Deflection of ProgressivelyCracking Partially PrestressedConcrete Flexural Members

Ali S. AlamehGraduate Assistant

Department of Civil EngineeringAmerican University of Beirut

Beirut, Lebanon

Muhamed H. HarajliAssistant Professor of Civil EngineeringAmerican University of BeirutBeirut, Lebanon

The no slip theory (based on the Ber-noulli-Navier hypothesis that plane

sections remain plane after bending) canbe used to predict fairly accurately thereinforcement stresses and bendingmoments of concrete members. Unfor-tunately, the theory does not reflect theactual mechanical behavior of suchmembers, especially in evaluating thebond-slip phenomenon.1.2

It is generally agreed that the transferof forces across the interface betweenthe reinforcement and concrete throughbond plays a significant role in the be-havior of concrete members. Aspects ofthis behavior include: location, dis-tribution and width of cracks," rein-forcement pull-out forces and develop-ment lengths,¢' anchorage characteris-tics, and slip of reinforcement inside

beam-cu]loOn joints and their importanteffect on the deformation response andductility of the joint under earthquaketype loading.'.8.9

Structural concrete members can beclassified as reinforced (RC), pre-stressed (PC) and partially prestressed(PPC). Whenever such members arecracked, during loading they undergo agradual increase in slip of their tensilereinforcement relative to the surround-ing concrete. This slip phenomenongives rise to a concentrated rotation atthe cracked sections which can heemployed (using the general principlesof mechanics and the second moment-area theorem) to compute deflections.An excellent review (together with de-sign recommendations) of partially pre-stressed members is given by Naaman.1°

94

The magnitude of the slip at thecracked sections and the correspondingconcentrated rotation represents, in ef-fect, a direct measure of the "exact" cur-vature integration between the cracks.There is no easy way to estimate thisquantity from conventional bendingtheory. Therefore, the effect of the un-cracked regions in the member and theircontribution towards the tension stif-fening in its load-deflection responseafter cracking can he measured directlyfrom the reinforcement slip mechanismand the magnitude of slip at the crackedsections.

The main objective of the current in-vestigation is to model analytically theservice load-deflection response of pro-gressively cracking partially prestressedmembers with due consideration to theeffect of tension stiffening. (This has notbeen undertaken before.) The model isderived based on the interaction be-tween steel and concrete through bondand slip of reinforcement at the crackedsections.

The two primary factors that contrib-ute to the tension stiffening phenome-non in concrete members are accountedfor, namely (1) the effect of uncrackedregions between the cracked sections,evaluated in this study from the rein-forcernent slip mechanism, and (2) theinfluence of tensile stresses below theneutral axis position at the cracked sec-tions which are not zero as convention-ally assumed in cracked sectionanalysis.

Failing to account for these two fac-tors in modeling the load-deflection re-sponse of concrete members results in adiscrepancy between the actual andcomputed deflections.

Before attempting to illustrate theformulation of the slip-deflection modeldeveloped in this investigation, a re-view of existing prediction methods forcomputing the deflection of PC and PPCmembers which take into account theeffect of tension stiffening are sum-marized briefly next.

SynopsisA theoretical model for computing

the load-deflection response of pro-gressively cracking partially pre-stressed concrete beams is de-veloped. The model is derived basedon the slip of reinforcement at thecracked sections and accounts for themain factors that contribute to the ten-sion stiffening phenomenon in con-crete members.

Based on an extensive parametricstudy using the developed slip-deflec-tion model, a simplified representationof the load-deflection response, simi-lar to the PCI Design Handbookbilinear model, is adopted, from whicha theoretically derived expression forthe effective moment of inertia i isproposed.

Existing prediction methods areevaluated and compared with the de-veloped slip-deflection model, theproposed !e method and experimentaldata. The agreement is quite good.

Numerical examples are included toillustrate the computation procedureusing the slip-deflection model andthe application of the proposed iequation.

BACKGROUND LITERATUREAt present, it is possible to compute

the service load deflection of reinforcedconcrete members using the effectivemoment of inertia formula (It ) and theconcept of secant modulus proposed byBranson. The expression forle (endorsedby ACI Committee 435 11.12 and adoptedin the ACI Code's ) was derived semi-empiricallye416 based on extensive ex-perimental data and showed very goodagreement with experimental results.

The equation for I,, originally pro-posed for RC members, is expressed asfollows.

PCI JOURNALMay-June 1989 95

`C

I-,t[g e

le

Mo 47 Moo _ Fe ec

p t Deflection

o p on ^1

Fig. 1. Idealization of load-deflection response of prestressedconcrete beam.

(M \3 r 'M 1 gIf

UfaJ (1)

where Mgt and Ma are the cracking andapplied moment at the beam criticalsection; 1. and 1, r are the gross (un-cracked) and cracked section moment ofinertia, respectively.

Although there is general agreementthat the concept of the [-effectivemethod should be extended to PC andPPC members, because of the presenceof the prestressing force, opinions differwidely concerning the definition of theterms in the !e equation. These are (1)the level of externally applied load Ma atwhich the 1t expression should be used,(2) the reference load or state of memberdeformation from which M^, is mea-sured, and (3) the section axis aboutwhich for is calculated, i.e., relative tothe centroidal axis of the cracked trans-formed section, or relative to the neutralaxis of bending assuming zero pre-stressing force (or equivalently infinitemoment). This difference in opinions

with regard to the application of the 1-effective method to PC and PPC mem-bers is reflected extensively in the dis-cussions of Refs. 16 to 18.

Using Branson's approach," the totalcentral beam deflection Ar correspond-ing to a given applied service load mo-ment Ma at the beam critical section iscalculated as the algebraic sum of theupward deflection A^, caused by the pre-stressing force acting alone (prestresslosses included) and the net positivedeflection A. (above zero) as follows(see Fig. 1):

A^ = Nos CFe + Ma — ,blob GF-

(2)EC 1U I3r 1,

where

je— Mer — Mu,a 1.+

s1_ (M,—M,. \ 1 1I (2a)

11R — .M08)

!11 . f* 1, + Fs 19 + FF e (2b)!b ' 9 yh

96

l e

o

Curvat•, }

Fig. 2. Idealization of moment-curvature relationship at criticalsection of prestressed concrete beam.

Mpa, = Fe (2e)

M0 = F, cc GFr/GF;, (2d)

in which GF., and GF^, are moment mul-tipliers corresponding to the gravityload and prestressing force, respectively;Moo, is the moment producing zero cur-vature at the critical section; M, is themoment at the critical section producingzero beam deflection or a fraction of theapplied moment to counterbalance thecamber deflection produced by the ef-fective prestressing force F, actingalone; ee is the eccentricity of the pre-stressing force at the critical section; andf is the concrete tensile strength.

In Eq. (2a), 4 . is calculated relative tothe neutral axis of bending of the criticalcracked transformed section assumingzero prestressing force, in a mannersimilar to the conventional methodadopted for RC members,

It should be mentioned that inanother investigation, Branson andTrost 1e interchanged ,llog by M k in the Ieexpression. Notice that interchangingMop by Monti or vice versa results in asmall discrepancy in the computed de-flection if the moment multipliers GFa

and GF, are different I see Eqs. (2c) and(2d)I; in other words, if the momentdiagram of the effective prestressingforce is not similar (but of opposite sign)to the moment diagram of the appliedgravity load.

In extending the I-effective method toPC and PPC members, Tadros" pro-posed the following expression for com-putingle (see Fig. 2):

_ 'cr — tWdec

1 e — `: — Mdec 1 I9 +

^ I T r Mcr — -L1dec. 3 l 1cr . lyM

(3)I1 e — '«dCC

where

Feld (3a)'N aec = + Fe er

AQTJb

in which V e is the decompression mo-ment which causes a zero stress at theprecompressed fiber of the critical sec-tion.

Unlike Branson's approach, ICr is cal-culated in Eq. (3) relative to the centroi-dal axis of the cracked section (as theo-retically it should) at a depth y from

PCI JOURNALJMay -June 1989 97

I- V ^i tc

TEIL--- dpiN.A

s fsbw

Fig. 3. Typical stress distribution in prestressed concrete section neglecting concretetension stress below the neutral axis position.

the top fiber (see Fig. 3). Using momentequilibrium and assuming linear stressdistribution across the depth of the sec-tion, the following equations can be de-rived for computing y and lc,.:19

(r, – b,,)h-I2+ b,,,c g/2+ rn,A5 d7,+ n.A. d,(b – b,)hf + cb,,+ n,,Ap + n2A,

(4)

1,., = b,,, y3/3 + b (c+(b–b,,)hf(y–h,12Y2

of inertia 1,, .In computing I,,., the PCI Design

Handbook (second edition) recom-mended the following approximate equ-ation:

Icr = n, A„ dF (1– J) (6)

where p„ is the prestressing steel ratio.Eq. (6) is applicable mainly to fully pre-stressed members. In order to accountfor partially prestressed members, Naa-manL° proposed the following modifica-tion:

+ (b – b,,,) h112 + np Ap (d,, – y ?+t A, ([ –j? (5) Irr= n ,ds)(1–SIP,+Ps)

where n,, and n, are the modular ratio ofthe prestressing and reinforcing steel,respectively; c is the depth to the neut-ral axis of bending calculated using alinear elastic cracked section analysis20corresponding to the given level ofapplied moment M. at which deflectionis being computed.

The PCI Design Handbook2I recom-mends the use of either a bilinear load-deflection model based on the require-inents of Section 18.4.2 of the ACICode13 or Branson's I-effective method.In the bilinear model, deflection beforecracking is computed using the grossmoment of inertia I., while the addi-tional deflection after cracking is calcu-lated using the cracked section moment

(7)

where pR is the ratio of reinforcing steel.More recently, in the PCI Design

Handbook (third edition), the cornputa-tion ofh. r has been modified to:

Icr = n,A,d2 (I – 1.67 v r1p pp) (8)

The I-effective method and the corre-sponding expression of 1c proposed byBranson represents a transition betweena fully cracked and a fully uncrackedbeam and, hence, is a direct method thataccounts empirically for the effect oftension stiffening. The same can benoted of Tadros' modified expression forIe [Eq. (3)1 as well as the bilinear modelof the PCI Design Handbook.

98

j ds0NA cn ^^`

I n 5;II

Fig. 4. Crack pattern and slip of reinforcement in a cracked concrete beam.

Tension stiffening can be also ac-counted fur using Branson's approach inan indirect manner by computing de-flection from curvature integration alongthe beam length. In this case, Bransonproposed that the curvatures at the vari-ous beam sections be computed usingEq. (1) in which the Mer /Ma ratio israised to the fourth power.

In a similar approach to Branson's in-direct method, Tadros et al.'" accountedfor the effect of tension stiffening byconsidering further the effect of changein eccentricity of the prestressing forceafter cracking. The curvature of the sec-tion into consideration is calculatedfrom:',

4)– Ma –FeecrEEIP (9)

Ie = R 4 I. + (1 – RA ) I (9a)cr

R = Mcr – Mal`. (9b)M. – Ma,

Other methods account indirectly fortension stiffening by interpolating be-tween curvatures of gross and fullycracked sections such as in the CEB-FIPmodel22 or by interpolating betweendeflections of uncracked and fullycracked members such as that proposedby Trost23 The phenomenon of tensionstiffening was also studied analyticallyby Bazant and Oh2t for RC members byaccounting solely for the effect of con-crete tension stresses at the crackedsections using a bilinear material stress-strain curve (tensile softening model) ofconcrete in tension.

SLIP-DEFLECTIONANALYTICAL MODEL

Consider the simply supportedcracked concrete member shown in Fig.4. The service load deflection of thebeam at midspan, measured from thecamber position, due to an externallyapplied symmetrical loading can he ex-pressed as:

e'er = d.n — ye (9c) A r = GFA if MQ a Mcr i LOa)Rc Iu

t/e = W+ (1–R) (9d) (9d)

where y' is as defined in Eq. (4) and yt is Ar = A, + X 8, 1; if M> Mcr (lob)the distance from the centroidal axis of i- 1the gross section to the top fiber, where

PCI JOURNAUMay-June 1989 99

ft

fr---

Et

Fig. 5a. Stress- strain behavior of concretein tension used in the analysis.

fC

c

fr

Fig. 5b. Typical predicted concrete stressdistribution in the beam cracked section.

= S. (11)

Llrn — C j

pP = Fee` GFu = M

°b GFa (12)

E19 E,Ig

in which the subscript i stands for thecrack number; n is the number of cracksin half the beam length; B and S are, re-spectively, the concentrated rotationand slip at the cracked section i ; d, and care the steel and neutral axis depths ofthe cracked sections and 1 1 is the dis-tance from cracked section i to the sup-port.

As indicated earlier, the primary ad-vantage of computing rotation from theslip of the tensile reinforcement [Eqs.(10) and (11)1 is that the magnitude ofrotation computed as such is theoreti-cally equal to the integral of "exact"curvature distribution between thecracked sections.

AssumptionsThe following is a summary of the as-

sumptions adopted in the current inves-tigation:

1. The stress-strain behavior of con-crete in tension follows the material lawshown in Fig. 5a.

2. The first primary crack initiates atmidspan whenever the f'lexiiral tensilestress in the bottom fiber exceeds the

concrete tensile strength f. The numberof primary cracks increases in a piece-wise manner at a constant crack spacingfrom the critical section location at mid-span towards the end support with in-creasing level of applied load. This as-sumption might appear to be in violationof the random crack formation close tothe maximum moment region along thespan. However, cracks whenever initi-ated tend to stabilize shortly after thecracking load. Also, a parametric evalu-ation of the effect of varying the crackspacing (within the practical limit) onthe predicted results and comparisonwith experimentally observed deflec-tions shows that this assumption is rea-sonable enough and does not violate theaccuracy of the model for deflectioncomputations.

3. Because of the large contribution ofbar deformation (lugs) in reinforcingsteel bars to bond resistance through themechanical interlock mechanism incomparison with the smooth prestress-ing steel surface,°- s the slip of rein-forcement is expected to be less inreinforcing steel bars in comparisonwith prestressing strands. Therefore, itcan be assumed that the crack openingin beams containing a combination ofprestressing strands and reinforcing bars(partially prestressed) is controlled bythe slip of reinforcing bars,

La !l

9max

Q ^ I IQ (O}

Vn

Qmax

wW I— — fso --^ (b)

to Ln^ I

O tZU

Fig. 6. Typical predicted bond stress, steel stress andconcrete tensile stress distribution between two cracks.

4. The externally applied moment isassumed constant between any two pri-mary cracks and is equal to the momentat the center of the cracked region, i.e.,the steel stresses at the left and rightboundary of the untracked concreteprism are equal.

Slip ComputationThe slip S and correspondingly the

concentrated rotation 0 at the crackedsections [Eq. (11)] are computed usingthe slip model developed in Ref. 2.

To illustrate the use of the slip modelin the current investigation, a concretetensile prism bounded by two primarycracks with typically predicted bond

stress, steel stress and concrete tensilestress distribution is depicted in Fig. 6.The reinforcement is subjected to twoequal boundary steel stresses (Assump-tion 4). These boundary steel stressesare obtained from moment equilibriumusing linear elastic cracked sectionanalysis taking into consideration theconcrete tensile stresses below theneutral axis.

As shown in Fig. 6a, the bond stressdistribution exhibits two portions: (1) anelastic portion extending a distance 2L,(to be determined) to the location wherethe peak bond strength q,,, is de-veloped inside the interior portion ofthe prism, and (2) two linearly de-

PCI JOURNALIMay-June 1989 101

scending portions, each extending adistance L,, from the crack to the locationwhere q,,, is developed.

Using the predicted bond stress dis-tribution, it is possible to compute thesteel stress and steel strain, the concretetensile stress (as shown in Fig. 6) andcorrespondingly the slip distributionalong the reinforcement embedmentlength between the two cracks. Particu-larly, the slip at the cracked section isexpressed as follows:

S = 2l sinh(A,L,)+ 2f'4,L

– q,^Qr q n,La/3 (13)

where

i'.= , k 0 in (13a)

Re = +

A,E,(13b)

k2 _ J,/Lm– fir /Lc (13c)Kl cosh (Jc., L,)

f r = f; – 0.5q kOLQ IA,,, (13d)

ffr = 0.5q.R.x4GLa/A, (13e)

Ar = A1 /n, (131)

in which K, and K Q are bond-slipcharacteristic coefficients; L is half thecrack spacing; k is the slope of the localbond stress-slip relationship; f and fare the steel stress and concrete tensilestress at the location of qm„..; Aso , / andE., are the area, circumference andmodulus of elasticity, respectively, ofone reinforcing bar or prestressingstrand; f. is the stress in the reinforcingbar or increase in stress above effectivein the prestressing strand at the crackedsection; A, is the effective area of theconcrete tension prism surrounding theprincipal reinforcement; and rt, is theequivalent number of bars (total area ofreinforcement divided by the area oflargest bar diameter). The definition ofA, and n is in accordance with Section10.6 of the ACT Code.'3

The distances L, and 1, shown sche-matically in Fig. 6 can be computed bytrial and error using the following trans-cendental equation:

K, qm,z 0.5 q,,, /1 m (L — Lt )] x

tanh (K, L t ) = 1.0 (14)

If L, = L, the left hand side of Eq. (14)is less than 1.0, then q m„, is not de-veloped and the bond stress distributionis elastic everywhere inside the tensileprism between two cracks. In this case,the slip S is computed from Eq. (14) bysubstituting Lr = L and I, = 0.0. Thecrack spacing a^. a and correspondingly Lcan be calculated using Gergely's em-pirical expression25 •26 as follows:

G,= 2L 7.6x10- s E.0 3^ d^.A, (15)

where d is the distance from the centerof the outermost steel layer to the con-crete tension fiber. This last expressionfor estimating the crack spacing wasfound to be in good agreement with ex-perimentally observed results of rein-forced, partially prestressed and fullyprestressed beam s. 27 21

The magnitude of qm. and k for rein-forcing steel bars can be estimated usingthe experimentally observed results ofbond-slip behavior and idealization ofthe local bond stress-slip relationshipproposed by Eligehausen et al l, asshown in Fig. 7. For unconfined con-crete, the magnitude of q,,,, x and k fordeformed reinforcing bars are expressedas a function off,' (ksi) :2

k = 145ff (k/in?) (16)

4mu.r – 0.72 mi fc 14.35 (ksi) (17)

For prestressing seven-wire strands,the magnitude of umar and k were esti-mated as 0.56 ksi (3.86 MPa) and 650kips/in3 (175 Nlmms ), respectively, cor-responding to f , = 6 ksi (40 MPa). Thesevalues are consistent with the range ofexperimental results reported by Ed-

102

a

5

4

C

m 3

0U021

C

f^ = 30 N/mm2 (4-35 ksi)

Ck

gmax 9max = 5 N/mm2(0.72ksi)

ligehausen et.al. (fief. 8)

ealized

0 0.1 0.2

Local Slip , mm

Fig. 7. Idealization of local bond stress-slip relationship usedin the analysis.

wards and Picard,5 Gyltoft g Stocker andSozen.'

The slip model described earlier wasshown2 to reproduce within a good de-gree of accuracy experimentally ob-served boned-slip results under serviceloads as illustrated in Figs, 8 and 9. Also,with little modification, the slip modelis able to predict experimentally ob-served results of both monotonic andcyclic load-slip response beyond theservice load and into the post-elasticrange .9

Method of Analysis

A computer program implementingthe above described model for comput-ing deflection was written in Fortranand used to conduct an extensiveparametric evaluation. The program iscapable of handling both uniformly dis-tributed and concentrated loads as wellas straight, draped and parabolic tendonprofiles.

During the computations, the load is

increased monotonically at small loadincrements. Before cracking, the beamdeflection is computed using Eq. (10a).At any load level after cracking, thenumber of cracks and their locationalong the beam length is monitored (As-sumption 2) and the applied bendingmoment at the center of the crackedregions is determined (Assumption 4).Then, the neutral axis position c is cal-culated for all the cracked sections usingcracked section analysis taking into ac-count the concrete tensile stressesbelow the neutral axis position (see Fig.5).

Considering the most general case of acracked T section (neutral axis in theweb), the computation of the neutralaxis position can he achieved by solvingiteratively the following two simultane-ous equations, derived from force andmoment equilibrium across the depth ofthe section:Force equilibrium:

Ace +Bc+C=0.0 (18)

PCI JOURNALMay-June 1989 103

0.3

0.2

E 0.1E

r ^

0.3

cv 0.2

0

+I

t

/ Beam PS1 PPIS2r

— Experiment (Ref. 27)---Slip model (Ref. 2)

PP2S1 •'^ RS3

•

0 20 40 W0 20 40 60

Steel Stress (fso ) , ksi

Fig. 8. Comparison of observed and predicted crack widthsversus steel stress.

o— Experiment (Ref. 4)--- Theory (Ref. 2)

12CC

1000to

800

_

600

200No.9 Bar, As_O . 875 in2

01 2 3 4 56 78(in)

Fig. 9. Comparison of analytical and experimental steelstress distribution.

104

where

A= fb2 [(T)

– 11 (18a)

JB = Apdp(epe+ece)

— E (Ar,Ep+AsE +A;E.)C

— (h — b,„) hr f (18b)

C=E d p + A.Es d,+ A; E.d;p ) E,

+ (b — b)f, h;/2 (18e)

Moment equilibrium:

Ma –A„E,,It, +ECe + (d — c `1c Ej

d –A, Ed ( d* –c} f d.C lJ Ec

_ f^^'^ 1 2 r l c2

2f 3.f l

– (c–h,)z(b b.)(hf+ ° 3h'12c )

+ f bc'16 +A;E,( –d$^ f d;c E^

= 0.0 (19)

where e, and Eye are, respectively, theeffective prestrain and concrete pre-compressive strain at the level of pre-stressing steel (see Notation in Appen-dix), f is the concrete compressive stressin the top fiber of the section.

Eqs. (18) and (19) contain two un-knowns c and j. The solution can beobtained by iteratively assuming f^ andsolving for c I Eq. (18)) until the momentequilibrium (Eq. 19) is satisfied with aspecified tolerance. It should he indi-cated that neglecting the tensile stressesbelow the neutral axis (f,. = 0), allowsthe combination of Eqs. (18) and (19)into a single cubic equation° inc.

Bycomputingf^, and c it would be pos-sible to calculate the stress fw in the ten-sion reinforcement. Consequently, theslip [Eq. (13)], the local rotation [Eq.(11)1 at each cracked section and the de-flection at midspan [Eq. (10b)I corre-sponding to the given level of appliedload are calculated, thus generating onepoint on the load-deflection response.The load is then increased and the com-putation procedure is repeated until themaximum elastic load level is attained.

The maximum load is chosen to satisfythe following two criteria: (1) the steelstress in the tension reinforcement isless than its elastic proportional limit,and (2) the compressive stress in theconcrete uppermost fiber at the criticalsection is less than 0.7f .

PARAMETRIC EVALUATION

An example beam with a 40 ft (12 m)span, designed for three types of beamsections as shown in Fig. 10 is used to

4ri 20'

11 3" 12.4'

1 P 4° 24"24 2y"

AsLJ A T2.4

Fig. 10. Types of sections studied.

PC! JOURNALJMay-June 1989 105

Table 1. Summary of parameters investigated.

I "Three types of beam sections: liectungu]ar, T and I sections

_'. Hour levels of Partial Prestressing Ratios (PPR): 0, 0.4, 0.7 and I

3_ Four levels of Reinforcing Index fw): 0.08, 0.12, 0.18 and 0.25

t. Three tendon profiles: straight, parabolic and sin .ade point depressed

conduct a parametric study of both thedesign and bond-slip parameters andalso to undertake an extensive evalua-tion survey of existing predictionmethods. The beam is simply supportedand subjected to uniformly distributedload.

A summary of design paralaetelS ispresented in Table 1. They includethree types of sections, three differenttendon profiles, lour levels of reinforc-ing indexer covering the practical rangeof design Q is defined in the AC! CodeSection 18.8) and four levels of partialprestressing ratio PPR. The partial pre-stressing ratio, originally proposed byNaaman and Siriaksorn, 20 is expressed asfollows:

PPR = AJ,,,(20)

Au1

r

w + Aelywhere f;,, is the stress in the prestressingsteel at nominal flexural strength and f,is the yield strength of the reinforcingbars. PPR equal to 0 and 1 representfilly reinforced and fully prestressedmembers, respectively, while PPRequal to 0.7 and 0.4 (see Table 1) re-present two different levels of partialprestressing. The reinforcement waschosen from commercially availablesizes and distributed in the beam sec-tion to satisfy AC! Code minimumspacing and cover requirements.

In addition to the parameters given inTable 1, different values of u, k andq ox and also different reinforcementsizes were used to study the effect ofcrack spacing, local bond stress-slipcharacteristics and bar diameter on thepredicted deflection results. It should

he noted that parameters such as k, q,,and bar diameter are not independent;rather, they are related directly or indi-rectly to crack spacing. The crack spac-ing for the various example beams var-ied between 6 and 14 in. (150 and 350mm) depending on the level of rein-forcing index and partial prestressingratio.

A comparison of load-deflectioncurves obtained using the slip-deflec-tion model for different crack spacings,together with those obtained by theconventional curvature integrationmethod neglecting the effect of tensionstiffening is shown in Fig. 11. Typicalresults describing the load-deflectionresponse for different q,, bar diame-ters and k are shown in Figs. 12, 13 and14, respectively. The following obser-vations can be made from these figures:

(a) Increasing the crack spacing (Fig.11) tends to decrease the predicted de-flections. Increasing the crack spacingincreases the contribution of the un-cracked concrete regions to the overallflexural rigidity (stiffness) of the beamand therefore reduces its deflection asexpected. However, although an in-crease or decrease in crack spacing af-fects the magnitude of slip and crackwidths proportionately, it does not havea significant effect on the load-deflec-tion response (compare solid lines inFig. 11). Also, it can he observed in Fig.11 that the stiffening in the load-deflec-tion response (compare solid anddashed lines) is a maximum followingthe cracking load and tends to decreaseprogressively with increasing level ofapplied load as expected. However, the

105

2.0

C 1-0

0X

4.0Lii

0

2.0

0

I /3PPR = 0.7

R=QO

7UujSlip- aes=10 PPR =7.DeflMod

IL acs=20„e

tensionshlFening

PPR 0.0

PPR = 0.7 PPR=1.0.' w=0.18

0 1 2 3 1 2 3 1 2MIDSPAN DEFLECTION A, IN.

Fig. 11. Effect of crack spacing (tension stiffening) on the load-deflection response ofpartially prestressed beams.

stiffening effect maintains almost a eon-stunt level for a large portion of theload-deflection response.

(b) Everything else staying constant,increasing the bond strength gmar (Fig.12) tends to decrease the computed de-flection. Increasing the bond strengthreduces the slip at the cracked sectionsand consequently reduces the beam de-flection. However, the overall effect ofincreasing or decreasing q,, on theload-deflection response appears to beinsignificant particularly with decreas-ing level ofrcinlorcement index.

(c) Varying the reinforcement diame-ter (Fig. 13) has a minor effect on thepredicted deflection results. Also, vary-ing the slope of the local bondstress-slip relation by more than 50 per-cent (Fig. 14) results in less than 5 per-cent difference in the predicted deflec-tion indicating that the linear idealiza-tion of the local bond stress-slip relation(see Fig. 7) is adequate.

EVALUATION OF EXISTINGPREDICTION EQUATIONS

AND PROPOSEDREPRESENTATION OF Ie

Based on the results of an extensiveparametric evaluation undertaken usingthe developed slip-deflection model(verified with experimental data as il-lustrated in the subsequent section), itwas found that the phenomenon oftension stiffening in computing serviceload deflection of PC and PPC memberscan be best simulated for design pur-poses using the load-deflection ideali-zation depicted in Fig. 15.

This idealization represents two pathsof load-deflection response: precrackingpath with Slope proportional to the grossmoment of inertia lu and post-crackingpath with "apparent" slope proportionalto the cracked moment of inertia Irn,which depends for prestressed and par-

PCI JOURNAL^May-June 1989 107

— m0x=9.{Eq.17)

40 0 9max = 0 67 90 w=0.25

©^mox =l.59.

2O —' ^Iu=012

N • PPR.0.4xI--z

W 1 2 3o 40 u+= 0.25

z

iI? Lu = 0.120

20

• PPR^O 7

00 1 2 3

MIDSPAN DEFLECTION (in)

Fig. 12. Effect of bond strength on the predicted load-deflection response.

108

4 ^ PPR=0 4w^0.25

M5.^ No.9

^^-- w=o-122:/u1

xF_ 0zW 40 u:0-25

No.PPRzfl.70 ^No 7

' w=0.12cJ No3

2 /I1:110

0 1 2 3

MIDSPAN DEFLECTION (in)

Fig. 13. Effect of bar diameter on the predicted load-deflection response.

PCIJOURNALJMay-June 1989 109

40

O_

I--Z

W

02

za9

0►9

20

—k=k.(Eq.16)

k=1.5k. iD.025

^2 067k,^

w=0.12

PPR,0.7

0 1 2 3

MIDSPAN DEFLECTION (in)

Fig. 14. Effect of linear idealization of the local bond stress-slip relationship on the predicted load-deflection response.

tially prestressed members on the levelof applied moment. The correspondingidealization can he l^tcilitated by deriv-ing a simple expression for I,* (see Fig.15) to be used in conjunction with theconcept of secant modulus proposed byBranson.

The expression for I, takes the fol-lowing form:

I r = I^^ if A4.1 Mcr

(21)

Ie = Icra ifMa > Mcr

I a 4 1– I17) (22)

where Ic,, is determined relative to thecentroidal axis of the cracked trans-formed section neglecting the concretetensile stresses below the neutral axisposition [Eq. (5)1; Me,. is the crackingmoment including the effect of the pre-stressing force [Eq. (2b)]; and M is thetotal applied service load moment (DL+ LL) at the critical section.

It should be mentioned that the pro-

posed idealization of the load-deflectionresponse shown in Fig. 15 is consistentwith Section 18.4.2 of the ACI Code andis also similar to the bilinear load-de-flection model of the PCI Design Hand-hook except that the slope of the post-cracking load-deflection response variesdepending on the magnitude of appliedload rather than constant as proposed inthe Handbook.

Typical comparison between load-deflection curves obtained using variousprediction methods (including the pro-posed 1* approach), and the more accu-rate slip-deflection model is shown inFig. 16. Comparison of results for all pa-rameters indicated in Table 1 with thedifferent prediction models presentedearlier are shown in Figs. 17a-17e. Incomparing with the PCI Design Hand-book bilinear model, Eq. (7) wasadopted for computing I,. since it ac-counts for the presence of additionalnonprestressed reinforcement in thebeam. Notice that Eq. (22) can also heused for computing deflections by thePCI Design Handbook bilinear model

110

H Deflection

^p An

Fig, 15. Proposed idealization of load-deflection response ofpartially prestressed beam.

40

30

C

° 20xzw0

10

0

— Slip-Deflection Model

A- Ie (Eq.22)

0- Branson (Eq.2a) W:0.18

q - Todros(Eq. 3) •'

_

fl U1= 0.12

u)= 0.08

PPR=0.7— — — — — -- Ef--

I0 1 2 3 4

DEFLECTION (in)

Fig. 16. Typical comparison of load-deflection curves obtained by various predictionmethods and analytical slip-deflection model.

PCI JOURNAUMay-June 1989 111

4

3N

LiJ

0J)c0L

W

I1

0 0Rect, T & I Section Q /•20/°w = 0.08-0.25 o Q/Stra. &parabolic profile QQ

O- PPR =1 00Q /Q -PPR=0.7 °/QEJ- PPR =0.4

L4 / /

o PPR = 0 o /0 /

,O ,/O O

A/O / p O

Fig. 17(a)

00 1 2 3

4

(Slip model)

4

r3crw00Lo 2

1

O-PPR= 1 QA A/ 20%A

A- PPR= 0 . 7 A

DR 1/110- PPR=0.4 Q

0Q-PPR=0 0 o /

OD

o / /

/ Q

E'cjDi Fig. 17(b)

1 2 3 4A, in (5f p model)

112

3

0UJ

0L

F-

I

O-PPR = o1 AD / °20%

A-PPR 7 07 A Aq -PPR

=

0 .4 /

G-PPR = 0 A // 20°%/

A A"^

A0 Q

Q ^0

UG 0• 0

D Dcp A Fig. 17(c)

/ IA Ai/,^

4

MJ 1 2 3

4

A,to (Slipmodel)

4

3L

C

2Un

<1

•20%O- PPR = 1 / D

D- PPR = 0.7 A

D-PPR =04 4/AA L//

G-PPR = 0 /0 0 /

/O/^ 0

p8 // Fig. 17(d)

1 2 3 4L , in (5I ip model)

PCI JOURNAUMay-June 1969 113

4

3

N

2

I

0

•20%O- PPR = 1 p

p ^- PPR = 0.7

O

d- PPR = 0 4 D ,,o- PPR = 0

cPg :J-

p p ^ oq i

os

pG

Fig. 17(e)

0 1 2 3 4A, in (Slip mode!)

Fig. 17. Comparison of deflection results obtained using various prediction methods andslip-deflection model: (a) Branson I Eq. (2a)]; (b) Tadros I Eq. (3)]; (c) Tadros (Eq. (9)1; (d)PCI Design Handbook bilinear model; (e) Proposed!, [Eq. (22)],

by simply substituting I.r [ Eqs. (6), (7),(8)) forlcra.

It can be observed in Fig. 17a thatwhereas the comparison between theslip-deflection model results and resultsobtained using Branson's I-effectivemethod [Eq. (2a)] is excellent for RCbeams ( = 0), the comparisonis not as good (hr PC and PPC members.Similar observation can be made (Fig.17b) using the 1, expression proposed byTadros [Eq. (3)[ with little larger dis-crepancy in the results particularly withincreasing level of partial prestressingratio.

Although Tadros' tension stiffeningmodel (Eq. (9)1 resulted in a better cor-relation with the slip-deflection modelresults in comparison with Eq. (3), it canbe seen in Fig. 17c that in some cases,the effect of tension stillening is largely

overestimated resulting in much lesscomputed deflections in comparisonwith the slip-deflection model results, Asimilar observation was reported byScanlon et al. in the discussion ofRel: 18.

On the other hand, it is interesting toobserve in Fig. 17d that with itssimplicity in application, the agreementbetween the PCI Design Handbookbilinear model and the slip-deflectionmodel results appears to he better thanthe I-effective method particularly withincreasing level of partial prestressingratio (PPR equal 0.7 and 1).

It can be shown in Fig. 17e that theproposed 1 equation reproduced con-sistently the slip-deflection model re-sults within ±20 percent at all levels ofpartial prestressing ratios. Furthermore,it is clear from the comparison of Figs.17a-17e that the proposed idealir.ation

114

00

Fiq 18(a)

EXPER (Ref.27)A _ SLIP MODEL

0— I (Eq.22) gP/

PS1 (PPR= 1) PS2 (PPR= 1 l PS3( PPR= 1)

F

i

rfPP2S1 (PPR=0. 7? PP252(PPR =0-7) r PP2$3(PPR=0.7)

0.5 1.0 0 5 1-0 0.5 1.0DEFLECTION L. in

Fig. 18(a) continued

rp

r

PP]S1 (PPR= 0.41 r PP1S2 (PPR=OA) 1 PPIS3{PPR=0.4)

RSI(PPR= 0) /S2)PPRZ 0) RS3(PPR= 0)

0.5 1.0 0.5 10 0-5 1.0DEFLECTION a , in

12

8

a

0

12

0

8

4

12

B

^ 4a

0

12

0

8

4

PCI JOURNAL/May-June 1989 115

a

C-

40J

Fig. 18(b)

— EXPER. (Ref. 29)

A—SLIP MODEL

20 O-1e (Eq.22)

R8.34.031 (PPR = 1 7,^^ 11

RB.34.093 ! R8.34.126(PPR= 1) i (PPR=1)

05 1.0 0.5 1.0 05 1 1

DEFL,ECTiON L.. in

Fig. 18(c)

— EXPER. (Ref. 30)

SLIP MODEL0— le (Eq . 22)

,^' SFr

83 (PPR = !) 1 1 84 (PPR = 1)

0 0.5 10 0 0.5 1.0DEFLECTION A, in

of load-deflection response shown in with other existing prediction methods.Fig. 15 and its corresponding derived J The next section, which compares theequation leads to a better simulation of various prediction inclhods with test re-the presumably more accurate slip-dc- stilts (pins also the numerical examnples),flection model res iii ts in comparison will make the above concepts clearer.

16aY

• 12

0 8

V

116

Fig. 18(d)

Fig 18(e)

120

zY

80

0J

40

L1652 P f(PPR Q) j i^

/ r

— EXPE. (Ref. 31)

1 A-SLIP MODEL0-1 (Eq.22)

2 4 6 8DEFLECTION mm

120

zY80

00J

40

/04) Pr

f. 32)EL

2)5 10 15

DEFLECTION, mm

Fig. 18. Comparison of analytically predicted and experimentally observed load-deflectioncurves: (a) Experimental results of Ref. 27; (b) Ref. 29; (c) Ref. 30; (d) Ref. 31; (e) Ref. 32.

COMPARISON OFPREDICTION METHODSWITH TEST RESULTS

Load-deflection results computedusing the developed slip-deflectionmodel and the derived expression forIare compared with representative ex-perimentally observed load-deflectioncurves of concrete beams with variouslevels of partial prestressing ratios. Theexperimental results include those re-ported by the aiithor et al. 2 7.21 Warwarn,Sozen and Siess,29 Nawv and Huang,3L,ovegrange,31 and Mansur 31 Results ofcomparisons are shown in Figs. 18a-18e.Comparison between analytical predic-tions obtained using the various predic-tion equations and all the experimentalresults reported in this text are shown inFigs. 19a-19f:

In the experimental program of Ref.27 and 28, four different sets of beamsdesigned for twelve different sets ofreinforcing parameters (three levels ofreinforcing index and, for each, fourlevels of partial prestressing ratios in-

eluding fully reinforced and fully pre-stressed) were tested. All teams were4.5 x 9 in. (115 x 230 min) sections, 9 ft(2.75 m) span and loaded in four-pointbending. A detailed description of theexperimental program and comprehen-sive set of data are found in Ref. 27. De-tails of material properties and rein-forcment layout of beam specimens byother investigators are found in theaforementioned references. In repro-ducing the experimental results, crack-ing in the beams is assumed to_com-mence corresponding to f,. = 7.5

It can be seen in Figs. 18a-18e that theslip-deflection model results and de-flections computed using I* are in verygood agreement with the experimentalresults far the entire range of partial pre-stressing. Both the slip-deflection modeland deflections computed using 1e [Eq.(22)1 capture the trend of load-deflectioncurves quite accurately with little dis-crepancy that can arise especially inunder or overestimating the crackingload due to larger or smaller concretetensile strength f, and effective prestress

PCI JOURNAL/May-June 1969 117

0EC-

0.5

O-PPR = 1 •20'/.-PPR=07 Ref.27 /

D-PPR = 04 /G-PPR = 0 / i/

0 -Other exper.

° Fig. 19(a)

s,

1.0

00 0.5 1.0

^,in (Exper.

1.0

C a5

°

•20 '1,

i

o ^a

I W i/

Fig. 19(b)

Le

05

1.0L\, in (Exper. )

fk, in the prestressing steel.Despite this discrepancy, the differ-

ence between the computed and ex-perimentally observed deflections aremostly within ±20 percent as illustratedin Figs. 19a and 191). On the other hand,it is interesting to observe from Figs.19b-19f that the correlation trend be-tween deflections obtained using thevarious prediction methods and the ex-perimentally observed results is quitesimilar to that made earlier with the

slip-deflection model. It is also clearfrom Figs. 19b-19f that compared toother existing design predictionmethods, deflections computed usingthe proposed I equation encounteredthe least scatter in comparison with theexperimentally observed results.

Three examples are given next to il-lustrate the application of the proposedt method and the computation proce-dure using the developed slip-deflec-tion model.

118

I6

uJc0C0

0.5

•20%

ZLZoLC

o^p Q

Fig. 19(c)

•20%,

0Q / /4O / /

A / - 20'/

o^zf

/ Fig. 19(d)

•

Q5 1 OSA. in (Exper.) ,, in (Exper'.)

I

L0

C

U 0.5

20%y

/ ^w/

/0/

I

od A

p

/ a

,^ Fig. 19(e)

+2 0'/.

// //

/ ' /-20'

,o

AOG

Fig. 19(f)

0 I/ t l^ I

0 0.5 1 0.5A. in (Exper.) A, in (Exper.)

Fig. 19. Comparison of deflections computed using various prediction methods withexperimentally observed results: (a) Slip-deflection model; (b) Proposed le methodI Eq. (22)]; (c) Branson [Eq. (2a)]; (d) Tadros [ Eq. (3);(e) PCI Design Handbook bilinearmodel; (f) Tadros I Eq. (9)].

PCI JOURNALJMay-June 1989 119

ILLUSTRATIVE EXAMPLES

EXAMPLE A

Consider the details of Beam PP2S2 ofRef. 27 with section dimensions andsteel depths as shown in Fig. 20. Thebeam is simply supported with a 9 ft(108 in.) span length I and loaded withtwo one-third point concentrated loads(four-point bending). The prestressingsteel is pretensioned with a straight ten-don profile.

Reinforcement and Section Properties:AD = 0.085 in.2 (one 3 -in, diameter,7-wire strand, Grade 270 ksi); A, = 0.15in.2 (three No. 2 bars, Grade 60); d, =6.25 in.; e, = ee = 1.75 in.; d, = 8.0 in.; A.= 50.5 in. 2 ; U, = yb = 4.5 in.; to = 273.4in.4

Material Properties: f, = 5.3 ksi; f, _0.55 ks i; E, = 4150 ksi; E,, = 28600 ks i; Es= 29000ksi; f= 167 ksi; Fe= A^,.f,^=14.2 kips.

Loading: Total applied load P = 7 kips;Mn = 0.0 (neglecting beam self weight);Dead load plus Live load moments:Ma = MD + M, = (P/2)(108/3)

= 126 kip-in.

Required: Determine the beam deflec-tion using (a) the slip-deflection model,and (b) the I e method,

Solution: Compute the gravity load andprestressing force moment multipliers(see Refs. 18 and 19).

GF„ _ (23/216)12 = (231216)(108)2= 1242 in.2

GFi, =1218 = (108)218 = 1458 in.2

(a) Slip-Deflection ModelCompute the cracking momentFrom Eq. (2h):

_ (0.55)(273.4)(14.2)(273.4)M`T 4.5 + (40.5)(4.5)

+ (14.2)(1.75)= 79.6 kip-in.

LnNl0 pp

aoAp ^'

As

Fig. 20. Typical section of Example Beam A.

Since Ma = 126 kip-in. is greater thanMci = 79.6 kip-in., a cracked sectionanalysis should be undertaken.

Because the computation of deflectionusing the slip-deflection model [Eq.(lob)] requires an integration process,relevant computations for only onesingle uncracked prism between twocracks in the constant moment regionare illustrated:

Using Eqs. (18) and 19), compute theneutral axis depthc and the compressivestress f^ in the top fiber corresponding toMa = 126 kip-in.:c = 3.1 in.;J = 3.3 ksi

The stress f„, in the No. 2 reinforcingsteel bars can be calculated assuminglinear strain (or stress) distributionacross the depth of the section:

fw= E. d. C !I E 1

= 29000(31 3.1 1 } 4150)

= 37 ksi

First, compute the crack spacing andthe bond-slip characteristics. From Eq.(15), with A, = 6.1 in, and d,. = 1.0 in.:a^a = (7.6)(10- 5 )(29000) 3.o)

= 4.0 in.L = 2.0 in. [from Eq. (15)]k = 769 kips/in.3 [from Eq. (16)]q^,a = 0.79 ksi [from Eq. (17)]q (No. 2 bar) = 0.78 in.

120

m'= 0.00073 kip-' I from Eq. (13h)1Rl = 0.66 in.-' [from Eq. (13a))Li = 1.2in.(fromEq.(14)]1,= L-Lt=0.8in,f t = 32.0 ksi [ from Eq. (13d)]

= 0.04 ksi [from Eq. (13e)10.0012 in. [ from Eq. (13c))

Therefore, the slip of mild steel rein-forcement at the cracked section is cal-culated from Eq. (13) as follows:

S = 2(0.0012) Binh (0.66 x 1.2)+ 2(37.0)(0.8)/29000

- (0.79) (0.78) (0.00073) (0.8)2/3= 0.004 in. (0.10 mm)

The rotation of the uncracked regionbetween the two cracks is:

0 = 0'^4 = 8.2 x 10-4 radians8 - 3.1

[from Eq. (11)1

Note that by computing the rotationusing the conventional curvature inte-gration method (i.e., by considering onlythe concrete tension stresses below theneutral axis), the angle becomes:

fc 1e = c F,,E

) c,,

_ 3.3(4,0)

(3.1)(4150)= 1.0 x 10-s radians

This angle is 22 percent larger thanthe actual angle (8.2 x 10-4 radians)computed using the slip-deflectionmodel with an almost similar discrep-ancy expected in the comported deflec-tion.

By conducting similar calculations forthe remaining uncracked regions alongthe member length results in:

a„ = £01 = 0.23 in, (5.8 mm)

From Eq. (12), compute the deflectiondue to the prestressing force (prestresslosses included):

0 = Fee` GF,D Fc 1g

- (14.2) (1.75) (1458.01)

' (4150) (273.4)= 0.023 in. (0.8mm)

Therefore, the total theoretical de-flection [from Eq, (loh)) measured fromthe camber position is:

A, = 0.032 + 0.23 = 0.26 in. (6.6 mm)

The total experimental deflection is:ar (exp) = 0.27 in. (6.8 mm)

(b) 1e Method

Calculate the position of the neutralaxis depth c, the depth of the centroidalaxis y and the cracked section momentof inertia 'era corresponding to Mg126.0 kip-in.

Using Eqs. (18) and (19) with . f,. - 0.0(see Refs. 20 and 18):c = 3.0 in,t/ = 2.1 it. [ from Eq. (4))1,, = 61.7 in. [front Eq. (5)1

Using Eq. (22), the effective momentof inertia is:

I * = 1CraC

1-- ,M l l jf oQ

61.71- 79.6 1- 61.7

126.0 273.4= 120.8 in!

Therefore, the total central beam de-Ilection (see Fig. 15) is:

qua

E, IP

= 126.0(1242.0)

(4150)(120.8)= 0.31 in. (7.9 mm)

a, (Experiment) = 0.27 in. (6.8 mm)

The deflection computation using theI, method can be simplified using a bi-linear load-deflection model such asproposed in the PCI Design Handbookin which Icra is calculated directly

PCI.OURNAL'May-June 1989 121

utilizing Eq. (7) without resorting to acracked section analysis:

Icra = Icr = 82.5 in. 4 [from E q. (7)l1 = 147.6 in.4 [from Eq. (22)1

Therefore, A, = (M. GFA )/(EE Ie) = 0.26in. (6.6 mm).

This value is closer to the experi-mentally observed deflection (0.27 in.)than using the proposed variable mo-ment of inertia method (I 61.7 in').

EXAMPLE B1

It is required to compute the midspandeflection of the 70 ft (840 in.) span,simply supported beam shown in Fig.21. The beam section is an idealizedsingle T taken from the PCI DesignHandbook. The prestressing steel issingle point depressed tendons withmidspan eccentricity ec = 17.05 in. andend section eccentricity ee = 8.0 in.

Reinforcement and Section Properties:

A,, = 2.75 in? (18 1/2-in. diameter, 7-wirestrands, Grade 270); A, = 0.0 in? (fullyprestressed beam); dp = 28 in. (midspansection); A. = 495.8 in 2; y j = 10.9.5 in.; yb

= 25.05 in.; la = 65338.4 in.4

Material Properties: ff = 5.0 ksi; ,Ec =4000.0 ksi; f f, = concrete compressivestrength at transfer = 3.5 ksi; E = 3370ksi; fr = 0.53 ksi; f, = transfer stress inprestressing steel = 175 ksi; ff = 150ksi; F1 =Apf,,= 481.3 kips; Fe=Apff=412.5 kips.Gravity load: w9 = 0.5 kips per ft; w,0,3 kips per ft; WL, = 1.0 kips per ft.

Solution:Midspan moments:419 = 3675.0 kip-in.; M8 = 2205 kip-in.;M, = 7350.0 kip-in.;Mb = 1. + M, = 5880 kip-in.;Ma = Mg + %I,+ ,,= 13230.0 kip-In.Load and prestressing force momentmultipliers:' 8,19

GF, = 5 (1 = 840)2148

= 73500.00 in,z

zGFn = [213+ eel(3et)j

= 72594.7 in.'Immediate camber deflection at trans-fer:

Amber = - --- (F, ee GFP WO GF„)Est 1,

eelCGS __^CGC

70 ft

^-^- -- 96"

2.361

H8"

Fig. 21. Reinforcement profile and section dimensions of Example Beams 81 and B2.

122

1ra,n r (3370)(65338.4) ><

[(481 .3)(17.05X72594.7)- (3675)(73500.0))

_ - 1.48 in. (- 37.6 mm)Service load deflection:Compute the cracking moment usingEq. (2b):

_ (0.53)(6.5338.4) + (412.5)(65338.4)

25.05 (495.68)(25.05)+ (412.5)(17.05)

= 10586.2 kip-in.Since Ma is greater than MC,., the result

iiEiplies that a cracked section analysisshould be used.

F'rorn a linear elastic cracked sectionanalysis, neglecting the concrete tensilestresses below the neutral axis position:c = 13.8 in. [using Eqs. (18) and (19)with fr = 0.0 or Refs. 18 and 201y = 4.62 in. [ from Eq. (4)]I„ ra = 15703.0 in .1 [from Eq. (5)]

Inserting the above values into Eq.(22., the eflective moment of inertia canhe found from:

= tcra

1 Llcr 1 Icra }

lla 1„

15703.0

105S2 15703.0- 13230.0 ^ 1 - 65;3,38.47

= 40044.0 in.4

'therefore, the total deflection corre-sponding to the given applied load is:

= tf * GF,, (see Fig. 15)

_

13230.0(4000)(40()44.0) (7:35110.0)

= 6.06 in. (152.7 mm)

Using the slip-deflection model:0 + = 5.82 in. (147.8 mm)

The deflection due to the prestress-ing force (prestress losses inclnrted) canhe calculated from Eq. (12):

Al, = Fe e' GF^,E, Ig= (412.5)( 7.05) (72594.7)

(4((X))(65338.4)= 1.95 in. (upward)

Therefore, the net positive deflection(above zero) is computed as the differ-ence between the total deflection A, andthe prestressing force deflection p,,:DR =A,-A,=6.01- 1.95

= 4.06 in, (103.1 nmm)The reader can verily that the net po-

sitive deflection r]„ can be computed inthe proposed idealization of the load-deflection response shown in Fig. 15using a modified expression of I, as fol-lows:D„ = M. - M°s GF„

E[IPwhere

I * = Irap 1- M- %;o6 11_ I'.a

M - Mos Ig

and!1Im = Fe ec GFQ/GFa

In order to calcu^late the live load rfe-flection, it is necessary first to computethe dead load deflection. Since VI, isless than 14c,., this implies that the deadload deflection AF should be calculatedbased on the gross moment of inertial,:

A„= tit° GFEr 1.

5)'ht).0 (73500.0)(4000)(653:38.4}

= 1.65 in. (41.9 mm)the live load deflection A, is the dif-

ference between the total deflection Qtand the dead load deflection Or,:

DL = At - OD= 6.01 - 1.65= 4,17 in. (105.9 mm)This deflection is about 1/200 of the

PCI JOURNALJ May-June 1989 123

span length. The corresponding liveload deflection is larger than theo c

vmo N maximum deflection (4)360) specified in

-a cll -^ 0 00 the ACl 318-83 Code for floors. How-' °1 ever, it is adequate (less than 1/180) for

r roofs not supporting or attached to non-'^, ^ 4 structural elements likely to be dam-SO te r', aged by large deflections.

EXAMPLE B2c d r N

& c^ This example is similar to ExampleBear' B1 except that the reinforcement

,^Ci q

oc is changed to have a partially pre-a - stressed beam with almost equal nomi-

nal moment capacity as Beam B1.Keeping everything else the same

4 r^ c^ oo (loading, material and section prop-N erties), the new area and depths of ten-

sile reinforcement are as follows:o a Ap = 1.84 in? (12 — r/a in, diameter,

r` ` 7-wire strands);N A, = 3.60 in .2 (6 No. 7 bars, Grade 60

with Ea = 29000.0 ksi);[4, (midspan) = 25.0 in.; dg = 31.75 in.;,n NMidspan eccentricityee = 14.04 in.;

m °' End section eccentricity remained un-changed (ee = 8.0 in.)

Summary of the deflection results forExample Beam 132 is given in Table 3.

x ° r')Ni Computed section moment of inertias

and characteristic moments relevant tovarious prediction methods are pre-sented for the three example beams (A,B1 and B2) in Table 2. Results of de-flections computed using the corres-ponding methods are summarized in

• o ono Table 3. It can be shown in Table 3 that de-

Elections computed using the proposeda ° 1* method and the PCI Design Hand-

,"',^ n book bilinear model (as a simplificationto the Ie approach) are in better agree-

G ;^ C6 N z ment with the experimentally observeddeflection (Beam A) and also the slip-

- deflection model results (Beams A, B1"^ and B2) in comparison with other exist-L

ing prediction methods [Eqs. (2a) andY (3)]. This close agreement further sup-

ports the observations made earlier inthe paper.

124

Table 3. Summary of deflections (in.) for example beams computed by variousanalytical methods.

PC:I Design Slit-Pranson Tadros Handbook Proposed deflection

Example Eq. (2a) Eq. (3) bilinear model If [Eq. (22)] model

Total deflec-tion A,(in.) 0.32 0.43 0.26 0.31 0.26Net positivedeflection A„ 0.29 0.39 0.23 0.28 0.23Dead load

_-^

deflection An 0.0 0.0 0.0 0.0 0.0Live loaddeflection A L 0.32 0.43 0.26 0.31 0.26

7.08 8.40 6.63 6.01 5.825.13 6.43 4.68 4.06 3.87

g 1 1.65 t .65 1.65 1.65 1.655.43 6.75 4.98 4.36 4.17

0r 7.10 7.33 5.52 6.25 5.715.98 6.21 4.40 5.13 4.59

4DB2

1.65 1.65 1.65 1.65 1.655.45 5.68 3.87 4.60 4.06

(experiment) = 0.27 in. (6.8 mm). Note: 1 in. - 25.4 nim.t0„ = 1.12 in. (28.4 mm).

SUMMARY ANDCONCLUSIONS

A theoretical model for predicting de-flections of progressivel y cracking par-tially prestressed concrete beams is de-veloped. The proposed model accountsfor the effect of tension stiffening bytaking into consideration the contribu-tion of the uncracked regions along themember length, evaluated from the slipof tensile reinforcement, and also theeffect of concrete tension stresses belowthe neutral axis position at the crackedsections. Deflections computed usingthe analytical model showed very goodagreement with experimental results.

A parametric evaluation undertakenusing the developed analytical modelhas shown that factors such as crackspacing and maximum bond strengththat could greatly influence the magni-tude of slip and crack widths at thecracked sections, have, in fact, little ef-

feet on the load-deflection response ofthe member after cracking.

Based on the extensive results ob-tained using the analytical model, it isfound that the phenomenon of tensionstiffening in predicting the deflectionresponse, particularly for prestressedand partially prestressed members, canbe best simulated for design purposesusing an idealization similar to the PCIDesign Handbook bilinear load-deflec-tion model. The proposed idealizationrepresents two paths of load-deflectionresponse: precracking path with slopeporportional to the gross nnoment of in-ertia 1 9 and post-cracking path with ap-parent slope proportional to the crackedsection moment of inertia 1,,., calcu-lated relative to the centroidal axis ofthecracked transformed section.

Unlike the PCI Design Handbook bi-linear model, the slope of the post-cracking path varies depending on thelevel of applied load at which deflection

PCI JOURNAL'May-June 1989 125

is being computed. The correspondingidealization is facilitated by deriving anexpression for 1, to be used in conjunc-tion with the concept of secant modulusproposed by Branson given as follows,

1 s _ 41.

P J tfa (1 I19

where M0. is the cracking moment (in-cluding the effect of prestress) and Ma isthe total applied gravity load moment atthe beam critical section.

The corresponding idealization ofload-deflection response is shown to be

better than other existing predictionmethods in reproducing the developedanalytical model results. Also, it showsbetter agreement with experimentaldata.

ACKNOWLEDGMENTThis research was supported by the

University Research Board (URB) at theAmerican University of Beirut. The au-thors are most grateful to the Faculty ofEngineering and Architecture at theAmerican University of Beirut far pro-viding the computer facilities.

REFERENCES

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2. Harajli, M. H., and Naaman, A. E.,"Cracking in Partially Prestressed BeamsUnder Static and Fatigue Loading," ACISpecial Publication, SP-1 13, Cracking inPrestressed Concrete Structures', Arueri-canConcrete Institute, Detroit, Michigan,1989, pp. 29-56.

3. Jiang, D. H., Shah, S. P., and Andonian,A. 1., "Study of Tensile Forces by Bond,"AC/ Journal, Proceedings, V. 81, No. 3,May-June 1984, pp. 251 -259.

4. Bresler, B., and Bertero, V., "Behavior ofReinforced Concrete Under RepeatedLoads," Journal of the Structural Divi-sion,ASCE, V. 94, No. ST6, June 1968, pp.235-252.

5. Edwards, A. D., and Picard, A., "BondProperties of Vs in. Strands," AC! Journal,Proceedings, V. 69, No. 11, November1972, pp. 684-689.

6. Gyltoft, K., 'Bond Properties of Stands inFatigue Loading," Division of StructuralEngineering, University of Lulea, Re-search Report No. LULEA 1972-22.

7. Stocker, M. F., and Sozen, M. A., "In-vestigation of Prestressed Concrete forHighway Bridges, Part V: Bond Char-acteristics of Prestressing Strands,"Bulletin No. 503, Engineering Experi-ment Station, University of Illinois, Ur-bana, 1970.

8. Eligehausen, R., Popov, E. P., and Ber-tero, V. V., "Local Bond Stress-Slip Re-lationships of Deformed Bars UnderGeneralized Excitations," EarthquakeEngineering Research Center, ReportNo. UCB/EERC 83-23, University ofCalifornia, Berkeley, California, 1983.

9. Harajli, M. H., and Mukaddani, M., "Slipof Steel Bars in Concrete Joints UnderCyclic Loading," Journal of the Struc-tural Division, ASCE, V. 114, No. 9,September 1988, pp. 2017-2035.

10. Naaman, A. E., "Partially PrestressedConcrete: Review and Recommenda-tions," PCI JOURNAL, V. 30, No. 6, No-vember-December 1985, pp. 30-71.

11. ACI Committee 435, "Deflections ofReinforced Concrete Flexural Mem-bers," American Concrete Institute, De-troit, Michigan, 1966,29 pp.

12. ACT Committee 435, Subcommittee 5,"Deflections of Prestressed ConcreteMembers," ACI Journal, V. 60, No. 12,December 1963, pp. 1697-1727.

126

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

14. Branson, D. E., "Design Procedures forComputing Deflections," ACI journal,Proceedings, V. 65, No. 9, September1968, pp. 730-742.

15. Branson, D. E., and Trost, H., "UnifiedProcedures for Predicting the Deflectionand Centroidal Axis Location of PartiallyCracked Non-Prestressed and Pre-stressed Concrete Members," ACI jour-nal, Proceedings, V. 79, No. 2, March-April 1982, pp. 119-130.

16. Branson, D. E., and Trost, II., "Applica-tion of the 1-Effective Method in Calcu-lating Deflections of Partially Pre-stressed Members," PCI JOURNAL, V.27, No. 5, September-October 1982, pp.62-77. Also, see Discussion by Tadrosand Suleiman, PCI JOURNAL, V. 28, No.6, November-December 1983, pp. 131-136.

17. Tadros, M. K., "Expedient Service LoadAnalysis of Cracked Prestressed Con-crete Sections," PCI JOURNAL, V. 27,No, 6, November-December 1982, pp.111. Also, see Discussion, PCI JOUR-NAL, V. 28, No. 6, November-December1983, pp. 137-158.

18. Tadros, M. K., Ghali, A., and Meyer, A.,"Prestress Loss and Deflection of PrecastConcrete Members," PCI JOURNAL, V.31, No - 1, January-February 1985, pp.114-141. Also, see Discussion, PCIJOURNAL, V. 31, No. 3, May-June 1986,pp. 144-156.

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PCI JOURNALI May-June 1989 127

APPENDIX - NOTATIONk = area of concrete surrounding

one reinforcing bar or pre-stressing strand

= area of gross concrete sectionarea of prestressing steel

= area of tension reinforcing steel= area of compression steel= area of one reinforcing bar or

prestressing strand= crack spacing= section flange width= web width= neutral axis depth

distance from outermost steellayer to bottom concrete ten-sion fiber

= depth to center of prestressingsteel

= depth to center of tensionreinforcing steel

= depth to center of compressionsteel

= depth to center of reinforcingbar or prestressing strand

= eccentricity of prestressingsteel

= eccentricity of prestressingforce at midspan section

= eccentricity of prestressingforce at end section

= concrete modulus of elasticity= modulus of elasticity of pre-

stressing steel= modulus of elasticity of rein-

forcing steel= modulus of elasticity of rein-

forcing steel or prestressingstrands

= concrete stress in top fiber= concrete compressive strength= effective prestress= concrete tensile strength

stress in reinforcing steel or in-crease in stress above effectivestress in prestressing strands

Fe = effective force in prestressingsteel

GF = moment multiplierh = beam heightht = flange thicknesslCr, = cracked section moment of in-

ertiaIp = gross section moment of inertiaL = half crack spacing1!a= gravity load applied momentder = cracking momentW,, = decompression moment?41ar = moment causing zero curvature

at critical sectionM,6= moment causing zero beam de-

flectionPPR = partial prestressing ratioS = slip of tensile reinforcement at

cracked sectionk = stiffness of local bond stress-

slip relationy,^^ = maximum bond strength

= distances from gross sectioncentroidal axis to top and bot-tom fiber, respectively

= distance from centroidal axis ofcracked transformed section totop fiber

w = reinforcing index= curvature

^, = circumference of one reinforc-ing bar or prestressing strand

= rotation at cracked sectionDca,nbe r = camber deflection

= net positive deflectionp p– upward deflection caused by

effective prestressing forcep t = total central beam deflection

due to total gravity load= concrete strain

E" = effective prestrain= f,1Ep

ECe = concrete precompressive strainat prestressing steel level

_ (Ap f,,.IA' + Ap f0. e211 )/EC

ApA,A;A.

acs

b

C

[4.

dp

d,

c(,c

e

C.

e,

E,EA

E,

E,,

f^fc,

f/xJrfm

NOTE: Discussion of this paper is invited. Please submityour comments to PCI Headquarters by February 1, 1990.

128