,

s t r a n d s p a c i n g = 1 0 00 x 1 7 0 / 2 4 0 = 7 0 8 m m c e n t r e s

P r o v i d e 1 5 . 2 m m s u p e r s tr a n d a t 7 0 0 m m c e n tr e s .

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1 2 D E T A I L I N G

This chapter i s concerned pr imar i ly wi th the product ion of pos t - tens ioning

drawings . I t a l so dea ls wi th some re la ted aspec ts which may need to be

considered but are not necessar i ly par t of the calculat ion process.

12 .1 D rawings and symbo ls

The spec ia l i s t pres t ress ing cont rac tor requi res shop drawings , showing the

informat ion requi red to assemble a l l rod re inforcement , pos t - tens ioning h ardw are

and tendons , and for s t ress ing the tendons . Of ten, separa te drawings a re

produced for the pres t ress ing i tems and rod re inforcement . The drawings

showing rod reinforcement conform with the pract ice for reinforced concrete .

The post - tens ioning shop drawings a re usua l ly prepared by the spec ia l i s t , to

sui t h is products and method of working. They should show:

anchorage loca t ions on plan and sec t ion

anchorage pocke t d imens ions

t endon l ayou t i n p l an

bundl ing o f t endons

horizontal and ver t ical prof i les

points of inf lexion

tendon he igh t s a t suppor t po in t s

order of assembly

tendon forces

ca lcula ted extens ions

Ancho rages sho uld be c learly ident if ied wi th regard to the num ber and types of

s t rand they a re des igned for . The ac tua l n um ber of s t rands be ing anch ored in an

anch orage assembly may , of course , be less than the capac i ty of the ancho rage .

The l ive end a nd the type of dead anch orage should be ident if ied.

Ten don s a re usua l ly colour coded for ident if icat ion. The drawing should show

the num ber a nd type of s t rands in a tendon, and i ts colour coding, length, jacking

force and extension. This info rma tion is given at the l ive end in case of tend ons to

be stressed from one end.

The sym bols , propos ed by the Con cre te Soc iety , for identi fying the num ber of

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272 POST- TENSIONED CONCRETE FLOORS

@

Tendon q uantity, length, colou r code and elongation

Placing sequence (where required)

0.40

i Dimensions from reference line to ~ of tendon group

- Symbol defining num ber of tendon s in group

0.80 C)

1.20 ~ _

1.60

D

2.00

@,,

2.40

One strand

Two strands

Three strands

Four strands

Five strands

Fixed end

I

,,---, 2.8 0 ~ Additiona l

1 x 12.500 ( th ru ' )9 ~ ~ I l L / - ~ - s trands I

Red A = 130 /

3 x 10 .5 00 ( a d d e d ) l

Blue A = 75 ~ Edge of slab

Stressed ends

Note: when more than one symbol appears on a tendon group,

the number of strands equals the sum of the symbol designations

Figure 12.1 Tendon notation

s t rands in a tendon, and the data to be given at the l ive end, are show n in Figure 12.1 .

When the des ign i s car r i ed ou t by o ther than the spec ia l i s t con t rac to r , the

in fo rm at ion g iven to h im by the des igner var ies accord ing to the local p rac t ice .

The min imum prac t i ca l in fo rmat ion requ i red by the spec ia l i s t to enab le h im to

produce h i s shop d rawings shou ld inc lude :

9 Draw ings ident i fy ing the s t ructur al e lements to be post - tens ione d and thei r s izes

9 Co ncre te qua l i ty , min im um covers and

e i ther : gu idance on the s t reng th requ i red a t s t res s ing

or: stressing stages

9 Ten d o n p ro f il e t y p e fo r each e l em en t - - h a rp ed o r p a rab o l i c

High and low po in t s o f the p ro fi le

9 Ei ther : f inal pres t ress in conc rete

or : num ber and type o f t endons and jack ing fo rces .

F igure 12.2 shows par t o f a typ ica l d raw ing g iv ing the con su l t an t ' s requ i rem ents

fo r the p rest ress an d rod re in fo rcement fo r a con t inu ous two-w ay spann ing so lid

f loor . Tendon he igh t s a re shown on a sec t ion in the long i tud ina l d i rec t ion ; a

s imi lar sect ion is required in the t ransverse di rect ion. Note that the s ingle s t rand

tendons a re no t marked wi th the symbols p roposed in F igure 12 .1

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F igure 12 .2 Tendon and reinforcement layout

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27 4

POST-TENSIONED CONCRETE FLOORS

Table 12.1 Minimum steel proportions in reinforced concrete (ACI 318)

Description

Percentage

Fo r grade 300 (45 ksi) deform ed bars

Fo r grade 400 (60 ksi) deform ed bars or we lded wire fabric

For steel with fr >40 0 N/m m 2 (60 ksi) at a strain of 0.35%

0.20

0.18

0.18 • 400/fy

1 2 .2 M i n i m u m r e i n f o r c e m e n t

In a pos t - t ens ioned f loor , the e l ements des igned as re in fo rced concre te , o r so

impl ied , rod re in fo rcement shou ld be p rov ided in accordance wi th the ru les fo r

re in fo rced concre te . Th i s app li es to beams wh ich ma y no t be pos t - tens ioned , and

to one-way s labs in the t ransverse di rect ion.

Fo r re in fo rced concre te m em bers BS 8110 specif ies a min im um of bon ded s tee l

qua n t i ty o f 0 .13% of the g ross concre te a rea fo r re in fo rcement wi th a y ie ld

s t reng th ( fy) o f 460 N/ rn m 2 (66000 ps i) . N o m in im um qua n t i ty o f s tee l i s,

however , spec i f i ed fo r pos t - t ens ioned members .

AC I 318 requ i res the min imu m area o f rod re in fo rcement to be no t l ess than

tha t g iven by the rat ios of gross concre te sect ion show n in the Table 12.1 , bu t no t

less than 0 .14% in any case . Th i s re in fo rcemen t is to be spaced no t fu r ther ap ar t

than f ive t imes the s lab th ickness , nor 500 mm (20 in) .

AC I 318 a l lows thi s m in im um am ou nt o f re in fo rcement to be rep laced by

p re s t r e s s ed t en d o n s p ro p o r t i o n ed t o p ro v i d e a mi n i mu m av e rag e co mp ara t i v e

s tress o f 1 .0 N/m m 2 ( 150 ps i) on the gross c oncre te are a a f ter a ll losses in pres t ress

have t ake n p lace . The m axim um spac ing o f such t endon s i s l imi ted to 2 .0 m (78

in) . Where thei r spacing exceeds 1 .4 m, addi t ional s teel , as per Table 12.1 , i s

requ i red be tween the t endons a t s l ab edges ex tend ing f rom the s l ab edge fo r a

d i s t ance equa l to the t endon spac ing .

Use o f pos t - t ens ion ing t endons in l ieu o f the min im um qua n t i ty o f rod

re in fo rcement in the non-s t ressed d i rec t ions shou ld be se r ious ly cons idered in

o n e -w ay s p an n i n g f l o o r s s u p p o r t ed o n p o s t - t en s i o n ed b eams . No mi n a l p o s t -

tens ioning of the are a of s lab be twee n the effective f langes of the be am s re duces

the long i tud ina l shear be tween the s l ab and the beam, and l imi t s the sp read o f

p res t ress f rom the beam in to the ad jacen t s l ab .

The fo l lowing d i scuss ion re la t es to the m in im um quan t i ty o f re in fo rcement in

the di rect ion of the pres t ress .

In mem b er s u s in g u n b o n d ed t en d o n s , ACI 3 18 r equ i re s b o n d ed r e i n fo rcemen t

to be p rov ided , d i s t r ibu ted un i fo rmly a long the t ens ion face. The a rea o f th is

re in fo rcement shou ld n o t be less than 0 .4% of the a rea o f the concre te on the

tension s ide of the sect ion centroid . T his reinforcem ent i s requ ired reg ardless o f

the service load s t ress condi t ion.

In two-way s l abs , ACI 318 a l lows the bonded re in fo rcement in the pos i t ive

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DETAILING

2 7 5

m om ent zone to be o m i t t ed i f the com puted t ens il e s tress in concre te a t se rv ice

loads does no t exceed [~ ' ) ~ I f th is l imi t is exceeded , then the m in im um

am o u n t o f b o n d ed r e in fo rcemen t r equ i red i s

A~

=

NJ(0.5fy) (12.1)

In the above express ion, Nr i s the tens i le force in concrete under service load,

ca lcu la ted on the bas i s o f an u ncrack ed sec t ion . fy shou ld n o t exceed 400 N /m m 2

(58 000 psi ) . This s teel should be uniform ly dis t r ibu ted over the tens ile zone as

close as pract icable to the tens ion surface.

In n eg a t iv e mo m en t a r ea s a t co lu mn s u p p o r t s , th e mi n i mu m a rea o f b o n d e d

re in fo rcement in each d i rec t ion shou ld be computed by

A~ = 0.00 075 DL (12/2)

where L is the sp an l eng th in the d i rec t ion o f the re in fo rcement be ing cons idered .

Th i s re in fo rcement shou ld be d i s t r ibu ted wi th in a s l ab wid th be tween l ines tha t

a re 1 .5D ou t s ide opp os i t e faces o f the co lumn. At l east four bars sho u ld be

prov ided in each d i rec t ion and the i r spac ing shou ld no t exceed 300 mm (12 in) .

L inks in pos t - t ens ioned beams are normal ly governed by requ i rements o f

shear capac i ty . Som e l inks a re , however , requ i red in the beam s a nd in the r ibs o f a

f loor to suppor t the t endons ; the normal spac ing fo r such suppor t s i s 1000 mm

(40 in) . F igure 12 .3 shows typ ica l a r rangem ents

In the case o f a num ber o f s lab pane l t endon s ge t t ing dam age d , the concre te

m ay be ab le to sus ta in som e of the load p ro v ided tha t a rch ac t ion ca n deve lop .

Th i s ma y be possib le in the inner spans o f a con t inuou s two-w ay f loor wi th a

ser i es o f spans in each d i rec t ion . The edge spans may a l so deve lop a rch ac t ion

a long the d i rec t ion o f the edge , bu t n o t across i t. The co rner pan e l s wo uld be very

un l ike ly to deve lop an y a rch ac t ion . In such f loors it m ay be p ru den t to p rov ide

the m in imu m qua n t i ty o f re in fo rcement in co rne r pane l s in bo th d i rec tions an d in

edge panels at r ight a ngles to the free edge, par t icu lar ly i f the pres t ress i s provide d

by unbo nde d t endons . A smal l am oun t o f t ie re in fo rcemen t wo uld , o f course , be

Tendons

Tendon suppor t

Sti rrup

Figure

12.3 Tendon supports in beams and ribs

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276

POST-TENSIONEDCONCRETE FLOORS

I

r , , 1 1 1 7 1

l l j

,

P

C

L L l i I / j

~ ' i " T ! t n 7

Iz-t'l:z'Ct~

I_ .~z ..,17

I-- . --I-',4-I

L !11Jl:J

i

I

r n F I T 7

L . L i ~ u

j

U

Figure

12,4 Sugges ted a r rangement o f min imum re in fo rcement

requ ired in the edge panels pa ral le l to the free edge. Figu re 12.4 indicates the

sugges ted pos i t ion o f such re in fo rcemen t .

1 2 .3 T e n d o n s p a c i n g a n d p o s i t io n

AC I 318 requ i res a m in im um pres t ress o f 1 .0 N /m m 2 (150 psi ) on the g ross

concre te sec t ion a f te r a ll l os ses . It recom m ends 2 .0 m as the max im um spac ing o f

t endons . I f the spac ing exceeds 1 .4 m, then add i t iona l b ond ed rod re in fo rcemen t ,

o f a rea g iven by Tab le 12.1, is requ i red be tween the t endon s a t s l ab edges

ex tend ing f rom the s lab edge for a d is t ance equa l to the t end on spac ing . BS 8110

does no t specify any u pper l imi t on the spac ing o f t endons .

The gap be tween ad jacen t t endon s shou ld a l low unh indered f low of concre te

th rou gh i t dur ing v ib ra t ion . A min imu m gap o f 5 m m ( 88 n ) la rger th an the

ag g reg a t e size is n o rma l l y ad equ a t e fo r u n b o n d ed t en d o n s ; f o r b o n d ed t en d o n s

the clear hor izon tal a nd v ert ical gaps sho uld n ot be less tha n the s ize of the duc t

measured in the same d i rec t ion .

In s l abs , two o r th ree unbonded t endons a re o f t en bund led toge ther a shor t

d i s t an ce aw ay f ro m t h e an ch o rag es . Th i s a l l o w s co mmo n s u p p o r t s t o b e u s ed ,

and a g rea te r gap is m ade ava i l ab le be tween the t en don g roups fo r se rv ice ho les.

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D E TA I L I N G 277

T he a ngu la r de v ia t ion , ove r t he d i s t a nc e be twe e n the a nc ho r a ge s a nd the po in t

whe r e t he y touc h e a c h o the r , shou ld be t a ke n in to a c c oun t i n c a l c u l a t i ng the

fr ic t ion loss in prestress .

I t is de s i ra b l e t o r un a f ew t e ndons th r ou gh the i n t e rna l c o lumn s , a nd to a nc h o r

som e wi th in the wid th of the ex te rna l co lu m n i f poss ib le . Such tend on s ac t a s tie s

be twe e n c o lum ns a nd t r a ns fe r some o f t he she a r l oa d d i r ec t ly t o t he c o lumn s . I n

prac t ice , however , the in te rna l co lumns a re of ten so conges ted wi th s tee l tha t

t e ndons c a nno t be a c c ommoda te d w i th in t he i r bounda r i e s . I t i s o f t e n no t

poss ib le t o p r ov ide a nc ho r a g e s i n the e x t e r na l c o lumn s , be c a use o f c onge s t ion ,

a nd b e c a use the lo s s o f c onc r e t e due to t he a nc ho r a ge poc k e t ma y no t be

acceptab le . In such c i rcum stances , the re i s no a l te rna t ive b u t to loca te a l l o f the

t e ndon s , a nd the a nc ho r a ge s , ou t s ide t he c o lum ns ; t he re q u i r e m e n t o f a de q ua te

rod re inforcem ent , to sa t i s fac tor i ly t r ansfe r the shea r load f rom the tend on s to the

c o lumns , shou ld be c he c ke d .

1 2 .4 D e f lec t ion and c ladd ing

I n no r m a l r e in f o r ce d c onc r e t e , t he de f le c tion i s a lwa ys dow nw a r ds , un l es s t he

f o r mwor k wa s p r ov ide d w i th a c a mbe r . I n a pos t - t e ns ione d f loo r , t he p r e s t r e s s

m a y induc e a n in it ia l c a m be r , wh ic h ma y c ha nge to a dow nw a r d de f l e c tion whe n

the pa r t icu la r e leme nt i s fu lly loaded . T he c reep e f fect wou ld , ov e r a long pe r iod ,

inc rease the de f lec t ion , o r the c am ber i f un der fu ll load the de f lec t ion was s t i ll

upw ards . This can cause d i f ficu lt ie s in fix ing precas t pane ls , p a r t icu la r ly i f heavy

precas t u n i t s a re sea ted on th in edges of the s lab , o r i f the c laddin g de r ives ve r t ica l

supp or t f r om suc h a s l a b . T he a l t e r na t ive s a nd po s s ib ly t he i r c om bina t ion s , t o be

cons ide red in such a case , inc lude :

9 Inc rease the s t i ffness of the s lab edge . Th is w ould no rm al ly requi re a loca l

inc rease in the th ickness or the provis ion o f a do w ns tan d; b o th a re in conf l ic t

with the buildabil i ty aspect of a f ia t soff i t .

9 Des ign the c ladd ing to be indep end ent of the s lab edge . The preca s t pane ls can

be de s igne d to b a y l e ng ths , suppo r t e d a t c o lum ns ; t he pa ne l s wo u ld , o f c ou r se ,

be la rge , heavy , and d i f f icu l t to handle and manoeuvre on s i te .

C u r t a in wa l l type o f c l a dd ing c a n be a t t a c he d to t he s l a b e dge f or ho r i zon ta l

sup por t bu t n o t ve r t ic a l. E i the r i ts l oa d mu s t be t ra ns f e r r ed to t he c o lumns by

s t r uc tu r a l me mbe r s i nc o r po r a t e d w i th in t he c l a dd ing sys t e m, o r i t mus t be

se l f - suppor t ing .

9 De s ign the c l a dd ing to a c c o m m od a te t he expe c t ed m ove m e n t o f t he s la b e dge .

1 2 .5 M o v e m e nt j o in ts

In pos t - tens ioned f loors i t i s o f ten necessa ry to de lay the connec t ion be tween a

slab and a s t if f ver t ica l e lem ent, su ch as a li ft core or sh ear w all . The delay a l low s

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278 POST-TENSIONED CONCRETE FLOORS

som e of the shor te n in g in the s lab length to take p lace be fore the conn ec t io n i s

es tab l i shed . Without th is de lay , the shor ten ing due to the ax ia l e la s t ic and c reep

s t r a in s w ou ld be r e s tr a ine d b y the ve r t ic a l el e me n t , w i th t he r e su lt t ha t so me o f

the ax ia l p res t re ss would be los t f rom the f loor to the ve r t ica l e lement .

A long de l a y o f s eve ra l mo n th s wou ld , o f cou r se , be de s i r ab l e , be c a use i t wou ld

a l lo w m o r e o f t h e m o v e m e n t t o t a k e p la c e, b u t t h e c o n s t r u c ti o n p r o g r a m m e is

un l ike ly t o a l l ow th i s ; a pe r iod o f one m on th is ge ner a lly c ons ide r e d to be t he

min imum. T he ne e d f o r de l a ye d c onne c t ion ha s be e n d i s c us se d in C ha p te r 5 .

F igu r e 12.5 shows two a r r a n ge m e n t s f o r j o in t s be twe e n a s l a b a nd a wa ll . B o th

a l low the wa l l to be con s t ruc ted abo ve the p a r t icu la r f loor leve l whi le leav ing the

f loor tempora r i ly f ree to move la te ra l ly .

I n a r r a ng e m e n t ( a) , a ga p o f a bo u t one m e t r e is l ef t unc o nc r e t e d be twe e n the

s lab and the wa l l , and i s concre ted a f te r a su i tab le de lay . The a r rangement i s

ca l led a gap or a n infill strip. The s lab anch orag es , w he the r live or dead , a re cas t in

the s l a b ; t he y a r e l oc a t e d a w a y f r om the wa l l a nd so do no t t r a ns fe r a ny o f t he

she a r t o t he wa l l . T he s l a b mus t r e ma in p r oppe d un t i l t he c onne c t ion w i th t he

wall is ful ly effec tive . T he r e info rcem ent in the g ap m us t be suff ic ient to enab le th e

ga p s t r i p t o a c t a s r e in f o r ce d c onc r e t e m e m be r f o r the c om bina t ion o f t he be nd ing

m om en t and res idua l ax ia l tens ion resu l t ing f rom the re s t ra in t o f fe red by the wa ll .

A r r a n ge m e n t ( b) c ons i s ts o f a t oo the d su pp or t t o t he s la b , p r ov id e d a t sho r t

lengths of the wa l l le f t unconcre ted . The s lab i s suppor ted d i rec t ly by the wa l l ,

t hus d i spe ns ing w i th t he p rops . M e t i c u lous a t t e n t io n to t he c on s t r uc t ion o f t he

s l id ing jo in t i s e ssen t ial to ensure f reedom of la te ra l m ove m ent . T he ve r t ica l tube

th r o ug h wh ic h the do we l pa s se s ma y c ons i s t o f a p ie c e o f p r e s t r es s ing sh e a th i f

bon de d t e ndo ns a r e be ing u se d . T he q u a n t i t y o f c onc r e te r e q u i r e d to f il l t he t ube s

a nd m a ke goo d the ga ps in t he wa ll is ve r y sma l l c omp a r e d w i th t ha t ne e de d in t he

ga p s t r ip . T he a nc ho r a ge s a r e loc a t e d in t he teeth of the s lab , a s nea r the w a l l a s

the y wo u ld ha ve be e n in t he a bsenc e o f t he j o in t . T h i s a r r a n ge m e n t is sl i gh tly

m or e d i f fi cu l t t o c on s t r uc t t h a n ( a) , bu t t he a bse nc e o f p r ops shou ld a l l ow a

longer de lay .

A gap s t r ip can be used in the mid dle of a la rge s lab a rea , to r educe the

m ove m e n t . I f t he s t ri p r uns a lo ng the d i r e c t ion o f spa n o f a one - wa y s l ab the n no

p r opp ing i s r e q u i r e d .

A pos t - t e ns ione d f loo r r eq u i r es t he s a me c ons ide r a t ions f o r a m ove m e n t j o in t

as a r e inforced concre te f loor , bu t wi th one major d i f fe rence . A re inforced

concre te f loor r educes in length because of the sh r inkag e of concre te ; the ac t io n i s

res is ted by the re inforcement con ten t o f the f loor . By com par iso n , a pos t - ten s ione d

concre te ha s a smal le r qu an t i ty of s tee l to r e s i st shr inkage , an d i t underg oes

fur the r long- te rm shor ten ing in length due to c reep . I t i s , the re fore , ex t remely

unl ike ly tha t a pos t - tens io ned f loor wi ll eve r in it s l i fe r each the sam e length as i t

ha d a t t he t ime o f i ts c ons t r uc t ion . T he r e fo r e , po s t - t e ns ione d f loo r s on ly ne e d

cont rac t ion jo in ts , where the gap i s l ike ly to widen . The jo in t f i l l e r i s l ike ly to

be c om e loose in t he j o in t a nd c om e o u t i f i t is no t r e s t r a ine d . T he se a l a n t c ho se n

f o r suc h a j o in t m us t be c a pa b le o f w i th s t a nd ing the t e ns ion wh ic h ma y de ve lop a t

t he c on t r a c t ion j o in t .

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DETAIL ING

279

I

r

I

!

Infill strip

_ l

- I

Say 1000 mm

(a) Infill or gap strip

Prop

I

~ Dowel

7 ~ , 1 Metal tube

~ . Slip strips

(b) Slotted support

Figure 12.5

Temporary release joints

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280

POST- TENSIONED CO NCRETE FLO ORS

Stressinganchorage Dead-end nchorage

[ lststage ~ / 2ndstage I

Slip strip

(a) Stepped oint

Stressinganchorage Dead-end nchorage

lststage __~~ / 2ndstage ~

Cast-indowel Cast-in ube

This end of dowelgreased

Figure

12.6 Contract ion jo in ts

(b) Dowel oint

Figure 12.6 shows two co n t rac t ion jo in ts w hich may be used in pos t - t ens ione d

floors.

I t is o f ten exped ien t to cons t ruc t the re in fo rced concre te w al ls ahe ad o f the

f loors . The po si t ions of the anc hor age s in relat ion to the wal l need to be careful ly

considered in such cases . The most des i rable i s the anchorage cas t in the wal l ; i t

t hen p rov ides a pos i t ive connec t ion be tween the wal l and the f loor . However ,

o f t en the wal l s a re no t th ick enough to fu l ly accommodate the anchorage , and

space may somet imes no t be ava i lab le ou t s ide the wal l fo r the s t ress ing opera t ion .

A dead anch orag e in a wal l is never a goo d cho ice, because i t requ ires the wh ole o f

the t endon l eng th to be co i l ed and suppor ted near the wal l un t i l t he f loor i s

cons t ruc ted . F igure 12.7 shows the con s t ruc t ion jo in t s a t the floor -wall con tac t .

Detai ls (a) and (b) show the dead ends , and (c) , (d) and (e) the l ive ends . In al l

cases the rod re in fo rcement a t the jo in t m us t be capab le o f cop ing wi th the

tens ion , and any f l exure , which may deve lop .

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DETAILING 281

Precast edge b eam or

~ J previously cast wall

Dead-end ~ h o r a g e

(a) - -Y - \ Starter bars

Precast edge b eam or

previously cast wall

S t ress ing~ ~ ] '1

Starter bars - U -

Stressing Concre te

anchorage cast

later

(b) e /

cast into wall Starter bars

Stressing

anchorage

and d uct cast into wall

(c)

(d)

Stressing

pocket

( e )

Figure

12.7 Wall-f loor construction joints

1 2 .6 De ta i l i n g fo r se i sm i c res i s tan ce

This sect ion out l ines some a ddi t ional detai l ing considera t ions for s labs const ruc ted

in areas of h igh seismici ty . I t was s tated in S ect ion 4 .13 that no special seismic

des ign p rocedures a re genera l ly requ i red fo r f loors . There a re , however , a few

issues relat ing to seismic res is tance which should be borne in mind during the

detai l ing of the f loor .

As was discussed in Sect ion 4 .13, the pr inc ipal ro le of a f loor un der seismic

load ing i s to ac t as a hor izon ta l d iaphragm. In o rder to ensure adequate

d iap hra gm ac t ion a nd p rev en t c rack ing , ho les in the f loor shou ld be f ramed w i th

add i t iona l uns t ressed rod re in fo rcement o r wi th edge beams . However , the

f raming necessary to ensure d iap hrag m ac t ion i s un l ike ly to exceed tha t re qu i red

for the res i s tance o f g rav i ty loads and the anc hora ge o f the t endons .

Cant i lever s labs , being very f lexible , are l ikely to undergo part icular ly large

d i sp lacemen ts unde r se ismic load ing . Bend ing re in fo rcem ent shou ld be p rov ided

in bo th the top and bo t tom faces o f such s l abs , s ince the com bina t ion o f f rame

bend ing and upw ards se ismic acce le ra t ion can cause a reversa l o f the norm al

curv ature . Ec centr ici ty of the pres t ress should be l imi ted so tha t no tens ion is

p rod uced in the absence o f g rav i ty loads.

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282

POST-TENSIONEDCONCRETE FLOORS

TResisting force

Inertia forcel

Slab

\

\

\

\

\

It"

I

Shear wall

Inertia force

/

/

//Possible failure plane in slab

/

J

Drag bars

F i g u r e

12.8

Provision of drag bars between slab and shear wall

In some cases , shear wal l s a re cons t ruc ted which a re on ly par t i a l ly embedded

in the s lab . In such ins tanc es , the r i sk of pul l -o ut o f the wal l un de r seismic loads

can be p reven ted by the p rov i s ion o f d rag bars , as shown in F igure 12 .8 .

Anchorages

"o~"

9

.#-

Strands

~

F i g u r e

12.9

Tendons as column ties

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DETAILING 283

T e n d o n s can b e d e t a i l ed s o as t o ti e co l u m n s i n t o t h e s l ab a s sh o w n i n F i g u re

1 2.9 . T h e d i s ad v a n t ag e o f t h is ap p ro ach is t h a t t h e an ch o rag e s t ak e u p a

co n s i d e rab l e a m o u n t o f s p ace a t t h e co l u m n ' s m o s t co n g es t ed s ec t io n ; F i g u re

1 2.9 sh o w s t w o m o n o s t r a n d a n c h o r a g e s i n a c o l u m n a p p r o x i m a t e l y 3 00 m m w i d e

(12 i n) . H o w ev e r , i f t h e ro o m can b e s p a red , t h e n t h i s m eas u re p ro v i d es a h i g h l y

benef ic ia l l eve l o f s t ruc tu ra l con t inu i ty .

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1 3 S IT E A C T I V I T I E S A N D

D E M O L I T I O N

This cha p ter dea l s wi th those aspec t s o f cons t ruc t ion and s it e ac tiv it ies o f

par t i cu la r re l evance to po s t - t ens ioned f loors . I t dea ls wi th the care o f the

prop r ie t a ry mater i a l s an d the ac t iv i ti es o f the specia li s t sub con t rac to r on a

pro jec t con ta in ing pos t - t ens ioned f loors . I t is no t m ean t fo r the spe c ia l i s t~ tho ug h

he ma y derive som e benefi t f rom i t ; i ts purp ose is to g ive som e ins ight in to the s i te

p rocedures to o ther d i sc ip l ines invo lved in the p ro jec t . Al so , recommendat ions

are g iven fo r the sa fe dem ol i t ion o f pos t - t ens ioned concre te s t ruc tu res .

The sequence o f opera t ions per fo rm ed in the cons t ruc t ion o f a pos t - t ens ioned

bu i ld ing i s g iven be low. The impor tan t opera t ions a re d i scussed under separa te

sections.

1 . Erect soff i t formwork

2 . Erec t edge boards

3 . At tach anchorages to ver t i ca l edge boards

4 . P lace and secure bo t tom re in fo rcement

5 . P lace and secure anchorage zone re in fo rcement

6 . P lace and secure t endons (o r shea ths )

7 . P lace and secure top re in fo rcement

8 . P lace concre te and cure

9 . R emo v e ed g e b o a rd s an d p o ck e t f o rmer s

10. Test concrete cube or cyl inder for s t rength required for s t ress ing

11. Th read bon ded t endons , i f on ly the shea ths w ere assembled in s t age 6

12. St ress tendons

1 3. S t ri p fo rmw o rk an d b ack p ro p if r equ i r ed

1 4 . G ro u t s h ea t h s fo r b o n d ed t en d o n s

15 . Cu t su rp lus s t rand l eng ths

16 . Apply rus t - inh ib i t an t , and p lace g rease caps

17. Fi l l anchorage pockets .

O the r tha n the specialis t pres t ress ing crew, the ma in w orkforce on a post - tens io ned

floor s ite consis ts of three m ajo r categ ories : for m w ork erectors , s teel f ixers an d

concre to rs . For maximum ef f i c i ency , the s i t e opera t ions would be so o rgan ized

t h a t t h e t h r ee c rew s h av e co n t i n u o u s w o rk , w i th n o o r mi n i m u m i dl e ti me . Th e

s it e p ro gra m m e w ould , therefo re , depe nd on a nu m ber o f fac to rs , inc lud ing the

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Table 13 1 A typ ica l programme cyc le

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286 POST-TENSIONED CONCRETE FLOORS

s ize and p lan shape o f the bu ild ing . Ta b le 13 .1 shows a cons t ruc t ion p ro gra m m e

cons i s ting o f a f ive-day cyc le , which m ay be a dap ted to su i t the p ar t i cu la r needs o f

a project .

1 3 . 1 S t o r a g e o f m a t e r ia l s

All o f the specia li zed p res t ressing co m pon en t s shou ld be han d led wi th care and

s to red under cover , awa y f rom chemica l s tha t m ay a t t ack s t ee l, and p ro tec ted

f rom excess ive mois tu re . Par t i cu la r care i s needed to avo id damage to p las t i c

ex t ru s i o n o n u n b o n d ed s t r an d .

The an chor ing wedges and the i r sea t ings in the cas tings shou ld no t be a l lowed

to get rus ty , d i r ty , or greasy. Rust or d i r t on the teeth of the wedges , or the co nical

su r faces o f the wedges and the an chorag es , w eakens the g r ip o f the anc hora ge on

the s t rand . Grea se m ay h ave a s imi la r weak en ing e f fect on the g r ip . Grease on the

conical anchoring surfaces also reduces fr ic t ion, which resul ts in an increase in

the l a t e ra l burs t ing fo rce on the anchorage cas t ing .

The s t rand shou ld be p ro tec ted f rom rus t ing and f rom a t t ack by chemica l s ,

par t i cu la r ly de le te r ious a re ch lo r ides , n i t ra t es , su lph ides , ac ids and hydrogen

re leas ing agen t s . The g rease-packed ex t rus ion a round the s t rand fo r use in

u n b o n d ed t en d o n s p ro v i d es ad equ a t e p ro t ec t i o n . Ca re s h o u l d b e t ak en i n

t ranspor t ing , hand l ing and s to r ing the t endons to avo id damage to the

pro tec t ion . The ex t rus ion can a l so be damaged by v ib ra to rs dur ing concre t ing .

The p las t i c ex t rus ion and g rease a re removed f rom the s t rand a t the anchorage ,

so that the wedges can gr ip i t . The s t rand in th is zone is , therefore, exposed to

poss ib le a t t ack , a nd an unb ond ed t endo n re li es comple te ly on the in tegr i ty o f the

anc hora ge which , o f course , is no t p ro tec te d in the ma nn er o f s t rand . I t is ,

therefo re , par t i cu la r ly impor tan t to ensure tha t the anchorage assembly i s wel l

p ro tec ted by a rus t - inh ib i t an t and a g rease cap , and tha t the anchorage recess i s

p ro p e r l y mad e g o o d .

Pre pare d t endons shou ld be c lear ly m ark ed to iden t ify the loca t ion where they

are to be used.

Ten don s and s t rand a re usua l ly supp l i ed in co il s; ex t reme care is necessary to

ensure that the coi l does not unwind uncontrol lably , as th is i s l ikely to cause

accidents .

M eta l shea th ing fo r use wi th bond ed t endon s i s usua l ly no t very s tu rdy , an d i s

p ro n e t o mech an i ca l d am ag e . A n y h o le s o r t e a r s may a ll o w i ng res s o f m o r t a r

pas te dur ing concre t ing w hich would h inder the s tres sing o f the t endons .

M ul t ip le hand l ing o f the shea th increases the r isk o f dam age . Op en ends o f the

shea th shou ld be p ro tec ted to avo id any mater i a l , ra inwater , o r water f rom the

s i te f inding it s way in . Wa ter in the she ath is l ikely to ren der the gr ou t ineffect ive

and cause rus t ing o f the t endon , a nd a ny de le te r ious ma t t e r wo uld a l so a t t ack the

t en d o n .

Anchorages , t endons and shea ths shou ld be s to red in the o rder in which they

are to be used . Th is p roced ure reduces mul t ip le hand l ing and the r isk o f dam age .

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SITE ACTIV IT IES AND DEMOLIT ION 287

All equ ipment shou ld be s to red in a c l ean and d ry loca t ion , and i t shou ld be

access ible on ly to the opera t ives . M ain ten ance and repa i r , which ma y af fec t the

ca l ib ra t ion o f the equ ipm ent , shou ld no t be a l lowed o n s it e.

1 3 . 2 I n s t a l l a t i o n

To avo id sp l i t respons ib i l i ty and poss ib le de lays , the same crew shou ld be

respons ib le fo r the assembly and ins ta l l a tion o f rod re in fo rcement , t endo ns and

ancho rages . T he ins ta ll e rs, the s t ress ing opera t ives a nd the personnel respons ib le

fo r t h e f i n i s h i n g o p e ra t i o n s - -g ro u t i n g , cu t t i n g s t r an d , mak i n g g o o d t h e

p o ck e t s - - s h o u l d b e co n t ro l l ed b y o n e s u p e rv i s o r .

The ins ta l l e rs shou ld s tudy the shop d rawings and o ther re l evan t documents

before s t a r t ing on s i te . They shou ld w ork ou t a sequence o f opera t ions , a l lowing

for the incorpo ra t ion o f any no n-s t ruc tu ra l e l ements in the s lab , such as condu i t s

or service runs .

Holes in ver t i ca l edge boards , fo r the s t rand to p ro jec t th rough , shou ld be

dr i l l ed accura te ly so tha t the l ive anchorage can be a t t ached in the spec i f i ed

pos i t ion . T he l ive anc hora ge shou ld be se t square to the board , un less o therwise

spec if ied . I t shou ld be f i rmly a t t ach ed to avo id be ing d i s lodged d ur ing concre t ing .

No m o r t a r p a s t e s h o u ld b e a ll o w ed t o r each an y o f t h e an ch o r i n g s u rf aces . D ead

anch orage s shou ld be secure ly suppor te d on cha i r s , wi th the requ i red end cover ;

they mus t no t be a t t ach ed to the ver t ica l edge boar d o f the fo rmw ork .

Poc ket fo rmers shou ld be rig id , they shou ld f i t i n to the anchor age wi tho u t any

gaps , and shou ld be f i rmly a t t ached so as no t to be dam age d o r d i sp laced dur ing

concre t ing . They shou ld be se t t rue in pos i t ion . Expanded po lys ty rene fo rmers

are of ten di ff icul t to remove. Burning or cleaning wi th chemicals leaves the

sur face o f the pocke t in too p oo r a s t a t e fo r the concre te p lug to m ake a good seal .

Anchorages shou ld remain access ib le fo r subsequen t opera t ions , unobs t ruc ted

by scaf fo ld ing o r o ther cons t ruc t ion .

Fo r b o n d ed t en d o n s , t h e s t r an d may b e t h r ead ed i n t h e s h ea t h b e fo re t h e

shea th i s l a id in pos i tion . Thre ad in g o f the s t rand a f t e r the shea ths have been

p laced in pos i t ion may cause them to be d i sp laced . The opera t ion a l so requ i res

access to the sheath end, which may not be avai lable on a bui ld ing s i te ,

par t icular ly at the upper f loor levels .

Ten don s m ay a l so be th reade d a f t er concre t ing . In th i s case it is im po r tan t to

prove tha t the duc t i s unobs t ruc ted by d rawing a do l ly th rough i t .

I t is im po r tan t tha t jo in t s be tween l eng ths o f shea th ing a re water t igh t . I f wa ter

ge ts in then i t may cause cor ro s ion o f the t endo n and , worse , if cem ent g ro u t ge t s

i n d u r i n g co n c re t i n g t h en t h e t en d o n w o u l d b eco me b o n d ed an d s t r e s s i n g may

not be poss ib le . Su bseque n t rem edy can on ly cons is t o f cu t t ing ou t the dam age d

por t ion and re -concre t ing which can be t ime consuming and expens ive .

M ak e sure that the requis i te length of s t rand is avai lable for jacking at the l ive end.

Af te r l ay ing the u nbo nde d t endon s in pos it ion , a l eng th o f the p las ti c ex t rus ion

shou ld be remov ed a t the l ive end and the s t rand inser t ed th roug h the anch orag e

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288 POST-TENSIONEDCONCRETE FLOORS

c a s ting . N o m or e tha n 25 m m ( one inc h ) o f ba r e s t r a nd shou ld be e xpose d be h ind

the a nc ho r a ge . A wa te r t i gh t s l ee ve shou ld be p l a ce d on th is l e ng th , so t ha t non e

o f t he ba r e s t r a nd r e ma ins e xpose d be h ind the a nc ho r a ge ; t h i s is r e q u i r e d to

p r e v e n t b o n d .

T e ndons shou ld be suppor t e d on p r ope r ly de s igne d c ha i r s , o r on r od

re inforcement , and secured in pos i t ion a t the spec i f ied he ights so tha t none ge t

d i sp l ac e d du r ing conc r e t ing . T he t e ndo n sup por t s a r e no r m a l ly spa c e d a bou t one

me t r e ( 40 in ) a pa r t . C a r e sh ou ld be t a ke n to e nsu re t ha t t he supp or t a r r a n ge m e n t

i s s t a b le a nd tha t t he t ie s a r e no t t oo t i gh t a r o un d a n un bon de d t e nd on ; e ls e t he

p la s t i c e x t r u s ion ma y ge t da ma ge d .

H o r i zon ta l de v ia t ion o f t he t e ndons f r om the in t e nde d pos i t i on s hou ld be

minimized as the wobble causes a loss in the pres t re ss ing force . This i s

pa r t i c u l a r ly im po r t a n t i n r i bbe d a n d w a ff le f l oo rs , whe re t he ho r i zon ta l de v ia t ion

of a tend on m ay se t up h igh la te ra l tens ile s tr e sses in the th in w eb. This m ay cause

loc al bu r s t i ng o f t he c onc r e te i f a de q u a te r e in f o rc e me n t ha s no t be e n p r ov ide d in

the a nc ho r a ge zone . T h e ho r i zon ta l wob b le i n a r ibbe d , o r wa f fl e f l oo r, ma y be

minimized by secure ly ty ing the tendons to a r e inforcement cage in the r ib .

T o e nsu r e t h a t a t e nd on de ve lops t he e xpec t e d p r e s tr e s s ing f o rc e , a nd wo r ks a s

envisaged in the des ign , i t i s e ssen t ia l tha t i t s h igh and low poin ts a re p laced

accura te ly . Ver t ical dev ia t io n of the tend on f rom i ts in tend ed prof i le be tw een the

h igh a nd the l ow po in t s shou ld be ke p t w i th in t he spec if ie d to l e ra nc e . I f no t

spec i f ied , the to le rances in Table 13 .2 may be used .

13.3 Concre t ing

T he c o nc r e t ing op e r a t ion f o r a pos t - t e ns ione d f loo r is ver y s imi l a r t o t ha t f o r a

re inforced concre te f loor ; the co ncre tor m us t be aware o f the fo l lowing"

9 S h e a th s f o r b o n d e d t e n d o n s a n d t h e p la s ti c e x tr u s io n o n u n b o n d e d t e n d o n s

c a n be e a s i l y da ma ge d du r ing c onc r e t ing , t h i s wou ld c a use p r ob le ms in

s t r e s s ing a nd g r ou t ing ope r a t ions .

9 A nc hor a ge s a nd t e ndons mus t no t ge t d i sp l a c e d du r ing c onc r e t ing . V ib r a to r s

a nd pump p ipe l ine shou ld no t be a l l owe d to c on ta c t t he t e ndons .

9 I t is e s sen t i al t ha t t he c onc r e t e imm e d ia t e ly be h ind a nd a r ou nd the a nc ho r a ge s

is we l l compac ted .

9 T he e dge boa r ds shou ld be r e mov e d a s soo n a s pos s ib le a f te r t he c onc r e te ha s

Table 13.2

Ver tical tolerances for tendon profile

Depth Tolerance

U p to 2 0 0 m m (u p t o 8 i n )

200-600 mm (8-24 in)

over 600 mm (ov er24 in)

+ 6 mm (+ 88 n)

+ 10 mm (+ 3 in)

+ 12 mm (+ 89 n)

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SITE ACTIVITIES AND DEMO LITION 289

ha rde ne d, bu t whi le i t i s s ti ll green. This al lows the p ock et form ers to be eas ily

removed , and the pocke t s to be inspec ted and any defec t s to be made good .

9 Th e co n c re t e s h o u l d b e p ro t ec t ed ag a i n s t d am ag e f ro m t h e en v i ro n m en t an d

s i t e opera t ions , and cured so as to avo id c racks f rom any cause inc lud ing

shr inkage .

Fo r po s t - t ens ioned f loors , it is the norm al p rac t ice to t es t the concre te s t reng th a t

abo u t th ree days , when some o r a ll o f the p res tress is app li ed . C oncre te samples

should be taken and cured as speci f ied . Typical s t ress ing sys tems al low 50%

prest ress to be a ppl ied at a cub e s t reng th of 15 N /m m 2 and full pres t ress ing at

2 5 N /m m 2 (1750 an d 3000 psi respect ively) . This m ust be veri fied for the

par t i cu la r sys tem used .

13.4 Stressing

St ress ing o f the t endons is the cen t ra l and m os t im por tan t ope ra t ion o f the

post - ten s ionin g proc ess . I t a lso carr ies an element of r i sk , in that po tent ial ly

dangerous s i tua t ions may ar i se when th ings go wrong . For these reasons , the

subject i s d iscussed in some detai l .

In co ld weather , the norm al g rease used in unbo nde d s t rand m ay be too s t if f fo r

the speci fied extension to be achieved , i t m ay be necessa ry to d elay full s tress ing

un t i l cond i t ions a re w arm er . I f co ld cond i t ions a re an t i c ipa ted then i t m ay be

poss ible for the s t rand suppl ier to provide a more viscous grease.

In ho t w eather , the reduced f r ic t ion resu l t ing f rom the more v i scous g rease m ay

resul t in h igher extensions and tendon forces .

13.4 . 1 Sa fety

All equ ipment mus t be checked to ensure i t s p roper opera t ion and ca l ib ra t ion .

Hy drau l i c hoses an d c onnec t ions shou ld be checked fo r signs o f l eakage .

Stress ing the tendons involves s t rain ing the tendons wi th high forces ; the

o p e ra t i o n ca r ri e s s ome r is k o f acc id en t. Th e re fo re , p ro t ec ti v e m ea n s - -b o a rd s o r

san db ag s - - sho u ld be p laced in line wi th the ancho rage to a r res t the p ro jec ti les in

case of an accident . O nly t rained personnel shou ld be al lowed in the operat ions area.

I t is adv i sab le no t to s t and near a j ack o r the pum p dur ing s t res s ing , o r in f ron t

o f an a nch orag e du r ing o r a f te r s tres sing , un ti l the pock e t has been m ade good .

Hy drau l i c j acks shou ld be t e thered du r ing s t res s ing to p reve n t the i r fa ll ing

dow n, po ss ibly off the f loor being s t ressed o n to the g rou nd below, in case of a

t en d o n b reak i n g .

Before s t res s ing , the concre te in the pocke t a nd near the a nchora ges shou ld be

inspec ted fo r any s igns o f weakness , such as c racks o r hone ycom bing . I f any such

defect i s foun d, or a ny of the project ing tend ons are not a t r ight angles to the face

of the ancho rage , then s t res s ing m us t no t p roceed . The defec t shou ld be inspec ted

by the eng ineer and the remed ia l works car r i ed ou t , i f necessary .

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2 9 0 PO ST - ENSIONED CONCRETE FLOORS

T he we dge se a t ings i n t he a nc ho r a ge s shou ld be in spe c t e d , a nd c l e a ne d i f

necessa ry . They sho uld be free of rus t , g rease , o i l, d i r t and o the r c on tam ina nts .

I n c ase o f a p r ob le m , t he c onc re t e a r o un d a n a n c ho r a g e ha s t o be c u t ou t . I n

cu t t ing the concre te , a t ten t ion must be g iven to i t s e f fec t on the ad jacent

anch orag es i f they a re s t re ssed . I t may be sa fe r to de - tens io n them .

Don ts of s t ress ing

9 Do no t s t re s s a ny a nc ho r a ge s w i th g r ou t i n side the c a s ti ng . Gr ou t i n the

cas t ing m ay p reven t pr op er sea t ing of the wedges . I t i s sa fe r , and less expens ive ,

to c l e a n ou t g r ou t t ha n to ha ve to de - t e ns ion , r e pa i r o r r e p l a c e t e ndons , o r

repa i r the j ack .

9 D o no t u se t he j a c k whe n i t doe s no t s e a t p r ope r ly i n to t he c a s t i ng .

9 D o no t ove r s tr e s s t e ndons to a c h ie ve p r ope r e longa t ion .

9 D o no t a l low o bs t r uc t ion s i n t he pa th o f t he j a c k e x t e ns ion .

9 D o no t c on t inue s t re s s ing i f som e th ing i s no t wo r k ing p r op e r ly .

9 D o no t de - t e ns ion w i th l oose p l a t es , spa c ing sh ims o r w i th two ja c ks i n t a nd e m .

9 Do no t s t a nd ne a r t he j a c k , be twe e n the j a c k a n d pu m p , o r ove r the a nc ho r a ge

du r ing s t r e s s ing o r de - t e ns ion ing .

9 D o no t ha m m e r o r b e a t on the j a c k o r j a c k c y l inde r s.

9 La s t ly , do n o t do a ny th ing i f you a r e no t su r e ; a sk som e one w ho kn ow s .

13 .4 .2 S t ress ing p roc edu re and m eas urem ent o f f o rce

The tendons should be s t r e ssed in the orde r agreed wi th the des igner . S t re ss ing

should take p lace as soon as poss ib le a f te r the concre te has reached the requi red

s t r e ng th .

F o r simultaneous s tr es s in g o f t e n d o n s f r o m b o t h e n d s , g o o d c o m m u n i c a t i o n

a n d c o o r d i n a t i o n b e t w e e n t h e t w o t e a m s a r e e x t r e m e l y i m p o r t a n t i n e n s u r i n g

s imu l t a ne i ty o f ope r a t ions . S im u l t a ne ous s t re s s ing , howe ve r , i s s e ldom use d .

T he t e n don f o rc e is r e a d f r om a c a l ib r a t e d p r e s su r e ga uge , a nd i s m on i to r e d b y

obse r v ing the t e ndo n e lon ga t ion . A p r e s su r e c el l o r a p r ov ing r ing ma y be u se d

f o r a m or e a c c u r a t e a nd d i r e ct me a su r e m e n t o f t he fo r ce .

T e ndon e longa t ion i s me a su r e d , a nd r e c o r de d , a t t he s a me t ime a s s t r e s s ing

ta ke s p l ac e . E lon ga t ion is me a su r e d to a n a c c u r a c y o f 2 m m ( to t he ne a r e s t ~ i n ).

B e fo r e s t re s s ing , the t e ndon s ha ve a n u nk no w n a m ou n t o f s l ac k . In m u l t i s t r a nd

tendons , each s t r and may have a d i f fe ren t s lack , bu t the d i f fe rence cannot be

a l lowed for i f a l l s t r and s a re s t r e ssed toge th e r . T o a l low for the s lack , the

fo l lowing procedure i s used in s t r e ss ing .

9 The s t r an d i s g r ippe d an d s t r e ssed to ab ou t 10% of the j ack i ng force.

9 T he s t r a nd is m a r ke d a s e t d i s t a nc e a w a y f r om the a nc h o r a ge f ac e u s ing a

re fe rence m easu r ing device (which ma y be a p iece of wo od) , usu a l ly by a pa in t

spray .

9 The ten do n i s s tr e ssed but n o t locked . T he force i s r ecorded . Th e re ference

me a su r ing de v ic e is p la c e d a ga in s t t he fa ce o f t he a nc ho r a ge , t he e long a t ion o f

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S T R E S S I N G R E C O R D

J o b n a m e

L o c a t i o n

Floor leve l

E q u i p m e n t

R e m a r k s

SITE ACTIVITIES AND DEM OLITION

291

Z o n e

D a t e

J o b N o .

Sheet

S t ressed by

Verif ied by

o f ~

T e n d o n

Reference

Specif ied

J a c k i n g E l o n g a t i o n

force

M e a s u r e d

F o r c e

In i t ia l Jack ing

E l o n g a t i o n

J a c k i n g D r a w - i n

Figure 13.1

Stressing record sheet

t he t e ndon i s m e a su r e d f r om the e nd o f t he dev ice t o t he pa in t sp r a y m a r k , a nd

recorded .

9 The te nd on i s locked , the re fe rence meas ur ing device is aga in p laced ag a ins t the

anc hora ge face , and the fina l e lon ga t io n me asured . The d i f fe rence be tween th is

a n d the p r e v ious m e a su r e m e n t is the we dge d r a w- in , w h ic h is a lso r e c o r de d .

F igu re 13 .1 show s the form at of a shee t for record ing the s t r e ss ing da ta .

M os t o f t he j a c ks f o r use w i th pos t - t e ns ione d f loo rs ha ve a r a m m ove m e n t o f

the o r de r o f 200 to 300 m m (8 to 12 in) . F o r l ong t e ndo ns , t he r e q u i r e d e lo nga t io n

m a y be mor e th a n the r a m s t r oke , a nd i t m a y ha ve to be s tr e sse d in two o r mo r e

s t rokes . Af te r the f ir s t s t roke , the ten do n i s anc ho red , the j ack re t rac ted , the

tendon re -gr ipped , and re - s t r e ssed .

1 3 .4 .3 S h o r t m e m b e r s

The re la t ive ly h igh loss in pres t re ss f rom draw- in , coupled wi th the unce r ta in ty

over the am ou nt of s t r an d s l ip , m akes i t d i ff icu l t to s t r e ss sho r t ten don s . A use ful

p r oc e du r e t o f o l low in s t r e s s ing sho r t t e ndons i s :

9 M ake a generou s a l low ance for draw - in in the des ign .

9 F o r t he f ir st fe w t e ndons , u se the m a x im um j a c k in g f o r ce ; mos t s t a n da r ds

a l low 80 % o f t he s t r a nd s t r e ng th .

9 M e a su r e t he a c tua l d r a w - in f o r e a c h t e ndo n .

9 C a lc u la t e t he f o rc e in e a c h s t r a nd f r om the m e a su r e d d r a w- in

9 Adjus t the j ack in g force for the nex t ten do n as necessa ry .

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292

POST-TENSIONEDCONCRETE FLOORS

The p ro cedu re i s s low, and requ i res acc ura te m easu rem ent o f fo rces and

ex tens ions , and know ledge o f the design . Bar sys tems are o f t en p refer red fo r sho r t

l eng ths , because they use a th readed anchor which has no d raw- in .

13.4.4 Stressing problems

Prob lems may ar i se dur ing the s t res s ing opera t ion main ly f rom two bas ic

defi c ienc ies : a poor con cre t ing op era t ion and mal fun c t ion ing o f the equ ipm ent o r

wedges .

The process of rect i fy ing the defects may require the force in a previously

s t ressed t endon to be conf i rmed . Th i s i s done , i f the p ro jec t ing s t rands have no t

been cut off, by gr ippin g the te nd on in a jac k a nd re-s t ress ing unt i l the wedges jus t

l if t off thei r seats . W hen the wedg es are in i t ia lly locked, p ar t of the te nd on force is

suppor ted by f r i c t ion be tween the wedges and the con ica l sea t ing . The fo rce

requ i red to l i f t t he wedges mus t overcome the res idua l t endon fo rce and the

fr iction. The refore, the jack ing force in th is ope rat io n is in i tia l ly h igh, an d i t drop s

when the wedges become f ree . The lower va lue represen t s the t endon fo rce .

The fo rce requ i red to f ree the wedges may no t be ach ievab le in a t endo n where

the o r ig ina l j ack ing fo rce was 80 % of the t endo n s t reng th .

Repeated re-seat ing of the wedges may cause thei r teeth to fai l .

Imp roper elongation

An y d i s c r ep an cy b e t w een t h e me as u red an d t h e ca lcu l a ted e l o n g a t i o n ex ceed in g

7% , o r as spec if ied , shou ld be no ted and repo r ted to the des igner . He m ay requ i re

some rem edia l measures to be t aken . I f the measu red e long at ion c ons i s t en tly

var i es f rom the ca lcu la ted va lue , then s t res s ing mus t be s topped , the reason

inves t iga ted , and s t eps t aken to rec t i fy the p rob lem.

A smal le r e longa t ion than expec ted sugges ts the poss ib i l ity o f the t end on

h av i n g b een i n ad v e r t en t l y b o n d ed . Th e b o n d may b e b ro k en b y r ep ea t ed

s t ress ing and de- t ens ion ing o f the t endon . Ca re shou ld , how ever , be t aken no t to

damage the t endon in the p rocess .

I f the bond can no t be b roken , then i t s loca t ion shou ld be assessed f rom the

observed extension. Several poss ibi l i t ies exis t in th is s i tuat ion.

9 I f the rem ainde r o f the t end on l eng th can be accep ted in it s uns t ressed s t a t e ,

then the t end on can be locked wi th on ly the par t l eng th s tres sed . I t shou ld be

borne in mind tha t , i n the even t o f the bon d b reak ing l a t e r , t he t endon fo rce wi ll

reduce to the p ro por t ion o f s t res sed to to ta l l eng th .

Accep tance o f a lower p res t ress ing fo rce in a t endon m ay imply p lac ing a

h igher re l i ance on the ad jacen t t endons , and s t res s ing them to a h igher l evel if

poss ible .

9 If th is i s no t ac ceptab le, an d the oth er en d o f the ten do n is access ible , then i t

m ay be p oss ible to s t ress from b oth ends . In th is case there i s no loss in the force.

9 I f ne i ther o f the abo ve i s accep tab le , then the t endo n mu s t be exp osed by

cu t t ing ou t the concre te and the shea th a t the b lockage . The t en don can then be

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S I TE A C T I V IT I E S A N D D E M O L I T I O N

2 93

f reed an d s t r e s s ed , t h e cu t co n c r e t e m ad e g o o d , an d t h e s h ea t h g r o u t e d i f t h e

t e n d o n is of t h e b o n d e d t y p e .

A n e l o n g a t i o n l a r g e r t h an ex p ec t ed i n d i ca t e s a s l i p p ag e a t t h e an ch o r ag e , o r a

b l o w o u t . T h e t w o a r e d e a lt w i t h b e lo w i n s e p a r a te s e ct io n s. T h e e l o n g a t i o n m a y

a l s o b e l a r g e i f t h e t en d o n i s o v e r s t r e s s ed ; s ee ' T en d o n b r e ak a g e ' b e l o w .

Wedg e sl ippage

I n ca s e o f ex cess i v e d r aw - i n a t t h e l iv e en d , t h e t en d o n s h o u l d b e d e - t en s i o n ed , t h e

w e d g e s r e p l a c e d , a n d t h e t e n d o n r e - s t r e s s e d . U n d e r n o c i r c u m s t a n c e s s h o u l d

mo r e t h an o n e s e t o f w ed g es an d b a r r e l s b e u s ed o n a s t r an d i n t r y i n g t o r e s t r ic t

t h e s l i p p ag e . A d e f ec t i v e w ed g e a s s emb l y s h o u l d a l w ay s b e r ep l aced .

Th e p ro b l e m i s m ore d i f fi cu l t t o r ec t ify i f i t occu r s on t he seco nd , o r

s u b s eq u en t , s t r o k e o f s t r e s si n g a l o n g t en d o n . I n t h is c i r cu m s t an ce ,

ne ve r u se a

s e c o n d j a c k o n t h e b a c k o f th e f i r s t o n e . T h i s i s a p o t en t i a l l y d an g e r o u s s i t u a t i o n

an d s p ec ia l eq u i p m en t o r p r o ced u r e s a r e ca l led f or , su ch a s us e o f d e - t en s i o n i n g

d ev i ce s c - d e - t en s i o n i n g a t t h e f a r en d .

La r g e r t h an ex p ec t ed e l o n g a t i o n may a l s o o ccu r d u e t o s l i p a t t h e d ead en d . I t

i s r a t h e r u n l i k e l y a t a b o n d ed d ead en d ; t h e m o r e l i ke l y i s t h e s l i p p ag e o f s t r an d i n

a p r e - l o ck ed an ch o r ag e . I n e i t h e r c a s e ,

9 If t here i s su f fi c i en t l a t i t u de i n the ad ja cen t t e ndo ns t o m ak e u p t he def i c i ency ,

t h e n t h e d a m a g e d t e n d o n m a y b e a b a n d o n e d a n d t h e a d j a c e n t t e n d o n s

s t r essed t o a h igher l eve l .

9 I f t h e t en d o n c an n o t b e ab an d o n ed , t h en t h e r e i s n o a l t e r n a t i v e b u t t o cu t t h e

c o n c r e t e o u t n e a r t h e d e a d e n d , t h r e a d a n e w t e n d o n t h r o u g h i f n e c e s s a ry ,

p r o v id e a n e w d e ad e n d a n c h o r a g e - - e i t h e r b o n d e d o r p r e - l o c k e d - - a n d s tr es s

t h e t en d o n .

Blowout

U n e x p e c t e d l y l a rg e e l o n g a t i o n a p p e a r s t o o c c u r i f t h e c o n c r e te , n e a r a p r e - l o c k e d

o r a l i v e an ch o r ag e , i s n o t s t r o n g en o u g h f o r t h e s t r e s s e s i n d u ced . I n ad eq u a t e

r e i n f o r c e m e n t i n t h e a n c h o r a g e z o n e m a y a l s o g i v e t h e s a m e i m p r e s s i o n . T h i s

c o n d i t i o n i s u s u a l l y a c c o m p a n i e d b y c r a c k s n e a r t h e d e a d e n d a n d s i g n s o f

m o v e m e n t o f t h e a n c h o r a g e a s s e m b l y ; it c a n l e a d to a b l o w o u t .

A b l o w o u t i s f a i lu r e o f co n c r e t e i n t h e an c h o r a g e z o n e d u r i n g s t re s s in g . I t i s

p o t en t i a l l y a d an g e r o u s p r o b l em a s t h e f a i l u r e may b e ex p l o s i v e . P o ck e t s o f

a g g r e g a t e , s a n d o r v o i d s i n t h e c o n c r e t e a r o u n d t h e a n c h o r a g e , a n d l a c k o f

a n c h o r a g e z o n e r e i n fo r c e m e n t a r e t h e m o s t c o m m o n c a u s es o f a b l o w o u t .

T h e p o o r c o n c r e t e m u s t b e c u t o u t , m a d e g o o d w i t h c e m e n t - s a n d m o r t a r

h av i n g a s t r en g t h n o t l e s s t h an t h e co n c r e t e , an d t h e t en d o n s t r e s s ed .

Tendon breakag e

T e n d o n b r e a k a g e c a n o c c u r f ro m m i s a l i g n m e n t o f w e d g e s , o v e r s tr e s s in g o r

i n t e r n a l d a m a g e t o t he t e n d o n . I t is i m p o r t a n t t o d e t e r m i n e t h e c a us e o f t h e

b r e a k a g e b e f o r e r e p l a c i n g t h e b r o k e n t e n d o n .

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2 9 4 POST-TENSIONED CONCRETE FLOORS

M isa l ign m ent of wedges occurs wh en w edges a re off se t p r io r to s t r e ss ing , in

wh ic h c a se t he y c a n p inc h one o r m or e o f t he w i r es o f t he s t r a nd . I f t he s t r a nd ha s

be e n s t r e s se d a nd the we dge s a r e ho ld ing the p r e s t r e s s t he n the t e ndon ma y be

accepted a t the d isc re t ion of the enginee r .

O ve r s t r e s s ing o f a t e nd on c a n o c c u r i f t he e q u ipm e n t i s no t p r op e r ly c a l ib r a t e d ,

or i f the d ia l gauge o n the s t r e ss ing p um p is mis read . I t is safe r to acce pt an

ove r s t re s se d t e ndo n i f t he we dge s ar e ho ld ing ; a n a t t e m p t a t de - t e ns ion ing m a y

br e a k the t e ndon .

D am ag e to a s t r an d m ay be the re su l t o f lack of ca re in si te ope ra t io ns . Care less

ha nd l ing o r s to r a ge ma y c a use p inc h ing o f t he s t r a nd loc a lly . D a m a ge m a y r e su l t

f r om im pr op e r u se o f c onc r e t e v ib r a to r s , a nd du r ing c u t t i ng the conc r e t e ne x t t o a

tendon. Care less dr i l l ing in ins ta l l ing f ix ings can a lso damage the s t r and .

13.5 Grouting

T he g r ou t f o r in j e ct ing she a th ing h ous ing the bo nde d t e n don s u sua l ly c ons i st s o f

a ne a t c e me n t pa s t e , o f t en c on ta in ing a p la s t ic i ze r , a r e ta r de r a nd a n e xp a nd in g

a dm ix tu r e . A sm a l l q ua n t i t y o f f ine s a nd is some t ime s inc lude d to r e duc e the

a m ou n t o f c e me n t , bu t t h i s is mo r e c o m m on in l a rge duc t s , wh ic h a r e un l ike ly t o

be used in a bu i ld ing . The p las t ic ize r a l lows a smal le r quan t i ty o f w a te r to be used ,

as a r e su l t a le sser quant i ty of wa te r r em ains un used by the chemica l r eac t ion in

the she a th . T he a ime d se t ti ng t ime shou ld be su ff ic ie n t t o c om ple t e t he g r ou t ing o f

a t e ndon a nd a l low f or pos s ib le mi sha ps , suc h a s b r e a k dow n o f e q u ip m e n t o r

b loc ka ge o f a she a th . A d mix tu r e s c o n ta in ing c h lo r ide s , f luo r ide s , su lph i te s a n d

n i t r a t e s s h o u l d n o t b e u s e d . E x p a n d i n g a d m i x t u r e s c o n t a i n i n g a l u m i n i u m a r e

be s t a vo ide d , be c a use the y li be r at e hyd r oge n , w h ic h ma y c a use e mb r i t t l e me n t o f

the s t r and .

T he g r ou t i s no r ma l ly de s igne d to p r oduc e a 28 - da y c ube s t r e ng th o f

35 N /m m 2 (4500 ps i) , and a seven day s t r ength o f no t le ss than 20 N /m m 2 (2500 ps i) .

Mix ing i s c a r r i e d ou t f o r a bou t two minu te s i n a spe c i a l l y de s igne d mixe r ;

m od ern mixers need a min ute or le ss to de l ive r a goo d mix . M ixing t ime i s c r i tica l

f o r ob t a in in g a go od un i f o r m m ix tu r e w i th t he de s i re d s e t t ing t ime a nd e a sy f low

charac te r i s t ic s . Too long a mix ing t ime resu l t s in unnecessa ry h ea t ing of the

g r ou t , wh ic h c ha nge s it s p r ope r t i e s . T oo sho r t a t ime ma y l ea ve lump s o f unm ixe d

c e me n t , wh ic h c a n c a use b loc ka ge in t he pump l i ne o r t he she a th ing .

W a te r shou ld no t be a l lowe d to r e m a in in t he she a th . I f a she a th i s to be f l u she d

f o r some r e a son the n u se o f l ime w a te r shou ld be c ons ide r e d be c a use o f i ts

a lka l in i ty . F lu sh ing shou ld be f o l lowe d by c ompr e s se d a i r b l a s t t o r e move the

wa te r a nd d r y the she a th .

T he t e n don s a r e no r m a l ly g r ou te d a s soon a f t er s tr e s sing as pos sib l e. H o we ve r ,

gro ut in g sho uld be avoide d in freezing wea the r ; i f the gro ut f reezes , i t s exp ans ion

m a y r up tu r e t he she a th a nd c a use c r a c k ing o f t he c ove r c onc r e te . G r ou t ing a t l ow

te m pe r a tu r e s shou ld be c a r r ie d ou t on ly i f t he me m be r i s f ir st he a t e d , a n d i ts

t e m pe r a tu r e c a n be ma in t a ine d a bove f r e ez ing fo r a t le a s t two da ys . T h i s a l l ows

the grout suf f ic ien t t ime to ha rden .

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S I TE A C T I V IT I E S A N D D E M O L I T I O N 295

G ro u t i n g is c a r r i ed o u t b y p u m p i n g t h e mi x u n d e r p r e s s u re , o f ab o u t

0 .5 N /m m a (75 ps i) , t h roug h the in let s p rov ided . G rou t is in jec ted a t low po in t s in

a t endo n p ro fi le . I t shou ld no t be a l lowed to f low dow n f rom the h igh po in t s in the

profi le as th is i s l ikely to leave ai r pockets in the sheath .

Progress o f g rou t ing is m oni to red a t the mon i to r ing tubes , cas t in the m em ber

a long the l eng th o f the shea th , usua l ly a t h igh p o in t s o f the p ro f il e . W hen the

g ro u t r each es a mo n i t o r i n g t u b e an d emerg es w i t h o u t a i r p o ck e t s an d u n d e r

pressure , the par t i cu la r moni to r ing tube i s capped . Wi thou t the moni to r ing

tubes , i t i s d i fficult to ensure tha t the wh ole of the she ath h as been p rope rly

grou ted . A usefu l check is to com pare the vo lume of g rou t pum ped wi th the

theore t i ca l ly requ i red vo lume.

In ca se o f a b r eak d o w n o r s t o p p ag e o f th e equ i p me n t d u r i n g g ro u t i n g , o r a

b lockage in the shea th , the ex ten t o f shea th rem ain ing ung rou ted shou ld be

de te rm ined by carefu lly d r i ll ing ho les th ro ugh the shea th so as no t to dam age the

tendon . These ho les can be used for in jec ting g rou t a nd m oni to r in g i t s p rogress .

Access to a spare grout ing set i s a useful safeguard agains t poss ible problems

ar i s ing ou t o f the b re akd ow n of the equ ipm ent in use.

13.6 F in ish ing ope rat ions

On sa t i sfac to ry comp le t ion o f the s tres s ing and g rou t ing opera t ions , su rp lus

leng ths o f the s t rands a re cu t o f f us ing a d isk cu t t e r , usua l ly wi th in ab ou t 25 mm

(an inch) o f the wedge faces. Ge nera t ion o f too m uch hea t , such as m ay occur i f a

f l ame cu t t e r is used , m ay ra ise the t em pera tu re o f the ancho rage assembly an d the

s t rand and can impai r the anchorage e f f i c i ency .

All exposed su r faces o f the anc hora ge assembly a re then g iven a rus t - inh ib i t an t

sp ray app l i ca t ion . W hen d ry , the ends o f the s t rand and the wedges in an

anchorage a re covered wi th a g rease f i l l ed cap . The pocke t i s then made good

w i t h a cemen t - s an d mo r t a r co n t a i n i n g a s u i t ab l e ex p an d i n g ag en t .

13.7 Demol i t ion

A rev iew of recen t research in to po ten t i a l dem ol i t ion p rob lem s shows tha t the

add i t iona l r i sk assoc ia ted wi th dem ol i t ion o f pos t - t ens ioned s t ruc tu res in

compar i son wi th re in fo rced concre te s t ruc tu res i s very smal l . The p recau t ions

needed a nd the cho ice o f su i tab le dem ol i t ion p ro cedure s a re d i scussed ; for the

mo s t par t these requ i re on ly minor m odi f i ca t ions o f the metho ds used fo r

re in fo rced concre te . In a dd i t ion , cons idera t ions fo r the cu t t ing o f ho les in ex i st ing

st ructures are d iscussed.

Demol i t ion methods fo r pos t - t ens ioned s t ruc tu res a re l a rge ly s imi la r to those

fo r re in fo rced concre te . The main d i f fe rences can be summar ized as fo l lows :

9 S ince the p res t ressing t en dons a re m ade o f ex t remely toug h , h igh-s t reng th

s teel , they are d i ff icul t to sever . However , once the tendons have been cut or

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2 9 6

POST-TENSIONEDCONCRETE FLOORS

otherwise de-tens ioned, post - tens ioned elements wi l l of ten be eas ier to

dem ol i sh th an re in fo rced concre te ones , as the to ta l am ou nt o f s tee l they

contain is less.

9 Loss o f p res tress due to the cu t t ing o f t endon s can c rea te a r i sk o f p rem atur e

co l lapse o f par t s o f the s t ruc tu re . Th i s m ay requ i re s ligh tly more subs tan t i a l

p rop p ing than would be used dur ing de mo l i t ion o f a re in fo rced s lab .

9 W hen a pos t - t ens ioned t endo n is severed , there i s a sudden re lease o f s to red

s t ra in energy . However , as wi l l be shown be low, in pos t - t ens ioned s l abs the

effec ts o f th i s energy re lease a re u sua l ly min im al and requ i re on ly very min or

changes to conven t iona l demol i t ion p rocedures . Trans fer beams have a much

h igher l eve l o f s to red energy an d they need care in demo l i tion .

Un t i l qu i te recen t ly , very few pos t - tens ioned s t ruc tu res had been dem ol i shed . As

a resu lt , concern pers i s t ed tha t the sud den re lease o f s t ra in energy caused by the

cut t ing of the pres t ress ing tend ons cou ld resul t in an explos ive fai lure of the

s t ruc tu re o r rap id , dan gerou s m ove m ent o f the tendo ns themselves. Th i s conc ern

was reflected in nu m ero us codes and guidel ines (BS 6187: 1982, F IP 1982, H eal th

and Safe ty Execu t ive 1984), which warne d o f the po ten t i a l dan gers bu t d id no t

r eco mmen d ap p ro p r i a t e d emo l i t i o n p ro ced u re s .

In the case of post - te ns ion ed f loors , since the av erage co m press ive s t ress in the

conc rete i s smal l , the l ikel ihood of explos ive failure due to the cu t t ing of a ten do n

i s min imal . Far g rea te r a t t en t ion has been pa id to the poss ib i l i ty tha t a severed

tendon migh t push ou t i t s end anchorage and be e jec ted f rom the s l ab a t h igh

veloci ty , becom ing a dan gero us p ro jec t il e . As the d i scuss ion be low dem ons t ra tes ,

th is i s ext remely unl ikely .

13.7. 1 R ev ie w o f researc h a nd f i e l d tes t i ng

In recen t years , cons iderab le l ab ora to ry research and in s itu m oni to r ing has been

per fo rm ed in o rde r to es t ab l ish the l ike ly l eve ls o f s t ruc tu ra l d am age o r t endo n

movement occur r ing when p res t ressed t endons a re cu t .

U n b o n d e d t e nd o n s

The g rea tes t concern ov er the poss ib il ity o f dan gerou s t en don m ove m ent has

been expressed wi th regard to s l abs pos t - t ens ioned by unbonded t endons . S ince

the t endons a re encased in g reased p las t i c shea ths , the end anchorages a re

requ i red to car ry the full t endon load th rou gh ou t the li fe o f the s t ruc tu re . Th e

lack o f any bo nd wi th the concre te a long the l eng th o f the t endons was equa ted

wi th a l ack o f res t ra in t to mov em ent , the assu m pt ion be ing tha t , if a t endon was

ab le to overco me the res is t ance o f i ts recessed end an chorag e , then i t migh t be

ejected from the s lab edge at h igh veloci ty .

A l ab o ra t o ry r e s earch p ro g ram m e (Wi ll iams an d W al d ro n , 1 98 9 a, 1 9 8 9b )

found tha t , fo r t endons in ex t ruded p las t i c shea ths wi th a good qua l i ty g rease

layer , the dam ping and f r i c t ion genera ted as the t endon moves w i th in i ts g reased

shea th p rov ide by fa r the g rea tes t res t ra in t to t endon movement . The res t ra in t

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SITE ACTIVITIES AND DEMOLITION 2 9 7

E

E

t -

G)

E

O

Q.

E3

120

100

80

6 0 -

4 0 -

2 0 -

9 4 .8 m long, 12 .5 m m dia

A 4 .8 m long, 15 .2 mm dia

O 7.8 m long, 12 .5 mm dia

A 7.8 m long, 15 .2 mm dia

0

9 A

O

~ o,

e

qD

e o

~ ! A

e e A ~ ~

0 I

5 0 1 0 0 1 5 0 2 0 0 2 5 0

Release load (kN)

Figure

3.2 Var iat ion of end d isp lac em ent w i th re lease load

prov ided by the mor ta r end p lugs i s in mos t cases qu i t e smal l , absorb ing on ly a

few per cent of the s t rain e nergy of a typical tendo n. Figu re 13.2 shows the

var i a t ion o f f ina l anch orag e d i sp lacem ent wi th re l ease load f rom la bora to ry t est s

on tendons up to 7 .8 m (26 f t ) in length . For these relat ively short lengths ,

d i sp lacements a re smal l , o f the o rder o f 100 m m (4 in) . No d i s t inc t ion has been

made in the f igure be tween t endons wi th and wi thou t p lugged ends , s ince bo th

gave s imi la r resu lt s . The sm al l e r d iameter t e ndon s show m arke d ly h igher level s o f

d i sp lacemen t a t a g iven load , th oug h th is t rend d i sappears i f the loads a re

expressed as f rac t ions o f the t endo n u l t imate t ens il e s t reng th ra ther th an as

abso lu te loads .

In add i t ion to these l abora to ry resu lt s , num erou s f i eld s tud ies o f dem ol i t ion o f

pos t - t ens ioned s t ruc tu res have been per fo rm ed recen tly , the m os t com prehens ive

of which a re those repor ted by B ar th a nd A alami (1989) . They m oni to r ed

m ove me nts o f over one thou sand 12.7 m m (0.5 in ) d iam eter t end ons o f l eng ths

varying between 1 .8 m (6 f t ) and 64 m (210 f t ) , wrapped in heat -sealed plas t ic

shea ths . Th i s me thod o f encas ing t endon s p rov ides a less tigh t fi t t han the more

wide ly used ex t ruded p las t i c shea th ing s tud ied by Wi l l i ams and Waldron , and

migh t , therefo re , be expec ted to resu lt in ra ther h ighe r d i sp lacements . B ar th a nd

Aalam i found tha t on ly 7% of the t endons d i sp laced by mor e than 75 m m (3 in )

and tha t 78% underwent no d i sp lacement a t a l l . They a l so no ted the t endency o f

the kern wi re to shoo t ou t ah ead o f the wou nd wi res , thou gh the kern wi re

d i sp lacement remained smal l .

In genera l , f ie ld s tud ies have sugges ted tha t l eve ls o f t end on m ove m ent when

severed dur ing dem ol i t ion a re low, tha t the o lder types o f unb ond ed t endon ,

where the shea th i s wrapped ra ther than ex t ruded , may g ive s l igh t ly h igher

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298

POST-TENSIONEDCONCRETE FLOORS

Table 13.3

Tendon d i sp lacement s in f i e ld t es ts (a ft er Barth and A alami ,

1989)

Released No . o f t endons w i th a d i sp lacement (mm) o f: Tota l

l ength number

(m) 0 < 75 76-150 151-300 301-4 50

1.8 248 102 0 0 0 350

15.8 89 12 5 1 0 107

16.5 17 1 5 0 0 23

20.6 36 2 4 7 1 50

22.6 32 2 5 9 0 48

28.7 15 7 7 4 1 34

40.8 21 2 11 1 0 35

50.3 17 4 7 0 1 29

64.0 324 1 1 0 0 326

d i s p l a c e m e n t s t h a n t e n d o n s i n e x t r u d e d s h e a t h s , a n d t h a t t e n d o n c u r v a t u r e h a s

l i t t l e e f f ec t on t he l i ke ly d i sp l acemen t .

O n e q u es t i o n t h a t a r i s e s is w h e t h e r t h e r e is an o p t i m u m l o ca t i o n f o r a cu t so a s

t o m i n i m i z e t h e p o s s ib i li t y o f d a n g e r o u s m o v e m e n t . T h i s is h a r d t o j u d g e

in tu i t i ve ly , s i nce an i ncrease i n l eng th w i l l r esu l t i n an i nc rease i n t he s t r a in e nerg y

r e l ea s ed , b u t a l s o i n an i n c rea s e i n th e d am p i n g an d f r ic t i o n a l r e s t r a i n t s . T h e r e i s

l i m i t ed ev i d en ce t o s u g g es t t h a t v e r y s h o r t an d v e r y l o n g t en d o n s g i ve ri s e t o t h e

l o w es t d i s p l acemen t s . F o r i n s t an ce , t h e t e s t r e s u l ts o f B a r t h an d A a l am i a r e

s u m m a r i z e d i n T a b l e 1 3.3 , w h i c h s h o w t h e n u m b e r s o f t e n d o n s e x p e r i e n c in g e n d

d i s p l acemen t s w i t h i n ce r t a i n r an g es . F r o m t h i s , i t i s c l e a r t h a t t h e l a r g e s t

d i s p l ace me n t s o ccu r r ed i n re l ea s ed l en g t h s b e t w ee n 2 0 m ( 66 f t) an d 5 0 m ( 16 4 f t) ,

w i t h v e r y s h o r t t e n d o n s m o v i n g o n l y a f ew mi l l i me t r e s an d n ea r l y a l l o f t h e 3 2 6

l o n g e r t e n d o n s e x p e r i e nc i n g z e r o d i sp l a c e m e n t . A s im i l a r t r e n d w a s o b s e r v e d b y

W i l l i ams an d W al d r o n ( 1 9 9 0 ) , u s i n g a f i n i t e e l emen t mo d e l v a l i d a t ed ag a i n s t t h e

l ab o r a t o r y t e s t s o u t l i n ed ab o v e . H o w ev e r , t h e s e r e s u l t s a r e t o o f ew an d t o o

d i s p a r a t e t o a l l o w a n y f i r m c o n c l u s i o n s t o b e d r a w n . T h e e x a c t m a g n i t u d e o f

t e n d o n m o v e m e n t w i ll v a r y w i th t e n d o n t y p e , l e n g t h , s tr es s le ve l a n d c o n d i t i o n ,

s o t h a t i t i s i mp o s s i b l e t o g i v e p r ec i s e g u i d an ce o n t h e ex ac t d i s p l acemen t t h a t

w o u l d b e ex p ec t ed i n a p a r t i cu l a r s i t u a t i o n . N ev e r t h e l e s s , i t i s c l e a r t h a t

d i sp l acemen t s a re l i ke ly t o be smal l i n a l l cases .

A r e l a t ed , b u t s l i g h tl y d i ff e ren t , p r o b l em w h i ch h a s b ee n o b s e r v ed i n a n u m b e r

o f s it e m o n i t o r i n g ex e r ci s e s i s a t en d en c y f o r t en d o n s t o b u r s t o u t o f t h e t o p o r

b o t t o m s u r face o f a s l ab a t p o i n t s w h e r e t h e co v e r is p a r t i cu l a r l y l o w ( S u a r ez an d

P o s t o n , 1 9 9 0 , S p r i n g f i e l d an d K ami n k e r , 1 9 9 0 , S ch u p ack , 1 9 9 1 ) . F r o m t h e

r ep o r t s p u b l i s h ed , i t ap p e a r s t h a t t h i s i s o n l y l ik e l y t o o ccu r a t l o ca t i o n s w h e r e t h e

co v e r t o t h e t en d o n is v e ry s ma l l , 2 0 m m ( 3 /4 i n) o r l es s. A s ens i b le p r eca u t i o n

ag a i n s t t h i s ev en t u a l i t y i s t o cu t t h e t en d o n a t a p o i n t w h e r e i t i s c l o s e t o t h e

m i d - d ep t h o f t h e s l ab ; i t i s l i k el y t h a t t h i s w i ll p r o v i d e a mo r e r a t i o n a l a p p r o a ch

t o c h o o s i n g c u t t i n g p o i n t s t h a n a t t e m p t s t o l i m i t t h e a x i a l m o v e m e n t .

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SITE ACTIVITIES AN D DEMO LITION

2 9 9

B o n d e d t e n d o n s

In general , s labs post - ten s ione d by bon ded tend ons are even less l ikely to g ive r ise

t o d emo l i t i o n p ro b l ems t h an t h o s e co n t a i n i n g u n b o n d ed t en d o n s . Th e ma i n

d if fe rence is tha t c u t t ing o f a bon ded t endo n usua l ly on ly causes loca li zed bo nd

fai lure, and hence release of pres t ress , in a short length of ten do n on ei ther s ide of

the cu t. There i s, t herefo re , no anch orage m ovem ent a nd no loss o f s t reng th in

ad jacen t spans .

Where t endons a re wel l g rou ted , the bond wi l l be su f f i c i en t to p reven t any

movement a t the anchorages dur ing demol i t ion o r a l t e ra t ion . Some loca l

debo nd ing is l ike ly a ro un d the cu t po in t , and poss ib ly some crac k ing o f the

concre te a long the l ine o f the t endon , ca used b y the increase in t endon

cross-sect ion as i t contracts longi tudinal ly . However , the resul t ing loss of

s t rength wi ll be confined to qu i te a smal l region, so that a catas t ro phic col lapse is

ex t rem ely un l ike ly .

The poss ib i li ty o f severe t endon m ovem ent on ly a r ises in cases where the

grou t ing i s ex t remely poor . Th e debo nd ing o f par t i a l ly g rou ted t end ons when cu t

is cu r ren t ly poor ly unde rs tood . W ha t l it tl e research has been car r i ed ou t (Belhad j

et al. , 1991) has conc en t ra ted on p rob lems assoc ia ted wi th very l a rge tendon s in

beam s , suggest ing tha t the me chan i sm s ac t ing a re ex t remely complex . I t is ,

therefore, not poss ible to g ive defin i tive guidan ce on the l ikely level of m ov em ent

of very poor ly g rou te d t endons .

13. 7 .2 De m ol i t ion p roce du res

The research a nd f ie ld observ a t ions d i scussed above suggest tha t pos t - t ens ioned

slabs can be safely and eas i ly demol ished. To ensure s i te safety , a number of

p rec au t ion ary m easures need to be t aken , bu t these a re un like ly to be much m ore

onerous than those requ i red fo r o rd inary re in fo rced concre te . There a re severa l

poss ib le metho ds o f dem ol i t ion , b u t a typ ica l p roce dure fo r a pos t - t ens ioned

floor is l ikely to consist of:

9 prop ping of the s lab being rem oved, and of any adjacent areas l ikely to be affected

9 prov is ion of shielding at locat ions whe re there i s a poss ibi l i ty of tendo ns ,

anchorages o r o ther debr i s be ing e jec ted (a l though , as d i scussed above ,

movement a t anchorages i s l ike ly to be smal l fo r unbonded t endons and zero

fo r b o n d ed t en d o n s )

9 cu t t ing o r de - t ens ion ing o f t endon s

9 "idemolit ion of the con cre te slab

The m ethod s ava i l ab le fo r de- tens ion ing a nd d em ol i t ion a re d i scussed in sec t ions

13.7.4 and 13.7.5 below.

Pr io r to commencing demol i t ion , i t i s es sen t i a l tha t a thorough p re l iminary

inves t iga tion i s car r i ed ou t , to ascer t a in the cond i t ion o f the s t ruc tu re and the

l ike lihood o f any p ro b lem s ar i sing dur ing d em ol i tion . Th i s will enab le a ra t iona l

dem ol i tion pro cedure to be devised and the necessary safety precaut ions to be taken.

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300 POST-TENSIONEDCONCRETE FLOORS

13.7.3 Planning a demolition job

A de mol i t i on c on t r a c to r shou ld a lwa ys s e e k gu ida nc e f r om a p r e s t r e s s ing

spe ci a li s t be f o re c om me nc ing the d e m ol i t i on o f a pos t - t e ns ione d s t r uc tu r e . T h i s

is p a r t ic u l a r ly i m p o r t a n t w h e n t ra n s fe r b e a m s o r u n b o n d e d t e n d o n s a re

invo lve d . If pos s ib l e , the o r ig ina l c on s t r uc t ion d r a wings f o r t he bu i ld ing shou ld

be ob ta ine d , t hough i t shou ld be bo r ne in mind tha t a l t e r a t i ons ma y ha ve be e n

m a de du r ing the l i fe o f the s t r uc tu r e .

A n im po r t a n t f i rs t st e p is to e s t a b l i sh t he t ype a nd loc a t ion o f t he t e ndon s . T h i s

c a n be done u s ing the d r a wings a nd c on f i r me d us ing a c ove r me te r on s i t e .

Loc a t ions w he r e the t e nd ons c ome ve r y c lo se t o t he t op o r bo t tom f ac e o f t he s l a b

can a lso be ident i f ied in th is way. Some sh ie ld ing should be provided a t these

po in t s t o r e s t r a in t he t e ndons f r om bu r s t i ng ou t a nd p r e ve n t f r a gme n t s o f

concre te f rom be ing e j ec ted ; a f ew p lanks he ld in p lace by props should be

su ff ic ie n t. C u t t i ng o f t e ndo ns shou ld no t be pe r f o r me d c lo se to t he se l oc a t ions ,

a nd a l l pe r sonne l shou ld be ke p t we l l a wa y f r om the m du r ing de - t e ns ion ing .

A n u m b e r o f o t h e r c h e ck s s h o u l d b e m a d e p r i o r t o c o m m e n c i n g d e m o l i ti o n .

T he type o f a nc ho r a ge s shou ld be de t e r mine d , s ince the l i ke l ihood o f t e nd on

m ove m e n t w i ll be i n flue nce d by the e xa c t wa y in wh ic h the s t r a nd is held . F o r

ins tance , i t shou ld be poss ib le to e s tab l i sh which a re dead- an d w hich a re l ive -end

a nc ho r a ge s , a nd w he the r t he de a d - e nd a nc ho r a ge s a r e bond e d o r p re - loc ke d . I t is

a l so impor t a n t t o i de n t i f y a ny c ons t r uc t ion j o in t s , i n f i l l s t r i p s , c oup le r s o r

s t r uc tu r a l a l t e r a t i ons , s i nc e the se ma y ha ve a n in f lue nc e on the de mol i t i on

p r oc e du r e . F o r e xa mple , c u t t i ng o f a n un bon de d t e ndo n o n one s ide o f a c oup le r

wi ll l ead to a loss of tens io n o nly on the s ide of the cu t , no t ove r the w hole te nd on

length .

F r om the num be r a nd s ize o f t he t e ndons , i t is pos s ib le t o m a ke a n e s t ima te o f

the s tress level in the f loor . As a ro ug h in dic at io n of the s ignif icance of the s tress

leve l , a r ed uc t io n in s t re ss of 1 N/ m m 2 (145 ps i ) g ives an ene rgy re lease equiv a len t

to a 25 m m (1 in ) d ro p in the leve l o f the s lab . Care fu l cons id e ra t io n m ust be g iven

to the m e tho d o f r e le a s ing th i s e ne rgy , e spec i al ly i f t he t e ndo ns a r e un bo nd e d o r

poor ly grouted ; th i s top ic i s d iscussed fur the r in sec t ion 13 .7 .4 .

Transfe r beams a re s t r e ssed to a much h igher leve l . They a re des igned to be

s t re ssed in seve ra l s tages a s inc reas ing dead load i s imposed by the cons t ruc t ion

of the s t ruc tu re ab ove . I f s t r e ssed in one o per a t io n , th en the pres t re ss wi l l induce

reverse tens i le and compress ive s t r e sses of a h igh magni tude which can resu l t in

f a il u re o f t he be a m. D e m ol i t i on m us t f o l low a s imi l a r mu l t i -s t a ge de - t e ns ion ing

p r oc e du r e ; a s pa r t o f t he s t r uc tu r e a bove i s de m ol i she d , some t e ndo ns m us t be

severed to reduce the pres t re ss .

I f unb on de d t e ndon s a r e p r e sen t , i t shou ld be e st a b l ishe d w he the r t he

she a th ing is o f t he e x t r ude d type wh ic h i s now use d a lm os t un ive r sa l ly , o r o f t he

o lde r a nd loose r wr a pp e d type . T he l a t t e r ma y g ive ri se t o s l igh t ly h ighe r t e nd on

d i sp l a c e me n t s . Gua r ds shou ld be p l a c e d a r ound the t e ndon e nds to c a t c h a ny

f ly ing de b ri s wh ic h m igh t be ge ne r a t e d a s t he a nc ho r a ge s a r e d i s lodge d . T h e se

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SITE ACTIVITIES AND DE MO LITION 301

need not be pa r t icu la r ly subs tan t ia l ; a f ew p lanks secure ly he ld over the

a nc ho r a ge s w i l l suffic e. N o pe r sonne l sh ou ld be a l l owe d to s t a nd ne a r t he t e n don

e nds du r ing de mol i t i on .

De - t e ns ion ing o f a n un bo nde d t e ndo n w i ll , o f c ou r se, c a use lo s s o f p r e s tr e s s

over the fu l l l ength of the ten do n, n o t jus t the sp an w here the cu t is m ade . W hi le a

f loo r m a y c o n ta in su ff ic ie n t rod r e in fo r c e me n t t o p r e ve n t c om ple t e c o l l apse w he n

the pres t re ss i s r emoved, i t i s s t i l l necessa ry to provide tempora ry suppor t to a

f loo r du r ing de - t e ns ion ing . I n ge ne ra l , t he a m ou n t o f p r op p ing r e q u i r e d f o r a

pos t - tens ioned f loor i s s l igh t ly grea te r than for a r e inforced concre te member . I t

shou ld be no te d th a t t he p r op s m a y be sub je c t ed to som e ho r i zon ta l f o r ce s due to

the e xpa ns ion o f t he f l oo r whe n the t e nd ons a r e c u t.

F o r b ond e d t e ndon s , i t is v i ta l to e s t a b l ish t he a de q u a c y o f t he g r ou t ing . T he

on ly r e li a b le wa y o f do ing th is is t o b r e a k ou t t he c onc r e te a r ou nd the t e ndo ns a t a

few loca t ion s an d p e r form a v isua l inspec t ion . I f the tend ons a re we l l -grouted ,

then cu t t ing i s un l ike ly to cause se r ious problems, the bond be ing suf f ic ien t to

p r e ve n t a ny move me n t a t t he a nc ho r a ge s . A s d i s c us se d e a r l i e r , on ly ve r y

loc al iz e d de bo nd in g o f the t e n don a nd c r a c k ing o f the c onc r e t e a r e l i ke ly, so t ha t

any loss of s t r eng th w i ll be conf ined to qu i te a smal l r eg ion . Som e pro pp ing of the

f loor i s advisab le , bu t aga in th is needs to be on ly s l igh t ly more subs tan t ia l than

tha t u se d whe n pe r f o r ming s imi l a r ope r a t ions on a r e in f o r c e d c onc r e t e f l oo r .

13 .7 . 4 De - tens ion ing o f tendons

W h i le some m e tho ds o f de m ol i t i on c a n be c a r ri e d o u t by a s ingle ope r a t ion ,

seve ra l r equi re pr io r cu t t ing o r de - tens ion ing of the tendo ns . T his ma y be for

sa fe ty reasons , in orde r to preve nt the sudd en , un con t ro l led re lease of a tendo n

la t e r i n the de m ol i t i on p r oc e s s , o r s imp ly be ca use the t oo l be ing u se d to de m ol i sh

the c onc r e te i s no t a b l e to c u t t h r ou gh the ha r de ne d s te e l f r om w h ic h the t e ndons

a r e ma nuf a c tu r e d .

F o r u n b o n d e d t e n d o n s , t h er e a re a n u m b e r o f m e t h o d s b y w h i c h t h e te n s i o n

can be re leased . These inc lude h ea t ing of the anc hora ges un t i l s l ip occurs ,

b r e a k ing ou t o f c onc r e t e be h ind the a nc ho r a ge s un t i l sl ip oc c u rs , a nd c u t t i ng o f

the t e ndons a t some po in t a long the i r l e ng th , a wa y f r om the a nc ho r a ge s . T h i s l a s t

op t ion c a n be done e i t he r by b r e a k ing ou t t he c onc r e t e a r ound a t e ndon , t he n

us ing a f la me to r c h o r d i s c c u t t e r, o r by m a k in g a s ingle c u t t h r o ug h the s l ab a nd

tendons us ing a saw or the rmic lance . Exper ience sugges ts tha t us ing a f lame

torch resu l t s in a mo re grad ua l loss of force , and , the re fore , in s l igh t ly smal le r

t e ndo n d i sp l a c e me n t s , t ho ug h the m ove m e n t i s un l ike ly t o be la r ge in a ny c a se.

W he n c hoos ing the c u t t i ng po in t s , pos i t i ons w he r e t he t e ndo n i s ve r y c lo se t o t he

top o r bo t to m f ac e o f t he s la b shou ld be a vo ide d . T he c ho ic e be twe e n the a bove

m e tho ds w i ll u sua l ly de pe nd on the pa r t i c u l a r c ond i t i ons e nc o un te r e d on a g ive n

s i te ; o ften , p roble m s of access wi ll gov ern wh ich m eth od i s the m ost su i tab le.

F o r we l l - g r ou te d bon de d t e ndons , t he r e i s l it tl e r isk o f a n unc o n t r o l l e d r el ea se

of ene rgy , so cu t t in g w i ll on ly be necessa ry i f the too ls used for dem ol ish ing the

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302 POST-TENSIONED CONCRETE FLOORS

concre te s lab a re unab le to cu t th rou gh the t endons . A ny o f the cu t ting m ethod s

a l ready ment ioned fo r unbonded t endons can be used , i . e . f l ame to rch , d i sc

cu t t e r , saw or thermic l ance .

While i t i s unl ikely that the relat ively smal l tendon forces in post - tens ioned

s labs would l ead to com ple te deb ond in g even o f very poor ly g rou te d t endons , i t is

r eco mmen d ed t h a t a cau t i o u s ap p ro ach i s ad o p t ed . P ro b ab l y t h e s a f e s t me t h o d

wou ld be a con t ro l led de- t ens ion ing p ro cedu re us ing open th roa t jacks , s imi la r to

the appr oac h ou t l ined in sec t ion 13 .8 fo r the a l t e ra t ion o f s l abs con ta in ing

u n b o n d ed t en d o n s .

As ment ioned above , a l l de- t ens ion ing p rocedures wi l l cause a reduc t ion o f

s t reng th in the s l ab , which wi l l , t herefo re , need to be p ropped t emporar i ly . For

bon ded t endons the loss o f s t reng th wi ll be qu it e loca l ized , whi le fo r unbo nde d

tendon s i t m ay occur on the fu ll leng th o f the s t ruc tu re .

13. 7 .5 D e m o l i t i o n m e t h o d s

Prov ided tha t the necessary p re l iminary measures ou t l ined above have been

fo l lowed , pos t - t ens ioned f loors can be dem ol i shed us ing mo s t o f the conv en t iona l

me t h o d s . Th e ad v an t ag es an d d raw b ack s o f t h e v a r io u s me t h o d s a r e br ie f ly

d i scussed be low. The cho ice o f m etho d i s depe nden t on nu m erou s fac to rs ,

inc lud ing s t ruc tu ra l fo rm, no i se and d us t l imi ta tions , space cons t ra in t s and cos t .

The t rad i t iona l wreck ing ba l l and c rane approach i s fas t and cheap . However ,

i t i s d i ff icul t to use in a control led way, and is const rained by height res t r ic t ions

and the need fo r c l ear space a round the bu i ld ing . I t genera tes l a rge amounts o f

no i se and dus t and i s no t ab le to sever the p res t ress ing t endons , which mus t ,

therefo re , be de- t ens ioned befo rehand .

The use o f l a rge c i rcu la r saws p rov ides a s low, con t ro l l ab le de mo l i tion p rocess

in which the s l ab and t endons a re cu t in a s ing le pass . The method can be

expens ive i f there a re a lo t o f t endons to cu t , as they cause cons iderab le wea r o f

the saw b lades . The b lade m us t be coo led by wa ter , c rea t ing a s lu r ry which d r ips

down the bu i ld ing ; th i s i s un impor tan t in demol i t ion bu t can be undes i rab le

when car ry ing ou t a l t e ra t ions .

Exp los ive demol i t ion t echn iques b r ing the s t ruc tu re down very qu ick ly and

econom ica l ly , bu t do no t b reak up the s t ee l t endons , which , therefo re , have to be

cu t whe n c lear ing the demo l i shed s t ruc tu re . The des ign o f the exp losive sys tem

m us t t ake accou n t o f the s to red energy in the s t ruc tu re due to the p res t ress ing .

Th e m e t h o d g en e rat e s p ro b l ems o f b l a st , g ro u n d -b o rn e v i b r a ti o n s an d f l yi ng

debr i s, bu t these shou ld be no w orse fo r pos t - t ens ioned bu i ld ings tha n fo r o ther

s t ruc tu ra l types .

The rmic l ances can be used to cu t th roug h the concre te and t endon s in a s ing le

opera t ion . Th i s method i s qu ie t and v ib ra t ion- f ree , bu t expens ive .

Percuss ion too l s such as d r i l l s p rov ide a cheap and con t ro l l ab le demol i t ion

m ethod , bu t c rea te p rob lems o f dus t an d no i se , and requ i re the p r io r de- t ens ion ing

of the t endons .

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303

13 .8 Cut t ing ho les

In man y s t ruc tu res , change o f use o r genera l m ain tena nce m ay requ i re the cu t t ing

of ho les in a f loor. Sm al l d iameter ho les a re in ma ny ins tances eas ie r to m ake in

post - tens io ned tha n in reinforced con crete f loors , s ince the s teel in a post - tens ion ed

floor i s l ikely to be more widely spaced. I t i s , therefore, qui te l ikely that smal l

ho les can be made wi thou t cu t t ing th rough any s t ee l . Obvious ly , t endon

loca t ions shou ld be ascer t a ined f rom cons t ruc t ion d rawings and checked on s i t e

us ing a covermeter p r io r to commencing cu t t ing opera t ions .

Fo r l a rger a l t e ra t ions , requ i r ing the cu t t ing o f t endons , c are shou ld be t ak en in

choos ing the ho le loca t ion . For ins tance , i t i s no t normal ly poss ib le to cu t

th ro ugh beam s , as th i s causes too g rea t a d i s rup t ion o f the s t ruc tu ra l sys tem.

Loca t ions w here the t endon s a re very c lose to the bo t to m of the s lab shou ld a l so

be avoide d, as th is resul ts in an eccentr ic appl ica t ion of pres t ress at the new ly

created free edge, and because i t i s d i ff icul t to inser t new anchorages at such

loca t ions . Somet imes a dow ns ta nd beam can be added a t the edge o f the open ing

in o rder to a l lev ia te the l a t t e r p rob lem , bu t the eccen t ri c app l i ca t ion o f the fo rce

remains . Cu t t ing th ro ugh bun ched t endon s can a lso p resen t p rob lems , s ince they

mus t be sp layed ou t in o rder to f i t new anchorages , requ i r ing the removal o f

concre te f rom ar oun d the t endons fo r some d i s t ance bey ond the edge o f the hole .

W h en ma k i n g a l t e r a t io n s i n v ol v in g t h e cu t ti n g o f u n b o n d ed t en d o n s ,

re - t ens ion ing is requ i red a f t e r the a l t e ra t ion , m ak in g s imple cu t t ing o f the

t en d o n s u n accep t ab l e , a s t hi s may cau s e d am ag e t o t h e t en d o n an ch o rag es . F o r

these instances , spec ia l open - th ro a t j acks a re av a i l ab le which a l low tend ons to be

de- t ens ioned in a g radua l and con t ro l l ed way . The normal p rocedure i s to b reak

ou t a ho le s l igh t ly l a rger than tha t requ i red , l eav ing the t endons in tac t .

Op en- th ro a t j acks a re then pos i t ioned over a t endo n a t e i ther end o f the open ing

and used to t ake up the load in the t endon , as shown in F igure 13 .3 (a) . The l eng th

of t endo n be tween the j acks becom es s l ack and is cu t , and the p ressure i s then

gradua l ly re leased f rom the j acks , l eav ing the t endon und am age d and f ree o f

t ens ion . N ew anch orage s can n ow be pos i t ioned a t the edges o f the ho le , and the

two halves of the ten do n re-s t ressed, Fig ure 13.3(b) .

Cu t t ing th rough bonded t endons p resen t s fewer p rob lems , as the p res t ress i s

main ta ined by the bond wi th the concre te , so tha t new anchorages need no t be

f i tt ed . Ins tead , the ind iv idua l wi res o f the t endon s can s imply be sp layed ou t and

concre ted over wh en m akin g go od the edges o f the ho le .

W hen m akin g l a rge ho les, i t is l ike ly tha t some add i t iona l re in fo rcemen t wi ll be

requ i red a round the per imeter , as shown in F igure 13 .3 (c) , in o rder to

co m pe nsa te for the loss of s t reng th c aused by the hole . This wi l l be the case in

bo th pos t - t ens ioned and re in fo rced concre te f loors .

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L i m i t o f c o n c r e t e b r e a k o u t

T e n d o n c u t h e r e a f t e r C e m e n t i t i o u s

. . . . . . Z p ~ s s u r i s i n g j a c k s ~ I r< o r e p o x y m o r t a r

- / / = _

\ I ~ j a c k s _ . . / / i { I i ~ i

I ' H ~ 1 7 6 p l a t e s i ' ~

/ / \ \ \ I

= i / " \ i l L - -

~ ~ , - - ~

( a ) D e - t e n s i o n i n g la y o u t

( b ) Ne w a n c h o r s la y o u t

F i g u r e 1 3 . 3

Cutting large holes

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SITE ACTIVITIES A ND DEMO LITION

305

Figure 13.4

VSL mon o jac k ( twin ram), stress ing slab tendon, Austra l ia

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R E F E R E N C E S

ACI 318-89 Bui ld ing co d e requirements for re in forced concre te , American Concrete

Institute, Detroit.

ACI Committee 435 (1974)

Deflect ion of Two-Way Reinforced Concrete Floor Systems:

State-of-the-Art Report ,

Rep ort No. AC I 435.6R-74 (Reapproved 1989), Am erican

Concrete Insti tute, Detroit .

ACI Commit tee 435 (1991)

State-of - the-Ar t Repor t on Control o f Two-Way S lab

Deflections, Repo rt No. A CI 435.9R-91, Am erican Concrete Insti tute, Detroit .

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308 POST- TENSIONED CONCRETE FLOORS

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I N D E X

Anchorages, 1, 47-54

burst ing forces arou nd, 191-4

couplers, 53, 58

dead anchorages, 51-3, 57

draw-in, 164-8

intermediate anchorages, 53

live anchorages, 47-51, 58, 94-6

pocket formers, 24

reinforcement, 193

Balanced load s e e Equivalent load, Load

balancing

Beams, 61, 66-9

downstand, 67

shell beam system, 67

strip, 61, 70

upstand, 68

Buckling, 21

Bursting force , 191-4

Cable

( s e e a l s o

Strand, Tendons), 6

Cladding, 199, 277

Coefficient of thermal expansion of

concrete, 26

Columns, 12, 81, 82

moments, 87-90, 107

punching shear, 230

Computer programs, 136

Concordant profi le, 114

Concrete, 26-39

air-entrained, 98

cover, 99-101

creep coefficient, 33-5

lightweight s e e Lightweight concrete

modulus of elasticity, 29-31

Poisson's rat io, 29

shrinkage, 31-3

strength:

compressive, 26-8

modulus of rupture, 28, 252

shear, 222

tensile, 28, 223

thermal propert ies, 26

Youn g's modulus, 29

Construction joints, 81, 92, 268

Corrosion, 42

Cover, 97, 99-101

Cracking, 91, 138

moment, 181,226, 228

Creep:

coefficient:

of concrete, 33-5

of lightweight co ncrete, 39

effect on axial movement, 90

effect on deflection, 199

losses s e e Prestress losses

Curling, 249

Curtai lm ent o f tendons , 93, 156

Cut t ing s e e Holes

Damping, 211

Deflections, 198-206, 213-19, 277

crossing beam method, 204

effect of load ing history , 199

formulae for one-way spanning slabs,

202

frame and slab method, 204

limits, 199

loads for deflection calculations, 199-201

plate formulae, 203

Demolition, 6, 295-302

Detailing, 271

Dispersion of axial prestress, 85-8

Drag theory, 258

Drop panels, 70

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310 I N D E X

Ducts , 14, 54

Durabi l i ty , 97-9

Ear thquakes s e e Seismic loading

Eccentricity, 4, 115, 130, 135

Equivalen t load, 109-12, 1 18, 135

Exposure condi t ions , 98-100

Loads :

balanced s e e Equivalent load

combinations, 143-4, 181

factors, 181

for deflection calculations, 199-201

on ground slabs, 251

Loa d balancing m ethod of des ign, 112,

146-59

Losses

s e e

Prestress losses

Failure:

of s trucutre, 21, 177-80

of tendons, 22

Finite elemen t analysis , 143, 205

Fire:

loss of strength at high temperatures, 18,

73

in rod reinforcement, 24

in tendons, 99

protect ion, 99-102

Flange width, 209

Flat slab, 61

Flexural s trength:

of anch orag e zone, 191-4, 197

of member , 1 77-91 ,194 -7

Formwork, 13, 21, 24, 73

Freezing/ thawing of concre te, 98

Minimum dimensions , 101

Minimum re inforcement , 274

Modulus of elasticity:

of concrete, 29-31

of l ightweight concrete, 38

of rod reinforcement, 24

of strand, 43

Mo dulus of rupture of concre te , 28, 252

M o m e n t :

cracking moment , 181,226, 228

pr imary and secondary mom ents ,

112-14

Movement joints , 277-81

Natura l f requency, 208-11

Grout ing, 294

Ground f loors , 249

loading of, 251

pos t- tens ioned f loors , 259-70

RC floors, 257

Holes, 18, 303

Jack, 55

Lightweight concrete, 35-9

creep coefficient, 39

modulus of elasticity, 38

Poisson's ratio, 39

shrinkage, 39

strength, 37-8, 223

Line of pressure, 113

One-way spanning f loors , 70

Partial safety factor, 183

Perceptibil i ty scales for vibration, 207-8

Permissible stresses:

in concrete, 138-41

in strand, 141-2

Plastic hinges, 178

Poisson ' s ra t io:

of concrete, 29

of l ightweight concrete, 39

of rod reinforcement, 24

of s t rand, 40

Post-tensioning, 1, 9, 14

bonded, 9, 14

unbonded, 9, 14

Prestress, 2

full, 137

initial and final values, 144

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INDEX 311

losses s e e Prestress losses

minimum, 102

part ia l , 137-8

Prestress losses, 7, 160

anchorage draw-in , 164-8

creep, 171-2

curvature, 163-4

elast ic shortening, 168-70

immediate , 160

friction, 163-4

long tendons, 162

long-term, 160

reducing losses, 161-2

relaxation, 172-3

shrinkage, 170-1

wobble, 163-4

Pre-tensioning, 9

Profile s e e Tendon profi le

Punching shear , 230-40

applied force, 230

cri t ical perimeter, 230, 232-7

decompression load method, 238-40

moment t ransfe r , 231-3 , 236

punching strength calculat ion, 232-40,

245-8

Radius of relative stiffness, 261-2

Refurbishment, 19

Reinforced concrete, 10, 81

Rod reinforcement, 24

minimum area , 274

modulus of e last ic i ty, 24

Poisson's rat io, 24

strength, 24

Ribbed slab, 61, 64-6, 73

Scheduling, 284-6

Secondary moments , 112-14, 148

Seismic loading:

design considerations, 103-6

detailing, 281

Shape factors, 145

Shear:

effect of inclined tendons, 12, 221-2

failure modes, 224

influence of f lexual cracking, 224-6

one-way shear s t rength , 226-9 , 224-5

punching

s e e

Punching shear

shear strength of concrete , 222-4

Shear reinforcement:

convent iona l , 229-30

shearheads, 241-4

shearhoops, 240

Sheathing, 14, 54

Shrinkage :

effect on axial movement, 90

losses

s e e

Prestress losses

strains,

in concrete , 31-3

in l ightweight concrete , 39

Slabs on grade

s e e

Ground f loors

Solid slab, 73

Span-depth rat io, 11, 74-7

Steelwork, 81

Strain compatibil i ty, 183, 189-91

Strand, 6, 40-6

compact, 41

corrosion, 42

drawn, 41

modulus of e last ic i ty, 43

normal , 40

Poisson's rat io, 40

proof load , 43

relaxation, 44

sizes, 45

strength, 45

tem pera ture effects, 46

transmission length, 46

Strength:

of concrete , 26-8

of l ightweight concrete , 37-8

of rod reinforcement, 24

of strand, 45

of structural e lements

s e e

Flexura l

strength, Shear

Strength reduction factor:

bending, 185

shear, 228, 238

Stress:

corrosion, 42

initial and final values, 8, 93, 13 4, 144

permissible stresses:

in concrete, 138-41

in strand, 141-2

reversal, 5

ult imate stresses: