Design Curves forTendon Profile in

Prestressed Concrete BeamsMohammad R. Ehsani

Assistant ProfessorDepartment of Civil Engineering and

Engineering MechanicsUniversity of Arizona

Tucson, Arizona

J. Russell BlewittGraduate Research AssistantDepartment of Civil andEnvironmentalEngineeringiverCornellUniversity UniversityIthaca, New York

T he preliminary design of prestressedconcrete beams is generally based

on working stress limitations. In order tosimplify the task of the designer, manytime-saving techniques for optimizingbeam cross section, prestressing force,and tendon eccentricity have been in-troduced.

One of the first and most popular de-sign aids is the Magnel diagram, shownin Fig. la, which was introduced in1948.' The Magnel diagram defines, fora known cross section, the acceptablecombinations of eccentricity and pre-stressing force at a particular locationalong the span. Although limited inscope, it has been sufficiently useful inits application to warrant continued useand revision, Current textbooks on pre-stressed concrete design continue torefer to the Magnel diagram as a usefuldesign tool The primary limitation ofthe original Magnel diagram is that the

designer must generate a separate dia-gram for each critical point along thespan.

Krishnamurthy has proposed a mod-ification of the Magnel diagram, 6'7 asshown in Fig. lb. This modificationgraphically represents the safe zone ofeccentricity and prestressing force as itrelates to the actual position on the crosssection. In particular, the method citedin Ref. 7 can be used to obtain non-dimensionalized design charts. Thecharts apply to a large number of crosssections subjected to arbitrary loadingand are easily generated by hand. How-ever, in order to arrive at the tendonprofile, the user determines the allow-able eccentricity at a number of pointsalong the span. These eccentricities canthen he plotted to give the accept-able zone for the entire length of thebeam.

The charts presented herein graphi-

114

catty represent the zone of acceptableeccentricity for the entire span. In addi-tion, a design procedure is proposedwhich enables the user to select thebeam cross section and the tendon pro-file. The method is applicable to pris-matic, symmetrical or unsymmetricalcross sections. Only uniformly loaded,simply supported beams are considered,and the effect of the presence of mildreinforcing steel is ignored.

The current ACI Code$ and AASHTOSpecifications' stipulate stress limitswhich form the primary basis for pre-liminary member selection. Theseworking stress designs often lead tomembers which satisfy ultimatestrength and serv iceability requirements.The move toward ultimate strength de-sign5 '° combined with computer-aidedoptimization techniques has resulted infewer advancements in working stressdesign methods. However, the need fora continuation in the refinement ofworking stress methods should not beoverlooked. The recent work by Oribsonis a step in this direction."

OBJECTIVESAs mentioned above, the limitations

of the Magnel diagram and its modifica-tions are as follows:

1. Each diagram can be used only forknown cross-sectional properties.

2. Each use of the diagram is limitedto a specific moment; therefore, newmoment calculations are required ateach critical location along the span.

Given these limitations, Magnel's de-finition of the safe zone cannot directlyrepresent the acceptable range of ec-centricity for the entire length of thebeam in a single application.

To overcome these Iimitations, amethod is proposed for developing de-sign charts which extend the idea ofMagnel's safe zone into a relationshipfor the entire length of a simply sup

-ported beam. The primary objectives in

SynopsisA proposed design procedure and

accompanying design charts for theworking stress design of uniformlyloaded, simply supported, prestressedconcrete beams is presented. Theproposed method is formulatedthrough an extension of Magnel's safezone into a relationship for the entirelength of the beam. A brief review ofthe Magnet diagram is followed by adetailed mathematical development ofthe proposed design curves.

The curves define the safe zone ofeccentricity of the prestressing steelas a function of the horizontal positionalong the span. A number of differentcurves, each representing combina-tions of span length, beam height, andapplied loads, are included in eachdesign chart. The design parametersapplicable to a given chart include theconcrete strength. the prestress lossratio, and the proportions of thecross-sectional dimensions. Exam-ples are included to demonstrate theuse of the proposed design method.

developing these design charts can hesummarized as follows:

(a) The charts will not be limited to aspecific cross section.

(b) The charts may be used to definethe zone of acceptable eccentricityalong the entire span.

(c) Generalized charts may be de-veloped in advance, with each chartbeing applicable to a large number ofpossible design situations.

In order to accomplish these tasks, thefollowing development is restricted tosimply supported, uniformly loadedbeams. The assumption of uniformloading is required since the ratio of theself weight of the beam to uniform deadplus live load is used to generalize thebasic design equations. The reason for

PCI JOURNAL'May-June 1986 115

assuming simple supports is due to thenecessity of establishing a moment re-lationship for the span (i.e., moment, M,as a function of the distance x from thesupport). It is, therefore, noted that dif-ferent support conditions could be ac-counted for if the appropriate momentrelationships were substituted in thefollowing equations.

BASIC DESIGNEQUATIONS

The development of the proposedmethod is based on the four stress con-straints which limit the allowablestresses in the top and bottom fibers of agiven section during initial and serviceload conditions. A possible form of these

P.1

zone of acceptable

combinations of P i and e

t kb e

(a)

e

kt

;g, C.A.

k b :f e

zone of acceptablecombinations of P.and e

(b)

Fig. 1. (a) Original Magnel diagram; (b) Krishnamurthy's modification.

116

requirements which defines the accept-able eccentricity of the prestressingtendons as a function of the position xalong the span may be written as fol-lows:

e (x) I- .f" S b — Sb + M' (x) (la)P, A P;

f St Sr i'vf (x)e{x}^– Pr + A + Pi (lb)

e(x).–fR+SA—`+^Rpx) (1c)r

e {x) . fra sb — Sb + M8 (x) (id)

RPr A RPi

In the above equations, the subscriptsI and s refer to initial and service loadconditions; subscripts t and b refer totop and bottom fibers; and B is the ef-fectiveness or prestress loss ratio. Notethat compressive stresses are taken aspositive and tensile stresses as negative.Additionally, the eccentricity, e, is takenas positive when below the centroidalaxis.

In order to generalize the above in-equalities so that they are not limited toa particular cross section, the followingdefinitions are introduced. Given theproportions of any section shown in Fig.2 (i.e., blb,n , hlhf , b2 /b), it is easily shownthat the properties of the cross sectioncan he expressed in the following form:

c = c'h (2a)

A = A'bh (2b)

S = S'bh2 (2c)

I = I'bh s (2d)

where c', A', S', and I' represent coeffi-cients for a specifically proportionedsection.

The initial prestressing force, Pr, asdefined by Nilson, , may he taken as:

P i = A ff (3)

where

f«t = J, – fi (j – .f;) (4)

Combining Eqs. (2a) and (4) gives:

f", = fr — Cb (J —fi) (4a)

It is noted thatf,,j is only a function ofthe section proportions and concretestrength. Combining Eqs. (la), (2), and(3) gives:

e(x)-_.f^rse(bh 2) – S b'(bh 2) +

A'(bh)f, A' (bh)

M f (x) (5)A' (b h) ff^,

which reduces to:

A'(bh)f{^ f

(6a)

Similarly, Eqs. (lb) to (Id) become:

e(x)'----h1 fr Mf(X) A' + A'(bh)f,,,(6b)

e {x) 7 $'jt 1 f^8 + Ms (x}

A' R f) A'(bh)f,

(6c)

e (x} -- s!

h 'x 1 + M, (x)b r fA' (R}eC , RA'(b h)f,,,

(6d)

The beam weight per unit length, w,,may be written as:

w, =A'(bh) wC(7)

where w, equals the unit weight of con-crete.

The load ratio, 'A75 , is defined as:

wrWR A

(8) (wx + wr)

in which wg is the sum of the uniformdead and live loads, wDL and wLL , re-

PC i JOURNAUMay-June 1986 117

^— b --*-I

hfl

Hbw^

spectively, and are usually expressed inunits of pounds per foot.

Using the definitions from Eqs. (7)and (8), the moments as a function of xfor a simply supported beam at initialand service conditions are:

Mr(x) =A' (bh) w (Lx — x2) (9)

2

M, (x) = A (bh) w, (Lx — x2) (10)2'H

Substituting Eqs. (9) and (10) intoEqs. (6a) to (6d), the four basic designequations may be written as:

Q (x) 1 S a h .f"i + t0, (Lx —x2)

A fcc• 2 feet

(11a)

e (x) Si h (i __iiL_. + we (Lx — X2)

A f 2f,

(11h)

k-- b 1

bw ,,

h

h fI ir`1 ':__

elf [..;:;;:;.Q.'1•

bW

2 h

h fl LaA. d

^— b — —1

bwh

e/

4...

b2

Fig, 2. Prestressed concrete cross sections.

118

B (x) st h (1 — lrs 1 + we (Lx — x2)

A' ft }^^ tJ 2 W, R fed

(lie)

e (x) ^ S e 1i r flr– 11 + we (Lx x2)

A ' (R f«, J 2WRRf•d

(lid)

DERIVATION OF ADESIGN COEFFICIENT

Dividing Eqs. (IIa) to (IId) by cb =cb h, then multiplying by L21L2, the fol-lowing equations (now expressed asequalities in e,,,ar ande,,,,) are obtained:

Pc62 I

L} A ca l erE – 1) +,, f

{L – 1x1 2] I2

(12a)2j,rb' h

em x S ` 1 – fti +CL ^^ ACb .fcce

r

r

L

()2

L2 (12b)2fct c n ^lz)

PCB s I L ) A Cb \1 — BfCB J +ccIxI – ( , L22R f,^,ca ^h 61R}

(12c)

miA (L)_ D ( J,s — 1+Fcn L A c

w` L (")L L' l (12d)

2Rff,ica \hWxl

Note that Eqs. (12a) to (12d) are di-mensionless when written in this form.It is apparent that plotting Eqs. (12a) to(12d) would result in nondimension-alized design curves which define themaximum and minimum ratios of elca at

any pointx along the span.It is convenient here to define cross

sections of the same group as thosehaving the same dimensional ratios(b1bw , h/hf, and b2 lb), concrete strength,unit weight, and the same prestress Iossratio, R, Thus, a closer look at Eqs. (12a)and (12b) reveals that within each groupof cross sections, defined in this manner,the quantity e mazlcb is only a function ofL2/h.

Similarly, the quantity emin lca is only afunction of L2I(h WR ). This leads to thedefinition of the eccentricity coefficient,CP, as:

L2 (13)Ce=hWR

It can be stated that the nondimen-sionalized safe zone of eccentricity (elcb)for uniformly loaded, simply supportedbeams that are in the same group, variesonly as Ce varies. Based on this observa-tion, design charts utilizing the curvesrepresented by Eqs. (12a) to (12d) canbe generated for any group of cross sec-tions, as defined above.

DEVELOPMENT OFTHE DESIGN CHARTS

Given a value of Ce , the controllingfunction, em,,Ic6, is chosen from Eqs.(12c) or (12d). This function defines theupper limit to the safe zone of eccen-tricity forthis value of C. Similarly, foragiven value of:

CQ WR = l (14)

the controlling function, emax lcb , is cho-sen from Eqs. (12a) or (12b). This func-tion defines the lower limit of the safezone of eccentricity for this value ofGe WR .

These curves of elc, versus x/L can beplotted for various values of P to obtainthe design charts shown in Figs. 3 to 7.Intermediate values of C, can be inter-

PCI JOURNAL/May-June 1986 119

o.00 0.10

X/L0.20 0.30

SECTIONPROPERTIES

f'. . 4000 psi.

f 5000 psi

R o. 80

P1 = 397.85 bh lbsA X 0.360 bh in2

I 0.01+92 bh3in4=

cb = 0.500 h in

8e°Ct

0

0.40 0.50

hf

°w h

thSECTION

PROPORTIONSbw/b = 0.2

hf/h = 0.1

0

S

C•e Soo

-3-5o

eC6

x`000

5 CeWR =50

^So

0

100 500

1508

0.00 0.10 0.20 0.30 0.40 0.50X/L

Fig. 3. Acceptable ranges of eccentricity for Example 1.

120

X/L0.00 0.10 0.20 0.30 0.40 0.50

F`— h

f ■1

i

0.30. 1 'I- 1

1

■^■■EEN N■W. ■■

■ ■■ .. ■• . ■■IMEMOOMMIMMININ. ,

■■■ ■

■ ■ ■^^ • ■:::■■■■IZ ■■■^ IN■►-_ ■^^\ ■■■_^ ■■■■^ ■■■\ ■■

_ ^_^■■15^f■̂_ ■ C! \■■ __■

■■O■M _

■N■ ■■ ■■ ■■■■ ■■ ■.•.■^ ■■- ■ ■■■■■■■■ ■^^■ ■■0.00 0.10 u.cu u..0X/LFig_ 4. Acceptable ranges of eccentricity for Section 1 of Example 2.

ecf

eC6

q .50

PCI JOURNAL/May-June 1986 121

0.00 0.10X/L

0.20 0.30

SECTIONPROPERTIES

fl i 4000 psi5000 psi

= 0.80P. = 194.71 bh lbs.

g A = 0.250 bh in

e t - 0.0289 bh 3 in4

Ct Cb - 0.626 h in

S

uuuuuuui_■■■■■■■III■■AU,

• ■■■■■■■^■_■■■■■■■■

:■■■■=a•. ''N_■■■■■■■■■ ■■■

0.40 0.0

bh 14b hw

T_1

h2 '^ hf

SECTIONPROPORTIONS

b2 /b 0.30

b /b - 0.15wh f /h = 0.10

0.00 0.10 q .20 0.30 0.40 0.50X/L

Fig, 5. Acceptable ranges of eccentricity for Section 2 of Example 2.

122

X/L

0.10 0.20 0.30

SECTIONPROPERTIES

f 4000 psi`

f l - 5000 psiCR = 0.80

P. - 240.14 bh lbs.iA - 0.270 bh in2

I 0.0350 bh 3in4=

'b = 0.5S33 h in

0.00

8

eCt

C

0.40 0.50

bw ^

b2 4 fig

SECT IONPROPORTIONS

b2 /b = 0.50

by/b = 0.15

h f lh = 0.10

I'

0

0

0

0e

C6EsC

I _ r=

N; I I t i l

0.00 0.10 0.20 0.30 0.40 O.50X/L

Fig. 6. Acceptable ranges of eccentricity for Section 3 of Example 2.

PCI JOURNAL/May-June 1986 123

X/a0.00 0.10 0-20 0.30 0.40 0.50

hSECTION ^— b ^ f

PROPERTIES

CL

0

= 4000 psi

f' = 5000 psiR - 0.80P. = 196.40 hh lbs.

A 0.230 bh in2- 0.0326 bh 3in4

cb0.597 h in

bW - h

__ _

b2 '^ h f

SECTIONPROPORTIONS

b2/b 0.50

bw/b = 0.10

h f /h = 0.10

0 iii:: _c11:cE111111ii iii iiio eC zso

o0 300

03sa

--- c ---

- -

yoo

--

B100

-

V•VW W.11•i U.c[7 U.90 0.40 0.50

X/L

Fig. 7, Acceptable ranges of eccentricity for Section 4 of Example 2.

eCt

eCb

124

polated to arrive at the required tendonprofile for any section to which the chartpertains. As shown in the followingexample, it is possible to achieve a com-bination of maximum eccentricity andminimum prestressing force by nar-rowing the depth of the safe zone atmidspan.

The following conventions areadopted for plotting purposes:

(a) For eccentricities below the cen-troidal axis, the values ofelcr, are plotted,while for eccentricities above the cen-troidal axis the corresponding values ofetc, are plotted.

(b) A conventional x-y plotting routineis used which requires that eccen-tricities below the centroidal axis beplotted as negative.

The following design values havebeen selected to generate the samplecharts. The allowable stress limits corre-spond to the current ACT Code . 8 How-ever, the allowable increase in tensilestresses at the end of the beam has notbeen included.

The value of the effectiveness ratio, R,is taken as 0.80. The values off,' E and ffare 4000 and 5000 psi,* respectively.The unit weight of concrete is taken as150 pcf for normal weight concrete.Note that in calculating Ce for Eq. (13),the span length is given in feet and thebeam height in inches.

The charts shown in Figs. 3 to 7 rep-resent a small sample of those thathave been generated. All of the com-putations and plotting of the curvesare intended to be performed by acomputer. Given the necessary soft-ware, charts for I-beams, T-beams, andbox girders, using various combina-tions of fG,, f, , and R can be easily pro-duced.

In addition, some of the aspects of thefollowing design procedure could becomputerized. For example, given the

• Note: All units are expressed in U.S. foot andpound units. For metric conversion factors, seeend of paper.

relative cost of placing the prestressingsteel and concrete per unit volume andsome initial architectural requirements,computer algorithms utilizing Eqs. (12a)to (12d) could be developed whichwould converge to an optimum crosssection. Depending on the availablecomputer facilities, this could be donein either an interactive or batch mode.

DETERMINATION OFTENDON PROFILE

In some cases the designer may wishto use the proposed design charts only toselect a tendon profile. To do this, thedesigner first calculates the eccentricitycoefficient, C,, from the known param-eters of the design. The design chart,applicable to this cross section im-mediately describes the acceptablerange of eccentricity along the entirespan. This procedure is demonstratedby the following example.

DESIGN EXAMPLESTwo numerical design examples are

presented to show the application of theproposed design method. The firstexample treats a symmetrical box beamwhile the second example covers an un-symmetrical I-beam.

EXAMPLE 1Consider a symmetrical prestressed

box beam. The tendon profile for thecross section of Fig. 8a will be deter-mined by the use of the design charts.The dimensional ratios of this cross sec-tion are:

h,p lb = 0.2 and h,Jh = 0.1The concrete strengths are specified

as:= 4000 psi and f f = 5000 psi

Normal weight concrete is to be used,and a prestress loss ratio (R) = 0.80 isassumed. Based on this information, thedesign chart of Fig. 3 is appropriate for

PCI JOURNALlMay-June 1986 125

4o in.

3 in. b 4 q i A 9

3 in.

D

Y

Ta L30 in.

II(a)

30f

zone of acceptableeccentricity

C.A.

Support(b) Midspan

Fig. 8. Box beam of Example 1 (a) cross section; (b) elevation showing zone ofacceptable eccentricity.

126

this cross section.The following data are also specified

for the design of the beam: the beamspan(L) = 60 ft and the unfactored deadplus live load (WDL + wL,) = 400 psf. Anacceptable tendon profile for this designis determined as follows.

SolutionThe load ratio, WR , from Eq. (8) is

equal to 0,252. This gives an eccentric-ity coefficient, Ce , from Eq. (13) equal to476. Given these values, the acceptablerange of eccentricity is easily deter-mined from the design chart of Fig. 3.The limits of eccentricity for this exam-ple are shown in Fig. 8b.

Once the designer has established thezone of acceptable eccentricity, the se-lection of the number and placement ofthe prestressing strands is a straightfor-ward procedure. Any tendon profile forwhich the center of gravity of the steel(cgs) lies within this zone will satisfy theworking stress requirements. A morecomprehensive design example (Exam-ple 2), including the selection and lay-out of the prestressing strands, followsthe proposed design procedure.

PROPOSED DESIGNPROCEDURE

For a given design problem, the val-ues of L, w, (from wr,1, and w), f/,, f' ,and R must be known or assumed. Inmany cases the values of b and/or h areknown or predetermined by architec-tural requirements. Although both b andh may be varied throughout the designprocedure, it is recommended that bremain constant. Then, if necessary, theprocedure can be repeated using a re-fined value of b. A flow chart of the de-sign procedure is shown in Fig. 9, and isdescribed in the following steps:

1. Select initial values of b and It,

2. Select a number of design chartsbased on dimensional ratios. Although aworking design can be determined

using a single design chart, it is gener-ally beneficial to obtain a number offeasible designs by using more than onechart. It may become apparent duringthe design iterations that one or more ofthe selected charts are inappropriateand are thus eliminated from considera-tion.

3. Tabulate the values ofA [Eq. (2b) l,w i [Eq. (7)1, WA [Eq . (8)1, and cb [Eq.(2a)] for the chosen sections. Based onminimum cover requirements, deter-mine a suitable value of elcb , usuallyaround 0.8. By interpolation, at thisvalue of elcb , obtain a preliminary C,from each design chart.

4. Determine the minimum h as fol-lows:

(, LL (15)rhnfn - C' WR

The optimum value of h, for each ofthe chosen sections, is between h,,,,,, andthe initial value ofh from Step. 1.

5. IC in the judgment of the designer,h., is not sufficiently close to the initialh, select a new h and repeat Steps 3 and4. In general, a value of h which iswithin 2 or 3 percent of is a suitablesolution. The reason that a value of hless than h,n ,, may be acceptable is thatthe ratio elcb chosen in Step 3 is an ap-proximation and often conservative.

6. Using the final value of h, calculateP, and determine the required numberof prestressing strands.

7. If the zone of acceptable eccentric-ity at midspan, based on the values of Ce

and Ce WR is too narrow to accommodatethe required number of strands whilemaintaining minimum cover require-ments, increase h and repeat Steps 3through 6.

8. Select the tendon profile. Any pro-file which lies between the applicableCe and Ce WR curves is acceptable.

9. Satisfy ultimate strength and otherestablished requirements.

It is important to note that a lowerlimit of eccentricity, as defined byC, W., which lies outside the beam crosssection (i.e., elca > 1) indicates that the

PCI JOURNALIMay-June 1986 127

LGiven : L , ws , fci , fc , H I

Select b and h

Select design chart(s)

Calculate A , Hi , W P I cbl

Select e/cb I

Obtain C e from chart(s)

Calculate hmin I

Select new h Isbetween h min N^ hmin appror,imatelyand initial h equal to h ?

YF5

Calculate P i and the required number

of prestressinq strands

Is the

Increseh No safe zone adequate fora the placement of

the strands ?

YE

Select strand profile

based on C e and CeWR

Satisfy ultimate strength and

other applicable requirements

Fig. 9. Flow chart of proposed design procedure.

128

Table 1. Steps 3 and 4 of Example 2.

Cross section 1 2

Fig. 5

3 4

Design chart Fig. 4 Fig. 6 Fig. 7

Iteration 1st 2nd 1st 2nd 1st 3rd 1st 2nd

b (in.) 60 60 60 60 60 60 60 60

h (in.) 30 32 30 29.5 30 28 30 27

A (in.') 378 403 450 443 414 386 486 437

w (lb/fl) 394 420 469 461 431 402 506 455

w, (lb/ft) 1375 1375 1375 1375 1375 1375 1375 1375

WR 0.223 0.234 0.254 0.251 0.239 0.226 0.269 0.249

cb (in.) 19.5 20.8 18.8 18.5 17.9 16.7 17.5 15.7

eleb 0.86 0.87 0.85 0.85 0.85 0.84 0.84 0.83

C, 325 330 340 338 390 385 385 380

h,,,i ,, (in.) 34.5 32.4 228.9 29.5 27.2 28.4 24.1 26.4

section selected is conservative. In thiscase, the designer may choose a differ-ent value of b or an alternate designchart.

EXAMPLE 2The proposed method will be demon-

strated by designing an unsymmetrical1-beam given the following specifica-tions:

Beam span (L) - 50 ftUnfactored dead plus live load(WDL + WLL) = 275 psff^a = 4000 psi, ff = 5000 psiPrestress loss ratio (R) = 0.8Top flange width (b) = 5 ftThe following step-by-step solution

parallels the proposed design procedureand the flow chart of Fig. 9.

SolutionStep 1- An L/h ratio of 20 yields h =

30 in. This will be assumed as a reason-able initial value of h.

Step 2 - The design charts of Figs. 4to 7 are selected as having appropriateproportions for this design. The selec-

tion is based on the thicknesses of theflanges and the web that will result fromthe initial selection ofb and h.

Step 3 - The results of the calcula-tions of A, w,, WR , and cb for each of thedesign charts is given in Table 1. Atmidspan (x/L = 0.5), the minimum coverrequirements" are such that the tendongroup eccentricity, e, minus c5, would bea minimum of about 2.5 to 3 in. There-fore, a value ofcn - e = 2.75 in. is used todetermine elec. By interpolation, thevalues of Cg are taken from each designchart and shown in Table 1.

Step 4 - Next, h,Rf„ is calculated fromEq. (15). The values obtained for eachcross section are included in Table 1.

Step 5 - It is apparent that, accordingto the values of h.mt,,, each of the foursections could he refined further, al-though Section 2, with h = 30 in. wouldappear to be a reasonably close design.Sections 1, 3, and 4, however, certainlywarrant further iteration. For the pur-pose of this example, only the seconditeration for Section 1 will be given.

Step 3 (second iteration) - An inter-mediate value ofh = 32 in. is chosen for

PCI JOURNALJMay-June 1986 129

Table 2. Step 6 of Example 2.

Cross section 1

Fig. 4

2 3 4

Design chart Fig. 5 Fig. 6 Fig. 7

h (in.) 32 29.5 28 27

A (in.') 403 443 386 437

P, (kips) 289 339 330 389

Number of'4 in. diameter strands,f„M = 270 ksi 10 12 12 14

Initial prestress, fa (ksi) 189 185 180 182

L 60"

3 2••

6 11

32"

Fig. 10. Preliminary cross section used in Example 2.

Section 1. The values ofA, w i , WR,, andcbare given in Table 1. Minimum coverrequirements indicate that a value ofelcb= 0.87 is appropriate, which leads to an

eccentricity coefficient (Ce ) = 330.Step 4 (second iteration) — The value

of hmtn for Section 1 is 32.4 in., as givenin Table 1. Values of h,,, 1„ for Sections 2,3, and 4 are also given for their final iter-ations.

Step 5 (second iteration) — The as-sumed value oflt = 32 in., for Section 1,is deemed adequate for continuation toStep 6. The final values of h = 29.5, h =28, andh = 27 in. were determined to he

appropriate for Sections 2, 3, and 4, re-spectively. Each of the four cross sec-tions will be retained and carried over toStep 6,

Step 6 — Having obtained the finalvalues ofh for the four possible sections,the values of P i are calculated from thesimplified form of Eq. (3) found on thedesign charts. In order to compare thesections further, the required number of1/a in. diameter Grade 270 strands, at amaximum allowable stress of 189 ksi, isalso calculated. These results are shownin Table 2.

At this point, the designer may decide

130

60"

',

611e—=090C b 32"

c.g.s.L5'gg

+ O 00 OD2.05" 1.75" ^.5"

__.._ 181

(a)

00

= 0.39 4.00"^ b Q 0

4.00"00

E12.70" c.g.s. 12.25"

D 0 0 G 7 75',

(b)

Fig. 11. Modified cross section used in Example 2 with strand pattern shown at(a) midspan and (b) ends.

to select a single cross section, or con- example, only one section will be cho-tinue with any number of the four possi- sen for Steps 7 to 9. Unless the beamhie designs. For the purpose of this depth is a controlling factor, Section I,

PCI JOURNAL/May-June 1986 131

which utilizes the maximum allowableinitial prestress of 189 ksi, would appearto be the best choice for the final design.This section is shown in Fig. 10. Notethat the flanges of the section may hetapered slightly without significantchanges in the section properties. In thiscase, the flanges in the final section,shown in Fig. 11, are tapered from 2.5 to4 in. , resulting in an average thicknessof 3.25 in., which is very close to theoriginal flange thickness of 3,2 in. Thechanges in the distance to the centroidalaxis and the moment of inertia as a resultof this modification are less than % per-cent of the original values.

Step 7 — The acceptable region forthe tendon profile is bounded by thecurves representing Ce = 334 and Ce Wk= 78 as shown in Fig. 4. This range rep-resents a 1.3 in. zone in which the cgsmust be placed at midspan. Note that astraight tendon profile corresponding toa constant eccentricity is not possiblegiven this optimized section. However,it would be a simple matter to find theminimum beam height required for anytendon profile.

Step 8 — The selected strand place-ment is shown in Fig. 11. For the center20 percent (10 ft) of the span, the strandsare placed such that the ratio elca isequal to 0.9. The end eccentricities areselected to give elcb = 0.39. The cgs isshown as a broken line in Fig. 3. It isclear that this strand profile results inratios ofeleb which are positioned withinthe acceptable region for the entirelength of the beam.

Step 9 — Check ultimate strength andsatisfy other applicable requirements.

PRACTICAL APPLICATIONSThrough the use of the proposed de-

sign charts, a designer may quickly andeasily arrive at a number of acceptableworking stress design alternatives. Thefact that the proposed charts define thesafe zone of eccentricity for the entirespan eliminates the need to check the

actual stresses at critical points along thebeam. As already noted, the values offs,f , and R may be varied, along with thedimensional ratios, to develop designcharts for any group of cross sections.Additionally, the designer may developcharts using any values of the allowableworking stresses (i.e., fCt , f,„ f^,, and f, }.Having generated the desired charts,considerable savings can be achieved byapplying the same charts to numerousdesign situations.

This method may be of particularinterest to the manufacturers of pre-stressed girders who generally utilize alimited number of cross sections. Insuch cases, the charts need only to begenerated for the available cross sec-tions. Regardless of the span length, de-sign charts could he used for a quickdetermination of the arrangement andnumber of prestressing strands.

Another advantage of this method isits potential use as a teaching aid. Thecharts can be a useful tool in allowingthe student to visualize the variation ofacceptable tendon profile over the en-tire span. Finally, the method could beeasily adapted to an interactive com-puter graphics format which wouldeliminate the need for preparation of thedesign charts in advance.

SUMMARY ANDCONCLUSIONS

A set of dimensionless design curveshas been developed which extends theapplicability of Magnel's safe zone froma single point on the span to the entirelength of the girder; the acceptablecombinations of P and e are expressedfor all points along the span. For easeof application, the curves have beenplotted, and design charts for severaltypes of cross sections have been in-cluded.

The design charts are dependent onthe concrete strength, the prestress lossratio, and the proportions of the cross

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section. Therefore, a single chart can beapplied to a number of cross sections allhaving the same relative proportions. Inaddition, the design charts are indepen-dent of the span length and load inten-sity. The two design examples demon-strate the use of the charts for deter-mining tendon profiles, and the time-saving potential of the proposed designmethod.

ACKNOWLEDGMENTThe work presented in this paper was

completed while the first author was en-rolled at the University of Arizona. Theuse of the computational facilities of theDepartment of Civil Engineering andEngineering Mechanics at the Univer-sity of Arizona is gratefully acknowl-edged.

NOTE: Discussion of this paper is invited. Please submit yourcomments to PCI Headquarters by January 1, 1987.

PCI JOURNAUMay-June 1966 133

REFERENCESI. Magnet, G., Prestressed Concrete, Con-

crete Publications Ltd., London, UnitedKingdom, 1948; also, McGraw-Hill BookCo„ New York, N.Y. 1954.

2. Khachaturian, N., and Gurfinkel, C., Pre-stressed Concrete, McGraw-Hill BookCo., New York, NY, 1960.

3. Lin, T, Y., and Burns, N. H., Design ofPrestressed Concrete Structures, ThirdEdition, John Wiley and Sons, Inc., NewYork, N.Y., 1981.

4. Naaman, A. E., Prestressed ConcreteAnalysis and Design, McGraw-Hill BookCo., New York, N.Y., 1982.

5. Nilson, A. H., Design of PrestressedConcrete, John Wiley & Sons, Inc., NewYork, N.Y., 1978.

6. Krishnamurthy, N., "Magnel Diagramsfor Prestressed Concrete Beams," ASCE,V. 109, No, 12, December 1983, pp.2761-2769.

7. Krishnamurthy, N., "Modified MagnelDiagram as Design Aid for Prestressed

Concrete Bridge Members," Proceed-ings, Second International Symposiumon Concrete Bridge Design, ACI Publi-cation SP-26, American Concrete Insti-tute, Detroit, Michigan, 1971, pp. 663-689.

8. ACT Committee 318, "Building CodeRequirements for Reinforced Concrete,(ACT 318-83)," American Concrete In-stitute, Detroit, Michigan, 1983.

9. AASHTO, Standard Specifications forHighway Bridges, 13th Edition, Ameri-can Association of State Highway andTransportation Officials, Washington,D.C., 1983.

10. PCI Design Handbook –Precast and Pre-stressed Concrete, Third Edition, Pre-stressed Concrete Institute, Chicago, Il-linois, 1978.

11. Orbison, J. G., "Generalized FlexuralDesign Equations for Prestressed Con-crete," PCI JOURNAL, V. 30, No. 2,March-April 1985, pp. 172-182.

Metric (SI) Conversion Factors1 ft = 0.305 m1 in. — 25.4 mm1 in. 2 = 645.2 mm2lib = 4.448N1 psi = 0.006895 MPa1 ksi = 6.895 MPa1 psf = 0.0479 kPaI pcf = 16.02 kglrn31 lb/ft = 14.59 Nlm

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APPENDIX - NOTATION

A = cross-sectional area of beam conditionsA' = coefficient for cross-sectional f = allowable tensile stress in ex-

area of beam treme fibers under service load-b = width of top flange of beam ing conditionsb2 = width of bottom flange of It = height of beam

unsymmetric I-beam hf = height of beam flangeb o = width of beam web I = moment of inertia of beam aboutC, = eccentricity coefficient centroidal axiscb = distance from extreme bottom I' coefficient for moment of inertia

fiber to centroidal axis of beam of beamc t = distance from extreme top fiber to kb = kern distance measured from

centroidal axis of beam centroidal axis to top of beamc =coefficient for distance from ex- k, = kern distance measured from

treme bottom fiber to centroidal centroidal axis to bottom of beamaxis of beam L = beam span

c,' = coefficient for distance from ex- M, = bending moment in beam undertreme top fiber to centroidal axis initial loading conditionsof beam MR = bending moment in beam under

CA = centroidal axis of beam service loading conditionscgs = center of gravity of prestressing P = prestressing force

steel P; = initial prestressing forcee = tendon eccentricity R = effectiveness or prestress loss"'max = maximum allowable tendon ec- ratio

centricity So = section modulus relative to bot-e„,;,, = minimum allowable tendon ec- torn fibers of beam cross section

centricity S, = section modulus relative to topf, = allowable compressive stress in fibers of beam cross section

extreme fibers under initial Sa = coefficient for section modulusloading conditions relative to bottom fiber of beam

f, = allowable compressive stress in cross sectionextreme fibers under service S; = coefficient for section modulusloading conditions relative to top fiber of beam cross

fey = concrete centroidal stress under sectioninitial loading conditions w, = unit weight of beam

f' = compressive strength of concrete w f = self weight of beam per unit= initial compressive strength of length

concrete w,, = uniform dead load per unit areafp„ = ultimate strength in prestressing Wu = uniform live load per unit area

steel wa = uniform dead plus live load onfn = initial prestress in prestressing beam per unit length

steel WR = load ratiofit = allowable tensile stress in ex- x = horizontal distance from end of

treme fibers under initial loading beam

PCI JOURNALIMay-June 1986 135