Ultimate analysis ofprestressed and partiallyprestressed sections bystrain compatibility

Antoine E. NaamanAssistant Professor of Structural DesignDepartment of Materials EngineeringUniversity of Illinois at Chicago CircleChicago, Illinois

T he analysis of prestressed concretesections at ultimate capacity is a

simple procedure if one follows theACh or the AASHTO2 specifications.

These specifications provide a num-ber of simplifying assumptions, namely,that the state of strain in the concretecompressive zone at ultimate is known,that the force in the concrete can beapproximated from the equivalent rec-tangular stress block and that the stressin the prestressing steel can be approxi-mated from materials and section prop-erties.

It has been shown3 ,4 that these as-sumptions lead to satisfactory predic-tions of ultimate moments for fully pre-stressed and somewhat partially pre-stressed normal weight concrete beams.

Partial prestressing implies the usein combination with prestressing steelof non-prestressed conventional rein-

forcement in the tensile and/or com-pressive zone of the section.

Prestressed concrete is today beingused in combination with substantiallyIarge amounts of non-prestressed rein-forcement. Newer concrete materialssuch as high strength, lightweight, fiberreinforced or polymer modified con-cretes, where all the Code assumptionsrelated to ultimate compressive strainand dimensions of the stress block maynot apply, are also being used.

Thus, there is an increasing need inthese types of applications for a tool topredict flexural capacity of the sectionand more importantly to predict withenough accuracy, curvatures, rotationsand deflections at ultimate. It should bebased on a more accurate analysis inwhich the actual stress-strain propertiesof the materials involved are taken intoconsideration. 5, 6

32

Such a procedure (referred to as"strain compatibility") is suggested bythe ACI specifications (Reference 1,Section 18.7) in order to determine thestress in the prestressing steel (fps) atultimate behavior; it is accepted in allcases in lieu of the more approximateCode formula for f.,s and required whenthe effective prestress is less than one-half the steel strength and/or when thesteel stress-strain curve does not con-form with specified ASTM standards.

Furthermore, the use of a more ac-curate analysis may lead to substantialsavings7 in the amount of prestressingsteel required which more than offsetthe additional cost in design.

The purpose of this paper is to pre-sent a simplified procedure to analyzethe behavior at ultimate of bondedprestressed and partially prestressedconcrete structural elements in whichthe non-linear behavior of the prestress-ing steel is fully accounted for. A com-puter program was written to imple-ment the above procedure.

For given conditions of reinforce-ment the program leads to the valuesof stress •and strain in the prestressingsteel at ultimate, the ultimate momentcapacity and the corresponding curva-ture of the section and other relevantinformation. Corresponding values ob-tained by strictly applying the ACIspecifications are also computed for di-rect comparison.

The program allows a quantitativeassessment of the influence on ultimatebehavior of important parameters suchas amount of non-prestressed reinforce-ment, effective prestress, type of pre-stressing steel, ultimate compressivestrain of the concrete, and stress blockdimensional factors.

Numerical results are applied to arectangular and a box (T) section anda typical design example is developedusing the ACI code approach, thestrain compatibility procedure, and acombination of prestressed and non-prestressed reinforcement.

SynopsisA non-linear analysis proce-dure which attempts to pre-dict the behavior at ultimateof prestressed and partiallyprestressed sections is pre-sented.

It takes into consideration thenon-linear characteristics ofthe steel but assumes thatthe strains and forces in theconcrete at ultimate are wellapproximated by the ACIspecifications.

The numerical techniquesand the computer programimplementing the proposedprocedure are rapidly re-viewed.

They help analyze the influ-ence on ultimate behavior ofmajor variables such as typeof prestressing steel, amountof non-prestressed reinforce-ment, ultimate compressivestrain and stress block di-mensional coefficients of theconcrete. Examples of appli-cations and typical resultsare presented.

It is shown that although theACI recommended procedureto predict ultimate momentcapacity is reasonably accu-rate and conservative, it maylead to substantial error inpredicting strain in the pre-stressing steel and the cor-responding curvature of thesection.

PCI JOURNAL/January-February 1977 33

Non-Linear Analysis

In the non-linear analysis of rein-forced concrete elements, two assump-tions are generally considered: firstthat the strain distribution in the sectionremains linear up to maximum momentcapacity and, second, that the actualstress-strain relations of the steel andconcrete are known and used.

In order to determine maximum mo-ment capacity the moment versus curv-vature curve is plotted and the peakvalue selected. 8 This can be achievednumerically in the most general casefirst by selecting a value of concreteextreme fiber compressive strain andthen finding by iteration a location ofneutral axis which simultaneously satis-fies compatibility and equilibrium.

The process is repeated for increas-ing values of the extreme fiber com-pressive strain. By calculating at everystep the internal moment and the cur-vature, the ultimate (maximum) mo-ment and the corresponding strain andcurvature at ultimate can be deter-mined. Practically speaking, a substan-tial computational effort is needed evenwhen a computer is used.

The above exact method was appliedto prestressed concrete sections andwas found (Appendix I) to lead to con-ditions at ultimate not significantly dif-ferent from a less exact method de-scribed below. This later method wasselected because first it reduces by anorder of magnitude the required com-putations, and second it accommodatesthe ACI specifications and is, there-fore, of more direct use to designers.

The proposed method takes into ac-count the actual stress-strain curve ofthe prestressing steel but considers thatthe ACI assumptions leading to theequivalent rectangular stress block inthe concrete are representative of theforces in the concrete at ultimate.4

The rationale of this approach stemsfrom the fact that in predicting the

stress-strain properties of concrete,»"°one usually encounters much more va-riabilities than for the steel.

The practically constant elastic mod-ulus of the steel as well as the require-ments of ASTM standards on propor-tional limit, yield strength, ultimatestrength and elongation make the pre-dicted stress-strain relation of a givensteel quite close to, and representativeof the actual relation. This approachwill be followed in this paper.

Stress and Strain inSteel at Ultimate

for Bonded Members

The design assumptions of the ACI-318-71 1 , 11 concerning the linear straindistribution and the concrete stressblock at ultimate are followed andcould be visualized by referring to Fig.1. It is also assumed that the stress-strain relation of the prestressing steelis known either graphically or numeric-ally.

It is further assumed that the strainin the top fiber of the concrete sectionunder effective prestress alone is negli-gible and that if non-prestressed tensilereinforcement is used it has the samecenter of gravity as the prestressed re-inforcement.

Note that in Fig. 1, and later in Fig.5, the non-prestressed tensile reinforce-ment for T sections is shown entirely inthe web while, in a particular design, itmay be partly associated with theflange.

The determination of the actualstress and strain in the prestressingsteel at ultimate requires also theknowledge of a relation between stressand strain as derived from compatibilityof strain and equilibrium of the section.

The derivation procedure for rec-tangular sections can be found in Refs.3, 12-15. It is generalized here to in-clude auxiliary non-prestressed rein-forcement and T-type sections.

34

RECTANGULAR TEE OR DOUBLE TEE I FORCES AT ULTIMATEIt o.esf

b I - b

c

w --- b-bw

J_ --- Asfcs fY, { fcs = JC^ As — —a^ ^_^hf ^IC

NA l - , - -------- ------------------- --- ---- o.eaf^bO,C or:GC—• — d 0.85f^(b-b^hf

e A s ^^f •0.85f^b.,,.....4

- -^ c w c^TENSILE

As AS FORCE'—Apsfps + ASfts [f,s i fYJ

TENSION ..l---j I I— . COMPRESSION0.003 IF ACI CODE

Cf--- STRAIN NCONCRETEUNDER fps

CONCRETE AT ULTIMATE dMOMENT CAFACITY^,^ECU Ece ---

`ps1 I TENSI%

`seSTRAIN IN

PRESTRESSED fps I 0STEEL

Fig. 1. Assumed forces and strains at ultimate capacity of prestressed beamsections.

Stress-strain relation as derivedfrom compatibility and equilibrium

Referring to the strain diagram inthe concrete at ultimate capacity (Fig.1), it can be shown 13"4 that the dis-tance from the top fiber to the neutralaxis is given by:

C=( Eu I d (1)\Cps + Eu — E.s, — Ecc/

whered = distance from top fiber to cen-

troid of tensile forceE,, = extreme fiber compressive strain

of concrete at ultimate (as-sumed equal to 0.003 in ACICode)

E,Ye = tensile strain in prestressingsteel under effective prestress

E,.e = compressive strain in concreteat level of steel under effectiveprestress

E 8 = tensile strain in prestressingsteel at ultimate capacity ofbeam

Note, that for a given beam crosssection, Eq. (1) is a relation betweenc and Eps as all other terms are known.In general, E,,, is small as compared tothe other terms and could be neglected.

An additional equation translatingthe equilibrium of tensile and compres-sive forces in the section at ultimatecapacity (Fig. 1) can be written, butits form depends on whether the sec-tion behaves as a rectangular sectionor as a T section.

If the section behaves at ultimateas a rectangular section the equilibrium

PCI JOURNAL/January-February 1977ҟ 35

condition leads to the following equa- It can be shown' that if the sectiontion: behaves at ultimate as a T section the

Apo fps + As Its —Ɂrelation between fp,, and Ep, takes the0.85 f' b 13 1 c + A' f' aɁ(2) following form:r. s c

where fps is the tensile stress in theprestressing steel at ultimate momentcapacity, f, is the tensile stress in thenon-prestressed tensile steel (Its G fy)and f' G, is the compressive stress in thenon-prestressed compression steel (f',,G fp)

Generally, f5 equals f, at ultimatewhile f',, can be substantially different.Other symbols of Eq. (2) are standardand can be found in Appendix III.

Note that this equation can easilybe modified to accommodate concretesfor which the dimensional factors ofthe stress block are different.

Eliminating c from Eq. (1) and (2)leads to the following relation betweenfps and Ep,:

0.85 f3 f'ebd Xfps Ap8

E,, +[Eps+Eu—E;re—Ece

(A'sfes — A;fts) (3)Aps

is less than their y

i

eld limit.

In this relation f' e, and f3 are func-tions of Eps provided their actual value

Referring to the strain diagrams inFig. I it can be shown that f',, and f isare given by:

d'f"'

—Es Eia+ d X

d'(e,+ Ece — Eu) —

(4)a EpS ?J

Its — Es(Eps — Ese'— Eee) G fii

where E3 is the modulus of elasticity ofconventional non-prestressed reinforce-ment.

fns=[ x 0.85 f3 1f' ebwd^ X

Aps

Euu +

[Eps+Eu—Ese—Ece

0.85 f c(b — bw)hf + A'sf cs — AsftsAps

(5)

where f' s, and ft5 can also be replacedby their values from Eq. (4).

Whether a rectangular section or aT section behavior exists at ultimate,the relation between fps and E 5 givenby Eq. (3) or (5) can be reduced to thefollowing most general form:

_ A

fps E -I- B + CEpa -I- D (6)

pa

where A, B, C, and D are parametersdepending on materials characteristics,effective prestress and geometric prop-erties of the section.

The graphical representations of Eq.(6) corresponding to various caseswhere C and D can take on zero ornon-zero values are shown in Fig. 2.

Steel stress-strainrelation

Eq. (6) relating the stress fp3 andthe strain Ep5 in the prestressing steelat ultimate capacity of the section hasbeen primarily derived from considera-tions of equilibrium and linear straindistribution in the concrete section.

Actual values of fps and E,s mustalso satisfy the stress-strain relation ofthe prestressing steel. The problem isthe same as having two equations withtwo unknowns and solving for theseunknowns.

As these equations are generally ofa high order, charts are developed for

36

TYPICAL RELATION AS DERIVED GRAPHICAL REPRESENTATION FROM EQUILIBRIUM

_ AfPS C B BI Equilibrium

A>0 Stress

B < 0 4 I Strain

• Rectangular beams with pre-stressed reinforcement only,

0Ags '

cps

ACfps a + B +

ps B I Equilibrium

C < 0ISat

Stres • Rectangular beam with Aps , As 4-p' Strain

and/or AS when fts = fy and/orfps = fl',

• TEE beam: with Aps only CTEE beam: with Aps, As and/or 0 EA' when f = f and/or f' = fy g ss y cs

fps = A B + Cc + D+cp

• Rectangular beam with Aps , Asand/or As when Its <fy and/or s-

ai

f' <f'Cs y

• TEE beam with Aps , As and/or ID_As' when fts < fy and/or 11

0 psf

p< fY s

Fig. 2. Typical relations between stress and strain in prestressing steel asderived from equilibrium and compatibility.

their solution such as, for example, inReferences 15 and 17 and in the PCIDesign Handbook (Reference 18, Fig.5.25); they lead, for purely prestressedbeams, to the stress in the prestressingsteel at ultimate; but they do not coverthe case where auxiliary non-pre-stressed reinforcement is used.

In this case a graphical solution (Fig.2) can, in general, be obtained4,13.15by determining the point of intersectionof a curve representing Eq. (6) with

the stress-strain curve of the prestress-ing steel.

The graphical solution, however, isnot very suitable if a repetitive analy-sis is to be performed with many cross-sectional shapes, types of prestressingsteel and different values of prestressedand non-prestressed reinforcement ra-tios. The use of a computerized nu-merical technique such as the one ex-plained below becomes a necessary toolfor solution.

PCI JOURNAL/January-February 1977 37

yof0)

IF

Numerical Techniqueof Solution

There are two aspects of the numer-ical technique as used here; the first isto find a numerical equation whichadequately represents the stress-straincurve of the prestressing steel espe-cially in its non-linear portion; the sec-ond is to find the point of intersectionof two curves, namely, Eq. (6) and thestress-strain relation.

This point of intersection leads tothe solution sought, i.e., the compatiblevalues of f,8 and ep,g at ultimate.

Curve fittingThe stress-strain curve of the pre-

stressing steel was approximated bythree equations, two of them linear,representing the initial and final por-tion of the curve, and one non-linearrepresenting the middle portion (seeFig. 3).

Using a polynomial fit 19,20 for themiddle portion, the stress-strain curveof a typical prestressing strand (diame-ter 0.50 in.; strength = 270 ksi; elasticmodulus = 27,890 ksi; ultimate strain= 0.070; produced by Kurt OrbanCompany) was represented by the fol-lowing equations:

f =27890€ for 0G €1-0.006f = 259 + 196.43(E — 0.014)

for 0.014<€1-0.070f = 167.34 + 254.68443

[100(€ — 0.006)] —37.19658 [100(€ — (7)0.006)] 2 — 419.53833[100(€— 0.006) ] 3 +308.89136[100(€ —0.006)]! for 0.006 <e G 0.014

The maximum error between actualstress value and the value estimatedfrom the polynomial equation isaround 0.4 percent.

PRESTRESSING STRAND(270 KS I)

?50 I LINEAR—

200

PRESTRESSINGNON LINEAR PORTION

1^ . • BAR (160 KSI)150 (POLYNOMIAL FIT]

100 NL

LINEAR REINFORCING BARf=EE

50

0.

0. 0.010 0.020 0.030

STRAIN

Fig. 3. Stress-strain curves of prestressed and con-ventional steel as used in numerical fits.

38

Another type of prestressing steel,namely, a prestressing bar (diameter= 1.25 in.; strength = 160 ksi; elasticmodulus = 28,750 ksi; ultimate strain-^- 0.055, produced by StressteelCorp.) was also used in the numericalexamples.

Its actual stress-strain curve was ap-proximated by the following linear andpolynomial equations:

f = 28750E for 0 G E G 0.004f = 144.90 + 503.33(E — 0.010)

for 0.010 < E G 0.040f = 115.0 + 244J2300{100(€

— 0.004)] — 896,22266[100(€— 0.004)] 2 +

1521.94995 [ 100(E -0.004)] 3 — 946.56592[100(€— 0.004)] 4

for 0.004 < e 0.010

(8)

GIVEN MATERIALS CHARACTERISTICS

AND CROSS SECTION PROPERTIES

6 1 = 0.85 YES f- 5 4000. psi NO

1 6 se f/E 5e s

A fc = s se (l + ±ce AE r2

a u = 0.003 if ACI Code

Design First as Rectangular Section

0.85B l f'bd cuIfps Aps cps + cu - cse - Ece

+ (As SfC -A s tsf)/Aps

Use Compatible Value of fps to Compute:

A f A f A'f'_ ps ps + s is _ s csq bdfc bdfc bdf,

where d is average for A As

1.18 qd/81 S hf ?--YES

0.855 f1 w

bd eu

fps Aps Eps c - E se - Ece

[0.85f b-b)h + A ' -A f* c w f s CS S t51/As

Plot: f = E A+ B • Ccps +D as forps Ips

rectangular section and select new fps

NO \ NEW VALUE OF L CLOSE

TO f INITIALLY USED /j- YESPS

Bl = 0.85

- 5x 10 -5 (f! -4000.)

Plot: fps= E+B +Ceps+Dps

on graph containing stress-strain curve of steel andselect the point of inter-section as compatible valuefor fps:

fps

dw 'peps

strain

I OR T SECTION

DESIGN AS

RECTANGULAR

SECTION

USE COMPATIBLE'

VALUE OF fps TO

COMPUTE ULTIMATE

RESISTING MOMENT,

Fig. 4. Flow chart to determine stress in prestressing steel at ultimate forbonded members.

PCI JOURNAL/January-February 1977 39

01

9 f > O.SfGET EXACT fps FROMNO

CIIseḏpuḏ FLOW CHART FIG. 4

11ḏp =A /bdḏ YESpḏps(Onḏfḏ= f5 • 10,000. +psḏe ḏ100pḏ

Pp = Aps/bdOḏ5 fp`,ḏpḏNOḏBONDED MEMBERḏYESḏ f

f = fp (1-O.SP ^)yḏ< fḏ+ 60,000.ḏ psḏuḏfC1ḏ seḏ p(Dḏ

c

n OḏA fḏA fḏA'f'-sḏ_ is sḏsḏs3 + bdf bdfḏNOḏPRESTRESSEDḏYESḏ=ḏAbdfAfyḏcḏcḏcḏ REINFORCEMENT ONLYḏ qḏpḏbdf'fDḏ0' W W

ḏ

+- ,p ḏ c

fp yḏUse here and in following stepsaverage d for Aps and As

CD

O

3 pḏ I OR T SECTIONḏNOC'

3ḏ1 =0.85(b-b )h f'/fḏb-bḏby - ' pfḏw f c Ps ; mow, / W ASḏYES

n

p1ḏ.A w =Aḏ-ḏ

hf

A f > 0 pḏ

paḏpḏ̂ḏ DESIGN AS

O yḏALL + Asfy ḏAsfYḏ̂f A •ḏNOḏ1.18ḏ< hḏYESḏRECTANGULARfl. ICḏqwḏbwdf'ḏbwdf''ḏbwdfc'p • AsḏBlḏfḏSECTION

3 3

(0 O

0

OḏNOḏqw 5 0.30ḏYESḏ NOḏq <_ 0.30ḏYES.H.

O coC W OVER REINFORCEDḏNORMALLY OR UNDER REINFORCEDḏOVER REINFORCEDḏNORMALLY OR UNDER REINFORCEDcD

-'CL Mu = I0.25fc'bwd 2Muḏ =ḏ d(1

(1

-O)ḏMu = 0I0.25f,'bd 2 Mu = $[A ps fps d ( I - 0.59q)

V Oḏ• 0.85

f'(b-b )h (d-h f /2)ḏ+ 0.SSf'(b-b )h (d-h /2)ḏ• A'f'(d-d')]ḏ• A f d(1 - 0.59q)]s O.ḏcḏw fḏcḏw fḏfḏs yḏs y•ḏ• A'f'ḏ s

ḏ

(d-d')]ḏ• A f d(1 - 0.59gw)]syt

fl1ḏNote: In the exact analysis fy and fY should be replaced by f ts < fy and fp s s fy .

Here the maximum error betweeninput stress and stress estimated fromthe polynomial is around 1 percent.

Intersection of two curvesThe point of intersection of the two

curves representing Eq. (6) and (7)or (8) was obtained using the Newton-Raphson's numerical technique.19,20

In this technique convergence is veryfast (quadratic), and only a few itera-tions are needed. For the results de-scribed below, the stress in the pre-stressing steel at ultimate was deter-mined with an accuracy of 0.1 ksi.

Test Examples

A computer program was written im-plementing the preceding analysis. Fora given prestressed and partially pre-stressed rectangular or T section, theprogram leads to the values of stressand strain in the prestressing steel atultimate behavior as well as the ulti-mate moment, curvature and other rele-vant parameters. Corresponding valuesusing the design procedure suggestedby the ACI Code are also computed fordirect comparison.

Two relevant flow charts of the com-puter program may be of interest to thereader and are shown in Figs. 4 and 5.

The first chart describes the pro-cedure to determine the stress in theprestressing steel at ultimate capacity

by strain compatibility and the secondone describes the steps for the ultimatemoment design of prestressed and par-tially prestressed sections accordingto the ACI specifications.)

This latter chart can also be usedto determine the ultimate moment ca-pacity according to the procedure de-scribed in this paper, provided thevalue of f s and the corresponding val-ues of in G f, and f', G f'„ as derivedfrom strain compatibility are used.

Two beam cross-sectional shapes, arectangular and box-type section(which behave as a T at ultimate) areused to illustrate the results obtainedby the above non-linear analysis. Theirdimensions are given in Fig. 6.

In the following figures, where ob-served results are described, referenceto Rectangular or T relates directly toone of these two beams and indicatetrends expected on similar shapes ofbeams.

Furthermore, reference to f p,,, = 270ksi and f,,, = 160 ksi relates to theprestressing strand and the prestressingbar described in the previous sectionon "curve fitting," for which the stress-strain relationships are respectivelygiven by Eq. (7) and (8).

Conventional non-prestressed rein-forcement is assumed to have a yieldstrength of 60 ksi and to have an elasto-plastic stress-strain relation as shown inFig. 3. A typical design example istreated in Appendix II.

d'= 2i

48'

d'2. IN.

d=45 IN.

A C 784 IN?["227549 IN45'1

Fig. 6. Cross sections of beams used in examples.

PCI JOURNAL/January-February 1977ҟ 41

Analysis of Results

Most relevant test results are de-scribed graphically in Figs. 7 to 12.The first series of results relates to thevariation of the stress in the prestress-ing steel at ultimate behavior of thesection, fps.

Fig. 7 shows the variation of fps

versus the prestressed reinforcementratio for two different values of con-crete compressive strength f', and twovalues of effective prestress, f se.

Also, the ACI recommended value offps for bonded members is plotted onthe same figure and corresponds tothe following equation:

fPS°feu[1-0.5p^f-1 (9)

It can be seen that the ACI equa-tion, as usually expected from a codespecification is quite conservative. Forexample, for p, = 0.4 percent a differ-ence of 7 percent in stress is observedbetween the value of fp. proposed byACI and the value predicted by thepresent analysis.

However, this relatively small differ-ence in stress leads to more than 35

percent difference in strain and a simi-lar large discrepancy may be expectedin computing curvature, rotation or de-flection.

Note, too, that a 7 percent differencein stress leads to about an equal dif-ference in the amount of steel requiredand the corresponding cost (see also theexample in Appendix II).

The influence on j a concrete

RECT. BEAM fpU=270.KSI, e10.003

cfSe=0.s5 xfpufc = 6000. PSI

fSe =0.55 xfpufC =5000.PSI

240y

Nn.

230

220

210

G ACICODE

FOR f >0.5xfpuf^ = 5000 PSI.

fse =0.45 x fpuL f^= 5000.PSI

0.002ҟ0.004ҟ0.006ҟ0.008ҟ0.010

PRESTRESSED REINFORCEMENT RATIO, PP

Fig. 7. Influence of concrete compressive strength andeffective prestress on f.

260

250

42

having a compressive strain at ultimatedifferent from the ACI recommendedvalue of 0.003 is explored in Fig. 8.

It can be seen that the code recom-mended value of f,1 allows for a rea-sonable range of variation of e,, with-out loss of safety. However, here too aslight difference in the value of flead to a substantial difference in strainand curvature.

Fig. 9 shows the influence of non-prestressed tensile and compressive re-inforcement on the value of f, for therectangular and the T section of Fig. 6.The ACI recommended value of Eq.(9) is also plotted for comparison.

It can be observed that the trendpredicted by the ACI equation for therectangular section is well followedover a wide range of variation of the

prestressed reinforcement ratio pp. Thisis not the case, however, for the Tsection at relatively large and not verycommon values of pp.

This result may be due to the defi-nition of the prestressed reinforcementratio for T sections, pp = Aps/bd,where b is the flange width. This ratiois the same as for an equivalent rec-tangular section of width b.

Actually if pP keeps increasing andthe section behaves as a T section, thecorresponding reinforcement ratio ofthe web alone A,,,/b,u d increases at amuch faster rate than pp and thus thestress and strain in the prestressingsteel decrease at a faster rate than forthe equivalent rectangular section.

Similar trends on T sections wereobserved when different prestressing

260

250

240

230

N

N220

210

200

PRESTRESSED REINFORCEMENT RATIO, pP

Fig. 8. Influence of ultimate concrete compressive strain on fps.

PCI JOURNAL/January-February 1977ҟ 43

260 =

240 ACI

N

- 220

Nd

200

.002

RECT. BEAM f = 5000. Sicҟpf = 27o ksifU

fse = 0.55 x fpUeu =0.003

/p=0.006p=0

p'= 0.p=0.

p==0,0.006

T BEAM ff =5000. psifps=27o ksifSe=0.55 x fpa

ACI CODE

(p' "0., p=oJ

(po, p=o.002)

0.001 0,002 0.003 0,004 0,005

PRESTRESSED REINFORCEMENT RATIO, pP

Fig. 9. Influence of non-prestressed reinforcement on fps.

260

N 240Y

Na-

220

200

steels or different effective prestresswere used, but they are not shownherein. It is interesting to note thatthis is a unique case where the ACICode is on the unsafe side.

The influence of prestressed and non-prestressed reinforcement ratios on thecurvature of rectangular and T sectionsat ultimate is shown in Fig. 10. Thecurvature has been computed from thefollowing equation:

= Eu = (Eps + Ell — Eve — Ece) (10)c d

which could also be derived fromEq. (1).

If on Fig. 10 one plots the curvatureat ultimate using E1)8 corresponding tothe fps predicted by ACI Eq. (9) it willalmost follow the lower curve for boththe rectangular and T section. How-ever, it was not plotted to keep thefigure clear.

44

RECT, BEAM fps =27o ksi

16ҟ

f' =5000.psi

fSe =0,55 X fpu

'c 127o

xw

8ҟ o- (p=0,p =o)

U

ҟ

ҟ o (p'=o,p= 0.006)

4

0ҟ.002ҟ.004ҟ•06ҟ.008ҟ.010

11

T BEAM

12

x 8wDI.-

o4

o- (p=o,002,p=o)

0- (p=O,P=o)

0 (p=0, p=0,002)

PRESTRESSED REINFORCEMENT RATIO, pp

Fig. 10. Influence of non-prestressed reinforcement oncurvature at ultimate.

It can be concluded from Fig. 10that the higher the prestressed rein-forcement ratio the smaller the influ-ence of non-prestressed reinforcementon curvature and the more accuratethe curvature predicted from ACI Eq.(9).

In the numerical tests carried in thisstudy with various beam cross-sectionalshapes and ratios of prestressed andnon-prestressed steel, it was generally

observed that the theoretical ultimatemoment capacity predicted by ACI wasless than 10 percent smaller than thevalue derived from the present analysis.

This conclusion confirms the 7 per-cent average difference observed in ac-tual tests as reported by Mattock.4However, as can be observed in Fig.11, a severe restriction on ultimate mo-ment is applied by ACI for "over-rein-forced" beams with prestressed and

PCI JOURNAL/January-February 1977 45

RECT. BEAM fp0= 270. KSI , fSe=0.55 X fpu

f8 = 5000 PSI8

6OVERREINFORC

p=0.0

0.002 0.004 0.006 0.008 0.010

6ҟp'=0.006

4

2

0.002 0.004 0.006 0.008 0.010PRESTRESSED REINFORCEMENT RATIO,pp

Fig. 11. Comparison of ultimate moment by ACICode and present analysis.

M0X

4zaYZW2aO7W

7 8

non-prestressed tensile reinforcement.In this figure a capacity modificationfactor of 0.9 has been used for all val-ues of moments.

In computing the reinforcement in-dex according to ACI, it is assumedthat the compressive steel yields; thisassumption does not seriously affect theultimate moment capacity. Fig. 12 illus-trates the actual variation of the stressin the non-prestressed compressivesteel at ultimate for the rectangularsection.

It can be seen that the higher the

amount of compressive reinforcementand the lower the strength of the pre-stressing steel, the higher the pre-stressed reinforcement ratio necessaryto assure yielding of compressive steel.

In order to illustrate the influence ofnon-prestressed tensile or compressivereinforcement we have used each ofthem separately in combination withprestressed reinforcement. Let us notethat if they are used simultaneouslythey tend to compensate each other'seffects on the stress in the prestressingsteel and the curvature at ultimate.

46

60

NO 40

20

I/0

RECT. BEAM

_fpu=270. KSI , fSe =o.58 xfpu

___fpu =160. KSI fSe =0.45 Xfpu

0.002 0.004 0.006 0.008 0.010 0.012

PRESTRESSED REINFORCEMENT RATIO , pP

Fig. 12. Typical variation of the stress in non-prestressedcompressive steel.

Conclusion

The use of computerized numericaltechniques to model and analyze thebehavior of prestressed and partiallyprestressed concrete beams in the non-linear range may become a useful andefficient design tool when such infor-mation is necessary.

Once a model has been proven satis-factory, the influence of many para-meters can be qualitatively and quanti-tatively analyzed, thus substantially re-ducing the amount of experimentalwork which would otherwise be re-quired.

From a practical viewpoint, the costof a more accurate method of analysismay be largely offset by the savings onthe amount of steel used.

NOTEThree Appendices follow:Appendix I presents anumerical comparison offour methods of analysis.Appendix II contains adesign example solvedusing three differentapproaches.Appendix III summarizesthe notation used.Lastly, a list ofreferences is given.

Discussion of this paper is invited.Please forward your discussion toPCI Headquarters by July 1, 1977.

PCI JOURNAL/January-February 1977ҟ 47

APPENDIX I-COMPARISON OF METHODSOF ANALYSIS

The exact method of non-linearanalysis was applied to prestressed con-crete beams using the stress-straincurve of prestressing steel say from Eq.(7), and that of concrete provided byPopovics9 • 1° and given by:

()(k)1;()fl (H)

where f„ is the peak stress in the con-

crete (i.e., f e,), E. the correspondingstrain, and n a material parameter.

The recommended values of E° andn for normal. weight conerete1° are,,riven by:

=2.7X10 - 4 X 4 f°

n=0.4X10-3Xf°+1 (12)

and were used in the example de-scribed in Table 1.

Table 1. Typical results using different methods of analysis.*

REINFORCEMENT METHOD OF ANALYSIS MOMENT STRESS STRAIN CURVATURERATIOS AT ULTIMATE $ Mu fps eps

kip-in. ksi in. x 10

Exact Nonlinear 2348 261.1 0.02472 10.6pp = 0.002 present Nonlinear 2332 260.9 0.02203 9.31P = 0 PCI Handbook (=261)p =

ACI *0 2286 255.4 0.01172 4.40

Exact Nonlinear 2859 260.2 0.02014 8.40P p = 0.002P = 0.002 present Nonlinear 2818 260.0 0.01843 7.59

p' = 0 ACI 2781 255.4 0.01172 4.40

Exact Nonlinear 2357 261.5 0.02676 11.6Pp = 0.002p = 0 present Nonlinear 2359 261.2 0.02425 10.36

p' = 0.002ACI 2320 255.4 0.01172 4.40

Exact Nonlinear 4374 258.5 0.01370 5.30P p = 0.004 Present Nonlinear 4310 257.5 0.01255 4.71

P = 0 PCI Handbook (=254)=

ACT 4067 240.8 0.00968 3.35

Exact Nonlinear 5188 256.1 0.01200 4.50p p = 0.004P = 0.004 present Nonlinear 5060 251.3 0.01084 3.90P' = 0

ACI 4917 240.8 0.00968 3.35

Exact Nonlinear 4501 260.0 0.01917 8.28

p p = 0.004 present Nonlinear 4464 259.6 0.01708 6.87P = 0p' = 0.004 ACI 4197 240.8 0.00968 ,35

*For the rectangular section of Fig. 4 with f, = 5000 psi; rpu = 270 ksi; fSe = 0.55 fpuand £y = f^, = 60 ksi.

For this method the strain was obtained from the corresponding stress on thestress-strain curve. The stress was computed from Equation (9).

48

In the less than exact non-linearanalysis described in this study it wasassumed that the forces and strains inthe concrete at ultimate are well ap-proximated by the ACI assumptions(Fig. 1) while the exact stress-straincurve of the prestressing steel [see Eq.(7)] is considered.

Using the ACI Code allows the ap-proximation of not only the forces inthe concrete at ultimate but also in theprestressing steel through the use ofEq. (9).

Typical results obtained by using thethree above described methods areshown in Table 1. Also shown are thestresses in the prestressing steel forpurely prestressed sections as derivedfrom the PCI Design Handbook, Fig.5.2.5.

It can be seen that although com-parable values of ultimate moments areobtained by the three procedures, sub-stantial differences in strains and curva-tures at ultimate exist between the ACImethod and the others.

It can also be observed that the ex-act non-linear analysis and the less thanexact present non-linear analysis lead tosubstantially similar results over a widerange of variation of prestressed andnon-prestressed reinforcement ratios.

Note that the results of the exactnon-linear analysis are dependent onthe equation predicting the stress-straincurve of the concrete. Eq. (11), forexample, may not be accurate enoughin predicting the descending branch ofthe stress-strain curve in the range ofinterest.

APPENDIX II—DESIGN EXAMPLE

Consider a prestressed concrete rec-tangular slab with dimensions given inFig. 6 (assuming a 1-ft wide strip) forwhich: f', = 5000 psi; fv ,, = 270 ksi;Ise = 0.55 f,, and f5 = f'y = 60 ksi.

Assume that the ultimate momentrequired at the critical section is equalto 425 ft-kips and that conditions atultimate control the design. Severalsolutions can be derived as follows:

(a) Using the ACI Code approach(see flowchart Fig. 5) for purely pre-stressed sections and after few itera-tions leads to:

pP = 0.0055 ep4 = 0.0890Ap, = 1.38 sq in. q = 0.253fps = 229.90Factorized moment capacity =(PM,, = 427 ft-kips (ok)

(b) Using the strain compatibilityprocedure described in the text leadsto:

pp = 0.005 e„s = 0.01085Ass = 1.26 sq in. q = 0.251fr, 251.3ksi

Factorized moment capacity == 425 ft-kips (ok).

Note that this solution which isalso accepted by the ACI Code resultsin about 9 per cent savings in theamount of prestressing steel required.This is due primarily to the more ac-curate estimation of the value of fp,g.

The value of fp,s can also be determinedin this case from the PCI Handbook,Fig. 5.2.5, which leads to f5, = 250 ksi.

(c) Using a combination of pre-stressed and non-prestressed tensile re-inforcement, the same ultimate mo-ment capacity can be achieved andleads, using the ACI Code approach,to:

pp = 0.004 p = 0.0049fps = 240.8 ksi A, = 1.24 sq in.A5, = 1.01 sq in. q = 0.253

while the more accurate analysis withthe same prestressed reinforcementratio leads to fP = 250.3 ksi and anarea of non-prestressed tensile steel

PCI JOURNAL/January-February 1977 49

As - 1.09 sq in. about 12 percentsmaller than in the previous case.

Note that the PCI Design Handbookdoes not include a chart for f pg when

auxiliary non-prestressed reinforcementis used. Note also that most of theabove results can be estimated fromFigs. 9 and 11.

APPENDIX III—NOTATION

a = B,c, depth of equivalent rec- to effective prestress f86

tangular stress block E,,, = concrete strain at ultimate onA58 = area of prestressing steel extreme compression fiberA8 = area of non-prestressed tension f = stress in general

reinforcement fp,g = average stress in prestressingA'8 = area of non-prestressed com- steel at ultimate load capacity,

pression reinforcement psib = width of flange of flanged mem- fp ,^ = ultimate strength of prestressing

ber or of rectangular member steel, psi= width of web of flanged mem- fse = effective stress in prestressing

ber steel after losses, psi= 0.85 if f', G 4000 psi f is = tensile stress in the non-pre-

i31 = 0.85 — 5 x 10 - 5 (f'0 — stressed tensile reinforcement,4000) if f' 8 > 4000 psi ksi, Its ' Iv

c = distance from extreme compres- fy = yield strength of non-prestressedsion fiber to neutral axis conventional reinforcement in

d = distance from extreme compres- tensionsion fiber to centroid of pre- f' 8 = specified compressive strengthstressing steel or to combined of concrete, psicentroid when non-prestressed f'6s = compressive stress in non-pre-tension reinforcement is in- stressed compression reinforce-eluded ment,'firs

d' = distance from extreme compres- f'„ = yield strength of non-prestressedsion fiber to centroid of com- conventional reinforcement inpressive reinforcement compression

E3 = modulus of elasticity of non- 4) = capacity modification factorprestressed steel if different = curvature of sectionfrom E55 hf = average thickness of flange of a

E, = modulus of elasticity of con- flanged membercrete q = top + co — co', reinforcing index

E53 = modulus of elasticity of pre- of a rectangular member or of astressing steel flanged member designed as a

E = strain in general rectangular memberECe = concrete strain at centroid of p = A$/bd, ratio of non-prestressed

prestressing steel due to effec- tension reinforcement

tive prestress p' = A's/bd, ratio of compression re-

= equivalent concrete strain at inforcementcentroid of prestressing steel, pp = A58/bd, ratio of prestressed re-at ultimate flexural capacity inforcement

Eps = strain in prestressing steel at co =ultimate flexural capacity co' =

Eye = strain in prestressing steel due wp = ppfp8/f'c

50

References

1. ACI Committee 318, "Building CodeRequirements for Reinforced Con-crete (ACI 318-71)," American Con-crete Institute, Detroit, 1971, Chap-ter 18.

2. AASHTO, Standard Specifications forHighway Bridges, Section 5—Rein-forced Concrete, American Associa-tion of State Highway and Transpor-tation Officials, Washington, D. C.,1973.

3. Janney, J. R., Hognestad, E., andMcHenry, D., "Ultimate FlexuralStrength of Prestressed and Conven-tially Reinforced Concrete Beams,"ACI Journal, Proceedings V. 52,January 1956, pp. 601-617.

4. Mattock, A. H., Kriz, L. B., and Hog-nestad, E., "Rectangular ConcreteStress Distribution in UltimateStrength Design," AC/ Journal, Pro-ceedings V. 57, February 1961, pp.877-928.

5. Abeles, P. W., "Design of Partially-Prestressed Concrete Beams," ACIJournal, Proceedings V. 64, No. 10,October 1967, pp. 669-677.

6. Heimdahl, P., and Bianchini, A. C.,"Ultimate Strength of Beams Rein-forced with Steel Having no DefiniteYield Point," AC/ Journal, Proceed-ings V. 71, No. 12, Dec. 1974, pp.600-603.

7. Aswad, A., "Strand Savings UsingStrain Compatibility in Strength De-sign," PCI JOURNAL, V. 21, No. 2,March-April 1976, pp. 78-81.

8. Burns, N. H., "Moment Curvature Re-lationships for Partially PrestressedConcrete Beams," PCI JOURNAL,V. 9, No. 1, February 1964, pp. 52-63.

9. Popovics, S., "A Review of Stress-Strain Relationships for Concrete,"ACI Journal, Proceedings V. 67, No.3, March 1970, pp. 243-248.

10. Popovics, S., "A Numerical Approachto the Complete Stress-Strain Curveof Concrete," Cement and ConcreteResearch, V. 3, No. 5, Sept. 1973,pp. 583-598.

11. ACI Committee 318, "Commentary onBuilding Code Requirement for Rein-forced Concrete (ACI 318-71),"American Concrete Institute, Detroit,1971, Chapter 18.

12. Abeles, P. W., Introduction to Pre-stressed Concrete, V. 2, ConcretePublications Ltd., London, 1966, pp.457-462.

13. Khachaturian, N., and Gurfinkel, G.,Prestressed Concrete, McGraw-Hill,New York, 1969, pp. 72-122.

14. Libby, J. R., Modern PrestressedConcrete: Design Principles and Con-struction Methods, Van NostrandReinhold Co., New York, 1971, pp.109-128.

15. Warwaruk, J., Sozen, M., and Siess,C. P., "Investigation of PrestressedReinforced Concrete for HighwayBridges: Part 3: Strength and Behav-ior in Flexure of Prestressed Con-crete Beams," Bulletin 464, Engi-neering Experiment Station, Univer-sity of Illinois, Urbana, 1962, pp. 105.

16. Naaman, A. E., "Computerized De-sign at Ultimate of Prestressed andPartially Prestressed Beams," Pro-ceedings, 5th Panamerican Sympo-sium on Structures, IMME, Universi-dad Central de Venezuela, Caracas,Dec. 8-12, 1975.

17. Beeby, A. W., and Taylor, H. P. J.,"Cracking and Deformations of Par-tially Prestressed Concrete Beams,"Cement and Concrete Association,Technical Report No. 42.465, Janu-ary 1972, 26 pp.

18. PCI Design Handbook—Precast andPrestressed Concrete, PrestressedConcrete Institute, Chicago, 1971,Fig. 5.2.5.

19. Acton, F. A., Numerical MethodsThat Work, Harper and Row, Pub-lishers, New York, N.Y., 1970.

20. Dahlquist, G., and Bjorck, A., Nu-merical Methods, Prentice-Hall, Inc.,Englewood Cliffs, N. J., 1974.

PCI JOURNAL/January-February 1977 51