MODELLING THE BEHAVIOR OF CONCRETE MEMBERS: …ecpfe.oasp.gr/sites/default/files/Fardis EC8-3 member models_0.pdf · Workshoppqgg ,,y: Advances in Earthquake Engineering for Urban

MODELLING THE BEHAVIOR OF CONCRETE MEMBERS:MODELLING THE BEHAVIOR OF CONCRETE MEMBERS:DEVELOPMENTS SINCE THE COMPLETION OF EN 1998DEVELOPMENTS SINCE THE COMPLETION OF EN 1998--3:20053:2005

Michael N. FardisMichael N. FardisUniversity ofUniversity of PatrasPatrasUniversity of University of PatrasPatras

Idealized skeleton curve - envelope to hysteresis loops

Yield Ultimate moment, Mu=My

MEffective elastic stiffness:

moment My

M < 0 8M

secant-to-yielding

Mres < 0.8Mu

member deformation:chord-rotation

δ yYield deformation

δ uUltimate deformation δ

chord rotationsection deformation:

curvature

Practical expressionsPractical expressionsact ca e p ess o sact ca e p ess o sfor the yield & failure properties for the yield & failure properties

developed for EN 1998developed for EN 1998--3:20053:2005theirtheir ad ancement sincead ancement since 20052005-- their their advancement since advancement since 2005 2005

D. BISKINIS, G. ROUPAKIAS & M.N. FARDIS, Cyclic Deformation Capacity of Shear-Critical RC Elements, Proceeding of fib Symposium, Athens, May 2003.M.N.FARDIS & D.BISKINIS, Deformation Capacity of RC Members, as Controlled by Flexure or Shear. Proceedings of International Symposium on Performance-based Engineering for Earthquake Resistant Structures honoring Prof. Shunsuke Otani University of Tokyo, Sept. 2003, pp. 511-530.D.BISKINIS, G.K. ROUPAKIAS & M.N.FARDIS, Resistance and Design of RC Members for Cyclic Shear, 14th Greek National Concrete Conference, Kos, Oct 2003 Vol B pp 363-374Oct. 2003, Vol. B, pp. 363 374.D.BISKINIS & M.N.FARDIS, Cyclic Strength and Deformation Capacity of RC Members, including Members Retrofitted for Earthquake Resistance, Proc. 5th

International Ph.D Symposium in Civil Engineering, Delft, June 2004, Balkema, Rotterdam, p. 1125-1133. D.BISKINIS, G.K. ROUPAKIAS & M.N.FARDIS, Degradation of Shear Strength of RC Members with Inelastic Cyclic Displacements, ACI Structural Journal, Vol. 101, No. 6, Nov.-Dec. 2004, pp.773-783.

S S S OS O O OS S O S S & S S S S f f C SM.N.FARDIS, D.BISKINIS, A. KOSMOPOULOS, S.N. BOUSIAS & A.-S. SPATHIS, Seismic Retrofitting Techniques for Concrete Buildings, Proc. SPEAR Workshop – An event to honour the memory of Jean Donea, Ispra, April 2005 (M.N.Fardis & P. Negro, eds.)S.N. BOUSIAS, M.N.FARDIS & D.BISKINIS, Retrofitting of RC Columns with Deficient Lap-Splices, fib Symposium, Budapest, May 2005.S.N. BOUSIAS, M.N.FARDIS, A.-S. SPATHIS & D.BISKINIS, Shotcrete or FRP Jacketing of Concrete Columns for Seismic Retrofitting, Proc. International Workshop: Advances in Earthquake Engineering for Urban Risk Reduction, Istanbul, May–June 2005.p q g g , , yD.BISKINIS & M.N.FARDIS, Assessment and Upgrading of Resistance and Deformation Capacity of RC Piers, Paper no.315, 1st European Conference on Earthquake Engineering & Seismology (a joint event of the 13th ECEE & the 30th General Assembly of the ESC), Geneva, September 2006.D.BISKINIS & M.N.FARDIS, Cyclic Resistance and Deformation Capacity of RC Members, with or without Retrofitting, 15th Greek National Concrete Conference, Alexandroupolis, Oct. 2006, Vol. B, pp. 495-506.D BISKINIS M N FARDIS Effect of Lap Splices on Flexural Resistance and Cyclic Deformation Capacity of RC Members Beton & Stahlbetonbau 102 2007D.BISKINIS, M.N.FARDIS, Effect of Lap Splices on Flexural Resistance and Cyclic Deformation Capacity of RC Members, Beton & Stahlbetonbau, 102, 2007D.BISKINIS & M.N.FARDIS, Cyclic Deformation Capacity of FRP-Wrapped RC Columns or Piers, with Continuous or Lap-Spliced Bars, 8th International Symposium on Fiber Reinforced Polymer Reinforcement for Concrete Structures (FRPRCS-8), Patras, July 2007.D.BISKINIS & M.N.FARDIS, Upgrading of Resistance and Cyclic Deformation Capacity of Deficient Concrete Columns, in “Seismic Risk Assessment and Retrofitting with special emphasis on existing low rise structures” (A.Ilki, ed.), Proceedings, Workshop, Istanbul, Nov. 2007, Springer, Dordrecht.D.BISKINIS & M.N.FARDIS, Cyclic Deformation Capacity, Resistance and Effective Stiffness of RC Members with or without Retrofitting, 14th World Conference on Earthquake Engineering, Beijing, paper 05-03-0153, Oct. 2008.D.BISKINIS & M.N.FARDIS, Chapter 15: “Upgrading of Resistance and Cyclic Deformation Capacity of Deficient Concrete Columns” , in “Seismic Risk Assessment and Retrofitting with special emphasis on existing low rise structures” (Ilki A et al, eds.), Springer, Dordrecht, 2009D BISKINIS & M N FARDIS Ultimate Deformation of FRP-Wrapped RC Members 16th Greek Nat Concrete Conference Paphos Oct 2009 paper 171106D.BISKINIS & M.N.FARDIS, Ultimate Deformation of FRP Wrapped RC Members, 16th Greek Nat. Concrete Conference, Paphos, Oct. 2009, paper 171106D.BISKINIS & M.N.FARDIS, Flexure-Controlled Ultimate Deformations of Members with Continuous or Lap-Spliced Bars, Structural Concrete, Vol. 11, No. 2, June 2010, 93-108.D.BISKINIS & M.N.FARDIS, Deformations at Flexural Yielding of Members with Continuous or Lap-Spliced Bars, Structural Concrete, Vol. 11, No. 3, September 2010, 127-138.D.BISKINIS & M.N.FARDIS, Effective stiffness and cyclic ultimate deformation of circular RC columns including effects of lap-splicing and FRP wrapping. Paper 1128, 15th World Conference on Earthquake Engineering, Lisbon, Sept. 2012.D.BISKINIS & M.N.FARDIS, Stiffness and Cyclic Deformation Capacity of Circular RC Columns with or without Lap-Splices and FRP Wrapping, Bulletin of Earthquake Engineering, Vol. 11, 2013, DOI 10.1007/s10518-013-9442-7.D.BISKINIS & M.N.FARDIS, Models for FRP-wrapped rectangular RC columns with continuous or lap-spliced bars, Engineering Structures, 2013 (submitted).

Experimental DatabaseExperimental Database1 Range/mean of parameters in tests for calibration of expressions1. Range/mean of parameters in tests for calibration of expressions

for the member chord rotation and secant stiffness at yielding1653 rectangular 214 rectangular 229 members of non 307 circular

Parameter 1653 rectangular beams/columns

214 rectangular walls

229 members of non-rectangular section

307 circular columns

min/max mean min/max mean min/max mean min/max mean

section depth or diameter, h (m) 0.1 / 2.4 0.31 0.4 / 3.0 1.19 0.2 / 3.96 1.4 0.15 /1.83 0.43

shear-span-to-depth ratio, Ls/h 1 / 13.3 3.78 0.45 / 5.53 1.92 0.45 / 8.33 2.03 1.1 / 10 3.77

section aspect ratio, h/bw 0.2 / 4 1.3 4 / 30 10.9 2.5 / 57 16.7 1.0 1.0

fc (MPa) 9.6 / 175 37.7 13.5 / 109 35.8 13.5 / 101.8 40.4 16.7 / 90 36.7

axial-load-ratio, N/Acfc -0.05 / 0.9 0.126 0 / 0.86 0.10 0 / 0.50 0.06 -0.1 / 0.7 0.137

transverse steel ratio, w (%) 0 / 3.54 0.62 0.05 / 2.18 0.54 0.04 / 2.44 0.7 0 / 8.83 0.86

total longitudinal steel ratio (%) 0 2 / 8 55 1 97 0 07 / 4 27 1 5 0 205 / 6 2 1 23 0 53 / 5 8 2 34total longitudinal steel ratio tot (%) 0.2 / 8.55 1.97 0.07 / 4.27 1.5 0.205 / 6.2 1.23 0.53 / 5.8 2.34

diagonal steel ratio, d (%) 0 / 1.68 0.027 0 / 0.25 0.005 - - - -

transverse steel yield stress fyw, MPa 118 / 2050 468 220 / 1375 443 160 / 1375 504 200 / 1728 454.2y fy

longitudinal steel yield stress fy, MPa 247 / 1200 440.2 276 / 1273 470 209 / 900 453 240 / 648 414.3

Experimental DatabaseExperimental Database2 Range/mean of parameters in tests for calibration of2. Range/mean of parameters in tests for calibration of

expressions for ultimate curvature in monotonic or cyclic loading

Parameter 415 rectangular beams/columns 59 rectangular walls

254 monotonic 160 cyclic 13 monotonic 46 cyclic

min/max mean min/max mean min/max mean min/max mean

section depth or diameter, h (m) 0.12 / 0.8 0.31 0.22 / 2.4 0.41 2.39 / 2.41 2.4 1.7 / 2.0 1.99

section aspect ratio h/bw 0 225 / 3 73 1 54 0 5 / 2 1 21 21 2 / 23 4 22 4 13 3 / 28 3 13 7section aspect ratio, h/bw 0.225 / 3.73 1.54 0.5 / 2 1.21 21.2 / 23.4 22.4 13.3 / 28.3 13.7

fc (MPa) 19.7 / 99.4 34.8 17.7 / 102.2 38.6 34.5 / 40.8 35.3 26.2 / 45.6 40.6

axial-load-ratio, N/Acfc 0 / 0.78 0.08 0 / 0.77 0.236 0.063 / 0.077 0.07 0.05 / 0.11 0.07

transverse steel ratio, w (%) 0 / 2.38 0.345 0.04 / 2.96 0.656 0.41 / 0.82 0.66 0.11 / 0.25 0.24

total longitudinal steel ratio tot (%) 0 / 3.68 1.4 0.37 / 4.19 1.82 0.88 / 1.76 1.35 0.07 / 0.77 0.63

t t l i ld t f MP 0 / 596 419 255 / 1402 477 2 440 / 483 443 3 465 / 562 502transverse steel yield stress fyw, MPa 0 / 596 419 255 / 1402 477.2 440 / 483 443.3 465 / 562 502

longitudinal steel yield stress fy, MPa 277 / 596 490 341 / 573 493.3 444 / 510 466.6 523 / 580 552

Experimental DatabaseExperimental Database3 Range/mean of parameters in tests for calibration of3. Range/mean of parameters in tests for calibration of

expressions for the member ultimate cyclic chord rotation1159 rectangular 95 rectangular 53 members of non 143 circular

Parameter 1159 rectangular beams/columns

95 rectangular walls

53 members of non-rectangular section

143 circular columns

min/max mean min/max mean min/max mean min/max meanmin/max mean min/max mean min/max mean min/max mean

section depth or diameter, h (m) 0.1 / 2.4 0.33 0.4 / 2.75 1.15 0.2 / 3.4 1.21 0.2 / 1.83 0.45

shear-span-to-depth ratio, Ls/h 1 / 13.3 3.7 0.5 / 5.53 2.15 0.65 / 8.33 2.85 1.77 / 10 4.22

section aspect ratio, h/bw 0.2 / 6 1.18 2.5 / 28.3 9.75 2.5 / 36 9.95 1.0 1.0

fc (MPa) 12.2 / 175 43.7 13.5 / 109 35.9 20.8 / 83.6 38.5 23.1 / 90 38.0

axial-load-ratio, N/Acfc -0.1 / 0.9 0.165 0 / 0.86 0.116 0 / 0.30 0.07 -0.09 / 0.7 0.15

transverse steel ratio, w (%) 0.015 / 3.37 0.82 0.05 / 2.18 0.63 0.04 / 2.09 0.59 0.1 / 8.83 0.94

total longitudinal steel ratio tot (%) 0 / 6.29 2.08 0.07 / 4.27 1.37 0.2 / 6.19 1.32 0.75 / 5.5 2.05

diagonal steel ratio, d (%) 0 / 1.68 0.028 0 / 0.25 0.004 - - - -

t t l i ld t f MP 118 / 1497 497 220 / 1375 435 3 178 / 1375 513 200 / 1569 482 5transverse steel yield stress fyw, MPa 118 / 1497 497 220 / 1375 435.3 178 / 1375 513 200 / 1569 482.5

longitudinal steel yield stress fy, MPa 281 / 1275 467.5 276 / 1273 471.2 331 / 596 438.7 240 / 648 428

Yield & failure properties of RC Yield & failure properties of RC e d & a u e p ope t es o Ce d & a u e p ope t es o Csectionssections

M-φ at yielding of section w/ rectangular compression zone (width b, effective depth d) –p ( , p )

section analysis• Yield moment (from moment-equilibrium & elastic σ-ε laws):

112111

2

3 116

123

15.02

Vyy

syycy

y EEbdM

• Yield moment (from moment-equilibrium & elastic σ-ε laws):

f

• 1, 2 : tension & compression reinforcement ratios, v: “web” reinforcement ratio, ~uniformly distributed between 1, 2 : (all normalized to bd); 1 = d1/d.

C t t i ldi f t i t l dE

f

ys

yy

1

2/1

• Curvature at yielding of tension steel:• from axial force-equilibrium & elastic σ-ε laws ( = Es/Ec):

ABAy 2/122 2

y

vy

v bdfNB

bdfNA 112121 15.0,

dEf

d yc

c

y

cy

8.1

yy ff• Curvature at ~onset of nonlinearity of concrete:

yy

11212121 15.0,8.1

vc

vsc

v Bbdf

NbdE

NA

Moment at corner of bilinear envelope to experimental moment-deformation curve vs yield moment from section analysisdeformation curve vs yield moment from section analysis

Left: 2085 beams/columns, CoV:16.3%; Right: 224 rect. walls, CoV:16.9% 10500

4000

di M 1 025M

9000

10500

median: My,exp= 0.99My,pred

3000

kNm

)

median: My,exp=1.025My,pred

6000

7500

Nm

)

2000

My,

exp

(k

3000

4500

My,

exp

(kN

0

1000

0

1500

M

Bias by +2 5% or 1% because corner of bilinear envelope of the

00 1000 2000 3000 4000

My,pred (kNm)

00 1500 3000 4500 6000 7500 9000 10500

My,pred (kNm)

Bias by +2.5% or -1%, because corner of bilinear envelope of the experimental moment-deformation curve ≠ 1st yielding in section. Same bias

considered to apply to predicted yield curvature.

Empirical formulas for yield curvature - section w/

f541 f751

rectangular compression zone

dEf

s

yy

54.1

hEf

s

yy

75.1for beams or columns:

1.37 yy

s

fE d

1.47 y

ys

fE h

for walls:

Empirical expressions don’t have a bias w.r.to experimental yield moment; but scatter is larger:y ; gIn ~2100 test beams, columns or walls: CoV: ~18%

Flexural Flexural failures failures -- columnscolumns

Flexural Flexural failures failures -- beamsbeams

Conventional definition of ultimate deformationConventional definition of ultimate deformationThe value beyond which, any increase in deformation cannot increase the resistance above 80% of the maximum previous p(ultimate) resistance.

250250 200Q RC

0

50

100

150

200

rce

(kN

)

0

50

100

150

200

rce

(kN

)

50

100

150

N)

Q-RC

-200

-150

-100

-50

0

a

App

lied

for

-200

-150

-100

-50

0

a

Appl

ied

for

-100

-50

0

Forc

e (k

N

conventional ultimate deformation coincides

-125 -100 -75 -50 -25 0 25 50 75 100 125-250

Deflection (mm)

100

150

200

250

N)

-125 -100 -75 -50 -25 0 25 50 75 100 125-250

Deflection (mm)

100

150

200

250

-120 -90 -60 -30 0 30 60 90 -200

-150

-100

Displacement (mm)

deformation coincides w/ real failure conventional ultimate deformation

before “real” failure

-100

-50

0

50

100

App

lied

forc

e (k

-100

-50

0

50

100

pplie

d fo

rce

(kN

) Displacement (mm)

-125 -100 -75 -50 -25 0 25 50 75 100 125-250

-200

-150b

Deflection (mm)-125 -100 -75 -50 -25 0 25 50 75 100 125

-250

-200

-150b

Ap

Deflection (mm)

Ultimate curvature of section with rectangular compression zone from section analysiscompression zone, from section analysis

C t l• Concrete σ-ε law:– Parabolic up to fc, εco, c co

– constant stress (rectangular) for εco< ε < εcu

• Steel σ ε law:• Steel σ-ε law:– Elastic-perfectly plastic, if steel strain rather low and

t f il fi tconcrete fails first;– Elastic-linearly strain-hardening, if steel strains are

relatively high and steel breaks at stress and strain ft, εsu.

Possibilities for ultimate curvature:

su

1. Section fails by rupture of tension steel, εs1= εsu, before extreme compression fibers reach their ultimate strain (spalling), εc < εcu →Ultimate curvature occurs in unspalled section due to steel rupture: dsu

su

1Ultimate curvature occurs in unspalled section, due to steel rupture:

2. Compression fibres reach their ultimate strain (spalling): εc = εcu →the confined concrete core becomes now the member section. Two possibilities:

i. The moment capacity of the spalled section, ΜRo, never increases above 80% of the moment at spalling Μ : Μ < 0 8Μ →

dcu

cucu

80% of the moment at spalling, ΜRc: ΜRo< 0.8ΜRc → Ultimate curvature occurs in unspalled section, due to the concrete:

ii. Moment capacity of spalled section increases abovep y p80% of the moment at spalling: ΜRo> 0.8ΜRc →The confined concrete core is now the member section and Cases 1 and 2(i) li d f th fi d th t ibiliti f tt i t2(i) - applied for the confined core - are the two possibilities for attainment of the ultimate curvature → φsu, φcu calculated as above but for the confined core; the minimum of the two is the ultimate curvature.

cu

co

cu

co

cus

ycRc

ffbdM

421

321

31)1)((

12))(1(

2

11

2112

Ultimate curvature of section w/ rectangular compression zone for steel rupture: d

susu

1

tt

cocuysutt

coysu ffff

1111 1133

1

Steel ruptures before concrete crushes, after compression steel yields, if ν: dsu1

cusu

cusu

y

tv

y

t

sucuysu

ysu

y

tv

y

t

ysu ff

ff

ff

ff

1

111212 1

11132

13

• ξsu calculated from axial force equilibrium for:

t

vy

t

su

co

y

t

suf

ff

ff

12

13

1 1211

vy

t

su

co

ff

1

311 1

ν=N/bdfc, ω1, ω 2 : tension & compression mech. reinforcement ratios, ωv: “web” mech. reinforcement ratio ~uniform distribution between ω1, ω 2 (ω= fy/fc); 1=d1/d.

ysutt

coysu ff

13

112

11

1, 2 ( y ); 1 1

Steel ruptures before concrete crushes or compression steel yields, if ν:

ysuyv

yysu ff

2112

• ξsu from: 1)1(23

1 2

1

y

su

sus

t

y

tv

su

co

Ef

ff

01)1(23

1)1(3

21

)(

21

11211

121

1

y

su

sus

t

y

t

y

su

y

t

su

co

y

su

sus

t

y

t

y

su

y

t

su

co

ysusysu

Ef

ff

ff

Ef

ff

ff

f

Ultimate curvature of section w/ rectangular compression zonefor concrete crushing:

dcu

cu

Concrete crushes after tension steel yields, w/o compression steel yielding, if ν :co

cuycuv

31

dcu

• ξcu from axial force equilibrium, for:ycuycu

yv

311 11

112

Concrete crushes w/ tension & compression steel yielding if ν :

012

11)1(23

1 1

1

12

1

121

22

1

y

cuv

y

cu

y

cu

ycu

ycuv

cu

co

11

Concrete crushes w/ tension & compression steel yielding, if ν :

ycu

cocu

ycu

ycuv

ycu

cocu

ycu

ycuv

3

131

1 11

12111

12

vcu

co

vcu

2

311

11

1

1211

• ξcu from:

Concrete crushes after compression steel yields w/o tension steel yielding if ν : Concrete crushes after compression steel yields, w/o tension steel yielding, if ν :

• ξcu from:

ycu

cocu

ycu

ycuv 31 1

112

0121)1(23

11

111

122

2

1

y

cuv

y

cu

y

cu

ycu

ycuv

cu

co

Confined core after spalling of concrete cover. Parameters are denoted by an asterisk and computed with: b, d, d1 replaced by geometric parameters of the core: bc, dc, dc1;

N li d t b d i t d f bd

Unconfined full section – Steel rupture

δ1 satisfies Eq.(3.38)?

yes no

Ruptureof tension steel

N, 1, 2, v normalized to bcdc, insteadof bd; σ-ε parameters of confined concrete, fcc, cc, used in lieuof fc, cu

no

yes su from

Eq.(3.41) ν<νs,y2 - LHS

Eq.(3.39)? ν<νs,c - RHS Eq.(3.39)?

yes

φsu from Eq.(3.36)

no

yes *su from Eq.(3.41)

ν*<ν*s,y2 - LHS Eq.(3.39)?

Unconfined full section – Spalling of concrete cover

ν<νs,c - RHS Eq.(3.39)?

yes su from Eq.(3.40)

q ( )

no

no

ν*<ν*s,c - RHS Eq.(3.39)?

yes *su from Eq.(3.40)

φsu from Eq.(3.36)

δ1 satisfies

Eq.(3.42a)? yes

ν<νc y2 - LHS cu from yes

no

ν< c y1 - LHSyes

Failure of compression zone (concrete)

no

ν<νc,y2 LHS Eq.(3.44)?

ν<νc,y1 - RHS Eq (3 44)?

Eq.(3.47)

cu from Eq (3 46)

no

no ν< c,y2 - RHS E (3 48)?

ν c,y1 LHS Eq.(3.48)?

no

no

ν*<ν*c,y2 - LHS Eq.(3.44)?

*cu from Eq.(3.47)

yes

Eq.(3.44)? Eq.(3.46)

cu from Eq.(3.45)

Eq.(3.48)?

cu from Eq.(3.49)

yes yes

no

noν*<ν*c,y1 - RHS Eq(344)?

*cu from Eq(346)

φcu from Eq.(3.37)

Compute moment resistances: MRc (of full, unspalled section) and MRo (of confined core, after spalling of cover).

MR < 0 8MR ? φcu from yes

*cu from Eq.(3.45)

Eq.(3.44)? Eq.(3.46)

yes

MRo< 0.8MRc? φEq.(3.37)

no

Ultimate curvature of confined core after spalling of concrete cover

Test results vs ultimate curvature w/ failure strains for cyclic flexure per EN 1998-3:2005 :

Steel: εsu: 2.5%, 5%, 6% for steel class A, B, C per Eurocode 2

ffαρε /5.0004.0 0 8 ccywsccu ffαρε /5.0004.0,

0,7

0,8

median: φu,exp=0.96φu,pred

cyclic test results0,5

0,6

0,4

,

exp

(1/m

) Cyclic all

Concrete crushing

Steel rupture

Median 0 96 0 90 0 98

0,2

0,3φu,

e Median 0.96 0.90 0.98

C.o.V. 46.7% 55.1% 38.0%

No 205 97 108

0,1 cyclic, slip

cyclic, no-slip

No. 205 97 108

00 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

φu,pred (1/m)

102• Monotonic flexure:

Test results vs ultimate curvature w/ failure strains for cyclic flexure per Biskinis & Fardis 2010 (adopted in fib MC2010):

Kmmhw

ccu

1285.0

)(100035.0*

10

2*

Monotonic flexure:•max. available steel strain: (7/12)εsu

• Cyclic flexure:

Kmmhw

ccu

12.0

)(100035.0* •max. available steel strain: (3/8)εsu

1.2

monotonic & cyclic data, no. 474, median=1 00

1median: φu,exp=φu,pred

median=1.00, CoV=49.7%

0 6

0.8

1/m

)

cyclic test data:0.4

0.6

φu,

exp

(1

Cyclic Concrete Steel

0.2cyclic, slipcyclic, no-slipmonotonic, slip

t i li

all crushing rapture

Median 0.99 0.99 1.01

00 0.2 0.4 0.6 0.8 1 1.2

φu,pred (1/m)

monotonic, no-slip C.o.V. 44.2% 52.6% 34.2%

No. 205 97 108

Yield & failure properties of RC Yield & failure properties of RC e d & a u e p ope t es o Ce d & a u e p ope t es o Cmembersmembers

Flexural Flexural behaviorbehavior at member level (at member level (MomentMoment--chord rotation)chord rotation)

Definition of chord rotations, θ, at member endschord rotation)chord rotation)

Elastic moments at ends A, B from chord rotations at A, B: • ΜΑ = (2ΕΙ/L)(2θΑ+θΒ), • ΜΒ = (2ΕΙ/L)(2θΒ+θΑ)

Fixed-end rotation of member end due to bar slippage from their anchorage zone beyond member end

Sli f t i b f i b d d ti ( f j i• Slippage of tension bars from region beyond end section (e.g. from joinor footing) →rigid body rotation of entire shear span = fixed end rotation θrigid-body rotation of entire shear span = fixed-end rotation, θslip

(included in measured chord-rotations of test specimen w.r. to base or joint; doesn’t affect measured relative rotations between any two member sections).affect measured relative rotations between any two member sections).

• If s = slippage of tension bars from anchorage → θslip= s/(1-ξ)d• If bond stress uniform over straight length lb of bar beyond section of maximumg g b y

moment → bar stress decreases along lb from σs (=fyL at yielding) at section omaximum M to zero at end of lb → s=σslb/(2Es) l b d f d d it l th ( A /( d ) d /4) di id d b b• lb= bond force demand per unit length (=Asσs/(dbL)=dbLσs/4), divided by ~bonstrength (assume =√fc)

• ε (=σ /E )/(1-)d = φεs( σs/Es)/(1 )d φ• At yielding of member end section

fd(fyL, fc in MPa )

c

ybLyslipy f

fd

8,

Fixed-end rotation of member end due to rebar pull-out from anchorage zone beyond member end, at member yieldingφy,measured/(φy,predicted+y,slip,/lgauge) no.160 measurements w/ slip:

median = 1.0, C.o.V = 34%

Ratio: 2.25

2.5

experimental-to-predicted yield

t ( /1.75

2

e to

slip

)

curvature (w/ correction for fixed-end-

1.25

1.5

inci

ples

+φ y

,due

fixed endrotation) in terms of gauge length 0.75

1

xp / (φ

y,1st

-pri

0.25

0.5φy,

ex

0

0 0.25 0.5 0.75 1 1.25 1.5lgauge / h

Chord rotation of shear span at yielding of end sectionper EN 1998-3:2005

slipyslVs

yy aLhzaL

,5.110014.03

• Rect. beams or columns:

R t t ll sL3

slipyslVs

yy azaL,0013.0

3

• Rect. or non-rect. walls:

“Shift rule”: 3Shift rule : Diagonal cracking shifts value of force in tension reinforcement to a section at a distance from member end equal to z (internal lever arm) • z = d d in beams columns or walls of barbelled or T section• z = d-d1 in beams, columns, or walls of barbelled or T-section, • z = 0.8h in rectangular walls. – av = 0, if VRc > My/Ls;

1 if V ≤ M /L– av = 1, if VRc ≤ My/Ls .VRc = force at diagonal cracking per Eurocode 2 (in kN, dimensions in m, fc in MPa):

dbANf

df

dV wcccR

15.02.012.0135,100180max 3/16/13/1

1, Add c

– asl = 0, if no slip from anchorage zone beyond end section; – asl = 1, if there is slip from anchorage zone beyond end section.

Test-model comparison – θy Beams/rect. columns, no. tests: 1653

3

3.5

median=1.01, CoV=32.1% 2.5

3

2%

)

median: θy,exp=1.01θy,pred

1.5

θ y,e

xp(%

1

θ

0

0.5

rect. beams / columns0

0 0.5 1 1.5 2 2.5 3 3.5

θy,pred (%)

Test-model comparison – θy Walls, no. tests: 386a s, o tests 386

1

1,2

1

1,2

0,8

1

0,8

1

0 4

0,6

θ y,e

xp(%

)


0 4

0,6

θ y,e

xp(%

)


0,2

0,4θ

Rectangular walls0,2

0,4θ

Rectangular walls

00 0,2 0,4 0,6 0,8 1 1,2

θ (%)

Non-rectangular Walls0

0 0,2 0,4 0,6 0,8 1 1,2

θ (%)

Non-rectangular Walls

Modified expression adopted in MC2010Expression in EN 1998-3:2005median=0.97, CoV=31.1% 50 00045 1s VL a z h

θy,pred (%)θy,pred (%)

median=1.01, CoV=30.9%,0.00045 1

3 3s V

y y sl y slips

aL

Test-model comparisons - θyCircular columns - not in EN 1998-3:2005: no. tests: 291

,20.0027 1 min 1;

3 15s V s

y y sl y slipL a z L a

D

3 15y y y pD

median=1.00, CoV=31.7%2,5

,2

median: θy,exp=θy,pred

1,5

p (%

)

1θ y,e

x0,5

circular0

0 0,5 1 1,5 2 2,5

θy,pred (%)

Effective elastic stiffness, EIeff(for linear or nonlinear analysis)(for linear or nonlinear analysis)

Part 1of EC8 (for design of new buildings): EIeff : secant stiffness at yielding =50% of

uncracked gross-section stiffness.- OK in force-based design of new buildings (safe-

sided for forces);- Unsafe in displacement-based assessment for

displacement demands).

More realistic:sy

eff

LMEI

3

secant stiffness at yielding of end of shear span L =M/V

y3

end of shear span Ls=M/V

Test-model comparison – EIeffBeams/rect. columns, no. tests: 1616

600


500 median: (MyLs/3θy)exp =MyLs/3θy)pred

400

MNm

2 )LM 300

3θy) e

xp (M

y

syeff

LMEI

3

200

(MyL

s/3y

0

100

rect. beams / columns0

0 100 200 300 400 500 600

(MyLs/3θy)pred (MNm2)

4000median: 1250

1500

median:

Test-model comparison – EIeff Walls, no. tests: 386sy LM

EIdetail

With i f θ

3000

MNm

2 )

median: (MyLs/3θy)exp=1.035(MyLs/3θy)pred

1000

1250

MNm

2 )


y

syeffEI

3

With expression for θy in EN 1998-3:2005median=1 035 1000

2000

yLs/3θ

y)exp(M

500

750

yLs/3θ

y)exp(M

median=1.035CoV=42.8%

0

1000

0 1000 2000 3000 4000(M

y

( / θ ) ( 2)

non-rectangular walls

rectangular walls

0

250

0 250 500 750 1000 1250 1500

(My

( / θ ) ( 2)


rectangular walls

4000median: (M L /3θ ) =0 98(M L /3θ ) d

1250

1500

median: (M L /3θ ) =0 98(M L /3θ ) d

detail(MyLs/3θy)pred (MNm2) (MyLs/3θy)pred (MNm2)

With new expression for θ of walls

3000

p(M

Nm

2 )

(MyLs/3θy)exp 0.98(MyLs/3θy)pred

750

1000

p(M

Nm

2 )


for θy of walls (adopted in MC2010)median=0.98 1000

2000

MyL

s/3θ

y)exp

non-rectangular walls 250

500

MyL

s/3θ

y)exp


CoV=41.1%0

0 1000 2000 3000 4000

(M



rectangular walls

0

250

0 250 500 750 1000 1250 1500(M



rectangular walls

Test-model comparison – EIeffCircular columns - not in EN 1998-3:2005, no. tests: 273


,sy

eff

LMEI

3 ed a 0 99, Co 3 %

3500

4000median: (MyLs/3θy)exp=0.99(MyLs/3θy)pred

175median: (MyLs/3θy)exp=0.99(MyLs/3θy)pred

y3

3000

3500

m2 )


125

150

m2 )


2000

2500

exp

(M

Nm

75

100

exp

(M

Nm

1000

1500

MyL

s/3θ y

) e

50

75

MyL

s/3θ y

) e

0

500

(M

circular0

25(Mcirculardetail

0 500 1000 1500 2000 2500 3000 3500 4000

(MyLs/3θy)pred (MNm2)0 25 50 75 100 125 150 175


Empirical secant stiffness to yielding, EIeff, independent of amount of reinforcement - not in EN 1998-3:2005

)](;50min[048.016.0;maxln8.0 MPa

AN

hLa

IEEI seff

AhIE ccc

(all variables known before dimensioning the longitudinal reinforcement)(all variables known before dimensioning the longitudinal reinforcement)

– If there is slippage of longitudinal bars from their anchorage zoneIf there is slippage of longitudinal bars from their anchorage zone beyond the member end: • a = 0.081 for columns;;• a = 0.10 for beams or non-rectangular walls (barbelled, T-, H-section);• a = 0.115 for rectangular walls;a 0.115 for rectangular walls; • a = 0.12 for members with circular section.– If there is no slippage of longitudinal bars: effective stiffness x 4/3If there is no slippage of longitudinal bars: effective stiffness x 4/3

Test-model comparison – Empirical EIeff, independent of amount of reinforcement - not in EN 1998-3:2005

Beams/columns

600of amount of reinforcement not in EN 1998 3:2005

no. tests: 1616median=1.00CoV=36 1%

500

median: (MyLs/3θy)exp =EIpredCoV=36.1%

400M

Nm

2 )

( y s y)exp pred

300

θ y) e

xp (M

200

MyL

s/3θ

100(M

00 100 200 300 400 500 600

EIpred (MNm2)

detail4000median: (MyLs/3θy)exp=EIpred

1500

median: (MyLs/3θy)exp=EIpred

Test-model comparison –Empirical EI ff

detail

2000

3000

xp(M

Nm

2 )

1000

xp(M

Nm

2 )

Empirical EIeffindependent of reinforcement , not

Wallsno tests: 386

1000

2000

MyL

s/3θ

y)ex


500

MyL

s/3θ

y)ex


in EN1998-3: 2005

no. tests: 386median=1.00CoV=44.6%

00 1000 2000 3000 4000

(M

EIpred (MNm2)

non rectangular walls

rectangular walls

00 500 1000 1500

(M

EIpred (MNm2)

non rectangular walls

rectangular walls

3000

3500

4000median: (MyLs/3θy)exp=EIpred 150

175median: (MyLs/3θy)exp=EIpred

EIpred (MNm ) EIpred (MNm )

detail

2000

2500

3000

xp

(MN

m2 )

100

125

xp

(MN

m2 )

Circular columnsno tests: 273

1000

1500

(MyL

s/3θ y

) ex

25

50

75

(MyL

s/3θ y

) ex

no. tests: 273 median=0.995 CoV=31.4%

0

500

0 500 1000 1500 2000 2500 3000 3500 4000

(

EIpred (MNm2)

circular0

25

0 25 50 75 100 125 150 175(

EIpred (MNm2)

circular

L50

Flexure-controlled ultimate chord rotation from curvature & plastic hinge length per EN 1998-3:2005

s

plplyuy

pluyu L

LLφφθθθθ

5.01)(

• y: yield curvature (section analysis);

dd su

su

cccu

ccuu )1(

,min,

,

Option 1: Confinement per Eurocode 2O ti 2 N EN 1998 3 2005

ywsxccc fαρff 5.2125.1 86.0/7.31 cywsxccc ffff

Option 2: New, per EN 1998-3:2005ywsxccc fαρff 5:05.0 cywsx ffαρ

:05.0 cywsx ffαρ

cywsxcoccu ff /2.0, y

ccywsxccu ff /5.0004.0,

yy

cybs ffdhLL 24.017.01.0pl

index c: confined;

cybs ffdhLL /11.02.030/pl : confinement effectiveness:

rectangular section:

ihh bssα 111

2

ρs: stirrup ratio; Ls=M/V: shear span at member end; h: section depth;

– rectangular section: – circular section & hoops:sh: centerline spacing of stirrups,

cccc hbhb

α6

12

12

12

21

c

h

Dsα

h: section depth;db: bar diameter; fy, fc: MPa

Dc, bc, hc: confined core dimensions to centerline of hoop; bi: centerline spacing along section perimeter of longitudinal bars (index: i) engaged by a stirrup corner or cross-tie.

Test-model comparison – ultimate chord rotation from curvatures & plastic hinge length per EN 1998-3:2005, no. tests: 1100

Option 1: confinement per EC2median=0.88, CoV=52.3% Option 2: new confinement per EC8-3

di 0 91 C V 52 2%

12 5Cyclic loading median:θ =0 91θ

median=0.91, CoV=52.2%17,5

Cyclic loading median:θ =0 88θ

10

12,5y g median: θu,exp=0.91θu,pred

12,5

15median: θu,exp=0.88θu,pred

7,5

exp

(%)

10

exp

(%)

5θ u,e

5

7,5

θ u,e

0

2,5beams & columnsrect. wallsnon-rect. sections

0

2,5 beams & columnsrect. wallsnon-rect. sections

00 2,5 5 7,5 10 12,5

θu,pred (%)

00 2,5 5 7,5 10 12,5 15 17,5

θu,pred (%)

Flexure-controlled ultimate chord rotation of rect. beams, columns, walls, non-rect. walls & circular columns from curvatures & plastic

hi l th b Bi ki i & F di 2010 2013 d t d i fibMC2010Flexure-controlled ultimate chord rotation, accounting separately for slippage in yield-penetration length, from yielding till ultimate deformation:

hinge length by Biskinis & Fardis 2010, 2013 - adopted in fibMC2010

slippage in yield penetration length, from yielding till ultimate deformation:

pl

plyuslipuslyu LL

La2

1)(, 43f

Confinement per fib MC2010:

sL2

2– Monotonic loading - Rect. beams, columns, walls, non-rectangular walls:

5.31

c

ywwccc f

fff

,57.0)(

100035.02

,cc

yww

occu f

fαρmmh

ε

g , , , g

L

hL s9i04011

alnosumonsu min,, 127

– Cyclic loading:

h

hL smonpl ;9min04.01.1,

4010003502

yww fαρε

3

hLhL s

cypl ;9min3112.0,

,4.0)(

0035.0,cco

ccu fmmhε

alnosucysu min,, 8

• Rect. beams/columns/walls, non-rect. walls:

hcypl 3,

• Circular columns:

DLDL s

circypl ;9min6116.0,,

Post-yield fixed-end rotation of member end due to bar slippage from yield penetration length beyond member end, from yielding

till lti t fl l d f ti Bi ki i & F di 2010 2013till ultimate flexural deformation per Biskinis & Fardis 2010, 2013 -adopted in fibMC2010 (not in EN 1998-3:2005)

2/1659 dd• Monotonic loading:

• Cyclic loading: 2/105.5 bLbLli dord

2/165.9, uybLubLsli pu dord

Cyclic loading: 2/105.5, uybLubLslipu dord

C l t ll t fComplete pull-out of beam bars, due to short anchorage inshort anchorage in corner joint← →

Test-model comparison – Cyclic ultimate chord rotation from curvatures & plastic hinge length per Biskinis & Fardis 2010, 2012

Rect. beams/columns/walls, non-rect. walls - no. tests: 1100 Circular columns – no. tests: median=1.00, CoV=43.2%

C li l di

16

143, median=1.00, CoV=30.3%

10

12,5Cyclic loading

median: θu,exp=θu,pred

12

14

median: θu,exp=θu,pred

7,5

10

p (%

)

8

10

(%

)

5θ u,e

xp

4

6

8

θ u,e

xp

0


2

4

circular00 2,5 5 7,5 10 12,5

θu,pred (%)

o ect sect o s0

0 2 4 6 8 10 12 14 16

θu,pred (%)

circular

Empirical cyclic flexure-controlled ultimate chord rotation of rect. beams/columns/walls, non-rect. walls in EN1998-3:2005 & fibMC2010

)25.1(25);01.0(max)';01.0(max)3.0(016.0 dc

ywsx

10035.0

s

225.0

cumρf

fαρ

ν

hLf

ωωθ

or:)275.1(25

);010(max)';01.0(max)25.0(0145.0 dc

ywsx

10035.0

s2.0c

3,0

yumplum

ρf

fαρ

ν

hLf

ωωθθθ

, ': mechanical ratio of tension (including web) & compression steel; : N/bhfc (b: width of compression zone; N>0 for compression);

);01.0(max hω

c ( p p )Ls/h : M/Vh: shear span ratio;α : confinement effectiveness factor :

ihh

hbb

hs

bsα

61

21

21

2

sx: Ash/bwsh: transverse steel ratio // direction of loading;d: ratio of diagonal reinforcement.

W ll 1st i di id d b 1 6 2nd lti li d b 0 6

cccc hbhb 622

• Walls: 1st expression divided by 1.6; 2nd multiplied by 0.6• Cold-worked brittle steel: 1st expression divided by 1.6; 2nd by 2.0

Non-seismically detailed members with continuous bars • Plastic part, pl

um=θum-y, of ultimate chord rotation: divided by 1.2.

Test-model comparison – Empirical cyclic ultimate chord rotation of members with seismic detailing per EN 1998-3:2005 – no. tests 1100

Model for total um Model with um=y+pl

median=1.00, CoV=37.8% median=1.00, CoV=37.6%

12,5Cyclic loading

median: θu,exp=θu,pred 12,5 median: θu,exp=θu,predmedian: θu,exp=θu,pred

Cyclic loading

10

, , p ,p

10

, p ,pu,exp u,pred

7,5

exp

(%)

7,5

,exp

(%)

5θ u,

5% fractile: θu,exp=0.5θu,pred

5

θ u,


0


0


u,exp u,pred

00 2,5 5 7,5 10 12,5

θu,pred (%)

00 2,5 5 7,5 10 12,5

θu,pred (%)

Test-model comparison – Empirical cyclic ultimate chord rotation of members without seismic detailing per EN 1998-3:2005 – no. tests 48

Model for total um Model with um=y+pl

median=1.00, CoV=30.6% median=0.99, CoV=31.8%

8

9

median:θ =θ d 8

9

6

7

8 median: θu,exp=θu,pred

7

8 median: θu,exp=0.99θu,pred

5

6

xp (%

)

5

6

xp (%

)3

4θ u,e

x

3

4θ u,e

x

1

2

1

2

00 1 2 3 4 5 6 7 8 9

θu,pred (%)

00 1 2 3 4 5 6 7 8 9

θu,pred (%)

General empirical ultimate chord rotation of rect. beams/columns/ walls & non-rect. walls per Biskinis & Fardis 2010, adopted in fib

MC2010 (not in EN1998 3:2005)MC2010 (not in EN1998-3:2005)

dc

yww

f

f

cs

slcyhbwst

plu f

hL

bhaa

1002.03

1

1

2 225.125;9min);010max();01.0max(2.0;10min;5.1max052.016.01)525.01(

αsthbw:

0.022 for hot-rolled or Tempcore bars; 12,5 median: θu,exp=θu,predmedian: θu,exp=θu,predcyclic loading

w hb 1);01.0max(

0.022 for hot rolled or Tempcore bars; 0.0095 for brittle cold-worked bars;αcy:

1 f li l di

10

= 1 for cyclic loading, = 0 for monotonic;αsl:

7,5

θ u,e

xp (%

)

sl= 1 if slippage of long. bars from anchorage zone is possible, = 0 otherwise; 2 5

5

θb & l


= 0 otherwise;bw: width of (one) web

0

2,5

0 2 5 5 7 5 10 12 5

beams & columnsrect. wallsnon-rect. sections

Non-seismically detailed members:• pl

um=θum-y divided by 1.2. no. tests: 1100, median=1.00, CoV=37.6%

0 2,5 5 7,5 10 12,5θu,pred (%)

Effect of lapEffect of lap--splicing the column splicing the column ect o apect o ap sp c g t e co usp c g t e co ubars in the plastic hinge regionbars in the plastic hinge region

Members with ribbed bars lap-spliced over length lo inside the plastic hinge region per EN 1998-3:2005 or other options

• EN 1998-3:2005:1. Both bars in pair of lapped compression bars count as compression steel.2. For the yield properties (My, y , θy), the stress fs of tension bars is:

fs = fy(lo/loy,min), if lo< loy,min=(0.3fy/√fc)db (fy, fc in MPa)

3. Ultimate chord rotationθu=y+pl

u(lo/lou,min), if lo < lou,min = dbfy/[(1.05+14.5αrsωsx)√fc],– fy, fc in MPa, ωsx=ρsxfyw/fc: mech. transverse steel ratio // loading, – αrs=(1-sh/2bo)(1-sh/2bo)nrestr/ntot (nrestr/ntot: restrained-to-total lap-spliced bars).

O Eli h & L tt 2007 f fib MC2010• Or, Eligehausen & Lettow 2007 for fib MC2010:

,202.511.0

max

3/12.025.055.0

tdcb fkKccflf

shls AnkkK 1

• cd = min[a/2; c1; c] ≥ db , cd ≤ 3db

,)20;max(20

2.51 ytrdbbb

s fkKcdmmdd

f

hbb

tr sdnkK

d [ ; 1; ] b , d b

• cmax = max[α/2; c1; c] ≤ 5db

• fy, fc in MPa

Test-model comparison: Yield moment & chord rotation, effective stiffness,ultimate cyclic chord rotation - lapped ribbed bars per EN 1998-3:2005

1,5

2median: θy,exp=1.05θy,pred

1500

1750

2000

median: My,exp=My,pred

1

θ y,e

xp (

%)

θno.tests: 92

median=1 05750

1000

1250ex

p (K

Nm

)

Mno.tests: 114

0

0,5θymedian=1.05

CoV=20%0

250

500My,e Mymedian=1.00

CoV=11.8%

250

median:


00 0,5 1 1,5 2

θy,pred (%)

00 250 500 750 1000 1250 1500 1750 2000

My,pred (KNm)

150

200

(MN

m2 )


8

10

(%)

50

100

MyL

s/3θ y

) exp

EIeff2

4

6

θ u,e

xp

θuno tests: 81no tests: 92

00 50 100 150 200 250

(M


0

2

0 2 4 6 8 10 12θu,pred (%)

uno.tests: 81median=1.035

CoV=39.3%

no.tests: 92median=0.955

CoV=24.8%

2000

di M 0 98M

Test-model comparison: Yield moment & chord rotation, effective stiffness, - lapped bars, steel stress per Eligehausen & Letow 2007

1250

1500

1750

m)

median: My,exp=0.98My,predsteel stress per Eligehausen & Letow 2007

no.tests: 114

500

750

1000

My,e

xp

(KN

mMedian=0.98CoV=12%

no tests: 92

0

250

500

0 250 500 750 1000 1250 1500 1750 2000

M

Myno.tests: 92


250

median: (M L /3θ ) =0 97(M L /3θ )


0 250 500 750 1000 1250 1500 1750 2000

My,pred (KNm)CoV 20.5%

150

200

(M

Nm

2 )

(MyLs/3θy)exp=0.97(MyLs/3θy)pred

1

1,5

p (%

)

50

100

MyL

s/3θ y

) exp

0,5

θ y,e

x

θ EIno tests: 92

00 50 100 150 200 250

(M(M L /3θ ) d (MNm2)

00 0,5 1 1,5 2

θy,pred (%)

θy EIeffno.tests: 92


Members with smooth hooked bars, with or without lap splice (of length lo) in the plastic hinge per EN 1998-3:2005Provided that lapping lo ≥ 15db:1 Yield properties M θ :1. Yield properties My, y, θy:

– As in members with continuous ribbed bars.2 Ultimate cyclic chord rotation:2. Ultimate cyclic chord rotation:

– For continuous bars:• 0.8θum (: ~0.95 for smooth bars /1.2 for no seismic detailing) or• θum=y+0.75pl

um (: ~0.9 for smooth bars /1.2 for no seismic detailing)T t d l ti f θ– For lapping lo ≥ 15db :

• um x 0.019(10+min(40, lo/db)) or

Test-model-ratio for θumno laps laps

M di 1 015 1 03• θum=y+pl

um x 0.019 min(40, lo/db)

– Ls for plum not reduced by lo

Median 1.015 1.03C.o.V. 33.3% 33.4%no tests 34 11s o um ot educed by o

(ultimate condition not necessarily controlled by region right above the lap)no. tests 34 11

Cyclic shear resistance of RC Cyclic shear resistance of RC Cyc c s ea es sta ce o CCyc c s ea es sta ce o Cmembersmembers

Cyclic shear resistance per EN 1998-3:2005 • Shear resistance in pl. hinge after flex. yielding, as controlled by stirrupsp g y g, y p(linear decay of Vc & Vw with cyclic plastic rotation ductility ratio θpl=(-y)/y>0

spl

θR VAfLρμfANxhV ;5min1601)100;50max(160;5min0501550;min

Vw=ρwbwzfyw, due to stirrups (bw: web width, z: internal lever arm; ρw: shear reinf. ratio)t t l l it di l i f t ti

wcctotθccs

R VAfh

ρμfANL

V ;5min16.01)100;5.0max(16.0;5min05.0155.0;min2

ρtot: total longitudinal reinforcement ratioh: section depthx : depth of compression zone at yielding

• Shear resistance as controlled by web crushing (diagonal compression)

p p y gAc= bwd

zbfhL

fANV wc

stot

plR

,2min2.01)100,75.1max(25.01,15.0min8.11,5min06.0185.0

Shear resistance as controlled by web crushing (diagonal compression) Walls, before flexural yielding (θpl = 0) or after (cyclic θpl > 0):

fhfA wctot

ccR

)(

Squat columns (Ls/h ≤ 2) after flexural yielding (cyclic θpl > 0):

4 Nl

δ: angle between axis and diagonal of column (tanδ=0.5h/Ls)

2sin40,min10045.0135.11,5min02.0174 zbf

fANV wctot

cc

plR

Test-model comparison: Cyclic shear resistance in plastic hinge (after flexural yielding) as controlled by stirrups, per EN 1998-3:2005

no. tests: 306

1000


800

600

N)

400

V exp

(kN

200

V

Rectangular

0

Circularwalls & piers

00 200 400 600 800 1000

Vpred (kN)

Test-model comparison: Cyclic shear resistance as controlled by web crushing (diagonal compression), per EN 1998-3:2005

Squat columnsWalls2500 800

2000600

1500

N)

400N)

1000

V exp

(k 400

V exp

(k

500200

00 500 1000 1500 2000 2500

V (kN)

00 200 400 600 800

V (kN)

no. tests: 62, median=1.00, CoV=19.3%

no. tests: 40, median=1.00, CoV=9.6%

Vpred (kN) Vpred (kN)

FRP FRP Jackets per EN1998Jackets per EN1998--3:20053:2005

Experimental DatabaseExperimental Database4 Range and mean values of parameters in the tests of4. Range and mean values of parameters in the tests of

FRP-jacketed rectangular columns in the databaseParameter 219 columns with continuous bars (145

with CFRP, 24 with GFRP, 27 with AFRP and 23 with other composite)

45 columns with lap-spliced bars (42 with CFRP, 3 with GFRP) p ) , )

min-max mean min-max mean effective depth, d (mm) 170-720 296 180-720 393 shear-span-to-depth ratio L /h 1-7 4 3 55 2-6 6 4 5shear span to depth ratio, Ls/h 1 7.4 3.55 2 6.6 4.5 concrete strength, fc (MPa) 10.6-90 31.8 11.7-55 31 vertical bar yield stress, fy (MPa) 295-816 431 331-617 492 stirrup yield stress f (MPa) 200 750 388 280 535 442stirrup yield stress, fyw (MPa) 200-750 388 280-535 442 axial-load-ratio, N/Acfc 0-0.85 0.255 0-0.4 0.143 transverse steel ratio, w (%) 0-1.18 0.24 0-0.445 0.212

0 81 6 2 08 0 81 3 9 1 88total vertical steel ratio, tot (%) 0.815-7.6 2.08 0.815-3.9 1.88 geometric ratio of FRP, ρf (%) 0.01-5.31 0.605 0.13-7.5 1.04 nominal FRP strength (MPa) 113-4830 2755 532-4430 2621 elastic modulus of FRP, Ef (GPa) 5.8-390 166 17.8-390 184 lapping-to-bar-diameter ratio, lo/db - - 15-45 30.4

FRP-wrapping of plastic hinges in rectangular members with continuous bars per EN 1998-3:2005

• MR, My: Enhanced by FRP jacket (by 9% w.r.to calculated w/o confinement)– EN 1998-3:2005: increase neglected.

• Effective (elastic) stiffness EI : unaffected by FRP; pre damage: 35% drop

p

• Effective (elastic) stiffness EIeff: unaffected by FRP; pre-damage: 35% drop• EN1998-3:2005: Flexure-controlled ultimate chord rotation, θu: Confinement by FRP increases that due to the stirrups by αfρfff e/fc, where: y p y fρf f,e c,

– ρf=2tf/bw : FRP ratio // direction of loading;– ff,e: FRP effective strength:

c

fffu,fu,ffu,fu,ef,

,min7.01;5.0min,min

fρEεf

Eεff

fu,f, Ef : FRP tensile strength & Modulus; εu,f: FRP limit strain. CFRP/AFRP: εu,f=1.5%; GFRP: εu,f=2%FRP confinement effectiveness:

bh

RbRhf

3221

22

– FRP-confinement effectiveness:

bh3b, h: sides of section; R: radius at section corner

Test-model comparison - yield properties for FRP-wrapping of rectangular columns with continuous bars per EN 1998-3:2005

no.tests: 203 (pre-damaged or not)median=1.09, CoV=19.4%

no.tests: 159 (no pre-damage)median=1.03, CoV=37.3%

2500

median: M =1 09M d

3 non-predamaged predamagedmedian (non-predamaged) median (predamaged)

2000median: My,exp=1.09My,pred

2

2,5

1000

1500

(K

Nm

)

1,5

y,exp

(%

) 500

My,e

xp

0 5

1

θ yθM

00 500 1000 1500 2000 2500

0

0,5

0 0 5 1 1 5 2 2 5 3

θyMy

EIeff (no pre-damage), no.tests: 159, median=1.02, CoV=30%My,pred (KNm)

0 0,5 1 1,5 2 2,5 3θy,pred (%)

Yield properties for FRP-wrapping of the plastic hinge and continuous bars – Biskinis & Fardis 2007, 2013

• Yield moment, My:– Strength of FRP-confined concrete fcc, instead of fc, in section-analysis

for φ M including the calculation of the concrete Elastic Modulus E ;for φy, My, including the calculation of the concrete Elastic Modulus, Ec; – Ec may be estimated per fibMC2010, using fcc instead of fc:

Ec=10000(fcc(MPa))1/3 fabf2

c 0000( cc( a))

– fcc from (widely used) Lam & Teng 2003:c

fuff

y

x

c

cc

ffa

bb

ff ,3.31

fu,f: effective strength of FRP: fu,f=Ef(keffεu,f)Ef: Elastic Modulus of FRP, εu,f: FRP failure strain, k ff: FRP effectiveness factor equal to 0 6 per Lam & Teng

(~same results using other FRP-confinement models: Teng et al 2009, Samaan et al 1998, Bisby et al 2005, Ilki et al 2008, Wang et al 2012)

keff: FRP effectiveness factor, equal to 0.6 per Lam & Teng

• Chord rotation at yielding, θy, effective stiffness, EIeff :– Yield curvature φ calculated with f (as above) & multiplied timesYield curvature φy calculated with fcc (as above) & multiplied times

correction factor 1.06– EIeff = MyLs/3y

Test-model comparison - yield properties for FRP-wrapping of rect. columns with continuous bars per Biskinis & Fardis 2007, 2013

no.tests: 159 (no pre-damage)median=1.01, CoV=37.3%

no.tests: 203 (pre-damaged or not)median=1.06, CoV=19.3%

2000

2500

median: My,exp=1.06My,pred 2.5

3 non-predamaged predamagedmedian (non-predamaged) median (predamaged)

1500

2000

m) 2

%)

1000

xp

(KN

m

1.5

θ y,e

xp

(%

500

My,e

x

0.5

1

θM θ

00 500 1000 1500 2000 2500

( )

00 0.5 1 1.5 2 2.5 3

θ (%)

My θy

EIeff (no pre-damage): no.tests: 159, median=0.99, CoV=30.4%

My,pred (KNm) θy,pred (%)

Extension of empirical ultimate plastic chord rotation of rect. columns with continuous bars & FRP-wrapping -

• Pre-damaged or not:f

Biskinis & Fardis 2007, 2013

defffc

u

c

yww

ρffαρ

ffαρ

sc

νslcy

plu h

Lfωωaaθ 100

35.02.0

3.0

275.125,01.0max

',01.0max25.06.1

152.010185.0 ,

with:

fuffuf fρfρfρa 40i50140i

c

fuf

c

fufff

efffc

u

ffρ

ffρ

caffρa ,,

,

;4.0min5.01;4.0min

cf=1 8 for CFRP or polyacetal fiber (PAF) sheetscf 1.8 for CFRP or polyacetal fiber (PAF) sheets, cf=0.8 for GFRP or AFRP.f =E (k ε ): effective FRP strengthfu,f=Ef(keffεu,f): effective FRP strength

Test-model comparison – ultimate cyclic chord rotation for FRP-wrapping of rect. columns with continuous bars: no. tests 128 (pre-damaged or not)

EN 1998 3:2005 v Biskinis & Fardis 2007 2013EN 1998-3:2005

median=1 09 CoV=30 6%Biskinis & Fardis 2013

median=1 025 CoV=30 4%

EN 1998-3:2005 v Biskinis & Fardis 2007, 2013

median=1.09, CoV=30.6% median=1.025, CoV=30.4%25

median (all): θu,exp=1.09θu,pred

25

median (all): θu,exp=1.025θu,pred

20 20

15

exp

(%) 15

exp

(%)

10θ u,e

CFRP jacket

10θ u,e

CFRP jacket5 AFRP jacket

GFRP jacketPAF jacket

5 AFRP jacketGFRP jacketPAF jacket

00 5 10 15 20 25

θu,pred (%)

PAF jacket0

0 5 10 15 20 25θu,pred (%)

PAF jacket

f

Alternative ultimate cyclic chord rotation for continuous bars & FRP-wrapping – per GCSI (KANEPE)

Test-to-predicted ultimate cyclic chord rotation, θu – no.tests: 128,

di 1 75 C V 54 5%

wdc

cc

ff 25.1125.1

FRP l t i ti median=1.75, CoV=54.5%25

ωwd: FRP volumetric ratioα: FRP-confinement effectiveness factor CFRP (adopted here also for

20median (all): θ e p=1 75θ pred

20035.0 ccccu ffCFRP (adopted here also for AFRP and PAF):

215

exp

(%)

θu,exp 1.75θu,pred 2007.0 ccccu ff

fuef ff ,, FRP effective strength:

GFRP:

10θ u,e

CFRP jacket

ψ: reduction factor for number of layers (k); ψ 1 for k<4

5

jAFRP jacketGFRP jacketPAF jacket

41 k for k≥4ψ=1 for k<4

00 5 10 15 20 25

θu pred (%)

PAF jacket

32 φδy

uθ μμ

θθμ

FRP-wrapped rectangular members with ribbed bars lap-splicedover length lo in the plastic hinge: EN 1998-3:2005 & other options

• EN 1998-3:2005:1. Both bars in pair of lapped compression bars count as compression steel.2. For the yield properties (My, y , θy), the stress fs of tension bars is:

fs=fy(lo/loy,min), if lo< loy,min=(0.2fy/√fc)db (fy, fc in MPa), loy,min: one-third shorter th ith t FRPthan without FRP.

3. Ultimate chord rotationθ = +pl (l /l ) if l < l = d f /[(1 05+14 5 ρ f )√f ]θu=y+pl

u(lo/lou,min), if lo < lou,min = dbfy/[(1.05+14.5 αlρf ff,e)√fc],– fc: MPa, ρf=2tf/bw: FRP ratio // loading, ff,e: effect. FRP strength (MPa), – α = α (4/n ) (n : total lap-spliced bars; only the 4 corner bars restrained)– αl = αf(4/ntot) (ntot: total lap-spliced bars; only the 4 corner bars restrained).

• Or, Eligehausen & Lettow 2007 for fibMC2010:20251

1.03/12.025.055.0

db fkKccflf

ffffhl EtnkAnk1

• cd = min[a/2; c1; c] ≥ db , cd ≤ 3db

,)20;max(

2020

2.51 maxytr

db

d

b

c

b

bs fkK

cc

dc

mmdf

dlf

s

ffff

h

shls

bbtr E

Etnks

Ankdn

kK 1

d [ ; 1; ] b , d b

• cmax = max[α/2; c1; c] ≤ 5db

• fy, fc in MPa

Test-model comparison - Yield properties for FRP-wrapping of rect. columns with bars lap-spliced over length lo per EN1998-3:2005, no.tests 45

My: median=1.075, CoV=10.5% θy: median=1.075, CoV=17.7%

1750

2000

median: My,exp=1.075y,pred 1.75


1500

1750

1 25

1.5

1000

1250

(KN

m)

1

1.25

exp

(%

) 500

750

My,e

xp

0.5

0.75θ y,

0

250

0

0.25

0 0 25 0 5 0 75 1 1 25 1 5 1 75 2

My θy

0 250 500 750 1000 1250 1500 1750 2000

My,pred (KNm)0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

θy,pred (%)

Test-model comparison - yield properties of FRP-wrapped circular

l ith l li d b4000

5000

median: My,exp=My,pred

columns with lap-spliced bars per EN1998-3:2005 (Biskinis & Fardis 2007, 2013)

3000

(KN

m)

2013)

1000

2000

My,e

xp

(

Myno tests: 42


C V 15 2%0

0 1000 2000 3000 4000 5000

Mypred (KNm)


CoV=22%

CoV=15.2%

1 5

2

median θ 0 98θ800

1000y,pred ( )CoV 22%

1

1,5

xp

(%)

median:θy,exp=0.98θy,pred

600

xp(M

Nm

2 ) EIeff

0,5

θ y,e

x

200

400

(MyL

s/3θ y

) ex

median:(MyLs/3θy)exp=0 96(MyLs/3θy)pred

θy


00 0,5 1 1,5 2

θy,pred (%)

00 200 400 600 800 1000



CoV=22.2%

Extension of empirical ultimate plastic chord rotation forrectangular members with lap-spliced bars & FRP-wrapping –

• Required lapping for no adverse effect of lap-splice on ultimate deformationfd

Biskinis & Fardis 2007, 2013

u

ybou

ffρa

fdl

2min,

25.1405.1

with:

c

efffctension

ffn

,

5.1405.1

fuffufu fρfρfρa 40i50140iwith:

f =E (k ε )

c

fuf

c

fufff

efffc

u

ffρ

ffρ

caffρ ,,

,

;4.0min5.01;4.0min

fu,f=Ef(keffεu,f)ntension: number of lapped bars on tension side of section

,i

min 1,pl plouu lap

ll

• If lo<lou,min, θupl is reduced as:

,minoul • (αρfu/fc )f,eff neglected in θu

pl if 2/ntension≤0.5

Test-model comparison – ultimate cyclic chord rotation of FRP-wrapped ofrectangular columns w/ lap-spliced bars, no.tests 44 (pre-damaged or not)

EN1998-3:2005 v Biskinis & Fardis 2007 2013

EN 1998-3:2005median=0 89 CoV=36 3%

Biskinis & Fardis 2013median=1 005 CoV=23 2%

EN1998-3:2005 v Biskinis & Fardis 2007, 2013

median=0.89, CoV=36.3% median=1.005, CoV=23.2%12

median: θu,exp=0.885θu,pred


8

10, p ,p

8

10

6

8

xp

(%)

6

8

xp

(%)

4θu,

ex

4θu,

ex

0

2

0

2

00 2 4 6 8 10 12

θu,pred (%)

00 2 4 6 8 10 12

θu,pred (%)

Extension of ultimate plastic chord rotation model with curvatures and plastic hinge length to circular columns with

l li d b & FRP i Bi ki i & F di 2013

slipuslsplplyuyu aLLL 5.01)( ubLslipu d 5.5,

lap-spliced bars & FRP-wrapping – Biskinis & Fardis 2013

Lpl, fcc, εcu as in members (FRP-wrapped or not) with continuous bars

slipuslsplplyuyu ,)( ubLslipu ,

6Steel strain at ultimate deformation of member depends on lap length, if lo< l i : f

5

6

median:θu,exp=0.97θu,pred

lou,min:

s

y

oy

osu

ou

olsu E

fl

ll

l

min,min,, 2.02.1

4

%)

with: ccyhsh

ybL minou,

fffa

fdl

/665

2

3

θ u,e

xp

(% ccyhsh fff

1

2

Test-to-prediction ratio, no.tests: 37 median=0.97, CoV=38.6%

00 1 2 3 4 5 6

θu pred (%)

Cyclic shear resistance of FRP-wrapped rectangular members as controlled by diagonal tension, per EN1998-3:2005

• Vf=min(εu fEu f, fu f)ρf bwz/2 : FRP-contribution to cyclic shear resistance:

fwccs

totpl

ccs

R VVAfhLfAN

LxhV

,5min16.01)100,5.0max(16.0,5min05.0155.0,min

2

f ( u,f u,f, u,f)ρf w y• ρf :FRP ratio, ρf = 2tf/bw;• fu,f:FRP tensile strength.

V b f t ib ti f b t l (b b idth i t l t l ti )• Vw=ρwbwzfyw: contribution of web steel (bw: web width, z: int. lever arm; ρw: steel ratio)• ρtot: total longitudinal steel ratio• h: section depth 1 2

1,4p

• x : depth of compression zone• Ac= bwd

0,8

1

1,2

V u,p

red

Test-to-prediction ratio vs μ, no. tests 12median=0.99, CoV=14.8%:

0,4

0,6

V u,e

xp /V

,

0

0,2

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5d tilit

• In a FRP-retrofitted member the shear resistance as controlled by diagonal tension cannot exceed the shear resistance of old member as controlled by web crushing

ductility

RC jackets per EN1998RC jackets per EN1998--3:20053:2005

Concrete Jackets (continued/anchored in joint; w/ or w/o lap splices in old member)Calculation assumptions:

F ll it ti f j k t & ld t d (j k t d b• Full composite action of jacket & old concrete assumed (jacketed member: monolithic”), even for minimal shear connection at interface (roughened interface,steel dowels epoxied into old concrete: useful but not essential);

• f of “monolithic member”= that of the jacket (avoid large differences in old & new f )• fc of monolithic member = that of the jacket (avoid large differences in old & new fc)• Axial load considered to act on full, composite section;• Longitudinal reinforcement of jacketed column: mainly that of the jacket. Vertical

bars of old column considered at actual location between tension & compressionbars of old column considered at actual location between tension & compression bars of composite member (~ “web” longitudinal reinforcement), with its own fy;

• Only the transverse reinforcement of the jacket is considered for confinement;• For shear resistance the old transverse reinforcement taken into account only in• For shear resistance, the old transverse reinforcement taken into account only in

walls, if anchored in the (new) boundary elements• The detailing & any lap-splicing of jacket reinforcement are taken into account.Then:Then: MR & My of jacketed member: ~100% of θy of jacketed member for pre-yield (elastic) stiffness: ~105% of Sh i t f j k t d b 100% f Shear resistance of jacketed member: ~100% of Flexure-controlled ultimate deformation θu: ~100% ofthose of “monolithic member” calculated w/ assumptions above.pIf jacket bars are not continued/anchored in the joint:The jacket is considered only to confine fully the old column section.

54 jacketed members w/ or w/o lap splices: test-to-calculated as monolithic1.2

1.6

1.8

0.8

1

M y,c

alc

1

1.2

1.4

y,Eq

.(1)&

(2)

0.4

0.6

M y,e

xp /

M

continuous bars plain15dblaps deformed15dblaps 0 4

0.6

0.8

θ y,e

xp / θ

y

0

0.2

continuous bars plain 15db laps deformed 15db laps

plain 25db laps deformed 30db laps deformed 45db laps

non-anchored jacket bars group average st. dev. of group mean

My

0

0.2

0.4

θy

2.5 1.6

1.75

2

2.25

1.2

1.4

(3) )

1

1.25

1.5

EI ex

p / E

I* eff

06

0.8

1

/ ( θ

*y + θ

uplEq

.(

0.25

0.5

0.75

EIeff0.2

0.4

0.6θ u

,exp

θu

a: no treatment, b: no treatment, predamage, c: welded U-bars, d: dowels, e: roughened, f: roughened / predamage, g: U-bars+ roughened, h: U-bars+roughened / predamage, i: roughened+dowels, j: roughened+dowels / predamage

0a b c d e f g h i j k

0a b c d e f g h i j

Steel jackets per EN1998Steel jackets per EN1998--3:20053:2005

J k t t b f th j i t ( l t j i t f )

Steel Jackets (not continued/anchored in joint)

Jacket stops before the joint (several mm gap to joint face)• Flexural resistance, pre-yield (elastic) stiffness & flexure-

controlled ultimate deformation of RC member is not enhanced by jacket (flexural deformation capacity ~same asi ld b i id j k t b fit f fi t)in old member inside jacket, no benefit from confinement);

• 50% of shear resistance of steel jacket, Vj=Ajfyjh, can be j j yjrelied upon for shear resistance of retrofitted member (suppression of shear failure before or after flexural

)yielding);• Lap-splice clamping via friction mechanism at jacket-

member interface, if jacket extends to ~1.5 times splice length and is bolt-anchored to member at end of splice region & ~1/3 its height from the joint face (anchor bolts at third-point of side)

Documents

MODELLING THE BEHAVIOR OF CONCRETE MEMBERS: …ecpfe.oasp.gr/sites/default/files/Fardis EC8-3 member models_0.pdf · Workshoppqgg ,,y: Advances in Earthquake Engineering for Urban