Time-Dependent Deflectionof Prestressed Beams bythe Pressure-Line Method

Antoine E. NaamanDepartment of Civil Engineering,

Mechanics and MetallurgyUniversity of Illinois at ChicagoChicago, Illinois

M ost prestressed concrete structuresare first designed and dimen-

sioned on the basis of allowablestresses or strength. If a sound designapproach is followed and if code re-quirements with respect to permissiblestresses, flexural strength, shear, andtorsion are satisfied, it is very likely thatthe design can be finalized without anyfurther modification.

There are, however, increasing situa-tions where the performance of thestructure during service, as related, forinstance, to long-term camber or de-flection, may control the design. This isparticularly true at the joints and con-nections of structures made of precastelements.

In reinforced, prestressed, and par-tially prestressed concrete members,deflection under sustained loadingcontinues to increase with time, mainly

due to the effects of creep and shrink-age of concrete and relaxation of pre-stressing steel. Excessive deflections,especially those developing with time,are common causes of trouble and mustbe limited.

In computing deflection, one maydifferentiate between camber and de-flection, short-term or immediate in-stantaneous deflection, and long-termor time-dependent deflection. The totaldeflection itself can be separated intotwo parts, an instantaneous part and anadditional time-dependent part. Fur-thermore, a different approach is fol-lowed whether the member is un-cracked, such as in fully prestressedmembers, or cracked, such as in rein-forced and partially prestressed mem-bers.

After a necessary review of availablemethods to estimate deflections in sim-

98

ply supported prestressed concretebeams, this paper proposes a newmethod to estimate long-term deflec-tion. The method combines the ration-ality of the incremental time steps pro-cedure with some simplified assump-tions that makes it readily useable inmany practical design situations.

The proposed method is called the"Pressure-Line Method" because itconsiders the beam subjected to a pre-stressing force following the profile ofthe pressure line along the beam in-stead of the profile of the prestressingsteel. The pressure line (also calledthrust line or C-line l •2) is the line of ac-tion of the concrete compressive stressresultant at each section along thebeam.

Note that in a statically determinatebeam the pressure line departs from thecentroid of the tendons by a distanceMIF, where M is the external momentand F is the prestressing force (Fig. 1).The pressure-line method, proposedhere to determine time-dependent de-flections, has the following characteris-tics whose justification is given later.

1. A single sustained loading com-prising the combined effects of sus-tained external moments (dead loadmoments) and prestressing moment isconsidered.

2. A lump sum estimate of totaltime-dependent prestress losses is as-sumed a priori. A reasonable number oftime intervals (which may be tailored tothe particular project) is selected andthe percents of total prestress loss at theend of these intervals are estimated.

3. Only two sections are considered,the midspan section and the supportsection. The eccentricities of the pres-sure line at these two sections at anytime t as influenced by creep and thepercent loss of prestress are computed.They are used to determine the deflec-tion at time t, from existing formulas inwhich the eccentricities of the pre-stressing steel are replaced by those ofthe pressure line.

PCI JOURNAL/March-April 1983

The proposed method applies to pre-stressed and partially prestressedbeams assumed uncracked under theirsustained load. Although it requiresminimal computational effort (a pocketcalculator is sufficient), design graphsare provided to simplify its implemen-tation.

99

e^C=F

eF CF F

(a)

(b)

(c)

PRESSURE LINE OR C-LINE—

F

PRESTRESSING STEEL

(d )

CGC

—F

:GC

PRESTRESSING STEEL

(e)

Fig. 1 Effect of an external moment on resulting stresses and pressure line in concrete.

INSTANTANEOUSSHORT-TERM DEFLECTION

Fully prestressed concrete membersare uncracked under service loads andare assumed linear elastic. Their in-stantaneous deflections can be deter-mined using general principles of me-chanics. Typical formulas are derivedfor simply supported beams with con-stant sectional properties. Commonlyused expressions for calculating deflec-

tion are shown in Fig. 2 for varioustypes of external loadings and variousprofiles of a prestressing force assumedconstant. Because superposition is validin computing deflections for uncrackedmembers, many combinations are prac-tically covered in Fig. 2. Note that twoexpressions are given for each case, oneas a function of the prestressing forceand the other as a function of the cur-vatures at the midspan section and atthe support.

100

CAMBER DUE TO PRESTRESSING FORCE DEFLECTION DUE TO LOADING

CGC

___ ___ +' o - Fe 1

Q _ £2

8EI 1 8

= 5wR" _ 592

384EI 1 48

CGCCGS P

_ - 8EI [e2 + (e l - e 2 )] PR3 RZ

2 2 48EI 1 12_ ^1 8 +(02 - l ) 48

CGCe2 ----- CGS P P

b b —'f'

4aa- 8EI [el + (e2 - e l ) 3 R 2]

---

24EI(3R 2 - 4b2

ARZ a2

8 + (^2 - l )6

A

_ 3k2 4b2

^1 24 )

Fig. 2. Typical midspan deflections for simply supported prestressed beams.

The deflection (positive downward,negative upward) expressed as a func-tion of the curvatures3 has the followingmost . general form:

A=(p , g +((D 2 —(D ^) 6 (1)

wherec 1 = curvature at midspan(D2 = curvature at supporta = length parameter that is a func-

tion of tendon profile used

PCI JOURNAUMarch-April 1983

The values of a have been directlyintegrated in the expressions shown inFig. 2. The curvature at any section canbe computed from the known expres-sion:

(D El

where M is the bending moment whichincludes the prestressing moment, E isthe modulus of elasticity, and I is themoment of inertia.

101

f ct—'cb

I^ Ect

wl

*cbi Irte-- ^ b

Fig. 3. Curvature representation for uncracked sections.

The curvature can also be computedfrom the strain distribution along thesection:

= E ct — Ecb(3)

h

wheree, t = strain on top fiber of sectionE lb = strain on bottom fiber of sectionh = depth of section

Eq. (3) is illustrated in Fig. 3 assum-ing uncracked sections.

In using all deflection expressionssuch as those given in Fig. 2, two im-portant properties must be defined,namely, the modulus of elasticity of theconcrete and the moment of inertia ofthe section (or of each section if a vari-able depth is used).

The modulus of elasticity of the con-crete material (secant modulus at 0.45f) can be estimated for design pur-poses from the expression recom-mended in the ACI Code, 4 i.e.,

E, = 33y 1.5 f (in psi)(4)

E, = 0.043yJ 5 V7 in MPa)

where y, is the unit weight of concretein lbs per cu ft (kg/m3 ) and f f is the de-sign specified compressive strength at28 days in psi (MPa).

Since the actual strength of concrete

varies with age, its actual modulus alsovaries. In computing initial camber ordeflection, it is common to use the ini-tial design modulus E,; while E0 is con-sidered for service load deflections.

The moment of inertia of the sectiondepends on whether the section iscracked or uncracked. When the sectionis uncracked, it is customary to use thegross moment of inertia Ig for preten-sioned members and the net moment ofinertia I. for post-tensioned memberswith unbonded tendons. In all caseswith bonded tendons, the moment ofinertia of the transformed section canbe used but the corresponding gain inaccuracy may not justify the lengthiercalculations.

LONG-TERM DEFLECTIONIn this section various methods are

given for computing long-term deflec-tions in prestressed concrete membersas follows: (1) Simplified predictionmethods; (2) Incremental time-stepsmethod; and (3) Proposed pressure-linemethod.

Simplified PredictionMethods

It was pointed out in the previoussection that the deflections calculatedfrom the various derived expressions of

102

t TIME

t= 0 t

/ /O•

/Ec i fEC.(t)

I E

Cci ' scC (t)l

cc (t) ''

(b)(a)

Fig. 4. Relationship between: (a) creep strain variation with time and(b) equivalent elastic modulus.

Fig. 2 are "instantaneous elastic" de-flections; that is, they are associatedwith a short-term loading. If the load issustained, such as in the case of deadloads, the deflection increases withtime, mainly because of the effects ofcreep and shrinkage of the concrete andrelaxation of prestressing steel. In sucha case, the deflection is called the totaldeflection.

The total deflection can be separatedinto two parts: an instantaneous elasticpart and an additional long-term part.The first part is calculated as describedin the previous section and is assumedto remain constant for a given load. Theadditional part increases with time.

Continuous research efforts are beingundertaken to predict time-dependentdeflection in reinforced and prestressedconcrete structures more reliably 3•s-21

It has been customary in everyday de-sign to estimate the additional long-term deflection by multiplying the in-stantaneous deflection A i by an appro-priate factor. The basic reason behindthis approach comes from analyzing theeffect of creep on the stress-strain re-sponse of concrete. The variation ofcreep strain under initial loading is il-lustrated in Fig. 4a. At any time t, the

PCI JOURNAUMarch-April 1983

creep strain assuming constant stress crcan be obtained by multiplying the in-stantaneous strain E, i under a givenstress by the creep coefficient Cc (t).Thus:

EcC (t) = Eci X CC (t) (5)

Looking simultaneously at thestress-strain response of the material inits linear elastic range (Fig. 4), one canseparate the instantaneous elastic strainfrom the creep strain. By "definition, theratio it/E,; is equal to the initial elasticmodulus E^ i (or the elastic modulus atage of loading). With time, however,the ratio of stress to total strain de-creases due to the creep strain. Thiseffect cam be simulated by using anequivalent modulus (Fig. 4b). Its valueis obtained from:

Ece (t) _ _ 0Ec (t) Eci + EcC (t)

Eci + Eci CC (t ) (6)

_ 0 __

Eci [I + Cc (t)] 1 + Cc (t)

Any deflection formula is an inversefunction of the modulus. In general,

103

one can write: mate creep coefficient. However, some

K other effects, such as those of shrink-

(7) = _ age, relaxation and the presence ofE non-prestressing steel, may be ac-

where K depends on dimensional and counted for in the equation predicting

loading properties. Applied to the ini- Addd• Three such prediction equations

tial instantaneous deflection, Eq. (7) are discussed next.gives:

In Section 95.2.5, the ACI Code4

suggests the following predictionK equation for non-prestressed concrete

_ (8) members:

If the loading, is sustained for a timet, the time-dependent total deflectionbecomes:

A (t) EcK (9)(t)

Replacing K in Eq. (9) by its valuefrom Eq. (8) and Ece (t) by its value fromEq. (6) leads to the total deflection attime t, assuming constant loading:

A (t) = D i [1 + CC (t)1 (10)

The above equation can be separatedinto two parts, a constant value 0; and atime-dependent value. Thus:

A(t)= At +C C (t)A; (11)

At the end of the life of the member,the deflection becomes:

D U = A,+ O j CCU = A,(1 +CCU)

(12)

= D E + Aadd

in which

Oadd = CCU 0{ (13)

and0 U = total ultimate deflection for the

sustained loading consideredA.dd = additional long-term deflection

Eq. (13) suggests that the additionallong-term deflection can be estimatedby multiplying the instantaneous de-flection by a factor. According to Eq.(13), the multiplier is equal to the ulti-

Ladd = XA i (14)

whereA j = instantaneous deflectionA = 2-1.2(Ag/A8)--0.6A8 = area of compression steelA 8 = area of tension steel

It can be seen that for A8 = 0, X = 2;that is, the multiplier A is about equal toan average value of the creep coeffi-cient.

Based on an extensive evaluation ofparameters influencing Dada for typicalprecast prestressed members, Martin17suggested the following equation:

EstDajd = XI = rl E kr CCU A t (16)

where7 = F/Fd , where F is the final pre-

stressing force and Ft the pre-stressing force immediately af-ter transfer

k,. = 1/(1 + A8 /A,,8 ) when AS /Ap82; A8 and A. are the areas ofthe reinforcing and prestress-ing steel, respectively

CCU = ultimate creep coefficientE, = elastic modulus of concrete at

time of transferE, = design elastic modulus of con-

crete [Eq. (4)]

Different values of the multiplier Awere recommended for prestressedcomposite and non-composite membersdepending on the effects of self weight,prestressing, and superimposed deadload, if any. These multipliers are also

104

given in Table 3.4.1 of the PCI DesignHandbook19 and generally vary be-tween 1.85 and 3.

Branson, et al. 9 '10 '13 •1 6 suggested thefollowing equation to predict thelong-term additional deflection in pre-stressed non-composite members:

O Q =I^– 1+( 1 2 ^

J

I kr CCUI (A,),,+

kkr Cev (At )c + KcA kr Ccu(Aj )sn

(17)

where the subscripts G and SD refer toself weight and superimposed deadload and KCA is the age', at loading factorfor creep (Table 1). Other notations arethe same as for Eq. (16). Note that thereduction factor k,. in Eqs. (16) and (17)should apply only when the net deflec-tion due to prestress and sustainedloading is negative (i.e.,', a camber).

Two observations can be made in re-lation to the above three simplifiedmethods: (1) they may yield signifi-cantly different results when applied tothe same problem22.23 and (2) they pre-dict ultimate deflection only, that is, thedeflection at the end of service life.

There are cases, however, wheremore accurate predictions are needednot only of the ultimate deflection butalso of the deflection at various pointsin time. For such cases the incrementaltime steps method or simplifiedmethods derived from it can be used.

incremental TimeSteps Method

The additional long-term deflectionis essentially caused by'the simultane-ous occurrence of creep, and shrinkageof concrete and relaxation of the pre-stressing steel. As these effects are alsoassociated with prestress losses, it isquite logical to combine the computa-tions of deflections with those of pre-stress losses. A similar approach wassuggested by Subcommittee 5 of ACICommittee 435 and is explained in

PCI JOURNAUMarch-April 1983

Reference 24. Theoretically, the com-putation procedure may include thefollowing steps:

1. Divide the span into several seg-ments (about 20 is more than adequate)to be each represented by its average ormidsection.

2. Divide the design life of thestructure into several time intervals.These are not equal intervals but haveincreasing lengths. A typical set is usedin the example of Appendix A.

3. Select a time interval, starting inorder by the first one. Determine thestrain distributions, curvatures and pre-stressing , force at each section at thebeginning of the time interval consid-ered. For the first interval, these valuescorrespond to the instantaneous effectsand the initial prestressing force. De-termine the incremental creep andshrinkage strains and relaxation lossduring the time interval. Compute newvalues of strains, curvatures and pre-stressing force at each section at theend of the time interval. These will beused as reference values at the begin-ning of the next time interval. This pro-cedure is then repeated for each timeinterval studied and at each section.Great care should be used to insure re-storation of both equilibrium and straincompatibility at the end of each timeinterval.

4. By integrating or summing upalong the beam the curvatures com-puted at the beginning of each timeinterval, the corresponding total time-dependent deflection can be deter-mined. It will include the instantane-ous deflection and the additionallong-term deflection defined earlier.

The above procedure is time con-suming and requires essentially a com-puter. In an attempt to simplify its im-plementation, Tadros l' accounted forthe interdependent effects of prestresslosses with creep, shrinkage and relax-ation by introducing two design param-eters: a creep recovery factor and arelaxation reduction factor. A design

105

00)

Table 1. Recommended relationships for some time-dependent properties of concrete (adapted from References 27 to 31).

Values of constantsProperty Relationship*

Moist-cured concrete Steam-cured concrete

Fort = 1 day Type I cement: Type I cement:Compressive t

(28)ff` (t ) `

b = 4 b = 1strength b + ct c = 0.85 c = 0.95

Type III cement: Type III cement:Equation is same for both normal b = 2.30 b = 0.70and lightweight concrete c = 0.92 c = 0.98

t o.00Cc (t) = CcUKcxKcaKcs t, to? 7 days t, to >– 1 to 3 days

10+ tO.60where and H > 40 percent and H % 40 percent

Creep KcA = age at loading factor KcA = 1.25 t° 11° KcA = 1.13 t°°°°coefficient Kcff = humidity correction factor Kc, = 1.27 – 0.0067 H KCX = 1.27 – 0.0067 H

Kcs = shape and size factor Kcs = 1.14 – 0.09 (V/S )t Kcs = 1.14 – 0.09 (V/S )tto = age at loading -- 0.68 > 0.68

* Valid in all systems of units; note that H is given in percent,t Equation fitting tabular data from Reference 31. Note that V/S is the volume-to-surface ratio of the

member in inches.

chart was suggested to estimate thecreep recovery factor.

Tadros also pointed out that for com-mon loadings in simply supportedbeams, the midspan deflection at anytime t can be obtained from the curva-tures at time t of only two sections: themidspan section and the support sec-tion [Eq. (1)]. This last simplification isalso used in the proposed pressure-linemethod described next.,

Proposed Pressure-LineMethod

The pressure-line method proposedin this paper to predict long-term de-flections combines the theory behindthe time-steps method 'with some sim-plified observations. It', has the advan-tage of being relatively simple and suf-ficiently accurate for a number of prac-tical cases, yet manageable in terms ofcomputational effort.

Since the deflection is calculated atdifferent time intervals, this methodallows for those special cases wherevalues of deflections at intermediatetimes are needed in addition to the de-flection at the end of the life of themember. The method applies to fullyprestressed members, and to partiallyprestressed members in which decom-pression is not exceeded under the ef-fect of sustained loads. The character-istics of the method and their justifica-tion are described below:

1. A set of appropriate time intervalsis selected; the total time-dependentloss of prestress (a lump sum value) andthe percent prestress loss at the end ofeach time interval are estimated apriori. Note that, in general, the totaltime-dependent prestress loss and thepercent loss with time can be estimatedwithin an acceptable margin. This isbecause the laws governing creep andshrinkage (and to a certain extent relax-ation) and their prediction equationslead to very comparable growth rates atcomparable times.

PCI JOURNAUMarch-April 1983

As shown in the example of AppendixA (Tables Al and A2), the total ultimatedeflection and to some extent the totaldeflection at various times are not sig-nificantly influenced by poor estimatesin the percent losses, provided the totalloss is kept the same. By estimating apriori the percent of prestress loss withtime, the effects of shrinkage and relax-ation are indirectly accounted for whilethe effect of creep on long-term deflec-tion is isolated and expanded upon.

2. Only one loading is consideredduring each time interval and com-prises the combined effect of the pre-stressing force and the sustained exter-nal load. This is contrary to othermethods where the loadings are sepa-rated into prestressing, self weight andsuperimposed dead load. Because ofthe combined loading, the beam is as-sumed equivalently prestressed by aforce following the trajectory of thepressure line instead of the trajectory ofthe prestressing steel (Fig. 5). The re-sulting compressive force is equal tothe prestressing force in magnitude. Itseccentricity at any section along thespan at any time t is given by:

e.(t) = eo – ) (18)

where M is the externally applied mo-ment and F (t) is the prestressing forcewhich varies with time, according tothe assumed prestress losses.

During any time interval (ti , ti ), Mand F(t) are assumed constant with thevalue of F(t) taken at the beginning ofthe interval, that is, F(ti ). Since in mostcommon cases the variation of eo alongthe span is either linear or parabolicand M is either linear or parabolic, theprofile of the trajectory of the pressureline will be either linear or parabolic.

3. Only two sections are consideredin the computations of time-dependentvariables, the midspan section and thesupport section. This is because thedeflection of the beam for commonlinear or parabolic trajectories of the

107

t

eo2 ^ r

pt

w

`'RESSURE 'eCE LINE eCl

CURVATURE CURVATURE(2 (si)

_ • sE +^–^1) 2

I

Fig. 5. Typical example of pressure line obtained undercombined effects of prestressing and external loads.

pressure line can be calculated as afunction of the curvatures of the mid-span and support sections [Eq. (1)].Such expressions are given in Fig. 2 forthe prestressing steel. They can beused directly here, noting that e1 and e2become the eccentricities of the pres-sure line and are time dependent [Eq.(18)1. In using these expressions, it isassumed that the resulting compressionforce, as well as its eccentricity, remainconstant during the time interval con-sidered.

Thus, according to the above de-scribed procedure, the deflection of thebeam (Fig. 5) during each time interval

can be calculated either directly fromthe formulas of Fig. 2 where el and eE

are the eccentricities of the pressureline or from Eq. (1). In the latter case,the curvature at any section can becomputed either from Eq. (2) in whichE is replaced by ECe (t) or from Eq. (3)in which the values of eat and ecb can bedetermined from:

Eet E F (t)Ac ^ l – e6 (t) ZJ (19)

Ecb ))AE,F(t l 1 + e^ (t) Za (20)

108

where Zt and Zb are the section modulifor the extreme top and bottom fibers,respectively, and ECe (t) is the equiva-lent modulus of elasticity of the con-crete as influenced by creep.

Assuming that the short-term instan-taneous modulus varies with time, theequivalent modulus of elasticity undersustained loading at any time t can beestimated similarly to Eq. (6) fromeither of the following two expressions:

Ece (t) = E, (tA) (21 a)1+C(T)

______ }ECe (t) – E (t) (21b)1 +Cc(T)

whereE, (t) = instantaneous elastic modu-

lus of concrete at time tE, (tA ) = instantaneous elastic modu-

lus of concrete at time tAt = age of concrete in days

Cc (r) = creep coefficient at time r-r= time after loading= t–to =0

tA = age of concrete at loading

Note that Eq. (21a) complies with Eq.(6) and Eq. (21b) differs from Eq. (21a)by its numerator E, (t) instead of E0 (tA).Eq. (21b) is proposed here to account,to a certain extent, for the effect ofaging of concrete. Its effect on theequivalent modulus of concrete is com-parable to that of the more exactmethod using the aging coefficient asproposed by Bazant25.26 and explainedlater.

The modulus of elasticity of concreteat any time t assuming short-termloading can be estimated according tothe ACI Building Code formula [Eq.(4)] from:

E (t) = 33yC'5 J7) (22)

where f, (t) is the compressive strengthof concrete at time t.

ACI Committee 209 27 suggests theuse of the following expression as pro-

posed in References 28 and 29:

R, (t)= b +ctf`(23)

where f f is the design specified com-pressive strength (at 28 days), and b andc are parameters which depend on thetype of cement used and the curingmethod.

These relationships are summarizedin Table 1. Thus, Eq. (22) becomes:

E, (t) = 33y.5 f^^+t

= E^ t (24)b+ct

where E0 is the design modulus of elas-ticity of concrete.

The creep coefficient at any time rcan be predicted using the time func-tion proposed in References 28 and 29,and recommended by ACI Committee20927 and the PCI Committee on De-sign Handbook 30

,ru.s

Cc (T) + To•s Ccv Kcx Kca Kcs (25)10

in whichT = t — tA = time after loading , 0

(26)Cc, ultimate creep coefficient of

the concrete materialKCH = correction factor accounting

for the average relative hu-midity of the environmentwithin which the structurewill perform

K = age at loading factorKcs = shape and size factor

Expressions for KCH and KCA , as sug-gested in References 27-29, are repro-duced in Table 1. Also shown is an ex-pression predicting Kcs which essen-tially fits the data given in tabular formby the ACI Committee on PrestressLosses 31 Replacing Cc (r) and E0 (t) by

PCI JOURNAUMarch-April 1983 109

their values from Eqs. (25) and (24) intoEq. (21b) leads to:

E, (t) = Ec G:+EC:t: x( )

10+ (t– to)0.6

[ 10 + (t — to )6•6 (I + CCU `NCH `ACA K2)

(27)

in which the constants b and c and thefactors KCH , KCA , and Kcs are given inTable 1.

For a given problem, Eq. (27) is re-duced to a time function multiplied bythe design modulus of elasticity of con-crete at 28 days, E.

The variation of concrete strain withtime due to creep and age at loading isa dominant factor in any long-term de-flection computation. In the above pro-cedure the equivalent modulus at anytime t is used instead of the creepstrain. Note that nondimensional graphs(or other design aids) can be developedto predict its value for common designapplications.

Eq. (27) can be rewritten as:

ECe (t) = ^

1,t

10 + 7°•6

E, [ 10 +T6.6(1 +II)]

= gi (t) g2 (T) (28)

where

11 = CcU KCH KCA Kes (29)

For a known value of H, the ratioECe (t)/E, is obtained from the product oftwo time functions, gl (t) and gE (T). Fig.6 provides a graphical solution for thesetwo functions. Note that if Eq. (21a)were used instead of Eq. (21b) to deriveEqs. (27) and (28), the function g l (t)would be reduced to a constant g l (tA).

An example illustrating the use ofthe pressure-line method to computethe time-dependent deflection in aprestressed beam is covered in Ap-pendix A.

AGE ADJUSTEDEFFECTIVE MODULUS

The effective modulus approach de-scribed in Eq. (21a) is the simplest andmost widespread approximate methodto analyze the effect of creep on thedeflection of a concrete structural ele-ment. However, it can be shown that,for a given creep law, it may lead to alarge error with respect to the theoreti-cally exact solution, if aging of concreteis significant.

Note that the term "aging" refers tothe change of the properties of concretewith the progress of its hydration orequivalently with time 25 Hence, for aconcrete member loaded at an earlyage, the effective modulus approachmay be very inaccurate.

Trost proposed a simple method toaccommodate the effect of concreteaging by adding an aging coefficient tothe effective modulus equations Themethod was later expanded and refinedby Bazant,25 following a rigorous for-mulation. The proposed age adjustedeffective modulus takes on the follow-ing most general form :25

E' ^e (t,tA ) =), (tA (30)/ E'

1 + ^( (t,tA ) CC (t,tA )

whereECe (t,t4 ) = age adjusted effective

modulus at time t whenloading occurs at time to

E (tA ) = instantaneous elasticmodulus at time tA

X (t,tA ) = aging coefficient at timet for a concrete memberloaded at time tA

Cc (t,t4 ) = creep coefficient at timet for a concrete memberloaded at time tA

Eq. (30) is valid only for a strain-timevariation that follows the variation ofcreep coefficient. Using a computerizedanalysis and a wide range of parame-ters, Bazant showed that the agingcoefficient varies significantly with the

110

M

a

1.0

7(3)TYP

EMENT MOIST CURED

CEMENT MOIST CURED

0.75 EMENT STEAM CURED

CEMENT STEAM CURED

a ARnECe(t)

E = 9 1 (t) 92(r)C

0110' 102 103

IOs

TIME t, DAYS, LOG SCALE

TIME r, DAYS, LOG SCALE

Fig. 6. Graphical chart to obtain the effective modulus at any timet assuming loading at time T.

I5

age at loading t,,, the value of the creepcoefficient, and whether a variable or aconstant modulus is considered in thecreep law. He also showed that whileother methods give exact solutions onlywhen the applied stress is constant, the

age adjusted effective modulus appliesfor the case of constant stress, constantstrain (as in a stress relaxation test), andthe case of straining a member by dif-ferential creep.

The observed range of the aging

PCI JOURNAUMarch-April 1983 111

coefficient was 0.71 to 1 when a con-stant modulus was assumed in thecreep law and 0.46 to 1 when a variablemodulus was considered. The lowervalues correspond to early ages atloading. As a first approximation andwhen loading occurs at normal ages anddeflections are estimated at a later age,a value of X — 0.85 is appropriate incommon design applications. In such acase, the aging coefficient has essen-tially a delaying effect on the value ofthe effective modulus at a time t. A de-tailed discussion of the effect of theaging coefficient X is given in Refer-ences 25 and 32.

The age adjusted effective modulusgiven by Eq. (30) can be used to re-place the effective modulus given byEq. (21) if a more accurate evaluation ofdeflection is desired. The method ofcalculation by the pressure-line ap-proach remains essentially the same.Note that the application of the age-adjusted modulus is important as com-pared to the effective modulus calcula-tions, especially if the stress variationin time is significant. This may be duenot only to the variation in prestress butalso external loadings.

CONCLUDING REMARKSThe pressure-line method proposed

in this paper to compute deflections atvarious times including ultimate de-flection is simple but still limited inreach. In its present form it appliesmostly to simply supported precast pre-stressed concrete elements for which amajor portion of the sustained load(self-weight) is applied at the time ofprestress. Because the pressure line incontinuous beams accounts automat-ically for the presence of secondarymoments, extension of the pressure-linemethod to compute long-term deflec-tion in continuous beams and staticallyindeterminate structures should be ofparticular interest.

Attempts are being currently madeto: (1) extend its applicability to pre-stressed continuous beams and crackedpartially prestressed concrete beams,(2) generalize the solution to accountfor a variable loading history where theage adjusted effective modulus will beused, and (3) provide a systematic com-parison of results predicted by the pres-sure-line method, other methods andactual data.

8' •0" x 32" Section Properties

Normal Weight ConcreteA = 567 in.'

= 55,464 in.'

7 314" 2" Yb = 21.21 in.Y b = 10.79Zb = 2615 in.'

32" Z1 = 5140 in.'

774,41

wt = 591 plf74 psf

VIS = 1.79 in.

Y, = 5000 psitPU = 270,000 psi 8DT32

Fig. Al. Cross section of example beam.

112

APPENDIX A - DESIGN EXAMPLE

Let us consider the double-tee pre-stressed pretensioned beam shown inFig. Al. It is assumed that the beam isplant precast, using Type III cement,steam cured and prestressed at one dayof age after casting.

The following information is given:Span = 80 ft, Ae = 567 sq in.Ig = 55464 in 4, f f = 5000 psi,Ee=4.287x106psi,Ap/g = 2.448 sq in.,F = 354.96 kips, ff = 145 ksi = F/A„8,

F, = F/0.83 = 427.663 kips,E 5 = 27 x 106 psi, E, = 29 x 106 psi,n,, = Epg /E, = 6.298,n8 = E5 /E = 6.765,wG = 0.591 klf, WL = 0.32 klf

Midspan moments:MD = 427.8 kip-ft, ML = 256 kip-ftMD + ML = 728.8 kip-fteo at midspan = 17.21 in.,eo at support = 8.21 in.,The prestressing steel is depressed at

the midspan section.Note that using an estimated value of= F/F2 = 0.83 is equivalent to a lump

sum estimate of the prestress losses thatoccur after transfer.

It is assumed that the dead load(beam weight) is applied at the time ofprestress and represents the sustainedload for which the long-term deflectionis to be calculated. Thus, the sustainedexternal moment is MD = 472.8 kip-ft.

Let us assume that the value of theultimate creep coefficient, C for theconcrete material used equals 2.5 andthe average relative humidity of the en-vironment is 50 percent. Referring toEq. (25) and Table 1, the following val-ues of the various constants are ob-tained:

b=0.70,c=0.98KCH = 1.27 – 0.0067H = 0.935KCA = 1.13 t;°°95 = 1.13Kcs = 1.14 – 0.09 (V/S) = 0.979

where V/S is the volume to surface ratio

of the beam and is equal to 1.79 in.

R = Ccu Kca Kca Kcs = 2.59Since the age of loading to is one day,

Eq. (27) becomes for this example:

E'ce(t ) = E, 0.7

+t0.98) 1

10 + (t - 1)0.610 + (t – 1)0.6 (3.59)

where Ee = 4.287 x 106 psi.Several time intervals are selected

(Table Al) and the percent of prestressloss at the end of each interval is as-sumed. Note: the end of the first timeinterval is one day, the end of the sec-ond interval is seven days, and so on.The computations are summarized inTable Al for each time interval. Let usfollow them for the first interval. Theprestressing force is equal to its initialvalue immediately after transfer, and isdirectly given by:

Ft = = 354.96 = 427.663 kips

i 0.83

Values of the prestressing force at theend of the other intervals are given inthe first line of Table Al and corre-spond to:

F, – (F, – F) x (percentage loss)

The equivalent modulus at t = 1 dayis obtained from:

Ece (1) = Ee 0.7 + 0.98

= 0.772 Ee = 3.307 x 106 psi

Moduli values for the other time in-tervals are given in Table Al and canbe computed directly.

The eccentricity of the pressure lineat midspan at the end of the first timeinterval and the corresponding ex-treme fibers' strains and section curva-ture are given by Eqs. (18), (19), (20)and (3) (see also Fig. 6):

PCI JOURNAL/March-April 1983 113

Tahle Al . Summary of time-dependent deflection computations for example beam.

End tj of time internal (ti , ti ), days

LifeParameters1 7 30 90 365 50 x 365

Estimated percentage oftime-dependent prestress loss 0 20 45 65 85 100

F (t), kips 427.66 413.12 394.95 380.40 365.86 354.96

ECe (t), 106 psi 3.307 2.599 2.025 1.695 1.439 1.230

eC1 (t), midspan, in. 3.943 3.476 2.844 2.295 1.702 1.226

ec2 (t), support, in. 8.21 8.21 8.21 8.21 8.21 8.21

z

= - 8El I f e,,,+ 6 e.l - ec2 ) in. -1.25 -1.408 -1.514 -1.529 -1.471 -1.432^e L

e,,,, 10-6 128.9 172.8 236.1 295.6 364.2 440.1

Midspan € ,, 10' 423.1 491.6 556.1 592.7 613.9 644.34, 10-' in. -9.19 -9.96 -10 -9.28 -7.80 -6.38

Eft 10 -6 21.5 26.4 32.5 37.3 42.3 48.0

Support Epp, 10 6 634.1 779.4 956.3 1100 1246 1415

cIa 2 , 106 in.-' -19.14 -23.53 -28.87 -33.2 -37.63 -42.2

lz l20 = cpl + (4) 2 - 4,)-,in. -1.25 -1.408 -1.514 -1.529 -1.471 -1.4328 8

NOTE: 1 kip = 4448 N; 1000 psi = 6.895 MPa; 1 in.' = 0.039 mm-', 1 in. = 25.4 mm.

ee (t)= 17.21- 472.8x12427.66

= 3.943 in.427.66 3.94_ _

Eet 3.307 x 103 x 567 ( 9.06= 128.9x10-e

427.66 3.94_Eeb 3.307 x 103 x 567 ( 1 + 4.61

= 423.1x106

-_ Ect Ecb

32_ -9.194 x 10-6 in.-'

The eccentricity of the pressure lineat the support section for the first timeinterval and the corresponding strainsand curvatures are given by:

ee (t) = e,, = 8.21 in. (since M = 0)

= 427.66 - 8.21

3.307 x 106 x 567 ( 1 9.06

= 21.5x10-6427.668.21

fcb- 3.307x106 x567 ( 1+ 4.61

= 634.1 x 10 -6

-_ E ct - Ecb

2 32_ -19.14 x 10-6 in. -1

Similar values at the end of othertime intervals are given in Table Al.

Note that the curvatures cI and (1)2

could have been obtained directly fromEq. (2) in which the moment M in-cludes the external moment and themoment due to the prestressing force.Hence, for the first time interval wehave at midspan:

_ M

Ece (t ) I_ 472.8 x 12000 - 427660 x 17.21

3.307 x 106 x 55464_ -9.19 x 10_6 in.-'

and at the support:

_ -427660 x 8.213.307 x 106 x 55464

_ -19.14 x 10-6 in.-'

These curvatures are the same as ob-tained earlier. They are not repeated inTable Al.

The trajectory of the prestressingsteel is linear and the external momentvaries parabolically. Thus the trajectoryof the pressure line [Eq. (18)] isparabolic. In computing the deflection,we can assume that the expressiongiven in Fig. 1 for a parabolic profileapplies. Thus, we could either use theformula given in function of the eccen-tricities of the pressure line or the for-mula given in function of the curvaturesat the midspan and support sections;they are:

Flt r

- -i-i(el_e2)]

or12 12

A = 1), 8 + (c2 - (D1 ) 48

At the end of the first time interval(one day) they lead to:

Q - 427.66 x 802 x 122X

8x3.307x106x55664

18.21 + 5 (3.943 - 8.21)1

L-1.25 in.

or

_ -9.19x10-6x (80x12P +8

(-19.14 + 9.19) (80x12)248

1.25 in.The computations for other intervals

are given in Table Al. Note that thefinal value of long-term deflection is --1.43 in. (camber). It can be shown that

PCI JOURNAL/March-April 1983 115

Table A2. Deflection computations for a different set of values of the percentages of time-dependent prestress losses.

End ti of time interval (ti, t5), days

LifeParameters1 7 30 90 365 50 x 365

Estimated percentage oftime-dependent prestress loss 0 40 60 75 90 100

F (t), kips 427.66 398.58 384.06 373.14 362.23 354.96

ECe (t), 106 psi 3.307 2.599 2.025 1.695 1.439 1.230

M –9.19 –8.23 –8.33 –7.96 –7.02 –6.38

–10-6 in.-'

Ece(t)I

4)2 = M , 10-6 in.-' –19.14 –22.7 –28.07 –32.58 –37.26 –42.71E. (t)

= (D1 12 + (4 – (D,)12

–1.25 –1.23 –1.34 –1.39 –1.39 –1.4328 48

Note: 1 kip = 4448 N; 1000 psi = 6.895 MPa; 1 in.-' = 0.039 mm'; 1 in. = 25.4 mm.

0)

it is significantly different from the val-ues otherwise obtained from the sim-plified methods [as derived from Eqs.(14), (16), and (17)].

Table A2 summarizes deflectioncomputations for the same exampleproblem assuming the same total time-dependent prestress loss but a differentset of values of percent loss of prestressat the various time intervals considered.A comparison between the results ofTables Al and A2 suggests that littleerror will result from reasonably poorestimates in the values of the percentloss of prestress. In Table A2 the cur-vatures were calculated from Eq. (2) inwhich the moment comprises the pre-stressing moment and the external mo-ment.

An example of application of thepressure-line method to the case of apartially prestressed beam can be foundin Reference 2.

APPENDIX B - NOTATION

a = length parameter that is func- Ftion of the tendon profile used

A,, = area of concrete section F (t)A,, = area of prestressed reinforce- F,

mentA $ = area of non-prestressed rein- g (t)

forcement IA$ = area of compression reinforce- k,.

mentC,, (t) = creep coefficient at time t KcACCU = ultimate creep coefficient KCHe. = eccentricity of the pressure

line Kcseo = eccentricity of the prestressing 1

steel ME,, = design modulus of elasticity of MD

concrete at 28 days MD +LECe (t) = equivalent modulus of elastic- ML

ity of concrete at time t npECZ = modulus of elasticity of con- n$

crete at time of initial prestress tE modulus of elasticity of pre- to

stressing steel ZiE. = modulus of elasticity of rein- Zb

forcing steel df^ = design specified compressive pj

strength of concrete

= final or effective prestressingforce

= prestressing force at time t= prestressing force at time of

transfer= function of time t= moment of inertia in general= reduction factor of additional

long-term deflection= age at loading factor for creep

humidity correction factor forcreepshape and size factor for creep

= span length of beam= bending moment= dead load moment

service moment= live load moment

modular ratio E,1E,= modular ratio E81E,= time= age at loading= section modulus top fiber= section modulus bottom fiber= deflection in general= initial instantaneous elastic

deflection

PCI JOURNAUMarch-April 1983 117

A = additional long-term deflectionA. = ultimate or life deflection for

the sustained loading consid-ered

I' = CCU KCH KCA Kc5= curvature= unit weight of concrete

e = strain in general

ecb = strain in concrete bottom fibere ct = strain in concrete top fiber71 = F/F;A = deflection multiplier [see Eq.

(14)]= aging coefficient

v = stress in generalr = time after loading = t — to

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NOTE: Discussion of this paper is invited. Please submityour discussion to PCI Headquarters by Nov. 1, 1983.

PCI JOURNAUMarch-April 1983 119