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Prestressed Loss and Deflectionof Precast Concrete Members
Maher K. TadrosAssociate ProfessorDepartment of Civil
EngineeringUniversity of NebraskaOmaha, Nebraska
Amin GhaliProfessorDepartment of Civil
EngineeringUniversity of CalgaryCalgary, Alberta, Canada
Arthur W. MeyerPhD Candidate and PEDepartment of Civil
EngineeringUniversity of NebraskaOmaha, Nebraska
rviceability, particularly deflection,is today becoming a more impor-
tant criterion than in the past due to theutilization of modern design proceduresand the use of high strength materialswhich result in slender members moresusceptible to large deflections. This isespecially true in prestressed concreteapplications, particularly when mem-bers are allowed to develop cracksunder service loads. Cracking couldcause a sizable drop in member stiffnessand increased deflections.
It is difficult to calculate member de-flections with a high degree of accuracyeven in a controlled testing laboratory.This is due to the random variations ofsome of the contributory factors such asthe concrete modulus of elasticity,creep, and shrinkage. In field conditions
not only are the variations greater, but inaddition the number of variables in-crease. Examples are the uncertaintiesabout levels and duration of loading andseasonal weather variations_ Calculateddeflection should therefore be viewedas an "estimate."
Acceptable deflection analysis shouldnot be highly complicated mathemat-ically, which would give a false impres-sion of exactness, nor should it be over-simplified, which would compound theprobable errors resulting from uncer-tainties in material properties andloading variations.
In this paper a simplified approach tocomputing instantaneous and long-termdeflections is proposed. The approachrationally accounts for all the importantparameters, yet it involves certain ap-
114
proximations which make it suitable formanual computations.
Long-term deflection multipliersbased on the Trost-Bazant l •$ aging coef-ficient are proposed. These multipliers,unlike those in the current PCI DesignHandbook, are shown to be valid forvarious environmental conditions andwhen nonprestressed steel is present.
Because prestressing produces de-flections that are opposite to and often ofthe same order of magnitude as the deadloads, it is important to determine pre-cise values of the prestressing force.Further, the presence of nonprestressedsteel tends to restrain creep and shrink-age of concrete, and its effect must beincluded when computing the compres-sion prestress force in concrete.
Since an accurate computation of pre-stress losses is necessary, a method ispresented here for their calculation. Itinvolves a simplification over existingmethods that account for environmentalvariations and for the presence of non-prestressed steel. It is important to note,however, that the proposed deflectionprocedure itself is independent of themethod of prestress loss calculation,and other loss estimates may be used ifthe designer so desires.
The deflection procedure describedherein accounts for possible live loadcracking. Two charts are presented forrapid evaluation of the cracked sectionmoment of inertia and centroidal depth.The charts include the effect which axialforce and the presence of nonpre-stressed steel have in increasing thecracked member stiffness. Their usethus results in avoiding unnecessaryoverestimation of live load deflection.
The live load deflection predictionmay further be refined by inclusion ofthe stiffening effects of uncrac^Ced con-crete in the tension zone, the so-called"tension stiffening." Tension stiffeningis generally accounted for empirically.Formulas are presented here for adjust-ment of the cracked section moment ofinertia and centroidal depth. The Euro-
SynopsisThis paper discusses the influence
of creep and shrinkage of concrete,relaxation of prestressed steel, andpresence of nonprestressed steel ontime-dependent deflection behavior ofprestressed concrete members.
Multipliers are developed for pre-dicting time-dependent deflections.Design aids are presented for deter-mining cracked section properties ig-noring concrete in tension. Empiricalconsideration of the stiffening effectsof concrete in the tension zone is dis-cussed.
The above factors are integratedinto a simple procedure for deflectionprediction which is suggested as amodification to the PCI Design Hand-book. Although application of themethod to precast concrete construc-tion is emphasized, it is equally appli-cable to cast-in-place post-tensionedmembers. Two numerical examplesare included to illustrate the applica-tion of the proposed design method.
pean Code method (CEB-FIP, 1978), aswell as other methods of considerationof tension stiffening, are also discussed.
The proposed method of deflectioncalculation is verified with test resultsand compared with other methods, in-cluding that in the PCI Design Hand-book. Five steps of calculation are pro-posed and illustrated with numericalexamples.
IMMEDIATE DEFLECTIONOF UNCRACKED MEMBERS
Immediate deflection, or camber, dueto the effects of initial prestressing Pcp
and member self weight is generallycalculated by basic principles of me-chanics of elastic structures, The value
PCI JOURNAUJanuary-February 1985 115
Table 1. Formula for midspan deflection of uncracked members.
O ^ ^ — l Oc - 0e l ( Multiplier) *8
Mldspan EndCause Curvature Curvature Multiplier
^c ^e
Two point depressed tendon
_ Pcoec _ coee a2
Eci19 Ecilg 6I °'I ^°—►^
One point depressed tendon
2 Poe Pae2
^_
Eci I g Eci Iq 24
C0
Straight tendon47
3 eaec coee 0
y t Eci 1 Eci Ig
EE Parabolic tendonE 4 Pcoec 0 t2
t Eci IQ 48
Gravity loadw Q2w 0 L2
SEci Ig 49
6 Prestress loss, OP due apc(e^lc nP(etsls ^2to creep, shrinkage and - U+XC ]v
- (I+7CCv 48
relaxation E 1ci 9E Ici g
If the two end curvatures are not equal, use an average value of the twocurvatures for d,.
Eccentricity to be used Is that of the prestressing steel.
$ Eccentricity to be used is that of the total steel, The term (1 + yC,) isvalid only for deflection at time infinity; for other multipliers see Table 2.
of PC, is equal to the jacking force lessthe initial prestress loss due to anchor-age set, elastic shortening, and relax-ation loss occurring between jackingand release time. References 3, 4, and 5discuss in detail the calculation of P.
A simplified form of calculating P,o is
presented in Appendix A. Curvature at asection is equal to bending moment di-vided by the section rigidity, EI; whereE = modulus of elasticity of concrete atthe time of prestress transfer and I =moment of inertia of the section, whichis simple to determine because cracking
116
is commonly not allowed to occur at thisstage.
In most cases, the moment of inertiamay he approximately taken equal tothat of the gross concrete area, I = l. Ifthere is an unusually large amount ofreinforcing steel present, the accuracywill be improved by use of the proper-ties of the transformed section.
The value of P,. includes the com-pression force picked up by the nonpre-stressed steel due to elastic shorteningat transfer. However, for simplicity thelocation of P assumed to coincidewith that of the prestressed steel, ratherthan the exact location of the resultant ofthe relatively large tension force in theprestressed steel and the much smallercompression force in the nonprestressedsteel.
Standard design aids, such as Fig.3.9.18 and Fig. 8.1.4 of the PCI DesignHandbook,{ may be used for calculatingthe immediate deflection of uncrackedmembers. Table I of the present paperincludes a deflection formula which,when used for initial prestress camber,should yield identical results to thosegiven in Fig. 3.9.18 of the PCI DesignHandbook for straight, single-point de-pressed, and two-point depressed ten-dons. The Table 1 formula is also usefulfor other cases including prestress lossdeflection as will be discussed in a sepa-rate section.
IMMEDIATE DEFLECTIONOF CRACKED MEMBERS
Discussion of the behavior of crackedprestressed members follows.
Cracked Section Properties
Two facts about the geometric prop-erties of cracked prestressed concretemembers should be clearly understood.
1. Prestressed concrete flexuralmembers are different from reinforcedconcrete members in that prestressed
members are simultaneously subject toboth axial force and bending moment.The axial f rce may be sufficiently highto permit no cracks orextremely low thatthe member behaves as a conventionallyreinforced member. Therefore, for agiven bending moment at a section, themoment of inertia varies from the gross Ito the cracked I of a conventionally re-inforced section depending on the mag-nitude of the prestressing force.
2. Furthermore, the neutral axiswhich is located at the bottom of theconcrete compression zone (assumingpositive bending moment), does notcoincide with the centroid of the trans-formed cracked section when an axialforce exists. For example, when the axialforce combined with the bending mo-ment produce zero stress at the bottomfibers, the neutral axis will be located atthe bottom edge ofthe section while thecentroid is, obviously, much higher. Themoment of inertia must be determinedabout the centroidal axis of the crackedsection and not the neutral axis.
Detailed discussions of the computa-tion of I, may be found in Refs. 6 to 12.Jittawait" developed formulas for thecentroidal axis depth, y r„ and thecracked moment of inertia, I. Theywere developed using the commonlyadopted assumptions of linear stressdistribution and simple equilibrium re-lationships, in a way similar to thetreatment of conventionally reinforcedconcrete columns, subject to combinedaxial force and bending.
Figs. I and 2 are approximations of theexact formulas. A program written forthe Hewlett Packard microcomputer,and available from the Prestressed Con-crete Institute," may be used to obtainmore exact values of the cracked sectionproperties as well as stresses before andafter cracking.
The empirical equation;
I TT = 1 s ^ P (1 — Ir pp ) (1)
given in the PCI Design Handbook isbased on a report by the PC I Committee
PCI JOURNAL/January-February 1985 117
I.^
I .:
k p I.'_ 0
I.^ 0.9
0.8
I,{
-- --,- •. .+ .+ -",, ".v"-v V,V C.J VV'.) U,VJJ U.V'+V
np
0.40
0.35
0.30
0.25
kc
0.20
0.15
0.10
0.05
Ia""Y k kpdCr
^°o
p \0 02
p 1p o
^o
Rectangular Section
b
INfService load Ymoment M d-
djtransformedSection
Pr^slre^sCentraid of bW force PCombinedSteel
0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040np
Fig. 1. Centroidal axis depth of cracked transformed section.
118
1.30
1.20
k pr1.1
I.0
Ides/M z
9
0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.04
np
0.35
0.30
0.25
kcr
0.20
0.15
0.10
0.05
1^0
Icr^ kcr kpr Section Rectangular Oi0
p Sect ion O ^Oo
b
Service load
TFps ^htmoment M Centroid ofd ransformed
section
PrestressCentroid of force Pcombined steel bw
____ H0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.04
np
Fig. 2. Moment of inertia of cracked transformed section.
PCI JOURNAUJanuary-February 1985 119
on Allowable Stresses.' 3 The formula issimple and appears to give reasonableresults in the absence of nonprestressedsteel, and when the effect of axial load isminor. Naaman 9 has suggested the fol-lowing modified version which accountsfor the two types of steel in the section:
z + /,rz)Icr = nnsAve ^A.9 T NAB"'!
{ 1 – p, Pe (2)
The subscripts ps and s refer to pre-stressed and nonprestressed steel, re-spectively. The symbols n, A, and d are,respectively, modular ratio, steel area,and its effective depth. The parameter pis the ratio A/bd, where b is the width ofthe compression face of the member.
Moment-Curvature Relationship
The moment-curvature relationshipfor a conventionally reinforced crosssection is illustrated in Fig. 3. The fig-ure also shows the relationship for thesame section if it were prestressed. Bothrelationships were prepared by thecomputer program given in Ref. 10 forthe cross section given in Example 1. Inorder to understand the cross sectionbehavior, it will be initially assumedthat, upon cracking, concrete is incapa-ble of resisting any tension. A moment,M. larger than the cracking moment,M.r, produces curvature:
/cr = (M a —Pe,.,)«cicr (3)
where P is the prestress force, and e cr isits eccentricity relative to the centroidof the cracked section. (Sign conven-tion is given in Appendix B.) For theconventionally reinforced section, P = 0and I,.,. is constant. Therefore, the M – 0diagram is a straight line passingthrough the origin point. The drop inrigidity due to cracking is representedby the horizontal line at the Me,. level.For the prestressed section, both I,, andy cr(and in turn e,.) are dependent on theloading level (see Figs. 1 and 2). Thus,the M – ¢ relationship is nonlinear.
It is important to note that the shift inthe centroid of the cross section uponcracking, results in larger prestress forceeccentricity, a , than the uncrackedmember eccentricity. This is particu-larly significant in prestressed flangedmembers, such as the commonly useddouble tees which are characterized byrelatively low steel area ratio, p. Forthese members, ignoring the increase ineccentricity upon cracking as implied inRefs. 14 and 15, could result in signifi-cant overestimation of deflection. 16
Because concrete tensile strength isnot zero, cracking does not extend to theneutral axis as assumed in standardcracked section analysis. In addition,uncracked concrete which exists be-tween cracks in the tension zone, con-tributes to the stiffness of the member.These effects are often called "tensionstiffening." If tension stiffening is ac-counted for, the M – 0 diagram be-comes continuous as indicated by thebold lines in Fig. 3. Empirical consid-eration of tension stiffening is discussedbelow.
Tension Stiffening
The following empirical equation isproposed here for the effective momentof inertia, I e to be used in calculatingcurvature:
J' = ''1+(1 = , – R 4) Icr (4)
with
1i = fr(f, (5)
where fr is the modulus of rupture ofconcrete and frm is the stress at the ex-treme tensile fiber assuming no crack-ing.
Eq. (4) performs an interpolation be-tween I, ant' I cy. Note that, in practicalapplications, f, cannot reach infinity.Therefore, IP is always larger than Icy,where I cr is the moment of inertia of thetransformed section ignoring concrete intension.
120
GRAVITY LOADMOMENT
SERVICE LOAD M a -
CRACKING Mcr
M = M o = 0
GRAVITY LOADMOMENT
SERVICE LOAD Md
CRACKING MCI
DECOMPRESSION MoM = Pecr
M = Pe'M = Pe
A — TENSION STIFFENING INCLUDED8 — TENSION STIFFENING IGNORED
M
oe ceI' is cracked moment of inertiaeconsidering tension stiffening,"effective I"
A
0e 0crCURVATURE
(a)
A
^g
IM a – Pe'
^e Ec I'eEc gI e' is eccentricity of
prestress consideringtension stiffening,"effective e'^
M = 0
0 Oe OcrCURVATURE
(b)
Fig. 3. Effect of tension stiffening in moment vs. curvature relationship:(a) Conventionally reinforced section; (b) Prestressed section.
PCI JOURNAUJanuary-February 1985 121
For prestressed concrete sections, itcan be shown that:
(frlftm) _ (M,.,. – M0)!(M8 – M^) (6)
where M. is the decompression moment,defined as the moment which wouldcause the stress to become zero at theextreme precompressed fibers. 10 Eq. (4)is an extension of the original Ie equa-tion developed by Branson 17 for con-ventionally reinforced sections.
To determine the change in eccen-tricity after cracking, it is here suggestedto employ the same interpolation ratio Rto calculate an effective centroidal dis-tance.
y e =R4y 0 + (I – R4) y n. (7)
where y e, y o , and y,,, are the effective,uncracked, and cracked (with tensionstiffening ignored) section centroidaldistances to the extreme compressionfibers (top surface in simple beams).
The values of I e and y e for varioussections along the member length arethen used to calculate the curvatures atthese sections. Eq. (3) may be used forthis purpose with Ie replacing I,,. and(d r, — Ye) replacing err.
Another way to account for tensionstiffening in reinforced concrete mem-bers is to interpolate between the cur-vatures 0g and (Acr of uncracked andcracked section instead of interpolationfor section properties. 18•' e The actualcurvature:
0 = R, 0a +(1 –R 1) 0e,. (8)
where R, is coefficient of interpolation:
R I = 1,,82 (f,r lf8) p (9)The symbols fr and f, are stresses in
the bottom steel in the cracked condi-tion due to M,,. and M., respectively,combined with the prestress force; Q, =1 for high bond bars and = 0.5 for plainbars, respectively; p2 = 1 for first load-ing and = 0.5 for loading applied in asustained manner or for a large numberof load cycles.
Trost2tl accounts for tension stiffeningby interpolation between deflectionvalues 8. and 5, r of uncracked andcracked structures:
8 =R, + (1 –H 2) Se, (10)
The interpolation coefficient:
R 2 =(Mn –Ma ) ? !{M„–Mcr) 2 (11)
where M. Ma , and M moments, re-spectively, corresponding to the theoret-ical (nominal) strength, applied loads,and cracking load.
It is possible to analytically accountfor tension stiffening assuming an ap-propriate stress-strain relationship ofconcrete in tension. For example, the"strain softening” computer model byBazant and Ohl, was reported to yieldsatisfactory results for conventionallyreinforced beams. It should be possibleto extend application of the same modelto prestressed members.
Eqs. (8) and (10) appear to representmore direct approaches to tension stiff-ening consideration than do Eqs. (4)and (7). A study is needed to correlate thevarious interpolation methods with ex-perimental results, especially on beamswith practical dimensions and rein-forcement content. The study shoulddetermine the method that offers thebest compromise of accuracy and sim-plicity. Until then, consideration of thissomewhat secondary effect, on pre-stressed concrete member deflection,may be done by any of the availablemethods.
Calculation of Deflectionfrom Curvature
Numerical integration of the curva-ture may be used to obtain the deflec-tion by any of the well known structuralanalysis methods, e.g., conjugate beam,moment area, or Newmark's integrationtechnique.
Consideration should be given to theinfluence of tendon profile on curvaturevariation along the member, especially
122
TENDON PROFILE MIDSPAN^
05L
(4)
MIDSPAN SECTIONUNCRACKED
TENSION STIFFENINGCONSIDERED, 0 _ oe
b) TENSION STIFFENINGIGNORED,0° 0Cr
ACTUALTIONOØ
PARABOLICAPPROXIMATION
IC) Z^^C +4R^
Fig. 4. Example of curvature distribution due to load combined withprestress in a cracked member with one-point depressed tendons:(a) Beam elevation; (b) Curvature due to total load plus prestress;(c) Live load curvature.
when one-point depressed tendons areused. Fig. 4 shows that, for this type ofmember, the most critical section is near0.41 rather than midspan.
Two more observations could bemade from Fig. 4. First, it is conserva-tive to ignore tension stiffening for pre-
liminary deflection estimates. Secondly,the curvature of simply supported pre-stressed members can reverse sign in asimilar manner to continuous ratherthan simply supported reinforced con-crete members.
Methods of averaging I e for the entire
PCi JOURNALJJanuary-February 1985 123
span of simply supported reinforcedconcrete beams, such as Eq. (9.7) of ACI318-83, are not applicable to prestressedmembers, without due consideration ofthe curvature reversals and the rela-tively small cracked portion of the span.
It is recommended that curvatures becomputed only at key sections along thespan, and integrated to obtain deflec-tion. As shown in Fig. 4c, and Example1, a good estimate of the live load de-flection of simply supported memberswith one-point depressed tendons canbe found, in terms of the curvatures atthe 0.41 and midspan sections by thefollowing simple formula:
=5 l4 00.4111 41 96 (12)
Eq. (12) is based on the assumptionsthat the live load curvature at memberends is zero, and that the curvature dis-tribution is approximated as a parabola.
As mentioned before, cracking ren-ders the problem nonlinear. Live loaddeflection cannot be obtained indepen-dently of consideration of dead load andprestress. First, curvatures due to totalload plus prestress must be obtained,then those due to dead load plus pre-stress, using the geometric propertiescorresponding to each load level. Thedifference between the two quantitiesyields the live load curvatures to beused in Eq. (12).
TIM E-DEPEN DENTDEFLECTION
The time-dependent deflection is de-fined as the part of the deflection causedby creep and shrinkage of concrete andstress relaxation of the prestressingsteel. Concrete creeps under prestressand sustained loads. This creep causescontinual increase in camber or deflec-tion. In the meantime, creep and shrink-age of concrete and stress relaxation ofprestressing steel cause prestress losswhich may be visualized as a negativeincrement of prestress.
The prestress loss, thus, contributes tothe deflection of the member. In fact, itis an important contributor since theprestress in many applications "bal-ances" a large portion of the dead load.It has been shown in a typical beam ex-ample, that computed deflection variesfrom 0.2 in. (downwards) to –1.6 in.(upwards) depending on the method ofconsidering the prestress loss deflec-tion. 12
The total long-term deflection istherefore equal to the algebraic sum of:
(a) Creep deflection due to sustainedload
(b) Creep camber due to prestress,the prestress force being assumedconstant and equal to its value attransfer
(c) Prestress loss deflectionThe deflections in items (a) and (h)
above are simply equal to the immediatevalue at load application multiplied bycreep coefficient of concrete, which isdefined as the ratio of creep strain to in-stantaneous strain clue to a constantsustained stress. Its value is sensitive tosuch factors as the relative humidity andtemperature of the surrounding air, theconcrete age at loading, and the creepperiod since loading. It also depends onthe concrete mix and curing method,and on member size.
Long-term deflections can be deter-mined using prediction methods such asthose reported in Refs. 18 and 23. Foraverage conditions and in the absence ofmore accurate information a value of theultimate (in time) creep coefficient C„ _1.88 is suggested in Refs. 4 and 23.
The creep coefficient is a function ofconcrete age at time of loading, and ofloading duration. Introducing correctionfactors to C. gives: creep coefficient Ca= 0.96 for age at loading 1 to 3 days andload duration 40 to 60 days; C,; = 1.5 forage at loading 40 to 60 days and infiniteduration. Erection of the member andapplication of superimposed dead loadsis assumed to occur 40 to 60 days afterrelease.
124
Table 2. Proposed time-dependent multipliers(time-dependent S = elastic 8 x multiplier).
A B C =B–AErection time Final time Long term
Loadcondition Formula Average • Formula Average* Formula Average*
Initialprestress 1 + C a 1.96 1 + C. 2.88 C1— C. 0.92
Pro
Prestress 1 — aa+ Xloss a.(1 + NC,) ]..00 1 + xC„ 2.32 (C„ — ca,C a ) 1.32
Memberweight i + Ce 1.96 1 + C. 2.88 Cy — C a 0.92
Superimposeddead load s 0 0 1 + C„ 2.50 1 + C, 2.50
Superimposeddead loads 1.00 1.00 1 + C. 2.50 C,; 1,50
• Assuming: Ca = 0.96, C, = 1.88 and C. = 1.50, x = 0.7 and a„ = 0.6 whichapproximately correspond to average conditions with relative humidity =70 percent, concrete age at release = 1 to 3 days and erection at 40 to 60 days.
t Applied after nonstructural elements are attached to member.
t Applied before nonstructural elements are attached to member.
The presence of prestressed or non-prestressed reinforcement in a crosssection restrains creep and shrinkage ofconcrete. However, creep deflectiondue to Causes (a) and (b) above are cal-culated as the product of C and the in-stantaneous deflection, using the prop-erties of the concrete section withoutreinforcement. The restraining effect ofthe reinforcement is accounted for inItem (c). An alternative approachadopted in Ref. 19 is to use a reductionmultiplier to the creep coefficient to ac-count for the restraining effect of steel.
Martin e' has proposed applying mul-tipliers to the elastic deflections in orderto obtain long-term deflections of pre-cast members. This approach greatlysimplifies the calculations, and is fol-lowed herein. Presented in Table 2 arethe proposed formulas for the time-de-pendent multipliers.
Stress due to prestress loss is grad-
ually introduced, and thus should pro-duce less creep compared to a stress in-troduced in its entirety at one instantand sustained thereafter. For this rea-son, the multiplier x C is used in Table 2in lieu ofC to calculate creep deflectionassociated with prestress loss; where xis a value less than unity called the"aging coefficient" to be further dis-cussed below.
PRESTRESS LOSSDEFLECTION
A valid prediction of deflection, dueto loss of the compressive force in con-crete, AP,, is dependent on the accuracyof the magnitude of that prestress loss.There are several methods that can beused in arriving at a value for prestressloss. The reader may refer, for example,to reports by Zia et al., E5 the PCI Corn-
PCI JOURNAUJanuary-February 1985 125
mittee on Prestress Loss,' Tadros etal., s -n and the CEB-FIP Model Code.'BIt should be noted that the deflectionprocedure presented herein is indepen-dent of the method chosen for estimat-ing the prestress loss. However, if non-prestressed steel is present, it shouldnot be ignored in calculating the pre-stress loss, AP,.
A method of prestress loss calculationis presented in Appendix A. It is a sim-plified form of the procedure given inRefs. 5 and 22, which was found byCurlitz r6 to he in close agreement withtest results in comparison with severalother prediction methods. One simplifi-cation of the method is replacement ofthe recovery parameter by the aging co-efficient, x.1.2
Both parameters introduce the creepeffects of the continually varyingstresses caused by prestress loss. Theyemploy numerical integration of as-sumed stress-time and creep-time func-tions. For the same time functions, bothparameters should yield the same re-sults. The aging coefficient, however, issimpler to use and is gaining wide ac-ceptance in Europe."'
A second simplification adopted inAppendix A is replacement of the relax-ation reduction factor chart s by a simpleformula. This has a somewhat secondaryeffect and use of an approximation forthe reduction factor should have a minoreffect on the prestress loss accuracy.The procedure in Appendix A is pre-sented in steps similar in format to thesimple steps given by Zia et al. 2 and byAASHTO. Er Yet it is more general in thatit includes the effects of the presence ofnonprestressed steel and allows for flex-ibility in selecting material properties.
Once the magnitude of prestress lossAP, is determined, the associated de-flection may be calculated by an equa-tion given in Table 1, which assumesparabolic variation of curvature due toAPE . It is sufficient to determine AP at0.41 for members with one-point de-pressed tendons or at midspan for other
cases and to consider that A P, is con-stant over the member length.
The value of the aging coefficientranges between 0.6 and 0.9. For 1 to 3day load application, as is common forprecast members, an approximate valueof x = 0.7 may be used (see Ref. 28). Forloading applications at a later concreteage x is larger. However, for the sake ofsimplicity and since x affects only theprestress loss deflection it is proposedthat a value x = 0.7 be used.
To account for the difference in themagnitude of prestress loss at erectionand final prestress loss the coefficient a,,is introduced in Table 2. It is the ratio ofthe time-dependent loss at erection tofinal prestress loss. It may be taken ap-proximately equal to 0.6 for average en-vironmental conditions, and for erectionat 40 to 60 days after release.
The prestress loss force AP, is as-sumed to act at the centroid of the pre-stressed and nonprestressed steel. Thus,the effect of the cross-sectional area ofthe nonprestressed steel and its locationin the section on the deformation, areaccounted for in the proposed method.
PCI DESIGNHANDBOOK METHOD
Table 3.4.1 of the PCI Design Hand-book e gives suggested multipliers as aguide to estimating long-term deflec-tions for typical members. These multi-pliers were derived by extending thefactor in Section 9.5.2.5 of ACI 318-77 toinclude the effect of prestress, thetime-dependent loss of prestress, andthe strength gain of the concrete afterprestress release.24
Examination of the method revealsthat the environmental condition themember is subject to has been takenconstant regardless of the geographic lo-cation, although it is known that mem-bers subject to a dry environment suchas Arizona (USA) or Alberta (Canada)could exhibit a considerably larger
126
Table 3. Comparison of proposed average time-dependent multipliers and PCI DesignHandbook multipliers.
Erection Final
Proposed PCI Proposed PCImethod method method method
C, Cs* C1 C2
A,IA a, A,IAp, A,IA p„ A,IA,Load ]P IP OP^IP,. A- P,/P,, APIP,,
conditions =0.15 =0.35 =0 =2 -0.15 =0.35 =0 =2
Initialprestress P o 1.96 1.96 - - 2.88 2.88 - -
Prestressloss AP, 1.00 1.00 - - 2.32 2.32 - -
Initial prestressless prestressloss 1.811 1.611 1.80 1.27 2,53$ 2.07t 2.4.5 1.48
Member weight 1.96 1.96 1.85 1.28 2.88 2.88 2.70 1.57
Superimposeddead load 1.00 1.00 1.00 1.00 2.50 2.50 3.00 1.67
C. = + A, /A P, fnj,o PCI Design Handbook.
I + A,JA,
Calculated from 1.96 - 1.00 App lPie
$ Calculated from 2.88 - 2.32 AP`}P,.,
creep than those built in a more humidenvironment such as Florida (USA).
A comparison of the PCI DesignHandbook multipliers and those inTable 2 shows that the multipliers arecomparable if environmental conditionsare average and if nonprestressed steelis not present (see for example Table 3).One exception is the multiplier for thesuperimposed dead load deflection atfinal time (2.50 versus 3.00). It wouldseem reasonable to have a relativelysmall multiplier for the case of superim-posed dead load which is applied laterthan the load due to member weight.
The PCI Design Handbook provides areduction factor, to be applied to thetime-dependent multipliers in order toaccount for the presence of nonpre-stressed steel, as shown in Table 3. Thereduction factor is a function of
and is applied in the Handbook to bothcamber due to prestress and deflectiondue to gravity loads. It is based on astudy by Shaikh and Branson 25 of mem-bers that exhibited camber under pre-stress combined with dead load. Thefactor was later adopted by ACI Com-mittee 435 30 for camber reduction only.
The accuracy of the correction factor,as given in the Handbook, is question-able for two reasons. First, the locationof the nonprestressed steel is as impor-tant as its area in determining its influ-ence. Bottom nonprestressed steel re-strains creep and shrinkage of the sur-rounding concrete, relative to the top fi-bers and, therefore, contributes an in-crease in net time-dependent downwarddeflection (or reduction in net camber).Secondly, the reduction factor does notrelate the steel area to the concrete area
PCI JOURNALiJanuary-February 1985 127
Table 4. Test beam details.
Beam Beam Tension
Prestressedsteel
Nonprestressedsteel
Totalarea,
Prestressforce atrelease,
Super-imposed
deadload.Amount, Area, Amount, Area,
No. type zone in. in.' in. in., in. 5 kips lb/fl
B-I Fully Un-pre- crackedstressed 1 – 0 ?4s 0.108 — — 0.108 19 100
B-2 Partially Un-pre- crackedstressed 2 – 3/a 0.160 1 -- ^ 0.080 0.240 12.5 100
0-3 Partially Crackedpre-stressed 1- 4,% 0.080 2- 0 e 0.160 0.240 6.3 150
B-4 Rein- Crackedforced — — 1 – 0 7/ 0.100 0.108 0 50
Notes:1. A]] hea,ns are simply supported, spar = 14 ft, and cross section = 6 x8 in.2, All tendons are seven-wire stress-relieved strands, f, = 250 ksi.3. Specified ultimate strength of concrete at age 28 days, (f. ).,w = 5000 psi.
restrained by its presence.It is interesting to note that this latter
point is applicable to the effect of com-pression reinforcement A; in conven-tionally reinforced members. For thesemembers, the deflection reduction fac-tor which was originally given in termsof (A;/A,), where A, is the tension steelarea, has recently been changed to amore rational factor, in terms of (A R/bd).See the ACI 318-83 Code, Section9.5.2.5.
The need for an empirical reductionfactor is eliminated in the proposed pro-cedure by rationally accounting for thepresence of nouprestressed steel in themagnitude and location of the prestressloss APr . Table 3 gives a comparison ofboth the proposed procedure and thePCI Design Handbook multipliers fortwo cases.
The first case is when A, = 0, andAPr /P,o is assumed = 0.15 as implied inthe Handbook multipliers. Addition ofnonprestressed steel of areaA, = 2 A,, atthe same level as the prestressed steel
roughly increases the prestress loss toAP,= 0.35P,o.
It may be noted in the comparison,that in the one case where the Hand-book reduced multiplier is appropriate(because it reduces camber), the correc-tion appeared excessive (2.45 versus1.48). This confirms Gurlitz' findings&that, for test beams with A,/A P, values inexcess of 2, the correction was exces-sive.
As indicated earlier, provision of non-prestressed mild steel generally in-creases the overall flexural stiffness ofthe member since a relatively large mildsteel area would be needed to substitutefor stronger prestressing strands. Theelastic deflections are thus reduced, cor-responding to an increase in the trans-formed moment of inertia.
This effect is often neglected if themember is uncracked since I Q is usuallyused in lieu of the uncracked trans-formed section moment of inertia. How-ever, the effect of nonprestressed steelon I,r is much more pronounced, as in-
128
dicated by Eqs. (1) and (2), and shouldbe taken into account. This is discussedfurther in Example 2.
SUPPORTING TEST DATATesting was conducted s to study de-
flection behavior of four simply sup-ported beams of equal rectangular crosssection, span length, and steel eccen-tricity. The beams ranged from fullyprestressed to conventionally reinforced(see Table 4). Deflection was measuredunder dead load, sustained for a periodof 180 days, as well as instantaneousload.
Measurements were also taken ofcreep, shrinkage, and strength devel-opment with time. Comparison betweentest results and values predicted by theproposed method indicate a closeagreement as represented by Fig. 5.Details of the experiment may be foundin Ref. 8.
PROPOSEDCALCULATION STEPS
The following is a summary of thesteps to be followed to determine de-flections in accordance with the pro-posed method. To simplify the organi-zation of data, it is suggested that valuesbe arranged in a chart form, such as thatshown in Table 5.
1, Estimate the prestress loss usingthe procedure outlined in Appendix A.
2. Determine instantaneous deflec-tions due to initial prestressing force,prestress loss, member weight, and su-perimposed dead load, using propertiesof uncracked concrete section withoutreinforcement. Table 1 or another de-sign aid may be used.
3. Using multipliers from Table 2,determine time-dependent effects ondeflection in Step 2. This will result inelastic-plus-creep deflections at "Erec-tion Time," and at time infinity "FinalTime," as well as- the difference be-
Total deflection, inch
-0.8
#0.6 Observed
4-04 j- - -• ^- Y —_—•._^ Calculafed — —.^.^----
+0.2
20 40 60 80 100 120 i40 160
Age after loading, days
0.2
Camber, inch
Fig. 5. Total time-dependent deflection of Test Beam B-3.
PCI JOURNAL/January-February 1985 129
Table 5. Deflection chart (Example 1).
Load condition
A B C D E=1)-C:
Elastic
S
Release
8
At erection Final "ultimate" Long-term
Multi-
plierS Multi-
plierS Multi-
plier6
Initial prestress -3.23 -3.23 1.96 -6.33 2.88 -9.30 0.92 -2.97
Prestress loss 0.84 - 1.00 0.84 2.32 1.95 1.32 1.11
Member weight 3.00 3.00 1.96 5.88 2.88 8.64 0.92 2.76
Superimposed
dead load* 0.48 - - - 2.50 1.20 2.50 1.20
Live load 2.73 - - - - 2.73 - 2.73
Total - -0.23 - 0.39 - - - 4.83
*For this example, the superimposed dead load is assumed to be fully applied after nonstructuralelements are attached to member.
Note: Deflection is in inches. 1 in. = 25.4 nim.
tween deflections at these two timeswhich is labeled "Long Term." SeeTable 5 for illustration.
4. If the member cracks under fullservice loads, determine the crackedsection properties from Figs. 1 and 2 andthe cracked member live load deflec-tion. If the total deflection, computedthus far, satisfies the design limitations,i.e., ACI 318-83, Table 9.5b, the de-signer may elect to terminate the calcu-Iations. Otherwise, a refined value forthe live load deflection may be obtainedfrom Step 5.
5. If the member is cracked, concretetension stiffening effects may be em-pirically taken into account using inter-polation Eqs. (4) and (7), or anothermeans.
NUMERICAL EXAMPLES
EXAMPLE ICalculate the central deflection of a
partially prestressed simply supportedbeam shown in cross section in Fig. 6.
One-point depressed pretensioningtendons are used as shown in Fig. 4. As-sume span = 70 ft. Other data are:
Concrete Propertiesfi = 3,500 psif, = 5,000 psifr = 530 psiE r a = 3,587 ksiE, = 4,287 ksiCreep coefficients:C a = 1.88,C= = 1.50Free shrinkage, E g g,,, = 560 x 10-e
Prestressing Steel
Ten ! -in, diameter, 270 ksi stressrelieved strandse, = 13.40 in., ee = 3.79 in.f, = 189 ksiA 92 = 1.53 in. 2, Ep„ = 28,000 ksiIntrinsic relaxation, L r = - 16.1 ksi
Nonprestressed Steel
Two #8 bars, e = 15.15 in.A, = 1.58 in. 2, E, = 28,000 ksi
Section PropertiesA. = 401 in. 2, 19 = 20,985 in.4yb = 17.15in.,yt=6.85in.
130
8'-0" _ _f -H--1
53/4r 2"
:.. 24„
4'-0"
- 33/4"
Fig. 6. Cross section of beam of Example 1.
Gravity Loads
Member dead weight = 418 plfSuperimposed dead load =80 plfLive load =200 plf
SolutionMoments at midspan: M d = 3,072 in.-kips, M, = 588 in.-kips, and M, = 1,470in.-kips. Moments at 0.41: M d = 2,949in.-kips, M. = 564 in.-kips, and M, _1,411 in.-kips.
Step 1 - Ultimate Prestress Loss, .Pr,at a Section 0.41 From SupportEccentricity of A,, at 0.41:
e p, = (0.8) (13.40) + (0.2) (3.79)= 11.48 in.
This corresponds to a total steel areaeccentricity:
=n (11.48) (1.53) + (15.15) (1.58)e, 1.53 + 1.58
= 13.34 in.
With an initial estimate of Pro = 0.9 P,= 260.3 kips, Eq. (A2) yields:
260.3+ (260.3) (11.48) (13.34)It,. = 401 20,985
_ (2,949) (13.34) = 0.674 ksi
20,985
Eq. (A1) yields the corresponding
elastic shortening loss, using:o f = 28,000/3,587 = 7.81,ES = (7.81) (0.674) = 5.3 ksi
Thus,f = f„$, - ES = 189 - 5.3 = 183.7 ksi
and ffp = - 5.3 ksi.An improved value of P,., may be
found from Eq. (A3):P,. = (183.7) (1.53) + (-5.3) (1.58)
= 272.7 kipsA second iteration gives:fe,. = 0.795 ksi and P,. o = 269.9 kips.These two values are used in further
calculations.By substitution into Eq. (A5) with:
n tA„IA Q = (7.81) (3.11)/401 = 0.0606,(1 + e 2 /r2 ) =
1 + (13.34)2 (401)/20,985 = 4.40,and (I + xC„) = 1 + (0.7) (1.88) = 2.32,the coefficient:
K = [1 + (0.0606) (4.40) (2.32)]-'= 0.618
Eq. (A4) gives the shrinkage loss:SH = (0.618) (560 x 10) (28,000)
= 9.69 ksiThe creep loss [Eq. (A6)), with:f^JF = (- 564) (13.34)120,985
-0.359 ksigives a value of:
CR = (0.618) [(7.81) (1.88) (0.795) +(6.53) (1 + 1.5) (-0.359)]
= 3.59 ksiEq. (A8) yields a relaxation reduction
factor:
PCI JOURNALIJanuary-February 1985 131
^=1- 3(9.69 + 3.59)18.3.7
= 0.783
Thus, the relaxation loss, using Eq.(A7):
REL = -(0.783) (0.618) (-16.1)= 7.79 ksi
The total time-dependent loss ofcompression in concrete, using Eq. (A9):
AP, = -(3.11) (9.69 + 3.59)- (1.53) (7.79)
= -53.2 kips
Step 2- Immediate Deflection
According to Table 1, the curvaturedue the initial prestress, P^.o , atmidspan:
_ (269.9) (13.4)& _ (3,587) (20,985)
_ -4.81 x 10- 5 in.-1
Similarly, the end section curvature:6P = -1.36x10-' in.-'
and the midspan deflection:
S = [_4.81___](840)Z 1
[81-+1.36) {840)2]10-s
= -3.23 in.
The immediate self weight deflection(using E,i ) can be found in a similar wayto he = 3.00 in. The superimposed deadload deflection (using E^) is 0.48 in.
Step 3 - Time-Dependent DeflectionsThe force AP, develops gradually
between prestress release and time in-finity. The change in curvature at anysection during this period is:( A PCeIMIE<<I.) (1 +
The first term, A P, e 1E 1 I, representsthe curvature which would instanta-neously occur if the entire value of APwere applied at prestress release. Thevalues of this hypothetical "elastic" cur-vature at midspan and at the ends are:1.01 x 10- ' and 0.68 x 10- s in.-'
These values reflect the assumption
that OPT, is located at the centroid of thecombined steel area. The correspondingmidspan and end eccentricities are14.29 and 9.56 in. The "elastic" deflec-tion, due to A P, according to Table 1 is0.84 in.
The multiplier (1 + xC q ) = 2.32 isthen applied to the elastic value to giveultimate deflection due to prestress Ioss= 1.95 in. Assuming a s = 0.6 and C. =0.96, the deflection at erection time (seemultipliers in Table 2) is:
0.6(1+0.96)0.84=0.99 in.The time-dependent deflections due
to initial prestress, dead load, and su-perimposed dead load are obtained byapplying the appropriate multipliers,according to Table 2, to the above cal-culated immediate deflections. The re-sults are shown in Table 5.
Step 4 - Immediate Live LoadDeflection
The bottom fiber concrete stress at the0.41 section using uncracked sectionanalysis, due to full load moment of:2,949 + 564 + 1,411 = 4,924 in.-kipsand effective prestress:P.,. = 269.9-53.2=216.7 kipscan be calculated:
216.7216.7 (11.48) (17.15f"m = 401 + 20,985
_ 4,924 (17.15) = _1,451 ksi
20,985
This tensile stress is in excess of themodulus of rupture, indicating crackingof the member. The cracked sectionproperties may he found from the chartsin Figs. I and 2, given the following pa-rameters:
Pdp,/M = (216.7) (18.33)14,924 = 0.81,rip = (6.53) (3.11)1(96) (20.2) = 0.010,h,,,lh=9.5196=0.1, andh,ld=2120.2=0.1.Note here that the decompression
force is taken approximately equal to theeffective prestress force, and that thedepth d is to the centroid of the totalsteel area.
Fig. 1 yields:
132
y,, = (0.143) (1.1) (20.2) = 3.18 in., andI,T = (0.11) (1.02) (96) (20.2)x/12
- 7,400 in.4The curvature due to full load plus
prestress, at 0.41, using Eq. (3):
4,924 - (216.7) (18.33 - 3.18)- (4,287) (7,400)
=5.17x10-5 in.-'
The live load curvature is equal to thiscurvature less the curvature producedby dead load plus prestress (using un-cracked section properties).
,,a, 5.17 x 10-s
_ 3,513 - 216.7 (11.48)(4,287) (20,985)
= 4.03x10-5 in.-'
Similarly, the live load curvature atmidspan = 3.45 x 10- 1 in.' Using Eq.(12), an estimate of the live load deflec-tion can be made:
8 = 5 [3.45 + 4.03] (10) '(840)196
2.75 in.
Step 5 — Consideration of TensionStiffening
Improved results may be obtained byempirical consideration of concrete ten-sion stiffening. One of the methods dis-cussed in this paper may he used. Forthis example, Eqs. (4) and (7) are usedto calculate adjusted "effective" mo-ment of inertia and centroidal depth.
At 0.41:R = (0.530/1.4.51) = 0.02,1, = 0.02(20,985) + 0.98(7,400)
= 7,670 in. 4, andy e = 0.02 (6.85) + 0.98(3.18) = 3.25The corresponding total load curva-
ture [Eq. (3)] is 5.04 x 10-5 in. - 1 and thelive load curvature is (5.04 - 1.14)10-5in.-' = 3.9 x 10- 5 in,-'
Similarly, the midspan live load cur-vature is 3.32 x 10-b in.-' and the corre-sponding live load deflection = 2.65 in.
Further improvement of the accuracywill result from computing the curva-tures at short intervals, and then numer-
ically integrating these curvatures overthe span length. This refinement resultsin a live load 8 = 2.73 in. A summary ofthe deflections of the beam is given inTable 5.
The calculated live load deflections isabout 1/300. This is slightly larger thanthe limit of (1/360) set by the ACI 318-83Code for floors not supporting or at-tached to nonstructural elements likelyto be damaged by large deflections. Themember would be satisfactory in similarroof construction (with 8,„ar = 11180) butnot in construction requiring attachmentof nonstructural elements, due to the ex-cessive long-term deflection (4.83 in. >1/240 = 3.50 in.)
EXAMPLE 2
In this example, comparison is madebetween three beams in order to illus-trate the influence of prestressing leveland of presence of nonprestressed steel.Table 6 gives the difference in prestressforce and reinforcement in the threebeams. Note that Beam A is identical tothat analyzed in the PCI Design Hand-book, Examples 3.2.8, and 3.4.1, exceptthat the live load is reduced from 35 to25 psf. Beam C is the beam of Example 1in this paper. The deflections are calcu-lated by both the proposed method andthe PCI Design Handbook method.
The results indicate that the total ini-tial and time-dependent deflections aresensitive to the magnitude of the pre-stress force. A slight change in the de-flection due to prestress results in amuch more pronounced total (net) de-flection because of the opposite effectsof prestress and gravity loads. The flatrate of prestress loss (10 percent elasticloss, and 15 percent time-dependentloss) results in about 1/4 in. error in theinitial deflection (camber) and a largererror in the final deflection, in Beam B.Errors in Beam A deflections are not aslarge.
Substitution of four in. strands (A =0.612 in. ,) with two #8 bars (A = 1.58
PCI JOURNALJJanuary-February 1985 133
Table 6. Influence of prestress level and reinforcement on deflection.
Input data: Beam A' Beam B Beam C
Prestress force just beforere leaseP, 405.0 289.2 289.2
Reinforcement 14 strands 14 strands 10 strands plusFor other data, see Example 1 2 #8 bars
PCI PCI PCIPro- Design Pro- Design Pro- Design
posed Hand- posed Hand- posed Hand-Results: method book method hook method book
375.3 364.0 274.6 26O.3 269.9 260.3Pre stre ss force just after release, P,.,
Deflection at release due to:Prestress -4.64 -4.49 -3.39 -3.15 -3.23 -3.15Self weight 3.00 3.00 3.00 3.00 3.00 3.00
Total deflection at release -1.64 -1.41 -0.39 -0.15 -0.23 -0.15
Time-dependent prestress loss, AP,. -71.7 -54.6 -36.8 -39.0 -53.2 -39.0
Final "ultimate" time-dependentdeflection due to:
Prestress -11.00 -11.00 -8.5.5 -7.72 -7.35 -5.39Self weight 8.64 8.10 8.64 8.10 8.64 5.52Superimposed dead load 1.20 1.44 1.20 1.44 1.20 0.96
Total time-dependent deflection -1.16 -1.46 1.29 1.82 2.49 1.13
Geometric properties underfull load:0.4 l INIoment of inertia
Prestress force eccentricity2098511.78
2098511.78
418016.06
- 740015.15
-
Mid- of inertia 20985 20985 5160 5541 7850 3890114oment
span Prestress force eccentricity 13.65 13.65 17.62 13.65 17.01 13.65
Live load curvature x 106:End section 0 0 0 0 0 00,41 section 15.7 15.7 53.8 47.2 40.3 65.3Midspan section 16.4 16.4 37.9 49.1 34.5 68.0
Live load deflection 1.21 1.21 2.76 3.61 2.73 5.00
*EL,d section eccentricity used ise = 4.29 in., which is consistent with the PCI DesignHandbook Example 3.2.8, but slightly diffe rent from e = 3.79 in. in Example 3.4.1.
Note: Forces are given in kips, and dimensions in inches. 1 kip = 4.45 kN; 1 in. = 25.4 mm.
in. ․) provides additional restraint to con-crete deformation in the bottom of thebeam. This results in higher initial andtime-dependent prestress losses inBeam C than in Beam B, and a corre-spondingly larger downward deflection(2.49 in. versus 1.29 in.). The PCI DesignHandbook empirical multipliers lead to
reduction of the time-dependent de-flection from 1.82 to 1.13 in., rather thanan increase from 1.29 to 2.49 in. in theproposed method.
One may pose the following question:since Beam C contains more steel thanBeam B, how could Beam C, which isstiffer, have more loss of prestress and
WE
long-term deflection than Beam B? Theanswer to this question becomes evi-dent if one remembers that the prestressloss, APE , is defined in the present workas the loss of compression force in con-crete, rather than the loss of tension inthe prestressed steel. The differencebetween the two quantities is equal tothe compression force developed in thenonprestressed steel. The force AP, isthe quantity that must he used in com-puting the concrete stresses and de-formation,
Examination of Table 6 shows thatBeam C has a larger prestress loss thanBeam B, due to the restraint caused bythe presence of nonprestressed steel.This increase in AP, has a significantinfluence in reducing the time-depen-dent prestress camber (-8.55 versus–7.35 in.). Since the gravity load de-flections are the same in both beams, thenet result is a significant increase in thenet downward time-dependent deflec-tion.
The live load deflection of Beam B ismuch larger than that of Beam A due tocracking as a result of reduced prestress.Cracking drops the moment of inertia toless than one-quarter of the gross section1. However, the deflection of Beam B isless than four times that of Beam A,partly due to the moderating effect ofthe increase in prestress force eccen-tricity upon cracking. Reasonably closeagreement between the two methodsexists for the uncracked beam, Beam A,and when a cracked beam does not con-tain nonprestressed steel, Beam B.However, a great difference exists forBeam C, 2.73 versus 5.00 in.
Note that the current PCI DesignHandbook equation for I Cr , [Eq. (1) ofthe present paper] does not account forthe area of nonprestressed steel. Con-sideration of the presence of nonpre-stressed steel, by using Eq. (2) ratherthan Eq. (1), results in a significant in-crease in I,,. and a drop in Beam C liveload deflection (from 5.00 to 2.46 in.).
It is interesting to note that the live
load deflections, by the proposedmethod, of Beams B and C are aboutequal although Beam C contains moresteel and would he expected to deflectless under live load. However, due tothe fact that the prestress loss A P r isgreater, and therefore the effective pre-stress is smaller, the beam begins tocrack under a lower load than Beam B.The live load thus causes cracking todevelop in Beam C over a greater por-tion of the span length than in Beam B.The increase in the amount of crackingappears to offset the increase in thecracked member stiffness resulting fromthe presence of nonprestressed steel.
CONCLUSIONS1. The method proposed here con-
tains simple steps in which the time-de-pendent effects, the presence of nonpre-stressed steel, and the effects ofcrackingare rationally accounted for. Designcharts are included which can be used todetermine cracked section moment ofinertia and centreidal depth.
2. The multipliers for time-depen-dent effects given here can be used forvarious environmental conditions, byselecting representative creep coeffi-cients and free shrinkage value.
3, It is shown that deflection due toprestress losses must he correctly con-sidered in order to obtain reliable de-flection estimates, especially when thedead load and the load "balanced" byprestress are nearly equal.
4. The presence of nonprestressedmild bar reinforcement causes a signifi-cant restraint to the surrounding con-crete and substantially affects thelong-term deflections. This correspondsto an increase in the loss, OPT, in thecompressive force in concrete. The pro-posed method for calculating A P, and itseccentricity, or another method that ra-tionally accounts for the effects of non-prestressed steel, should be used in ac-curate deflection calculations.
5. The geometric properties of
PCI JOURNAUJanuary- February 1985 135
cracked sections are influenced by thepresence of axial force combined withbending moment, particularly forflanged members where the steel arearatio is relatively low. Also replacementof high strength tendons with a largeramount of nonprestressed mild rein-forcing bars can considerably increasethe cracked section rigidity.
6. The stiffening effects caused byconsideration of uncracked tensionedconcrete in deflection analysis requiresthe determination of two quantities, theeffective moment of inertia, and the ef-fective eccentricity of prestress force,These two parameters must, by defini-tion, lie between the uncracked sectionproperties, and those of prestressed
cracked sections, with concrete in ten-sion ignored.
7. Test data show a close correlationbetween observed deflections and thosecalculated by the proposed method.
S. When the proposed method iscompared to the PCI Design Handbookmethod, it is seen that there is a closeagreement in the deflection multipliersfor average environmental conditionsand when nonprestressed steel is ab-sent.
ACKNOWLEDGMENT
Initial stages of the research reportedin this paper were sponsored by a PCIResearch Fellowship Grant.
136
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14, Branson, Dan E., and Trust, H., "UnifiedProcedures for Predicting the Deflectionand Centroidal Axis Location of PartiallyCracked Nonprestressed and PrestressedConcrete Members," ACI Journal, Pro-ceedings V. 79, No. 2, March-April 1982,pp. 119-130.
15. Branson, D. E., and Trost, H., "Applica-tion of the I-Effective Method in Calcu-lating Deflections of Partially Pre-stressed Members," PCI JOURNAL,V. 27, No. 5, Sept.-Oct. 1982, pp. 62-77.
16. Tadros, M. K., and Sulieman, H., Discus-sion of Ref. 13 by Branson and Trost, PCIJOURNAL, V. 28, No. 6, Nov.-Dec. 1983,pp. 131-136.
17. Branson, D. E., "Instantaneous andTime-Dependent Deflections of Simpleand Continuous Reinforced ConcreteBeams," HPR Report No. 7, Part 1, Ala-bama Highway Department, Bureau ofPublic Roads, August 1965, p. 78.
18. CEB-FIP, Model Code for ConcreteStructures, 1978, Comite Euro-Interna-tional du Btxton, 6 Rue Lauriston, F-75116, Paris, France.
19. Favre, R., Beeby, A. W., Falkner, H.,Koprna, M., and Schiessl, P., "Crackingand Deformation," Bulletin d'Informa-tion No. 143, Comite Euro-Internationaldu B6ton, 6 Rue Lauriston, F-75116,Paris, France, Sept. 1981.
20. Trost, H., "The Calculation of Deflec-tions of Reinforced Concrete Members— A Rational Approach," Special Publi-cation SP-76, American Concrete Insti-tute, Detroit, Michigan, 1982, pp, 89-108.
"Editor's Note: The third edition of the PCI DesignHandbook (which will soon be published) containsessentially no changes relative to the material pre-sented in the article on "Prestress Loss and Dc flee-lion of Precast Concrete Members."
PCI JOURNAL/January-February 1985 137
21 Bazant, Z. P., and Oh, B. H., "Deforma-tion of Progressively Cracking Rein-forced Concrete Beams", ACI Journal,V. 81, May-June 1984, No, 3, pp. 268-278.
22 Tadros, M. K., Ghali, A., and Dilger,W. H., "Time-Dependent Prestress Lossand Deflection in Prestressed ConcreteMembers," PCI JOURNAL, V. 20, No. 3,May -June 1975, pp. 86-98.
23. ACI Committee 209, "Prediction ofCreep, Shrinkage and Temperature Ef-fects in Concrete Structures," SpecialPublication SP-76, American ConcreteInstitute, Detroit, Michigan, 1982, pp.193-300.
24. Martin, L. D., "A Rational Method forEstimating Camber and Deflection ofPrecast Prestressed Members," PCIJOURNAL, V. 22, No. 1, Jan.-Feb. 1977,pp. 100-108.
25. Zia, P., Preston, H. K., Scott, N. L., andWorkman, E. B., "Estimating PrestressLosses," Concrete International, V. 1,No. 6, June 1979, pp. 32-38.
26. Gurlitz, Carey Lang, "Loss of Prestress(Compression Loss) in Partially Pre-stressed Concrete," MSCE Thesis di-rected by F. Shaikh, Department of CivilEngineering, Univeristy of Wisconsin-Milwaukee, 1979, 135 pp.
27. AASHTO, Standard Specifications forHighway Bridges, American Associationof State Highway and Transportation Of-ficials, Washington, D,C., 1977.
28. Dilger, Walter H., "Creep Analysis ofPrestressed Concrete Structures UsingCreep Transformed Section Properties,"PCI JOURNAL, V. 27, No. 1, Jan.-Feb.1982, pp. 989-117.
29. Shaikh, A. F., and Branson, D_ E.,"Non-Tensioned Steel in PrestressedConcrete Beams," PCI JOURNAL,V. 15, No. 1; Jan.-Feh. 1970, pp. 14-36.
30. ACT Committee 435, "Proposed Revi-sions to ACT Building Code and Com-mentary Provisions on Deflections,"ACIJournal, Proceedings, V. 75, No. 6, June1978, pp. 229-238.
APPENDIX A - PRESTRESS LOSS CALCULATION
The method presented here is a sim-plified form of that given in Refs. 5 and22. The Trost-Bazant aging coefficient xis utilized in evaluating creep of con-crete due to variable stress. The methodis equally applicable to pretensionedand single-stage post-tensioned mem-bers, except that Step 1, which is givenhere for pretensioned members, can heapplied to post-tensioned members onlyif(ES) in prestressed steel is taken equalto zero, yielding ff ,o = fPB,. If multistagepost-tensioning is used, further adjust-ments would he needed,
In some of the following terms, thegross concrete section properties areused as approximations of net, or trans-formed section properties. This is an ac-cepted simplification unless an unusu-ally large amount of steel is used. Thetime-dependent losses given are ulti-mate values. When losses at an interme-
diate time are needed, use adjusted mate-rial properties € ,,, C, C, and L,..
Step I — Stress Loss at Transfer
ES = n ifer (Al)
where n, = E SIE Ct, and fir is concretestress at centroid of total steel area dueto initial prestress and member weight.
fcr = (Pc0fA 0) + (F ea e ps e t,11d (Mde, 11 Q) (A2)
Since Pro is yet to be determined, it isreasonable for initial calculation ofES toassume that Pro = 0.9 P. Initial stressesimmediately after transfer are. fpm = fp,i– (ES), and fgQ = –(ES) in the pre-stressed and nonprestressed steel, re-spectively.
Initial force in concrete, can now becalculated more accurately;
Pao = fpcoA p, + f"A, (A3)
Substituting this value of P C , back intoEq. (A2) results in an improved value offor , for use in Step 3.
138
Step 2 — Stress Loss Due toShrinkage
SH = K e15E a (A4)
where
K = 1I[1 + (E A Ar,/E c A))(1 + e2,lr 2) (1 + xC„)l (A5)
The symbols Ar, and er g are the totalsteel area and its eccentricity, respec-tively.
Step 3 — Stress Loss Due to Creep
CR = K Jn,C ufcr + n(1 + Cv)fc&l (AG)
where
f = stress at centroid of total steelarea due to superimposed deadloads not used in computing f,.
nC u = ultimate creep coefficient for
concrete at loading equal totime of release of prestressing
C = ultimate creep coefficient forconcrete at application of thesuperimposed dead loads
It is here assumed that time of appli-cation of the superimposed dead loadscoincides with time of attachment ofnonstructural elements. Note that theinstantaneous change in stress due toapplication of superimposed loads is in-cluded in Eq. (A6), while the corre-sponding change due to initial dead loadis already accounted for in Step 1.
Step 4 — Stress Loss Due toRelaxation
REL = - i/iKL r (A7)
where L r is the intrinsic (constantlength) relaxation loss.
See Ref. 3 for estimates of L r for stressrelieved and for low relaxation steel.The symbol fr designates a factor thataccounts for the reduction in relaxationloss resulting from the continual short-ening of tendon embedded in concrete
due to creep and shrinkage. Values of iimay be obtained from a chart presentedin Ref. 22 or from the following formulawhich is a straight line approximation ofthat chart.
tP = 1 - 3[(SH + CR)lf,,] (A8)
Step 5— Total Time-DependentLoss of Compression Force inConcrete
A? = -A,, (SH + CR) -A D ^ (REL) (A9)
Note that the negative sign designatesa tensile force increment. For light-weight concrete, appropriate values ofthe material properties, such as E, t, E.
e,h, C,,, and C,, should be used.For precast concrete members, Eqs.
(A4) — (A7) reduce to Eq. (A10) — (A13)when the following representative pa-rameters are adopted:Relative humidity = 70 percentVolume to surface ratio, V/S = 1.5Concrete unit weight = 150 pefAccelerated curing, release strength
(at concrete age 1 to 3 days)f = 3,500 psi
E,.= 3600 ksi, f, = 5000 psiE, = 4300 ksiC,, = 1.88, C, = 1.50co,=550x10- 6,x= 0.7f^olf„u = 0.65 and 0.7 for stress relieved
METRIC (Si) CONVERSIONFACTORS
1ft=0.305mI in, =25.4mm1 in. 2 = 645.2 mm'I in. = 42,077 mm'1 kip = 4.45 kN1 plf = 14.56 NlmI ksi =6.9MPa1 psi = 0.0069 MPa1 in-kips = 0.113 kN-m
PC{ JOURNALJJanuary-February 1985 139
and low relaxation strands,respectively
Lr _ –18.8 ksi and –4.8 ksi for the twotypes of strand
E, = E, = 28,000 ksi, f,,= 270 ksi
CR = K[14.6frr + 16.3fcaa] (Al2)
REL = 18.8K – (4.9 + 4.7 fcr+ 5.2f 4 )K 2(A13a)
or
SH = 15.4 K (A10) REL = 4.8K – (1.17 + 1.1 fcr+ 1.2f)K 2(A13b)
whereEq. (13a) applies to stress relieved
K = 11[1 + 18.0{A,/A,) (1 +e2 Blr2)] strands while Eq. (A13h) applies to low(All) relaxation strands.
APPENDIX B - NOTATION
Sign convention — Compressivestress in concrete and tensile stress insteel are positive. M is positive when itproduces tension at the bottom fiber of amember. Positive curvature correspondsto positive M. Downward deflection ispositive.
Aa = gross area of concrete sectionA, = area of prestressed reinforce-
mentAg = area of nonprestressed tension
reinforcementA, = area of total steel = A 8 +Ar,,b = width of compression face of
member= web width
C = creep coefficient, defined ascreep strain divided by initialstrain due to constant sustainedstress
Cc = creep coefficient for age of load-ing at release and loading dura-tion up to time of erection
Cu = ultimate creep coefficient forconcrete at loading equal totime of release of prestressing
Cu = ultimate creep coefficient forloading at time of attachmentof nonstructural elements, as-sumed to occur at time of super-imposed dead load application
CR = stress loss due to creep of con-crete
140
d = distance from extreme compres-sion fiber to centroid of tensionreinforcement
da, = distance from extreme compres-sion fiber to centroid of pre-stressing steel
e, = eccentricity of prestress forcefrom centroid of section at cen-ter of span
err = eccentricity of prestress forcefrom centroid oferacked section
e, = eccentricity of prestress forcefrom centroid of section at endof span
e P, = eccentricity of prestressed steelmeasured from centroid of un-cracked section
e t , = eccentricity of total steel areameasured from centroid of un-cracked section
E c = modulus of elasticity of con-crete at service
Eci = modulus of elasticity of con-crete at time of initial prestress
E, = modulus of elasticity of steelES = stress loss due to elastic short-
ening of concretef concrete compressive stress at
center of gravity of A A + Ap, dueto all permanent (dead) loadsnot used in computing fer
fcr = concrete stress at center of grav-ity of combined steel immedi-ately after transfer
= stress in prestressed reinforce-ment immediately prior to re-lease
= stress in prestressed reinforce-ment immediately after release
= modulus of rupture of concrete= stress in nonprestressed steel
immediately after release= apparent tensile stress at ex-
treme bottom fiber due to totalload moment and effective pre-stress using uncracked sectionproperties
hr = depth of flangeIcr = moment of inertia of cracked
section transformed to concreteIe = effective moment of inertia for
computation of curvature at asection
1. = moment of inertia of gross sec-tion
K = coefficient defined by Eq. (A5)= span length
Lr = intrinsic relaxation stress loss,for a condition ofconstant strain
MQ = total moment at sectionMcr = cracking momentM, = decompression bending mo-
mentMd = moment due to member weight
dead loadM, = moment due to service live loadMV = moment due to superimposed
dead loadn = modular ratio E,/ECn, = modular ratio ES/Ed
Pee = effective prestress force in con-crete
Pro = prestress force in concrete, im-mediately after release
P t = prestress force before releaseAP, = time-dependent loss of com-
pressive prestress force in con-crete, assumed, to be applied atcentroid of combined steelarea
R = (M,, – M5)1(M 5 – M Q), see Eqs.(4) and (5)
r = radius of gyration = pREL = stress loss due to relaxation of
tendonsSH = stress loss due to shrinkage of
concreteTier = distance from extreme compres-
sion fibers to centroid of crackedsection
Ye = effective distance from extremecompression fibers to centroidof cracked section, for curvaturecalculation; see Eq. (7)
y Q = distance from extreme compres-sion fibers to centroid of un-cracked section
aQ = ratio of prestress loss in con-crete at erection to final pre-stress loss
S = midspan deflection or camber0, = curvature at midspan section0, = curvature at cracked section(A, = curvature at end sectionx = aging coefficiente el, = free shrinkage of concretep = A,,l(bd), to be used in Figs. 1
and 2ip = relaxation reduction factor, Eq.
(A8)
J Fx1
Ipso
Jr
foe
.ftm
NOTE: Discussion of this paper is invited. Please submityour comments to PCI Headquarters by September 1, 1985.
PCI JOURNAWanuary-February 1985 141
