ACI 20913-92(Reapproved 1997)

Prediction of Creep, Shrinkage,and Temperature Effects in

Concrete Structures

Reported by ACI Committee 209James A. Rhodest

Chairman, Committee 209

James J. Beaudoin John R. KeetontDan E. Branson* Clyde E. KeslerBruce R. Gamble William R. NormanH.G. Geymayer Jack A. Means?Brij B. Goyalt Bernard L Meyers?Brian B. Hope R.H. Mills

Corresponding Members: John W. Dougill, H.K. Hilsdorf

Domingo J. Carreira’tChairman, Subcommittee II

K.W. NasserA.M. NevilleFrederic Roll+John ‘IimuskMichael A. Ward

Committee members voting on the 1992 revisions:

Marwan A. DayeChairman

Akthem Al-Manaseer Chung C. FuJames J. Beaudoin Satyendra K. GhoshDan E. Branson Brij B. GoyalDomingo J. Carreira Will HansenJenn-Chuan Chern Stacy K. HirataMenashi D. Cohen Joe HutererRobert L Day Hesham Manouk

l Member of Subcommittee II, which prepared this reportt Member of Subcommittee IISD--d

This report reviews the me&a!9 for predicting creep, shrinkage and temper-ature effects in concrete structures. It presents the designer with a unifiedand digested approach to the problem of volume changes in concrete. Theindividual chapters have been written in such a way that they can be usedalmost independently from the rest of the report.

The report is general& consistent with ACI 318 and includes materialindicated in the Code, but not specifical& dej?ned therein.

Keywords: beams (supports); buckling camber; composite construction (concreteto concrete); compressive strength; concretes: concrete slabs; cracking (frac-turing): creep properties; curing: deflection; flat concrete plates; flexunl strength;girders: lightweight-aggregate concretes; modulus of elasticity; moments of inertia;precast concrete; prestressed concrete: prestress loss; reinforced concrete; shoring;shrinkage; stnins; stress relaxation; StNCtUnt design: temperalure; thermalexpansion; two-way slabs; volume change; warpage.

AC1 Committee Reports, Guides, Standard Practices, andCommentaries are intended for guidance in designing, plan-ning, executing, or inspecting construction and in preparingspecifications. References to these documents shall not bemade in the Project Documents. If items found in thesedocuments are desired to be a part of the Project Docu-ments, they should be phrased in mandatory language andincorporated into the Project Documents.

Bernard L MeyersKarim W. NasserMikael P.J. OlsenBaldev R. SethKwok-Nam ShiuLiisa Panulat

CONTENTS

Chapter l-General, pg. 209R-2l.l-Scope1.2-Nature of the problem1.3-Definitions of terms

Chapter 2Material response, pg. 209R-42.1-Introduction2.2-Strength and elastic properties2.3-Theory for predicting creep and shrinkage of con-

crete2.4-Recommended creep and shrinkage equations

for standard conditions

The 1992 revisions became effective Mar. 1, 1992. The revisions consisted ofminor editorial changes and typographical corrections.

Copyright 8 1982 American Concrete Institute.All rights reserved including rights of reproduction and use in any form or by

any means, including the mating of copies by any photo process, or by any elec-tronic or mechanical device, printed or written or onI. or recording for sound orvisual reproduct ion or for use in any knowledge or retr ieval system or device ,unless permission in writing is obtained from the copyright proprietoa.

2J-Correction factors for conditions other than thestandard concrete composition

2.6-Correction factors for concrete composition2.7-Example2.8-0ther methods for prediction of creep and

shrinkage2.9-Thermal expansion coefficient of concrete2.1O-Standards cited in this report

Chapter PFactors affecting the structural response -assumptions and methods of analysis, pg. 209R-12

3.1-Introduction3.2-Principal facts and assumptions3.3--simplified methods of creep analysis3.AEffect of cracking in reinforced and prestressed

members35Effective compression steel in flexural members3.6-Deflections due to warping3.7-Interdependency between steel relaxation, creep

and shrinkage of concrete

Chapter 4-Response of structures in which time -change of stresses due to creep, shrinkage and tem-perature is negligible, pg. 209R-16

4.1-Introduction4.2-Deflections of reinforced concrete beam and slab4.1Deflection of composite precast reinforced beams

in shored and unshored constructions4.ALoss of prestress and camber in noncomposite

prestressed beams4.5-Loss of prestress and camber of composite pre-

cast and prestressed-beams unshored and shoredconstructions

4.6Example4.7-Deflection of reinforced concrete flat plates and

two-way slabs4.8-Time-dependent shear deflection of reinforced

concrete beams4.%Comparison of measured and computed deflec-

tions, cambers and prestress losses using pro-cedures in this chapter

Chapter &Response of structures with signigicant timechange of strtss, pg. 209R-22

U-ScopeU-Concrete aging and the age-adjusted effective

modulus methodU-Stress relaxation after a sudden imposed defor-

mation5.4-Stress relaxation after a slowly-imposed defor-

mationSJ-Effect of a change in statical systemU-Creep buckling deflections of an eccentrically

compressed member5.7-%o cantilevers of unequal age connected at time

t by a hinge 5.8 loss of compression in slab anddeflection of a steel-concrete composite beam

5.9-0ther cases5.10-Example

Acknowledgements, pg. 209R-25

References, pg. 209R-25

Notation, pg. 209R-29

Tables, pg. 209R-32

CHAPTER l-GENERAL

l.l-scopeThis report presents a unified approach to predicting

the effect of moisture changes, sustained loading, andtemperature on reinforced and prestressed concretestructures. Material response, factors affecting the struc-tural response, and the response of structures in whichthe time change of stress is either negligtble or significantare discussed.

Simplified methods are used to predict the materialresponse and to analyze the structural response underservice conditions. While these methods yield reasonablygood results, a close correlation between the predicteddeflections, cambers, prestress losses, etc., and themeasurements from field structures should not be ex-pected. The degree of correlation can be improved if theprediction of the material response is based on test datafor the actual materials used, under environmental andloading conditions similar to those expected in the fieldstructures.

These direct solution methods predict the response be-havior at an arbitrary time step with a computational ef-fort corresponding to that of an elastic solution. Theyhave been reasonably well substantiated for laboratoryconditions and are intended for structures designed usingthe AC1 318 Code. They are not intended for the analy-sis of creep recovery due to unloading, and they applyprimarily to an isothermal and relatively uniform en-vironment.

Special structures, such as nuclear reactor vessels andcontainments, bridges or shells of record spans, or largeocean structures, may require further considerationswhich are not within the scope of this report. For struc-tures in which considerable extrapolation of the state-of-the-art in design and construction techniques is achieved,long-term tests on models may be essential to provide asound basis for analyzing serviceability response. Refer-ence 109 describes models and modeling techniques ofconcrete structures. For mass-produced conciete mem-bers, actual sire tests and service inspection data willresult in more accurate predictions. In every case, usingtest data to supplement the procedures in this report willresult in an improved prediction of service performance.

1.2-Nature of the problemSimplified methods for analyxing service performance

are justified because the prediction and control of time-dependent deformations and their effects on concretestructures are exceedingly complex when compared withthe methods for analysis and design of strength perfor-mance. Methods for predicting service performance in-volve a relatively large number of significant factors thatare difficult to accurately evaluate. Factors such as thenonhomogeneous nature of concrete properties caused bythe stages of construction, the histories of water content,temperature and loading on the structure and their effecton the material response are difficult to quantify even forstructures that have been in service for years.

The problem is essentially a statistical one becausemost of the contributing factors and actual results are in-herently random variables with coefficients of variationsof the order of 15 to 20 percent at best. However, as inthe case of strength analysis and design, the methods forpredicting serviceability are primarily deterministic innature. In some cases, and in spite of the simplifjlingassumptions, lengthy procedures are required to accountfor the most pertinent factors.

According to a survey by AC1 Committee 209, mostdesigners would be willing to check the deformations oftheir structures if a satisfactory correlation between com-puted results and the behavior of actual structures couldbe shown. Such correlations have been established forlaboratory structures, but not for actual structures. Sinceconcrete characteristics are strongly dependent on en-vironmental conditions, load history, etc., a poorer cor-relation is normally found between laboratory and fieldservice performances than between laboratory and fieldstrength performances.

With the above limitations in mind, systematic designprocedures are presented which lend themselves to acomputer solution by providing continuous time functionsfor predicting the initial and time-dependent averageresponse (including ultimate values in time) of structuralmembers of different weight concretes.

The procedures in this report for predicting time-dependent material response and structural service per-formance represent a simplified approach for designpurposes. They are not definitive or based on statisticalresults by any means. Probabilisitic methods are neededto accurately estimate the variability of all factors in-volved.

M-Definitions of termsThe following terms are defined for general use in this

report. It should be noted that separability of creep andshrinkage is considered to be strictly a matter of defin-ition and convenience. The time-dependent deformationsof concrete, either under load or in an unloaded speci-men, should be considered as two aspects of a singlecomplex physical phenomenon”

13.1 ShrinkageShrinkage, after hardening of concrete, is the decrease

with time of concrete volume. The decrease is due tochanges in the moisture content of the concrete andphysico-chemical changes, which occur without stress at-triiutable to actions external to the concrete. The con-verse of shrinkage is swellage which denotes volumetricincrease due to moisture gain in the hardened concrete.Shrinkage is conveniently expressed as a dimensionlessstrain (in/in. or m/m) under steady conditions of relativehumidity and temperature.

The above definition includes drying shrinkage, auto-genous shrinkage, and carbonation shrinkage.

a) Drying shrinkage is due to moisture loss in theconcrete

b) Autogenous shrinkage is caused by the hydrationof cement

C) Carbonation shrinkage results as the variouscement hydration products are carbonated in thepresence of CO,

Recommended values in Chapter 2 for shrinkagestrain (Q), are consistent with the above definitions.

13.2 CreepThe time-dependent increase of strain in hardened

concrete subjected to sustained stress is defined as creep.It is obtained by subtracting from the total measuredstrain in a loaded specimen, the sum of the initial in-stantaneous (usually considered elastic) strain due to thesustained stress, the shrinkage, and the eventual thermalstrain in an identical load-free specimen which is sub-jected to the same history of relative humidity and tem-perature conditions. Creep is conveniently designated ata constant stress under conditions of steady relativehumidity and temperature, assuming the strain at loading(nominal elastic strain) as the instantaneous strain at anytime.

The above definition treats the initial instantaneousstrain, the creep strain, and the shrinkage as additive,even though they affect each other. An instantaneouschange in stress is most likely to produce both elastic andinelastic instantaneous changes in strain, as well as short-time creep strains (10 to 100 minutes of duration) whichare conventionally included in the so-called instantaneousstrain. Much controversy about the best form of “prac-tical creep equations” stems from the fact that no clearseparation exists between the instantaneous strain (elasticand inelastic strains) and the creep strain. Also, the creepdefinition lumps together the basic creep and the dryingcreep.

a) Basic creep occurs under conditions of nomoisture movement to or from the environment

b) Drying creep is the additional creep caused bydrying

In considering the effects of creep, the use of either aunit strain, 6, (creep per unit stress), or creep coefficient,v, (ratio of creep strain to initial strain), yields the same

results, since the concrete initial modulus of elasticity,Ed, must be included, that is:

v, = sJ!& (l-1)

This is seen from the relations:

Creep strain = u 8,= q v,, and

hi = u/q

where, u is the applied constant stress and l i is the in-stantaneous strain.

The choice of either of S, or v, is a matter of con-venience depending on whether it is desired to apply thecreep factor to stress or strain. The use of v, is usuallymore convenient for calculation of deflections and pre-stressing losses.

133 RelaxationRelaxation is the gradual reduction of stress with time

under sustained strain. A sustained strain produces aninitial stress at time of application and a deferred neg-ative (deductivedecreasing rate.l !I

stress increasing with time at a

13.4 iuodulus of dasticityThe static modulus of elasticity (secant modulus) is the

linearized instantaneous (1 to 5 minutes) stress-strainrelationship. It is determined as the slope of the secantdrawn from the origin to a point corresponding to 0.45f,’ on the stress-strain curve, or as in ASTM C 469.

13.5 Contmctiof and eqansionConcrete contraction or expansion is the algebraic sum

of volume changes occurring as the result of thermal var-iations caused by heat of hydration of cement and byambient temperature change. The net volume change isa function of the constituents in the concrete.

CHAPTER 2-MATERIAL RESPONSE

2.1-IntroductionThe procedures used to predict the effects of time-

dependent concrete volume changes in Chapters 3,4, and5 depend on the prediction of the material responseparameters; i.e., strength, elastic modulus, creep, shrink-age and coefficient of thermal expansion.

The equations recommended in this chapter are sim-plified expressions representing average laboratory dataobtained under steady environmental and loading con-ditions. They may be used if specific material responseparameters are not available for local materials andenvironmental conditions.

Experimental determination of the response para-meters using the standard referenced throughout thisreport and listed in Section 2.10 is recommended if anaccurate prediction of structural service response isdesired. No prediction method can yield better resultsthan testing actual materials under environmental and

loading conditions similar to those expected in the field.It is difficult to test for most of the variables involved inone specific structure. Therefore, data from standard testconditions used in connection with the equations recom-mended in this chapter may be used to obtain a moreaccurate prediction of the material response in thestructure than the one given by the parameters recom-mended in this chapter.

OccasionalIy, it is more desirable to use materialparameters corresponding to a given probability or to useupper and lower bound parameters based on the expect-ed loading and environmental conditions. This predictionwill provide a range of expected variations in the re-sponse rather than an average response. However, prob-abilistic methods are not within the scope of this report.

The importance of considering appropriate water con-tent, temperature, and loading histories in predictingconcrete response parameters cannot be overemphasized.The differences between field measurements and the pre-dicted deformations or stresses are mostly due to the lackof correlation between the assumed and the actual his-tories for water content, temperature, and loading.

2.2-Strength and elastic properties2.2.1 Concrete compre.wive strength versus timeA study of concrete strength versus time for the data

of References l-6 indicates an appropriate general equa-tion in the form of E . (2-l) for predicting compressivestrength at any time. 69*

P-1)

where a in days and JI are constants, (fc’)2s = 28-daystrength and t in days is the age of concrete.

Compressive strength is determined in accordance withASTM C 39 from 6 x 12 in. (152 x 305 mm) standard cyl-indrical specimens, made and cured in accordance withASTM C 192.

Equation (2-l) can be transformed into

(2-2)

where al’ is age of concrete in days at which one half ofthe ultimate (in time) compressive strength of concrete,cf,‘), is reached.%

The ranges of a andp in Eqs. (2-l) and (2-2) for thenormal weight, sand lightweight, and all lightweight con-cretes (using both moist and steam curing, and Types Iand III cement) given in References 6 and 7 (some 88specimens) are: 4 = 0.05 to 9.25,jI = 0.67 to 0.98.

The constants a_ and j? are functions of both the typeof cement used and the type of curing employed. The useof normal weight, sand lightweight, or all-lightweightaggregate does not appear to affect these constantssignificantly. Typical values recommended in References7 are given in Table 2.2.1. Values for the time-ratio,OVK)28 or VX(f,‘)/(t;‘). in Eqs. (2-l) and (2-2) aregiven also in Table 2.2.1.

“Moist cured conditions” refer to those in ASTM C192 and C 511. Temperatures other than 73.4 -C 3 F (23* 1.7 C) and relative humidities less than 95 percent mayresult in values different than those predicted when usingthe constant on Table 2.2.1 for moist curing. The effectof concrete temperature on the compressive and flexuralstrength development of normal weight concretes madewith different types of cement with and withoutaccelerating admixtures at various temperatures between25 F (-3.9 C) and 120 F (48.9 C) were studied in Ref-erence 90.

Constants in Table 2.2.1 are not applicable to con-cretes, such as mass concrete, containing ‘I)pe II or ?LpeV cements or containing blends of portland cement andpozzolanic materials. In those cases, strength gains areslower and may continue over periods well beyond oneyear age.

“Steam cured” means curing with saturated steam atatmospheric pressure at temperatures below 212 F (100Cb

Experimental data from References l-6 are comparedin Reference 7 and all these data fall within about 20percent of the average values given by Eqs. (2-l) and(2-2) for constants u and /3 in Table 2.2.1. The tem-perature and cycle employed in steam curing may sub-stantially affect the strength-time ratio in the early daysfollowing curing.“’

2.2.2 Modulus of rupture, direct tensile strength andmodulus of elasticity

Eqs. (2-3), (2-4), and (2-5) are considered satisfactoryin most cases for computing average values for modulusof rupture, f,, direct tensile strength,f,‘, and secant mod-ulus of elasticity at 0.4(f,‘),, EC, respectively of differentweight concretes.‘*“l*

For the unit weight of concrete, w in pcf and the com-pressive strength, vi), in psi

& = 0.60 to 1.00 (a conservative value of g, = 0.60may be used, although a value g, = 0.60 to0.70 is more realistic in most cases)

8 = 1%g Cl = 33

For w in Kg/m3 and (&‘), in MPa

& = 0.012 to 0.021 (a conservative value of g, =0.012 may be used, although a value of g, =0.013 to 0.014 is more realistic in most cases)

& = 0.0069gCl = 0.043

The modulus of rupture depends on the shape of thetension zone and loading conditions. Eq. (2-3) corres-ponds to a 6 x 6 in. (150 x 150 mm) cross section as inASTM C 78. Where much of the tension zone is remotefrom the neutral axis as in the case of large box girdersor large I-beams, the modulus of rupture approaches thedirect tensile strength.

Eq. (2-5) was developed by Pauw** and is used in Sub-section 8.5.1 of Reference 27. The static modulus of e-lasticity is determined experimentally in accordance withASTM C 649.

The modulus of elasticity of concrete, as commonlyunderstood, is not the truly instantaneous modulus, buta modulus which corresponds to loads of one to fiveminutes duration.86

23-Theory for predicting creep and shrinkage of con-crete

The principal variables that affect creep and shrinkageare discussed in detail in References 3,6, 13-16, and aresummarized in Table 2.2.2. The design approach pre-sented6*7 for predicting creep and shrinkage refers to“standard conditions” and correction factors for otherthan standard conditions. This approach has also beenused in References 3, 7, 17, and 83.

Based largely on information from References 3-6, 13,15, 18-21, the following general procedure is suggestedfor predicting creep and shrinkage of concrete at anytime.7

r*“I= d+@% (2-6)

where d and f (in days), $ and cu are considered con-stants for a given member shape and size which definethe time-ratio part, v,, is the ultimate creep coefficientdefined as ratio Of creep Strain t0 initial Strain, (&& iS

the ultimate shrinkage strain, and t is the time afterloading in Eq. (2-6) and time from the end of the initialcuring in Eq. (2-7).

When 1p and a are equal to 1.0, these equations arethe familiar hyperbolic equations of Ross” and Lorman*’in slightly different form.

The form of these equations is thought to be conven-ient for design purposes, in which the concept of theultimate (in time) value is modified by the time-ratio toyield the desired result. The increase in creep after, say,100 to 200 days is usually more pronounced than shrink-age. In percent of the ultimate value, shrinkage usuallyincreases more rapidly during the first few months. Ap-propriate powers oft in Eqs. (2-6) and (2-7) were foundin References 6 and 7 to be 1.0 for shrinkage (flatterhyperbolic form) and 0.60 for creep (steeper curve for

larger values of t). This can be seen in Fii. (2-3) and(2-4) of Reference 7.

Values of #, d, vu, a, fi and (eJU can be determinedby fitting the data obtained from tests performed inaccordance to ASTM C 512.

Normal ranges of the constants in Eqs. (2-6) and (2-7)were found to be?*’

ds = = 0.40 6 t o 3 0 to 0.80,days,

% = 1.30 to 4.15,

; = = 0 . 9 0 2 0 t o t o 1 3 0 1 . 1 0 , d a y s ,khh = 415 x lod to 1070 x 10d ht./in., (m/m)

These constants are based on the standard conditionsin Table 2.2.2 for the normal weight, sand lightweight,and all lightweight concretes, using both moist and steamcuring, and Types I and III cement as in References 3-6,13, 15, 1820,23,24.

Eqs. (24% (2-9). and (2-10) represent the averagevalues for these data. These equations were comparedwith the data (120 creep and 95 shrinkage specimens) inReference 7. The constants in the equations were deter-mined on the basis of the best fit for all data individually.The average-value curves were then determined by firstobtaining the average of the normal weight, sand light-weight, and all lightweight concrete data separately, andthen averaging these three curves. The constants vu and(Q,& recommended in References 7 and 96 were approx-imately the same as the overall numerical averages, thatis v%= 2.35 was recommended versus 2.36; (Q,),, = 800x 10 in/in. (m/m) versus 803 x 10d for moist cured con-crete, and 730 x 10” versus 788 x 10d for steam curedconcrete.

‘I&e creep and shrinkage data, based on 20-year mea-surements” ’ for normal weight concrete with an initialtime of 28 days, are roughly comparable with Eqs. (2-8)to (2-10). Some differences are to be found because ofthe different initial times, stress levels, curing conditions,and other variables.

However, subsequent worp9 with 479 creep datapoints and 356 shrinkage data points resulted in the sameaverage for v, = 2.35, but a new average for (& =780 x 10d in/in. (m/m), for both moist and steam curedconcrete. It was found that no consistent distinction inthe ultimate shrinkage strain was apparent for moist andsteam cured concrete, even though different time-ratioterms and starting times were used.

The procedure using Eqs. (2-8) to (2-10) has also beenindependently evaluated and recommended in Reference60, in which a comprehensive experimental study wasmade of the various parameters and correction factorsfor different weight concrete.

No consistent variation was found between the dif-ferent weight concretes for either creep or shrinkage. Itwas noted in the development of Eq. (2-8) that moreconsistent results were found for the creep variable in the

form of the creep coefficient, v, (ratio of creep strain toinitial strain), as compared to creep strain per unit stress,6,. This is because the effect of concrete stiffness is in-cluded by means of the initial strain.

2.4-Recommended creep and shrinkage quations forstandard conditions

Equations (2-8), (2-9), and (2-10) are recommendedfor predicting a creep coefficient and an unrestrainedshrinkage strain at any time, including ultimate values.67They apply to normal weight, sand lightweight, and alllightweight concrete (using both moist and steam curing,and ‘Apes I and III cement) under the standard condi-tions summarized in Table 2.2.2.

Values of v,, and (Q)” need to be modified by thecorrection factors in Sections 2.5 and 2.6 for conditionsother than the standard conditions.

Creep coefficient, v, for a loading age of 7 days, formoist cured concrete and for 1-3 days steam cured con-crete, is given by Eq. (2-8).

t0.60v , =

10 + co.60 vu(2-Q

Shrinkage after age 7 days for moist cured concrete:

Shrinkage after age 1-3 days for steam cured concrete:

In Eq. (2-8), t is time in days after loading. In Eqs.(2-9) and (2-lo), t is the time after shrinkage is con-sidered, that is, after the end of the initial wet curing.

In the absence of specific creep and shrinkage data forlocal aggregates and conditions, the average values sug-gested for v,, and ((Fsts,, are:

vu = 2.35 yc and

wu = 780~~ x 10d in/in., (m/m)

where y, and y& represent the product of the applicablecorrection factors as defmed in Sections 2.5 and 2.6 byEquations (2-12) through (2-30).

These values correspond to reasonably well shapedaggregates graded within limits of ASTh4 C 33. Aggre-gates affect creep and shrinkage principally because theyinfluence the total amount of cement-water paste in theconcrete.

The time-ratio part, [right-hand side except for v, and(e&l of Eqs. (2-8), (2-9), and (2-lo), appears to beapplicable quite generally for design purposes. Valuesfrom the standard Eqs. (2-8) to (2-10) of v,lvU and

(Q,),/(Q), are shown in Table 2.4.1. Note that v is usedin Eqs. (4-ll), (4-20), and (4-22), hence, ‘v,/vU = vs/vUfor the age of the precast beam concrete at the slabcasting.

It has also been shown” that the time-ratio part ofEqs. (2-8) and (2-10) can be used to extrapolate 28-daycreep and shrinkage data determined experimentally inaccordance with ASTM C 512, to complete time curvesup to ultimate quite weII for creep, and reasonabIy weIIfor shrinkage for a wide variety of data. It should benoticed that the time-ratio in Eqs. (2-8) to (2-10) doesnot differentiate between basic and drying creep norbetween drying autogenous and carbonation shrinkage.Also, it is independent of member shape and size,because d, f, #, and a are considered as constant in Eqs.(2-Q (2-9), and (2-10).

The shape and size effect can be totally considered onthe time-ratio, without the need for correction factors.‘Ihat is, in terms of the shrinkage-half-time TV,,, as givenby Eq. (2-35) by replacing t by UT*,, in Eq. (2-9) and byO.lr/~~,, in Eq. (2-8) as shown in 2.8.1. Also by taking $= a = 1.0 and d = f = 26.0 [exp 0.36(v/s)] in Eqs. (2-6)and (2-7) as in Reference 23, where v/s is the volume tosurface ratio, in inches. For v/s in mm use d = f = 26.0exp [1.42 x 10s2 (v/s)].

References 61, 89, 92, 98 and 101 consider the effectof the shape and size on both the time-ratio (time-dependent development) and on the coefficients affectingthe ultimate (in time) value of creep and shrinka e.

AC1 Committee 209, Subcommittee I Report 1% is re-commended for a detailed review of the effects ofconcrete constituents, environment and stress on time-dependent concrete deformations.

2.Lorrection factors for conditions other than thestandard concrete composition’

AI1 correction factors, y, are applied to ultimatevalues. However, since creep and shrinkage for anyperiod in Eqs. (2-8) through (2-10) are linear functionsof the ultimate values, the correction factors in thisprocedure may be applied to short-term creep andshrinkage as well.

Correction factors other than those for concrete com-position in Eqs. (2-11) through (2-22) may be used inconjunction with the specific creep and shrinkage datafrom a concrete tested in accordance with ASTM C 512.

2.5.1 Loading ageFor loading ages later than 7 days for moist cured

concrete and later than 1-3 days for steam cured con-crete, use Eqs. (2-11) and (2-12) for the creep correctionfactors.

Creep yc, = 125(r,)-O*“s for moistcured concrete (2-11)

Creep yea = 1.13 (tcJo*o94 for steam curedconcrete (2-12)

where tea is the loading age in days. Representative val-ues are shown in Table 25.1. Note that in Eqs. (4-ll),(4-20), and (4-22), the Creep yea correction factor mustbe used when computing the ultimate creep coefficient ofthe present beam corresponding to the age when slab iscast, vW. That is:

%J = vu Wreep YCJ (2-13)

2.5.2 Differential shrinkageFor shrinkage considered for other than 7 days for

moist cured concrete and other than 1-3 days for steamcured concrete, determine the difference in Eqs. (2-9)and (2-10) for any period starting after this time.

That is, the shrinkage strain between 28 days and 1year, would be equal to the 7 days to 1 year shrinkageminus the 7 days to 28 days shrinkage. In this examplefor moist cured concrete, the concrete is assumed to havebeen cured for 7 days. Shrinkage y,, factor as in 2.5.3below, is applicable to Eq. (2-9) for concrete moist curedduring a period other than 7 days.

2.53 Initid moist curingFor shrinkage of concrete moist cured during a period

of time other than 7 days, use the Shrinkage y,, factorin Table 2.5.3. This factor can be used to estimate differ-ential shrinkage in composite beams, for example.

Linear interpolation may be used between the valuesin Table 2.5.3.

2.5.4 Ambient relative humkiityFor ambient relative humidity greater than 40 percent,

use Eqs. (2-14) throughage correction factors.’ ;L6

2-16) for the creep and shrink-J2

Creep y1 = 1.27 - 0.00671, for R > 40 (2-14)

Shrinkage y1 = 1.40 - 0.0101, for 40 s rl I 80(2-15)

= 3.00 - O.O30R, for 80 > R 5 100(2-16)

where R is relative humidity in percent. Representativevalues are shown in Table 2.5.4.

The average value suggested for R = 40 percent isWU = 780 x 10” in/in. (m/m) in both Eqs. (2-9) and(2-10). From Eq. (2-15) of Table 2.5.4, for A = 70 per-cent, (qJu = 0.70(780x 10”) = 546 x 10” i&m. (m/m),for example. For lower than 40 percent ambient relativehumidity, values higher than 1.0 shall be used for Creepy1 and Shrinkage yr.

2.5.5 Average thickness of member other than 6 in. (l50mm) or volume-su@ke ratio other than 1.5 in. (38 mm)

The member size effects on concrete creep and shrink-age is basically two-fold. First, it influences the time-ratio(see Equations 2-6,2-7,2-8,2-9,2-10 and 2-35). Second-ly, it also affects the ultimate creep coefficient, vW andthe ultimate shrinkage strain, (Q),.

TWO methods are offered for estimating the effect of

member size on v,, and (cd,,. The average-thicknessmethod tends to compute correction factor values thatare higher, as compared to the volume-surface ratiomethod,” since Creep yA = Creep y, = 1.00 for h = 6in. (150 mm) and v/r = 1.5 in. (38 mm), respectiveIy; thatis, when h = 44s.

2.5.!5.a Average-thickness methodThe method of treating the effect of member sixe in

terms of the average thickness is based on informationfrom References 3, 6, 7, 23 and 61.

For average thickness of member less than 6 in. (150mm), use the factors ’

6ven in Table 2.5.5.1. These cor-

respond to the CEB values for small members. Foraverage thickness of members greater than 6 in. (150mm) and up to about 12 to 15 in. (300 to 380 mm), useEqs. (2-17) to (2-18) through (2-20).

During the first year after loading:

Cr=p rh = 1.14-0.023 h, G-17)

For ultimate values:

Creep yII = 1.10-0.017 h, (2-18)

During the first year of drying:

Shrinkage y,, = 1.23-0.038 h, (2-19)

For ultimate values:

Shrinkage y,, = 1.17-0.029 h, (2-20)

where h is the average thickness in inches of the part ofthe member under consideration.

During the first year after loading:

Creep yh = 1.14-0.00092 h, (2-17a)

For ultimate values:

Creep oh = 1.10-0.00067 h, (2-18a)

During the first year after loading:

Shrinkage y,, = 1.23-0.00015 h, (2-19a)

For ultimate values:

Shrinkage y,, = 1.17-0.00114 h, (2-20a)

where h is in mm.Representative values are shown irL Table 2.5.5.1.

2.SJ.b Volume-swjke ratio methodThe volume-surface ratio equations (2-21) and (2-22)

were adapted from Reference 23.

Creep yW = %[1+1.13 exp(-0.54 v/s)] (2-21)

Shrinkage yW = 1.2 exp(-O.12 v/s) (2-22)

where v/s is the volume-surface ratio of the member ininches.

Creep y, = 4;[1+1.13 exp(-O.O213 v/s)] (2-21a)

Shrinkage yW = 1.2 exp(-O.O0472 v/s) (2-22a)

where v/s in mm.Representative values are shown in Table 2.5.5.2.However, for either method y& should not be taken

less than 0.2. Also, use ysl (QJ,, 2 100 x 10” in/in.,(m/m) if concrete is under seasonal wetting and dryingcycles and yscl (Q)” 2 150 x 10” in/in. (m/m) if concreteis under sustained drying conditions.

2.56 Temperature other than 70 F (21 C)Temperature is the second major environmental factor

in creep and shrinkage. This effect is usually consideredto be less important than relative humidity since in moststructures the range of operating temperatures is srnaRaand high temperatures seldom affect the structuresduring long periods of time.

The effect of temperature changes on concrete creep@and shrinkage is basically two-fold. First, they directIyinfluence the time ratio rate. Second, they also affect therate of aging of the concrete, i.e. the change of materialproperties due to progress of cement hydration. At 122F (50 C), creep strain is approximately two to three timesas great as at 68-75 F (19-24 C). From 122 to 212 F (50to 100 C) creep strain continues to increase with tem-perature, reaching four to six times that experienced atroom temperatures. Some studies have indicated an ap-parent creep rate maximum occurs between 122 and 176F (50 and 80 C).M There is little data establishing creeprates above 212 F (100 C). Additional information ontemperature effect on creep may be found in References68,84, and 85.

2.6-Correction factors for concrete compositionEquations (2-23) through (2-30) are recommended for

use in obtaining correction factors for the effect ofslump, percent of fine aggregate, cement and air content.It should be noted that for slump less than 5 in. (130mm), fine aggregate percent between 40-60 percent,cement content of 470 to 750 lbs. per yd3 (279 to 445kg/m3) and air content less than 8 percent, these factorsare approximately equal to 1.0.

These correction factors shall be used only in con-nection with the average values suggested for v,, = 2.35and (Q,, = 780 x 10” in/in. (m/m). As recommended in2.4, these average values for vu and (e;rS, should be usedonly in the absence of specific creep and shrinkage datafor local aggregates and conditions determined in accord-ance with ASTM C 512.

If shrinkage is known for local aggregates and con-ditions, Eq. (2-31), as discussed in 2.6.5, is recommended.

The principal disadvantage of the concrete compo-sition correction factors is that concrete mix charac-teristics are unknown at the design stage and have to beestimated. Since these correction factors are normally notexcessive and tend to offset each other, in most cases,they may be neglected for design purposes.

2.6.1 Slump

Creep ys = 0.82 + 0.067s (2-W

Shrinkage ys = 0.89 + 0.041s (2-24)

where s is the observed slump in inches. For slump inmm use:

Creep Y, = 0.82 + 0.00264s (2-23a)

Shrinkage ys = 0.89 + 0.00161s (2-24a)

2.6.2 Fine aggregate percentage

Creep Y# = 0.88 + 0.0024$ (2-25)

For $ I 50 percent

Shrinkage y, = 0.30 + 0.014$ (2-26)

For $ > 50 percent

Shrinkage = 0.90 + 0.002g (2-27)

where $ is the ratio of the fine aggregate to total aggre-gate by weight expressed as percentage.

2.63 Cement contentCement content has a negligiile effect on creep co-

efficient. An increase in cement content causes a reducedcreep strain if water content is kept constant; however,data indicate that a proportional increase in modulus ofelasticity accompanies an increase in cement content.

If cement content is increased and water-cement ratiois kept constant, slump and creep will increase and Eq.(2-23) applies also.

Shrinkage y, = 0.75 + 0.00036c (2-28)

where c is the cement content in pounds per cubic yard.For cement content in Kg/m3, use:

Shrinkage yC = 0.75 + 0.00061~ (2-28a)

2.6.4 Air content

Creep ra = 0.46 + 0.09qbut not less than 1.0 (2-29)

Shrinkage y= = 0.95 + 0.008~~ (2-30)

where a’ is the air content in percent.

2.65 Shrinkage ratio of concretes with equivalent pastequa@’

Shrinkage strain is primarily a function of the shrink-age characteristics of the cement paste and of the ag-gregate volume concentration. If the shrinkage strain ofa given mix has been determined, the ratio of shrinkagestrain of two mixes (eJI/(e&, with different content ofpaste but with equivalent paste quality is given in Eq.(2-31).

Wrl_ 1 - (vJy)kh L 1 - (vy

(2-31)

where vl and v2 are the total aggregate solid volumes perunit volume of concrete for each one of the mixes.

2.7-ExampleFind the creep coefficient and shrinkage strains at 28,

90, 180, and 365 days after the application of the load,assuming that the following information is known: 7 daysmoist cured concrete, age of loading t(, = 28 days, 70percent ambient relative humidity, shrinkage consideredfrom 7 days, average thickness of member 8 in. (200mm), 2.5 in. slump (63 mm), 60 percent fine aggregate,752 Ibs. of cement per yd3 (446 Kg/m3), and 7 percent aircontent.’ Also, find the differential shrinkage strain,(eJIE for the period starting at 28 days after the appli-cation of the load, tta = 56 days.

The applicable correction factors are summarized inTable 2.7.1. Therefore:

v, = (2.35)(0.710) = 1.67

(es/Ju = (780 x 10”)(0.68) = 530 x 10”

The results from the use of Eqs. (2-8) and (2-9) orTable 2.4.1 are shown in Table 2.7.2.

Notice that if correction factors for the concretecomposition are ignored for v, and (es,,),, they will be 10and 4 percent smaller, respectively.

2.8-Other methods for predictions of creep and shrink-age

Other methods for prediction of creep and shrinkageare discussed in Reference 61, 68, 86, 87, 89, 93, 94, 95,97, and 98. Methods in References 97 and 98 subdividecreep strain into delayed elastic strain and plastic flow(two-component creep model). References 88,89,92,99,100, 102, and 104 discuss the conceptual differences be-tween the current approaches to the formulation of thecreep laws. However, in dealing with any method, it isimportant to recall what is discussed in Sections 1.2 and2.1 of this report.

2.8.1 Remark on refined creep fomulas needed forspecial structure~3*94,95

The preceding formulation represents a compromisebetween accuracy and generality of application. More ac-curate formulas are possible but they are inevitably notas general.

The time curve of creep given by Rq. (2-8) exhibits adecline of slope in log-t scale for long times. This prop-erty is correct for structures which are allowed to losetheir moisture and have cross sections which are not toomassive (6 to 12 in., 150 to 300 mm). Structures whichare insulated, or submerged in water, or are so massivethey cannot lose much of their moisture during theirlifetime, exhibit creep curves whose slope in log-r scale isnot decreasing at end, but steadily increasing. Forexample, if Eq. (2-8) were used for extrapolating short-time creep data for a nuclear reactor containment intolong times, the long-term creep values would be seriouslyunderestimated, posstbly by as much as 50 percent asshown in Fig. 3 of Ref. 81.

It has been found that creep without moisture ex-change (basic creep) for any loadin

Qdescribed by Equation (2-33).86mgo”39age tt, is betterThis is called the

double power law.In Eq. (2-33) #1 is a constant, and strain c is the sum

of the instantaneous strain and creep strain caused byunit stress.

where Z/E,, is a constant which indicates the lefthandasymptote of the creep curve when plotted in log t-scale(time t = 0 is at - oo in this plot). The asymptotic valuel/E, is beyond the range of validity of Eq. (2-33) andshould not be confused with elastic modulus. Suitablevalues of constants are $I = 0.97~~ and l/E, = 0.84/E,,,being EC, the modulus of concrete which does not under-go drying. With these values, Eq. (2-33) and Eq. (2-8)give the same creep for rt,, = 28 days, t = 10,000 daysand 100 percent relative humidity (yl = 0.6), all othercorrection factors being taken as one.

Eq. (2-33) has further the advantage that it describesnot only the creep curves with their age dependence, butalso the age dependence of the elastic modulus EC, inabsence of drying. EC, is given by l = Z/E,, for f = 0.001day, that is:

Eq. (2-33) also yields the values of the dynamic modu-lus, which is given by Q = lfE,+, when t = 10” days issubstituted. Since three constants are necessary to de-scribe the age dependence of elastic modulus (E,, $, and%), only one additional constant (i.e., V,) is needed todescribe creep.

In case of drying, more accurate, but also more com-plicated, formulas may be obtained94 if the effect of crosssection size is expressed in terms of the shrinkage half-time, as given in Eq. (2-35) for the age r, at which con-crete drying begins.

(2-35)

where:

characteristic thickness of the cross section,or twice the volume-surface ratio2 v/s ill mm)

Drying diffusivity of the concrete (approx.10 mm/day if measurements are unavail-able)

age dependence coefficient

c&(0.05 + +/icqiJ

f - 12, if C’ < 7, set C’ = 7if C, > 21, set C, = 21

coefficient depending on the shape of crosssection, that is:

1.00 for an infinite long slab1.15 for an infinite long cylinder1.25 for an infinite long square prism1.30 for a sphere1.55 for a cube

temperature coefficient

concrete temperature in kelvin

reference temperature in kelvin

water content in kg/m3

By replacing t in Eq. (2-9) rich,,, shrinkage is expressedwithout the need for the correction factor for size in Sec-tion 2.5.5.

The effect of drying on creep may then be expressedby adding two shrinkage-like functions vd and vP to thedouble power law for unit stress.” Function vd expressesthe additional creep during drying and function vP, beingnegative, expresses the decrease of creep by loading afteran initial drying. The increase of creep during dryingarises about ten times slower than does shrinkage and sofunction vd is similar to shrinkage curve in Eq. (2-9) witht replaced by 0.1 r/r& in Eq. (2-8).

This automatically accounts also for the size effect,without the need for any size correction factor. The de-crease of creep rate due to drying manifests itself onlyvery late, after the end of moisture loss. This is apparentfrom the fact that function Ts,, is similar to shrinkagecurve in Eq. (2-9) with t replaced by 0.01 t/75/1. Roth v,,and vP include multiplicative correction factors for rela-tive humidity, which are zero at 100 percent, and func-tion vd further includes a factor depending on the timelag from the beginning of drying exposure to the begin-ning of loading.2.GThermal expansion coeflicient of concrete

2.9.1 Factors affecting the qansion coefficientThe main factors affecting the value of the thermal

coefficient of a concrete are the type and amount ofaggregate and the moisture content. Other factors suchas mix proportions, cement type and age influence itsmagnitude to a lesser extent.

The thermal coefficient of expansion of concrete usu-ally reflects the weighted average of the various constitu-ents. Since the total aggregate content in hardened con-crete varies from 65 to SO percent of its volume, and theelastic modulus of aggregate is generally fiie times thatof the hardened cement component, the rock expansiondominates in determining the expansion of the compositeconcrete. Hence, for normal weight concrete with asteady water content (degree of saturation), the thermalcoefficient of expansion for concrete can be regarded asdirectly proportional to that of the aggregate, modifiedto a limited extent by the higher expansion behavior ofhardened cement.

Temperature changes affect concrete water content,environment relative humidity and consequently concretecreep and shrinkage as discussed in Section 2.5.6. Ifcreep and shrinkage response to temperature changes areignored and if complete histories for concrete water con-tent, temperature and loading are not considered, theactual response to temperature changes may drasticallydiffer from the predicted one.”

2.9.2 Predktion of thermal expansion coe@ientThe thermal coefficients of expansion determined

when using testing methods in ASTM C 531 and CRD 39correspond to the oven-dry condition and the saturatedconditions, respectively. Air-dried concrete has a highercoefficient than the oven-dry or saturated concrete,therefore, experimental values shall be corrected for theexpected degree of saturation of the concrete member.Values of emc in Table 2.9.1 may be used as correctionsto the coefficients determined from saturated concretesamples. In the absence of specific data from localmaterials and environmental conditions, the values givenby Eq. (2-32) for the thermal coefficient of expansion erl,may be used.76 Eq. (2-32) assumes that the thermal co-efficient of expansion is linear within a temperaturechange over the range of 32 to 140 F (0 to 60 C) andapplies only to a steady water content in the concrete.

For et,, in 10%

eth = =inc + 1.72 + 0.72 e, (2-32)

For et,, in 10d/Cz

where:

%I = %c + 3.1 + 0.72 e,, (2-32a)

cmc = the degree of saturation component as givenin Table 2.9.1

1.72 = the hydrated cement past component (3.1)

%I = the average thermal coefficient of the totalaggregate as given in Table 2.9.2

If thermal expansion of the sand differs markedly fromthat of the coarse aggregate, the weighted average bysolid volume of the thermal coefficients of the sand andcoarse aggregate shall be used.

A wide variation in the thermal expansion of the ag-gregate and related concrete can occur within a rockgroup. As an illustration, Table 2.9.3 summarizes therange of measured values for each rock group in theresearch data cited in Reference 76.

For ordinary thermal stress calculations, when the typeof aggregate and concrete degree of saturation areunknown and an average thermal coefficient is desired,eh = 5.5 x 10dlF (e* = 20.0 x 10d/CJ may be sufficient.However, in estimating the range of thermal movements(e.g., highways, bridges, etc.), the use of lower and upperbound values such as 4.7 x 10d/F and 6.5 x 10d/F (8.5 xlO-6/C and 11.7 x 10%) would be more appropriate.

2.1~Standards cited in this reportStandards of the American society for Testing and

Materials (ASTM) referenced in this report are listedbelow with their serial designation:

ASTM A 416

ASTM A 421

ASTM C 33

AsTMc39

ASTMC78

AC1 C 192

ASTM C 469

ASTM C 511

ASTM C 512

ASTM c 531

“Standard Specification for UncoatedSeven-Wire Stress-Relieved Strand forPrestressed Concrete”“Standard Specification for UncoatedStress-Relieved Wire for PrestressedConcrete”“Standard Specifications for ConcreteAggregates”“Standard Test Method for CompressiveStrength of Cylindrical Concrete Speci-mens”“Standard Test Method for FlexuralStrength of Concrete (Using SimpleBeam with Third-Point Loading)“Standard Method of Making AndCuring Concrete Test Specimens in theLaboratory”“Standard Method for Static Modulus ofElasticity and Poisson’s Ratio of Con-crete in Compression”“Standard Specification for Moist Cabi-nets and Rooms Used in the Testing Hy-draulic Cements and Concretes”“Standard Test Method for Creep ofConcrete in Compression”“Standard Method for Securing, Pre-paring, and Testing Specimens fromHardened Lightweight Insulating Con-crete for Compressive Strength”

A!J’lM E 328 “Standard Recommended Practice forStress-Relaxation Tests for Materials andStructures”

The following standard of the U.S. hy Corps of En-gineers (CRD) is referred in Section 2.9 of this report:

CRD C39 “Method of Test for Coefficient ofLinear Thermal Expansion of Concrete”

CHAPTER 3-FACTORS AFFWTING TEESTRUCTURAL RESPONSE-ASSUlKITIONS AND

METHODS OF ANALYSIS

3.1-IntroductionPrediction of the structural response of reinforced

concrete structures to time-dependent concrete volumechanges is complicated by:

a) The inherent nonelastic properties of the con-crete

b) The continuous redistriiution of stress9 The nonhomogeneous nature of concrete proper-

ties caused by the stages of constructiond) The effect of cracking on deflectione) The effect of external restraintsf) The effect of the reinforcement and/or pre-

stressing steelg ) ‘lhe interaction between the above factors and

their dependence on past histories of loadings,water content and temperature

‘The complexity of the problem requires some simplify-ing assumptions and reliance on empirical observations.

3.2-Principal facts and assumptions32.1 Principal facts

4

b)

Cl

4

Each loading change produces a resulting defor-mation component continuous for an infiniteperiod of time”Applied loads in homogeneous statically indeter-minate structures cause no time-dependentchange in stress and all deformations are pro-portional to creep coefficient v, as long as thesupport conditions remain unchanged”The secondary, statically indetermined momentsdue to prestressing are affected in the sameproportion as prestressing force by time-depen-dent deformations, which is a relatively smalleffect that is usually neglectedIn a great many cases and except when instabilityis a factor, time-dependent strains due to actualloads do not significantly affect the load capacityof a member. Failure is controlled by very large

strains that develop at collapse, regardless of pre-vious loading history.” In these cases, time-dependent strains only affect the structure ser-viceability. When instability is a factor, creep in-crement of the eccentricity in beamcolumusunder sustained load will decrease the membercapacity with time

e) Change in concrete properties with age, such aselastic, creep and shrinkage deformations, mustbe taken into account

33.2 Assumph

4

b)

4

4

4

fl

g)

Concrete members including their creep, shrink-age and thermal properties, are considered ho-mogeneousCreep, shrinkage and elastic strains are mutuallyadditive and independentFor stresses less than about 40 to 50 percent ofthe concrete strength, creep strains are assumedto be approximately proportional to the sustainedstress and obey the principle of superposition ofstrain histories.70S0

However, tests in References 105 and 106have shown the nonlinearity of creep strain withstress can start at stresses as low as 30 to 35 per-cent of the concrete strength. Also, strain super-position is only a first approximation because theindividual response histories affect each other ascan be seen with recovery curves after unloadingShrinkage and thermal strains are linearlydistributed over the depth of the cross section.lhis assumption is acceptable for thin andmoderate sections, respectively, but may result inerror for thick sectionsThe complex dependence of strain upon the pasthistories of water content and temperature isneglected for the purpose of analyzing ordinatystructuresRestraint by reinforcement and/or prestressingsteel is accounted for in the average sense with-out considering any gradual stress transferbetween reinforcement and concreteThe creep time-ratio for various environmenthumidity conditions and various sixes and shapesof cross section are assumed to have the sameshape

Even with these simplifications, the theoretically exactanalysis of creep effects according to the assumptionsstateda is still relatively complicated. However, more ac-curate analysis is not really necessary in most instances,except special structures, such as nuclear reactor vessels,bridges or shells of record spans, or special ocean struc-tures. Therefore, simplified methods of analysisMbO arebeing used in conjunction with empirical methods to ac-count for the effects of cracking and reinforcementrestraint.

33-!Simplified methods of creep analysisIn choosing the method of analysis, two kinds of cases

are distinguished.33.1 Cases in which the gradual time change of stress

due to creep and shrinkage in small and has little effectThis usually occurs in long-time deflection and pre-

stress loss calculations. In such cases the creep strain isaccounted for with sufficient accuracy by an elastic analy-sis in which the actual concrete modulus at the time ofinitial loading, is replaced with the so-called effectivemodulus as given by Eq. (3-l).

4 = EJ(l + VJ (3-l)This approach is implied in Chapter 4. To check if the

assumption of small stress change is true, the stresscomputed on the basis of E,i should be compared withthe stress computed on the basis of E,.

33.2 Cases in which the gradual time change of stressdue to creep and shrinkage iv significant

In such cases, the age-adjusted effective modulusmethod67*68V69 is recommended as discussed in Chapter 5.

3.AEffect of cracking in reinforced and prestressedmembers

To include the effect of cracking in the determinationof an effective moment of inertia for reinforced beamsand one-way slabs, Eq. (3-2)toz* has been adopted bythe AC1 Building Code (AC1 318).27

4 = PfJM,,,,,J3 I, + [l - (MJN,,d31Z,, (3-2)

where M, is the cracking moment, M,,,, denotes themaximum moment at the stage for which deflection isbeing computed, Is is the moment of inertia of the grosssection neglecting the steel and I, is the moment ofinertia of the cracked transformed section.

Eq. (3-2) applied only when M,,,, 2 MC,; otherwise,I, = Is

I, m Eq. (3-2) has limits of I, and I,,, and thusprovides a transition expression between the two casesgiven in the AC1 318 Code.‘2f7 The moment of inertiaZ, of the untracked transformed section might be moreaccurately used instead of the moment inertia of thegross section I in Eq. (3-2), especially for heavilyreinforced mem$e rs and lightweight concrete members(low EC and hence high modular ratio E,/E,,).

Eq. (3-2) has also been shown2 to apply in thedeflection calculations of cracked prestressed beams.

For numerical analysis, in which the beam is dividedinto segments or finite elements, it has been shownz thatI, values at individual sections can be determined bymodifying Eq. (3-2). The power of 3 is changed to 4 andthe moment ratio in both terms is changed to M,,/M,where M is the moment at each section. Such a numeri-cal procedure was used in the development of Eq.(3-2)?

The above cracking moment is given in Eqs. (3-3) and

(3-9

For reinforced members:

Mcr = fi Idy P-3)

For noncomposite prestressed members:

(ML,), = Fe + (FIJIAs y, + ct; I&y, - MD (3-4)

The cracking moment for unshored and shored com-posite prestressed beams is given in Eq. (41) and (42) ofReference 63.

Equation (3-2) refers to an average effective Z for thevariable cracking along the span, or between the in-flection points of continuous beams. For continuousmembers (at one or both ends), a numerical proceduremay be needed although the use of an average of thepositive and negative moment region values from Eq.(3-2) as suggested in Section 9.5.2.4 of Reference 27should yield satisfactory results in most cases. For spanswhich have both ends continuous, an effective averagemoment of inertia I, is obtained by computing an aver-age for the end region values, I#1 and I,, and then av-eraging that result with the positive moment region valueobtained for Eq. (3-2) as shown in Eq. (3-5).

I, + I& + I&2 + ‘epw (3-5)In other cases, a weighted average related to the

positive and negative moments may be preferable. Forexample, the weighted avera e moment of inertia I,would be given by Eq. (3-6). 7!

where, Z6P is the effective moment of inertia for the posi-tive zone of the beam and/l is a positive integer that maybe equal to unity for simplicity or equal to two, three orlarger for a modest increase in accuracy.

For a span with one end continuous, the (I,I + I,,)/2in Eqs. (3-5) and (3-6) shall be substituted for I for thenegative end zone.

For a flat late and two way slab interior panels, it hasbeen shownB that Eq. (3-2) can be used along with anaverage of the positive and negative moment regionvalues as follows:

Flat plate-both positive and negative values for thelong direction column strip.

‘&o way slabs-both positive and negative values forthe short direction middle strip.

The center of interior panels normally remains un-cracked in common designs of these slabs.

For the effect of repeated load cycles on crackingrange, see Reference 63.

3sEt&ctive compmKdon steel in fkxuml membersCompression steel in reinforced flexural members and

nontensioned steel in prestressed flexnral members tendto offset the movement of the neutral axis caused bycreep. The net movement of the neutral axis is theresultant of two movements. A movement towards thetensile reinforcement (increasing the concrete com-pression zone, which results in a reduction in themoment arm). This movement is caused by the effect ofcreep plus a reduction in the compression zone due tothe progressive cracking in the tensile zone.

The second movement is produced by the increase insteel strains due to the reduction of the internal momentarm (plus the small effect, if any, of repeated live loadcycles). As cracking progresses, steel strains increasefurther and reduce the moment arm.

The reduced creep effect resulting from the movementof the neutral axis and the presence of compression steelin reinforced members A,‘, and the inclusion of non-tensioned high strength or mild steel (as specified below)in prestressed members is given by the reduction factorr, in Eqs. (3-7) and (3-9).

The approximate effect of progressive cracking undercreep loading and repeated load cycles is also included inthe factor &. Eq. (3-8) refers to the combined creep andshrinkage effect in reinforced members.

For reinforced flexural members, creep effect only:10

t, = 0.85 - 0.45 (A,‘IA,), but not less than 0.40 (3-7)

For reinforced flexural members, creep and shrinkageeffect:=s

<, = I - 0.60 (A,‘IA,), but not less than 0.30

For prestressed flexural members?m

WV

E, = 1111 + A;IAJ o-9

Approximately the same results are obtained in Eqs.(3-7), (3-Q, and (3-9) as shown in Table 3.5.1. It isassumed in Eq. (3-9) that the nontensioned steel and theprestressed steel are on the same side of the section cen-troid and that the eccentricities of the two steels are ap-proximately the same. See Reference 28 when the eccen-tricities are substantially different.

Eqs. (3-8) and (4-3) are used in AC1 318n with atime-dependent factor for both creep and shrinkage, TV= 2.0. As the ratio, A,‘/A,, incremes, these two sets offactors approach the same value, since shrinkage warpingis negligible when the compression reinforcement is high.

The effects of creep plus shrinkage are arbitrarilylumped together in Eq. (3-8).

In Reference 74, Branson notes that Eq. (3-8), as usedin AC1 318*’ is likely to overestimate the effect of the

compression steel in restraining time-dependent deflec-tions of members with low steel percentage (e.g. slabs)and recommends the alternate Eq. (3-10).

E, Tu = TJ[l + 50 $1 (3-10)

where r, TV is a long time deflection multiplier of theinitial deflection and p’ is the compressive steel ratioA,‘lbd. He further suggests that a factor, T,, = 2.5 forbeams and T” = 3.0 for slabs, rather than 2.0, would giveimproved results.

The calculation of creep deflection as [, I, times theinitial deflection Up yields the same results as that ob-tained using the “reduced or sustained modulus of elast-icity, E,, method,” provided the initial or short-timemodular ratio, n, (at the time of loading) and the trans-formed section properties are used. This can be seenfrom the fact that E, used for computing the initialdeflection, is replaced by E, as given by Eq. (3-l), forcomputing the initial plus the creep deflection. Thefactor 1.0 in Eq. (3-l) corresponds to the initial de-flection. Except for the calculation of I in the sustainedmodulus method (when using or not using an increasedmodular ratio) and I,lf, in the effective section method,the two methods are the same for computing long-timedeflections, exclusive of shrinkage warping.

The reduction factor <,, for creep only (not creep andshrinkage) in Eq. (3-7) is suggested as a means of takinginto account the effect of compression steel and the off-setting effects of the neutral axis movement due to creepas shown in Figure 3 of Ref. 10. These offsetting effectsappear normally to result in a movement of the neutralaxis toward the tensile reinforcement such that:

(3-11)

in which E, from Eq. 3-7 is less than unity. (See Table35.1). Subscripts cp and i refer to the creep and initialstrains, curvatures 4, and deflections a, respectively.

The use of the long-time modular ratio, n, = n(Z +v,), in computin the transformed section properties hasalso been show&* to accomplish these purposes and toprovide satisfactory results in deflection calculations.

In all appropriate equations herein, v,, vu, T,, TV, arerepla=d by Er v,, E, vu, E, TV, E, 7” resp=+ely, whenthese effects are to be included.

3.6-Deflections due to warping3.6.1 Warping due to shrinkugeDeflections due to warping are frequently ignored in

design calculation, when the effects of creep and warpingare arbitrarily lumped together.*’ For thin members, suchas canopies and thin slabs, it may be desirable to con-sider warping effects separately.

For the case in which the reinforcement and eccen-tricity are constant along the span and the same in thepositive and negative moment regions of continuous

beams, shrinkage deflections for uniform beams arecomputed by Eq. (3-12).

a& = tw (b,A e2 (3-12)

where & is a deflection coefficient defined in Table 4.2.1for different boundary conditions, and &i is the curva-ture due to shrinkage warping. For more practical cases,some satisfactory compromise can usually be made withregard to variations in steel content and eccentricity, andfor nonuniform temperature effects.

3.6.2 Methods of computing shrinkage curvatureThree methods for computing shrinkage curvature

were compared in References 10 and 25 with e“p

eri-mental data: the equivalent tensile force method,313 3a27Miller’s method% and an empirical method based onMiller’s a roach extended to include doubly reinforcedbeams.” PThe agreement between computed and mea-sured results was reasonably good for all three of themethods.

The equivalent tensile force method (a fictitious elasticanalysis), as modified in References 10 and 25 using EC/2and the gross section properties for better results, isgiven by Eq. (3-13).

(3-13)

where T, = (A, + A,‘)gross section.

l shEs, and es and Is refer to the

Miller’s method% assumes that the extreme fiber ofthe beam furthest from the tension steel (method refersto singly reinforced members only) shrinks the sameamount as the free shrinkage of the concrete, Q. Fol-lowing this assumption, the curvature of the member isgiven by Eq. (3-14).

4L-9 %h 1 5-=- - -

sh = d d I 1sh(3-14)

where es is the steel strain due to shrinkage. Miller sug-gested empirical values of (c&h) = 0.1 for heavily rein-forced members and 0.3 for moderately reinforced mem-bers.

The empirical method represents a modification ofMiller’s method. The curvature of a member is given byEqs. (3-15) and (3-16) which are applicable to both singlyand doubly reinforced members. The steel percentage inthese equations are expressed in percent (p = 3 for 3percent steel, for example).

For (p - p’) I 3.0 percent:

4sh = coe7)

For (p - p’) > 3.0 percent:

cp sh = %hlh (3-16)

where h is the overall thickness of the section.For singly reinforced members, p’ = 0, and Eq. (3-15)

reduces to Eq. (3-17).

4& = (0.7) 2 PIP P-17)

which results in:

4 sh = 0.56 (es&h, w h e n p’ = 0.5 percat0.70 1.00.88 2.01.01 3.0

Eqs. (3-H), (3-16), and (3-17) were adapted fromMiller’s approach. For example, his method results in thefollowing expression for singly reinforced members:

4 s h = 0.7 l sh/d for “moderately” reinforced beams

4s h = 0.9 l sh/d for “heavily” reinforced beams

which approximately correspond to p = 1.0 and p = 2.0in Eq. 3-17.

The use of the more convenient thickness, h, insteadof the effective depth, d, in Eqs. (3-15), (3-16), and (3-17) was found to provide closer agreement with the testdata.

3.6.3 Warping due to temperature changeSince concrete and steel reinforcement have similar

thermal coefficients of expansion (i.e., 4.7 to 6.5 x lO”/Ffor concrete and 6.5 x 106/F for steel), the stresses pro-duced by normal temperature range are usually negli-gible.

When the temperature change is constant along withthe span, thermal deflections for uniform beams aregiven by Eq. (3-18).

where & is the deflection coefficient (Table 4.2.1). Thecurvature +I!, due to temperature warping is given by Eq.(3-19).

4th = (et), t,Jlh (3-19)

where et,, is the thermal coefficient of expansion and th isthe difference in temperature across the overall thicknessh .

The values of v,, es,,, and erk usually correspond tosteady state conditions. A sustained nonuniform changein temperature will influence creep and shrinkage. As aresult, significant redistribution of stresses in staticallyindeterminate structures may occur to such an extent thatthe thermal effects caused by heating may be completelynullified. A nonuniform temperature reversal may causea stress reversal.79

3.7-Interdependency between sttel relaxation, creep andshrinkage of concrete

The loss of stress in a wire or strand that occurs atconstant strain is the intrinsic relaxation CfrLS,. Stress lossdue to steel relaxation as shown in Table 3.7.1 and assupplied by the steel manufacturers (ASTh4 designationsA 416, A 421, and E 328) are examples of the intrinsicrelaxation. In actual prestressed concrete members, aconstant strain condition does not exist and the use ofthe intrinsic relaxation kiss will result in an overest-imation of the relaxation loss. The use of ct’,), and VW)),,as in Table 4.4.1.3, is a good approximation for most de-sign calculations because of the approximate nature ofcreep and shrinkage calculations. In Reference 78, arelaxation reduction factor, #, is recommended to ac-count for conditions different than the constant strain.Values of $ in Table 3.7.2 are entered by the&‘fpl ratioand the parameter 0 given in Eq. (3-20).

where (A), is the total prestress loss in percent for a timeperiod (tl - r) excluding the instantaneous loss at transfer.

Prestress losses due to steel relaxation and concretecreep and shrinkage are inter-dependent and also time-dependent. lo3 To account for changes of these effectswith time, a step-by-step procedure in which the timeinterval increases with age of the concrete is recom-mended in Ref. 78. Differential shrinkage from the timecuring stops until the time the concrete is prestressedshould be deducted from the total calculated shrinkagefor post-tensioned construction. It is recommended thata minimum of four time intervals be used as shown inTable 3.7.3.78

When significant changes in loading are expected, timeintervals other than those recommended should be used.It is neither necessary nor always desirable to assumethat the design live load is continually present. The fourtime intervals in Table 3.7.3 are recommended for mini-mum noncomputerixed calculations.

CHAPTER 4-RESPONSE OF !3TRUCTURES INWHICH TIME-CHANGE OF STRESSES DUE TO

CREEP, SHRINKAGE AND TEMPERATURE ISNEGLIGIBLE

4.1-Introduction4.1.1 AssumptionsFor most cases of long-time deflection and loss of pre-

stress in statically determinate structures, the gradualtime-change of stresses due to creep, shrinkage and tem-perature is negligible; only time changes of strains aresignificant. In some continuous structures, the effects ofcreep and shrinkage may be approximately lumped to-gether as discussed in this chapter. Shrinkage inducedtime-change of stresses in statically indeterminate struc-tures is discussed in Chapter 5.

While deflections and loss of prestress have essentiallyno effect on the ultimate capacity of reinforced and pre-stress members, significant over-prediction or under-prediction of losses can adversely affect such service-ability aspects as camber, deflection, cracking and con-nection performance.63 The procedures in this chapterare reviewed in detail in Reference 83.

4.13 Pmsentation of equationsIt should be noted that Eqs. (4-8) through (4-24) can

be greatly shortened by combining terms and substitutingthe approximate parameters given herein. These equa-tions are presented in the form of separate terms inorder to show the separate effects or contributions, suchas prestress force, dead load, creep, shrinkage, etc., thatoccur both before and after slab casting in compositeconstruction.

4.2-Deflections of reinforced concrete beam and slab43.1 Dejktion of noncomposite reinforced concrete

beams and one-way slabDeflections in general may be computed for uniformly

distributed loadings on prismatic members using Eq.(4933”

I(4-l)

where a,,, is the deflection at midspan (approximate maxi-mum deflection in unsymmetrical cases), and the mo-ments M,,,, MA, and MS, refer to the midspan and twoends respectively. This is a general equation in which theappropriate signs must be included for the moments, usu-ally (+) for M,,, and (-) for MA and iU&

When idealized end conditions can be assumed, it isconvenient to use the deflection equation in the form ofEq. (4-2), where Q and M are the deflection coefficientsgiven in Table 4.2.1 for the numerically maximum bend-ing moment. Eqs. (4-2) and (4-3), which describe an “1,- E,- rru or “Z, - f, - T” procedure for computing de-flections, are used in this chapter.

Short-time deflections:

Additional long-time deflections due to creep or creepplus shrinkage:

a, = t, V, cli or a, = (, 7 ai, (4-3)

when the creep and shrinkage effect is lumped together.Equations for <, M, EC,., Z,, 6, and v, are as given in

this report. The AC1 318 Code*‘specifies f, as in Eq. (3-8), but not less than 0.3 and an ultimate value of r = 2.0.

Since live load does not act in the absence of deadload, the following procedure must be used to determinethe various deflection components:

(4-4)

frequently (I,) for MD equals Is,

(ah = i,v,faJD (4-5)

a fictitious value

(4-6)

and then for live load,

fai)L = fai)D+L - (ai)D F (d-7)

The ACI-318 Codes l”’ refer to (a,)o + (ailL in cer-tain cases for example.

In general, the deflection of a noncomposite rein-forced concrete member at any time and including ulti-mate value in time is given by Eqs. (4-8) and (4-9)respectively.n

(1) (2) (3) (4)- - - -

af = fai)D + fat)D + ash + cai)L (d-8)

a, = [Eq. (4-8) except that vu and (& shall beused in lieu of v, and es,, when computingterms (2) and (3) respectively.] (4-9)

where:

Term (1) is the initial dead load deflection asgiven by Eq. (4-4)

Term (2) is the dead load creep deflection as givenby Eq. (4-S)

Term (3) is the deflection due to shrinkage warp-ing as given by Eq. (3-12)

Term (4) is the live load deflection as given by Eq.(d-7)

4.3-Deflection of composite precast reinforced beams inshored and unshared construction48.49.77

For composite beams, subscripts 1 and 2 are used torefer to the slab or the effect of the slab dead load andthe precast beam, respectively. The effect of compressionsteel in the beam (with use of &) should be neglectedwhen it is located near the neutral axis of the compositesection.

It is suggested that the 28-day moduli of elasticity forboth slab and precast beam concretes, and the gross I(neglecting steel and cracking), be used in computing thecomposite moment of inertia, Z,, in Eqs. (4-10) and(4-12), with the exception as noted in term (7) for liveload deflection. Note that shrinkage warping of the pre-cast beam is not computed separately in Eqs. (4-10) and(4-12).

43.1 Defiction of unshared composite beamsThe deflection of unshored composite beams at any

time and including ultimate values, is given by Eqs.(4-10) and (4-11) respectively.

(1) (2) (3)--

a, = (42 + V,faih + (VQ - V,) fail2 :

(4) (5) (6) (‘1h h

I2+ (ai)1 + V&a&l I, + ai3 + aL (4-10)

(1) (2)-rHh

&

4a” = (ai) + V,f& + fV,-VJ (ai)2 i

e

(4) (5) (6) ('1--A-

4+ (41 + VJai)l+ i + a8 + aL (4-11)

cwhere:

Term (1) is the initial dead load deflection of theprecast beam, (ai) = t M, tT2/ECi12. See Table 4.2.1 fort and A4 values. For computing I2 in Eq. (3-2), M,,,, re-fers to the precast beam dead load and MC, to the precastbeam.

Term (2) is the creep deflection of the precast beamup to the time of slab casting. v, is the creep coefficientof the precast beam concrete at the time of slab casting.Multiply vs and v,, by [, (from Eq. 3-8) for the effect ofcompression steel in the precast beam. Values of v,/v,,= vJv,, from Eq. (2-8) are given in Table 2.4.1.

Term (4) is the initial deflection of the precast beamunder slab dead load, (ai) = rMl e2/E,12. See Table4.2.1 for t and M values. For computing Z in Eq. (3-2),M,, refers to the precast beam plus slab dead load andM,, to the precast beam.

Term (3) is the creep deflection of the compositebeam for any period following slab casting due to theprecast beam dead load. v12 is the creep coefficient ofthe precast beam concrete at any time after slab casting.Multiply this term by f, (from Eq. 3-8) for the effect ofcompression steel in the precast beam. The expression,12/IC, modifies the initial value, in this case (aJ2, andaccounts for the effect of the composite section in re-straining additional creep curvature after slab casting.

Term (5) is the creep deflection of the compositebeam due to slab dead load. viz is the creep coefficientfor the slab loading, where the age of the precast beamconcrete at the time of slab casting is considered. Mul-tiply vIl and vu by i, (from Eq. 3-8) for the effect ofcompression steel in the precast beam. See Term (3) forcomment on Z2/Zc. v, is given by Eq. (2-13).

Term (6) is the deflection due to differential shrink-age. For simple spans, as = QY, e2f8 E-I,, where Q =SA,E,,/3. The factor 3 provides for the gradual increasein the shrinkage force from day 1, and also approximatesthe creep and varying stiffness effects.6*48 In the case of

continuous members, differential shrinkage produces sec-ondary moments (similar to the effect of prestressin but

$8opposite in sign, normally) that should be included.Term (7) is the live load deflection of the composite

beam, which should be computed in accordance with Eq.(4-7), using EC,. For computing I, in Eq. (3-Z), M,,,,refers to the precast beam plus slab dead load and thelive load, and A$, to the composite beam.

Additional information on deflection due to shrinkagewarping of composite reinforced concrete beams of un-shored construction is given by Eq. (2) in Ref. 77.

4.33 Lkjkction of shored composite beamsThe deflection of shored composite beams at any time

and including ultimate values is given by Eqs. (4-12) and(4-13), respectively.

a, = Eq. (4-lo), with Terms (4) and (5) modified asfollows. (4-12)

a” = Eq. (4-ll), except that the composite moment ofinertia is used in Term (4) to compute (ai)t, andthe ratio, I,/I,, is eliminated in Term (5). (4-13)

Term (4) is the initial deflection of the compositebeam under slab dead load, (ai) = f Mt e2/E,I,.

Term (5) is the creep deflection of the compositebeam under slab dead load, v,t(ai)t. The composite sec-tion effect is already included in Term (4).

4.Abss of pnetress and camber in noncomposite pre-stressed beams6~495863

4.4.1 Loss of prestres in prestresed concrete beamsLoss of prestress at any time and including ultimate

values, in percent of initial tensioning stress, is given byEqs. (4-14) and (4-15).

(1) (2)--

4 = I(nfc) + (nf,) v,U - 2) +

(3) (4), .

(1) (2)--

12, = [(rife) + (nfc,) vu (1 - &j +

(3) (4)\

(4-14)

(4-U)

Term (1) is the prestress loss due to elastic shortening,in which

Pi lp’ h&ef,= x+r,-I, and n is the modular ratio atthe time of prestress$g. Frequently F,, A and I areused as an approximation instead of F,, A,, %nd I,, kingFo = Fi(l - np). Only the fust two terms for f, apply atthe ends of simple beams. For continuous members, theeffect of secondary moments due to prestressing shouldalso be included. Suggested values for n in are given inTable 4.4.1.1.

Term (2) is the prestress loss due to the concretecreep. The expression, v, (1 - F,l2F& was used inReferences 50 and 53 to approximate the creep effectresulting from the variable stress history. Approximatevalues of F,IF, (in the form of FJF,, and FJF,,) v thissecondary effect as given in Table 4.4.1.2. To considerthe effect of nontensioned steel in the member, multiplyvl, v,,, (Q), and @,,,I, by E, (from Eq. 3-g).

Term (3) is the prestress loss due to shrinkage.M Theexpression, (Q,), E,, somewhat overestimates this loss.The denominator represents the stiffening effect of thesteel and the effect of concrete creep. Additional infor-mation on Term (3) is given in Ref. 63.

Term (4) is the prestress loss due to steel relaxation.Values of &) and cf,),, for wire and strand are given inTable 4.4.1.3,‘3 where t is the time after initial stressingin hours and f, is the 0.1 percent offset yield stress.Values in Table 4.4.1.3 are recommended for most designcalculations because they are consistent with the approximate nature of creep and shrinkage calculations.Relaxation of other types of steel should be based onmanufacturer’s recommendations supported by adequatetest data. For a more detailed analysis of the inter-dependency between steel relaxation, creep and shrink-age of concrete see Section 3.7 of this report.

4.4.2 Camber of noncomposite prestresed concretebeams

The camber at any time, and including ultimate values,is given by Eqs. (4-16) and (4-17) respectively. It is sug-gested that an average of the end and midspan loss beused for straight tendons and 1-pt. harping, and the mid-span loss for 2-pt harping.

(1) (2) (3)-.

a, = - (a& + (ai)D -I

F,- F + (’ -

0Gvf

I(ot)P,

(4) (3--

+ VtCai)D + q. (4-16)

[F,

a , = - (a,),. + (U&D - - F + (’ - Gp’) v 1”0

WF,

where:V-17)

Term (1) is the initial camber due to the initial pre-stress force after elastic loss, F,. See Table 4.4.2.1 forcommon cases of prestress moment diagrams with form-ulas for computing camber, (&,

Here, F, = Fi(l - nfclfsi), where f, is determined as inTerm (1) of Eq. (4-14). For continuous members, the ef-fect of secondary moments due to prestressing shouldalso be included.

Term (2) is the initial dead load deflection of theham, (a&D = rMe2/E,iI . I is used instead of Z, forpractical reasons. See Table l2.1 for 6 and M values.

Term (3) is the creep (time-dependent) camber of thebeam due to the prestress force. This expression includesthe effects of creep and loss of prestress; that is, thecreep effect under variable stress. F, refers to the totalloss at any time minus the elastic loss. It is noted that theterm, F,/F,, refers to the steel stress or force after elasticloss, and the prestress loss in percent, I as used herein,refers to the initial tensioning stress or force. The twoare related as:

and can be approximated by:

(4-18)

(4-18a)

Term (4) is the dead load creep deflection of thebeam. Multiply vt and vpI by 6 (from Eq. 3-9) for theeffect of compression steel (under dead load) in themember.

Term (5) is the live load deflection of the beam.Additional information on the effect of sustained loads

other than a composite slab or topping applied sometime after the transfer of prestress is given by Terms (6)and (7) in Eqs. (29) and (30) in Ref. 63.

4SLoss of pnstnss and camber of composite precastand pnstnsscd beams, unshorcd and shored construc-tionr5.49-S8.63.77

4.5.1 L.0.w of prestress of composite precast-beams andprestmed beams

The loss of prestress at any time and including ulti-mate values, in percent of initial tensioning stress, is

given by Eqs. (4-19) and (4-20) reqectively for unshoredand shored composite beams with both prestressed steeland nonprestressed steel.

(4) (9 (6),-.+ (~~)tE,l(l+npE~ + UJ, - (mf,) -

(7) (8)- -

(nf,) (vu - v,) (1 4 F.+F.2F)a i +c

(4) (5) (6).- b

&;A),, E,l(l + npE$ + K,), - (mf=) -

(7) (8)-e

(4-19)

(mf,)v, + - mf,d/ y WYc d

where:Term (1) is the prestress loss due to elastic shortening.

See Term (1) of Eq. (4-14) for the calculation off,.Term (2) is the prestress loss due to concrete creep up

to the time of slab casting. v, is the creep coefficient ofthe precast beam concrete at the time of slab casting. SeeTerm (2) of Eq. (4-14) for comments concerning the re-

duction factor, (2 - &). Multiply vs and vu by & (fromEq. 3-9) for the effect 8f nontensioned steel in the mem-ber. Values of v,lvU = v,/v,, from Eq. (2-8) are given inTable 2.4.1.

Term (3) is the prestress loss due to concrete creepfor any period following slab casting. v, is the creep co-efficient of the precast beam concrete at any time after

slab casting. The reduction factor, (l-v), with theincremental creep coefficient, (v12 - v,), &mates the

effect of creep under the variable prestress force thatoccurs after slab casting. Multiply this term by 6, (fromEq. 3-9) for the effect of nontensioned steel in the pre-cast beam. See Term (3) of Eq. (4-10) for oomment onI#,.

Term (4) is the prestress loss due to shrinkage. SeeTerm (3) of Eqs. (4-14) and (4-15) for comment.

Term (5) is the prestress loss due to steel relaxation.In this term t is time after initial stressing in hours. SeeTerm (4) of Eqs. (4-14) and (4-15) for comments.

Term (6) is the elastic prestress gain due to slab deadload, and m is the modular ratio at the time of slab cast-

W&&ing.f, = I , Mw refers to slab or slab plus dia-phragm dead foad; c and I refer to the precast beamsection properties for uns ored construction and thegcomposite section properties for shored construction.Suggested values for n and m are given in Table 4.4.1.1.

Term (7) is the prestress gain due to creep under slabdead load. vrl is the creep coefficient for the slab load-ing, where the age of the precast beam concrete at thetime of slab casting is considered. See Term (5) of Eq.(4-10) for comments on 6, and I& For shored con-struction, drop the term, Z,/Ic vus is given by Eq. (2-13).

Term (8) is the prestress gain due to differentialshrinkage, where fed = Qy,e,/l, is the concrete stress atthe steel c.g.s. and Q = (S Abt EC,)/3 in which A I

Pand

E,, refer to the cast in-place slab. See Notation or ad-ditional descriptions of terms. Since this effect results ina prestress gain, not loss, and is normally small, it mayusually be neglected.”

453 Camber of composile beams-precast beams pre-stressed unshared and shored construction

The camber at any time, including ultimate values, isgiven by Eqs. (4-21), (4-22), (4-U), and (4-24) for un-shored and shored composite beams, respectively. It issuggested that an average of the end and midspan loss ofprestress be used for straight tendons and 1-pt. harping,and the midspan loss for 2-pt. harping.6

It is suggested that the 28-day moduli of elasticity forboth slab and precast beam concretes be used. For thecomposite moment of inertia, Z, in Eqs. (4-21) through(4-24), use the gross section I’ except in Term (10) forthe live load deflection.

a) Uitshored construction

(1) (2) (3)---e

12 I2+(Q- VJ (ai)2 y + (ai)1 + vtl(ai)l I, + a8 + “L

(4-21)

(4)

q-y +(I - F) (V” - vJ](u,),, +0 c

+ V,(aih

4 4+ (vu-v,) (ai)27 + (ai) + vw(aih I, + a8 + “L

c

(4-22)

where:Term (1) See Term (1) of Eq. (4-16).Term (2) is the initial dead load deflection of the pre-

Cast beam, (ai) = Q M2e2/E,.I, See Term (2) of Eq. (4-16) for additional comments.

Term (3) is the creep (time-dependent) camber of thebeam, due to the prestress force, up to the time of slabcasting. See Term (3) of Eq. (4-16) and Terms (2) and(3) of Eq. (4-19) for additional comments.

Term (4) is the creep camber of the composite beam,due to the prestress force, for any period following slabcasting. See Term (3) of Eq. (4-16) and Terms (2) and(3) of Eq. (4-19) for additional comments.

Term (5) is the creep deflection of the precast beamup to the time of slab casting due to the precast beamdead load. See Term (2) of Eq. (4-10) for additionalcomments.

Term (6) is the creep deflection of the compositebeam for any period following slab casting due to theprecast beam dead load. See Term (3) of Eq. (4-10) foradditional comments.

Term (7) is the initial deflection of the precast beamunder slab dead load, (a,)1 = 6 MI e21E&. See Table4.2.1 for Q and M values. When diaphragms are used, forexample, add to this term:

where MID is the moment between two symmetrical dia-phragms, and a = 414, 413, etc., for the diaphragms at thequarter points, third points, etc., respectively.

Term (8) is the creep deflection of the compositebeam due to slab dead load. vII is the creep coefficientfor the slab loading, where the age of the precast beamconcrete at the time of slab casting is considered. SeeTerm (5) of Eq. (4-10) for additional comments. vW isgiven by Eq. (2-13).

Initial Camber = 1.93 - 0.80 = 1.13 in (28.7 mm)

Residual Camber = 0.13 in (3.3 mm), Total in Table4.6.4

Live Load Plus Impact Deflection = -0.50 in (-12.7mm), (Girder is untracked)

Term (9) is the deflection due to differential shrink-age. See Term (6) of Eq. (4-10) for additional comments.

Term (10) is the live load deflection of the compositebeam, in which the gross section flexural rigidity, E,I,, isnormally used. For partially prestressed members whichare cracked under live load, see Term (7) of Eq. (4-10)for additional comments.

Residual Camber + Live Load Plus Impact Deflection= 0.13 - 0.50 = -0.37 in, (3.3 - 12.7= -9.4 mm)

AASHTO (1978) Check:

Live Load Plus Impact Deflection = -0.50 in, (-12.7-1t/800 = (80) (12)/W = 1.20 > 0.50 in, (30.5 >12.7mm), OK.

b) Shored construction

a, = Eq. (4-21), with terms (7) and (8) modifiedas follows: (4-23)

Term (7) is the initial deflection of the compositebeam under slab dead load, (ai)l = fMlt2fE,Ic. SeeTable 4.2.1 for [ and M values.

Term (8) is the creep deflection of the compositebeam under slab dead load = ~,,(a,)~. The composite-section effect is already included in Term (7). See Term(5) of Eq. (4-10) for additional comments.

The detailed calculations for the results in this ex-ample can be seen in Ref. 83.

4.7-Deflection of reinforced concrete flat plates andtwo-way slabs

A state of the art report on practical methods forcalculating deflection of the reinforced concrete floorsystems, including that of plates, beam-supported slabs,and wall-supported slabs is given in Ref. 74.

au = Eq. (4-22) with Terms (7) and (8) modifiedas follows: (4-24)

Term (7), use composite moment of inertia to com-pute (%-)I +

Term (8), eliminate the ratio Z,/I,.For additional information on composite concrete

members partially or fully prestressed, see Refs. 62 to 64.

Although creep and shrinkage effects may be higher inthin slabs than in beams (time-dependent deflections aslarge as 5 to 7 times the initial deflections have beennoted 2%~~ the same approach for predicting time-depeident beam deflections may, in most cases, be usedwith caution for flat plates and two-way slabs. Theseinclude Eqs. (3-7), (3-8), and (3-10) for the effect ofcompression steel, etc., and Eq. (4-3) for additionallong-time deflections. The effect of cracking on theeffective moment of inertia Z,, for flat plates and two-wayslabs is discussed in Section 3.4 of this report.

4.6-Example: Ultimate midspan loss of prestress andcamber for an unsbored composite AASHTO Type IVgirder with prestressing steel only, normal weight con-crete63

The initial deflection for uniformly loaded flat plates,azd;yEvay slabs are given by Eqs. (4-25) and (4-&“).

Material and section properties, parameters and con-ditions of the problem are given in Tables 4.6.1 and 4.6.2.The ultimate loss of prestress is computed by the (Eq. 4-20) and the ultimate camber by (Eq. 4-22). Results aretabulated term by term in Tables 4.6.3 and 4.6.4.

The loss percentages in Table 4.6.3 show the elasticloss to be about 7.5 percent. The creep loss before slabcasting about 6 percent and about 2 percent followingslab casting. The total shrinkage loss about 6 percent.The relaxation loss about 7.5 percent and the gain in pre-stress due to the elastic and creep effect of the slab deadload plus the differential shrinkage and creep of about4.5 percent. The total loss is 24.3 percent.

Flat plates ai = E~q@/E~iZ~ (4-25)

Two-way slabs ai = r,,qe4/E,iI, (4-26)

where I, and q refer to a unit width of the slab. ThePoisson-ratio effect is neglected in the flexural rigidity ofthe slab. Deflection coefficients E

RIand [,, are given in

Table 4.7.1 for interior panels. ote that these coef-ficients are dimensionless, so that q must be in load/length (e.g. lb/ft or kN/m). These equations provide forthe approximate calculation of slab initial deflections inwhich the effect of cracking is included.

The following is shown in Table 4.6.4 for the midspan

Reference 44 presents a direct rational procedure forcomputing slab deflections, in which the effect ofcracking and long-term deformation can be included.

camber: An approximate method based on the equivalent

frame method is presented in Reference 75. This methodaccounts for the effect of cracking and long-term defor-mations, is compattble in approach and terminology withthe two alternate methods of analysis in Chapter 13 ofAC1 318n and requires very few additional calculationsto obtain deflections.

4.&-Time-dependent shear deflection of nlnforced con-crete beams

Shear deformations are normally ignored when com-puting the deflections of reinforced concrete members;however, with deep beams, shear walls and T-beamsunder high load, the shear deformation can contriiutesubstantially to the total deflection.

Test results on beams with shear reinforcement and aspan-to-depth ratio equal to 8.7 in Ref. 73 show that:

Shear deformation contributes up to 23 percent of thetotal deflection, although the shear stresses iu thewebs of most test beams were not very high.

Shear deflections increase with time much more rapid-ly than flexural deflections.

Shear deflection due to shrinkage of the concrete websis of importance.

4.8.1 Shear dejlection due to creepyThe time-dependent shear stiffness G, for the, initial

plus creep deformation of a cracked web with verticalstirrups can be expressed as given by Eq. (4-27).

4, ld E,G, = (1-1.1 v,/v,)/pw+ 4n (1 + v,) (4-W

where:vx = nominal shear stress acting on sectionvc = nominal permisstble shear stress carried by

concrete as given in Chapter 11 of AC1 318*’b,,, = web width

= A,,fb,,+v$;= area of shear reinforcement within a distance

sS’ spacing of stirrups

Eq. (4-27) is based on a modified truss analogy as-suming that the shear cracks have formed at an angle of45 deg to the beam axis, that the stirrups have to carrythe shear not resisted by concrete and that the concretestress in the 45 deg struts are equal to twice the nominalshear stresses vX.

4.8.2 Shear defrcction due to ~hrinkage’~In a truss with vertical hangers and 45 deg diagonals,

a shrinkage strain csr, results in a shear angle of 2 c,,,radians. The shear deflection due to shrinkage of a mem-ber with a symmetrical crack pattern is given by Eq.(4-28).

(4-28)

Eq. (4-28) may overestimate the shrinkage deflectionbecause the length of the xone between the inclinedcracks is shorter than 4.

4.9-Comparison of measured and computed deflections,cambers and pnstnss losses using proceduns in thischapter

The method presented in 4.2,4.3,4.4,4.5,4.7, and 4.8for predicting structural response has been reasonablywell substantiated for laboratory specimens in the refer-ences cited in the above sections.

The correlation that can be expected between the act-ual service performance and the predicted one is reason-ably good but not accurate. This is primarily due to thestrong influence of environmental conditions, load his-tory, etc., on the concrete response.

In analyzing the expected correlation between the pre-dicted service response (i.e., deflections, cambers andlosses) and the actual measurements from field struc-tures, two situations shall be differentiated: (1) The pre-diction of their elastic, creep, shrinkage, temperature,and relaxation components; and (2) the resultant re-sponse obtained by algebraically adding the components.

In the committee’s opinion, the predicted values of thedeflection, camber, and loss components will normallyagree with the actual results within 215 percent whenusing experimentally determined material parameters.Using average material parameters given in Chapter 2will generally yield results which agree with actualmeasurements in the range of 230 percent. With someknowledge of the time-dependent behavior of concreteusing local concrete materials and under local conditions,deflection, camber, and loss of prestress can normally bepredicted within about 220 percent.

If the predicted resultant is expressed in percent, widerscatter may result; however, the correlation between thedimensional values is reasonably good.

Most of the results in the references are far moreaccurate than the above limits because a better cor-relation exists between the assumed and the actual lab-oratory histories for water content, temperature andloading histories.

CHAPTER S-RESPONSE OF STRUCI’URES WITHSIGNIFICANT TIME CHANGE OF STRESS

5.1-ScopeIn statically indeterminate structures, significant re-

distribution of internal forces may arise. This may becaused by an imposed deformation, as in the case of adifferential settlement, or by a change in the staticalsystem during construction, as in the case of beamsplaced first as simply supported spans and then subse-quently made continuous.

Another cause may be the nonhomogeneity of creep

properties, which may be due to differences in age,thickness, in other concrete parameters, or due to inter-action of concrete and steel parts and temperature re-versal. Large time changes of stress are also produced byshrinkage in certain types of statically indeterminatestructures. These changes are relaxed by creep. Incolumns, the bending moment increases as deflectionsgrow due to creep and this further augments the creepbuckling deflections.

As stated in Chapter 3, creep in homogeneous stat-ically indeterminate structures causes no change in stressdue to sustained loads and all time deformations areproportional to v,.

5.2-Concrete aging and the age-adjusted effectivemodulus method

In the type of problems discussed in Section 5.1 above,the prediction of deformation by the effective modulusmethod is often grossly in error as compared with the-oretically exact solutions.66 The main source of error isaging of concrete, which is expressed by the correctionfactor Creep ycp in Eqs. (2-11) or (2-12), and by the timevariation of & given by Eqs. (2-l) and (2-S). Gradualstress changes during the service life of the structureproduce additional instantaneous and creep strains, whichare superimposed on the creep strains due to initialstresses and to all previous stress changes. Because ofconcrete aging, these additional strains are much lessthan those which would arise if the same stress changesoccurred right after the instant of first loading, tC,,. Thiseffect can be accounted for by using the age-ad’ustedeffective modulus method, originated by BTrost6’v andrigorously formulated in Ref. 65 and Ref. 69. Furtherapplications are given in References 66, 81, and 82. Re-ferences 66 and 82 indicate that this method is better intheoretical accuracy than other simplified methods ofcreep analysis and is, at the same time, the simplest oneamong them. In similarity to the effective modulusmethod, this method consists of an elastic analysis witha modified elastic modulus, E,, which is defined by Eq.(5-l), and is called the age-adjusted modulus.

4, = E,if(l + X VI)

The aging coefficient, X, depends on age at the timeI~,, when the structure begins carrying the load and onthe load duration t - t(,. Notice that t - t(,, as used inChapter 5, represents the t used in Eq. (2-8) and inChapter 4.

In Table 51.1, the X values are presented for thecreep function in Eq. (2-8). For interpolation in thetable, it is better to assume linear dependence on log rCaand log (r - rta).

The values in Table 5.1.1 are applicable to creep func-tions for different humidities and member sixes that havethe same time shapes as Eq. (2-8) when plotted as func-tions of r - rta, that is, mutually proportional to Eq. (2-8).An empirical equation for the approximation of the age-

adjusted effective modulus E,, that is generally applic-able to any given creep function is given by Eq. (16) inReference 108. The percent error in E,, is usually below1 percent when compared with the exact calculations bysolving the integral equations.

The analysis is based on the following quasi-elasticstrain law for stress and strain changes after load appli-cation:

where:

P-2)

(5-3)

(5-4)

Here (~~)a represents a known inelastic strain changedue to creep and shrinkage and is treated in the analysisin the same manner as thermal strain. a,, in Eqs. (5-4)and (5-5) represents shrinkage differential strain. If grad-ual thermal strain occurs, it may be included under (e;h)&

Some applications of the age-adjusted effective meth-od are discussed in the following sections. Equations (S-6) through (5-13) are theoretically exact for a given linearcreep law, only if the creep properties are the same in allcross sections, i.e., the structure is homogenous. In mostpractical situations, the error inherent to this assumptionis not serious.

Q-Stress relaxation after a sudden imposed defor-mation68bs

Let (S)i be the stress, internal force or momentproduced by a sudden imposed deformation at time r(,(such as short-time differential settlement or jacking ofstructure). Then the stress, internal force or moment (S),.at any time r > rt, is given by Eq. (5-6).

P-6)

The creep coefficient v, in this equation must includethe correction by factor 6, in Section 3.5 of this report.

!!A-Stress relaxation after a slowly imposed defor-mation

Let (s)$h be the statically indeterminate internal fOrCe,

moment or stress that would arise if a slowly imposed de-formation (e.g., shrinkage strain or slow differential set-

tlement) would occur in a perfectly elastic structure ofmodulus E,i (at no creep). Then the actual statically in-determinate internal force, moment or stress, (S),, attime t caused by a slowly imposed deformation includingthe relaxation due to creep is given by Eq. (S-7).

(53,6% = -1+x v, V-7)

M-Effect of a change in statical system695.S.l Stress relaxation afler a change in statical systemConsider that statical System (1) is changed at time tJ

to statical System (2).If Subscripts 1 and 2 refer to the stress, internal force

or moment computed according to the theory of elasticityfor statical Systems (1) and (2), respectively, the actualstress, internal force or moment after a sudden change inthe statical system at time t > tJ, is given by Eq. (5-8).v, - @,I,w, = SJ + cs, - Sl, 1 +x y1 1 (5-8)fand by Eq. (5-9), after a progressive change in the stat-ical system.

(s), = SJ + (s,- SJ, I 1I+:;: )I 1

P-9)It is assumed that the structure begins carrying the.load at tune tt, < tJ and v, and (v,)~ are creep coef-

ficients at time t and tJ, respectively.The value of X is to be read from Table 5.1.1 for ar-

guments tea and t - tta. Equation (5-8) is exact only if theload is applied just before time tJ, that is, for tJ = tta,and (v,)~ = 0, but, in most other cases, it is good approx-imation.

5.53 Long-time dejlection due to creep afer a change instatical system

The long-time deflection due to creep a,, after staticalsystem (1) is changed into statical system (2) at time tJ isgiven by Eq. (5-10).

at = v, al + (v, - v,J)(q - 01) (5-10)

where at and a2 are the elastic (short-time) deflectionscorresponding to statical systems (1) and (2). Term v, uIrepresents the usual creep deflection without the effectof the change of statical system (1). The second term isthe creep deflection (positive or negative) due to thechange in the statical system at time tJ 2 tta.

Typical examples are beams which are first cast assimply supported spans and carry part of the dead loadbefore time t at which the ends of the beams are rigidlyconnected, without changing the stress and strain state attime tJ. Also, a cantilever which carries the load beforeits free end is placed on a support. ‘Ihis is a typicalsituation in segmental bridge construction.

S&-Creep bucklin6d

deflections of an eccentrically com-pressed member

The creep deflection in excess of the elastic (short-time) deflection for a symmetric cross section is given byEq. (5-11).

whereYP

a, = 1 -@/P&J

where y is the maximum distance of the cross-section?fcentrot from the axis of axial load P prior to its ap

plication. P,i or P,, is the buckling load of an elasticcolumn with concrete modulus E,i or E,,, respectively. Z,is the moment of inertia of steel and Z, is the moment ofinertia of the whole transformed cross section with con-crete modulus Ea. Coefficient v, in this equation mustinclude correction by factor [, in Section 3.5 of thisreport.

Equation (5-11) is theoretically exact if creep pro-perties are the same in all cross sections and if thecolumn has initially a sinusoidal curvature. The error isusually small for cases other than sinusoidal curvature.Similar equations hold for creep Suckling deflections ofarches, shells, plates, and for lateral creep buckling ofconcrete beams or arches.

5.7--o cantilevers of unequal age connected at time t,by a binge66*69

The statically indeterminate shear force S, in the hingeat time t > tJ is computed from the compatibility relationin Eq. (5-12).

{[I + x, (v,)Jl NJ + [I + x, btj2.J cn,) s, =

h$)2 - fv,J)21a2 - h,)J - (v,J)J/aJ (5-12)

Subscripts 1 and 2 refer to cantilevers (1) and (2) re-spectively. (v,)J, (v,)J , and Xl are determined using tt,= (t&)J in which (t(,,)J is the age of cantilever (1) whenit starts carrying its dead load or prestress. aJ is the elas-tic deflection at the end of cantilever (1) due to its deadload or prestress, considering concrete modulus as E, atage (t(,)J . cf3J is the elastic flexibility coefficient Of canti-lever (1) which is the relative displacement of cantileverend in the sense of S, due to load S, = 1, using modulus4 at age (tto)J.

S.&-Loss of compression in slab and deflection of asteel-concrete composite beam6’

Compression loss (NC), in a steel-concrete compositestatically determinate simple supported beam is given inEq. (5-13).-. I

09, = -Yf NC, + 6, EC,4

1 + xv, + n 2 (1eZA (5-W

+Ls z )*

where Nti is the initial compressive force carried by theslab at the time tt,, of dead load, application and 8, asgiven by Eq. (S-5). A, and I, are the area and the mo-ment of inertia of the steel girder about its centroidalaxis, c is the eccentricity of slab centroid with regard tosteel girder cent&l, n = EJE, and A, is the area of aconcrete slab. I, is assumed negligible.

The moment change in the steel girder equals e(N,),.The creep deflection of a composite girder can be com-puted from the moment in the steel girder e(N&,.

M-other casesSimilar equations of greater theoretical accuracy are

possible f, prestress 10ss,~ but here the difference be-tween the results using such equations and those of thischapter is normally less than 2 percent and thus negli-gible.

For a general creep analysis of nonhomogeneous crosssections and nonhomogeneous structures, see Ref. 66, forexample. An application of the age-adjusted effectivemodulus method to the creep effects due to the nonuni-form drying of shells has been made in Reference 81.Bruegger, in Reference 82, presented a number of otherapplications.

5.1~Example: Effect of creep on a two-span beamcoupled after loading**

Find the maximum negative moment at the support ofa two-span beam made continuous by coupling two 90 ft(27.43 mm) simple supported beams.

Data: 4, = $ = 4? = 90 ft (27.43m)4 = 4 k/ft (58.4 KN/m) (sustained load appliedbefore coupling)

For coupling at tta = 30 daysAverage thickness = 8 in. (200 mm)Relative humidity, A = 60 percentvu = 2.35

Since the rotation at the support resulting from creepis prevented after coupling of the two single span beam,Eq. (S-9) applies.

Yea = 0.83 yA = 0.87 y,, = 0.96

hence, v, = 2.35 x 0.83 x 0.96 = 1.63

in which, X,, = 0.83, (for tea = 30 and (v,)~,, = 2.35)

since s, d= 0 and$= -7

= -4050 ft-kips, (-5492KNm)

therefore,

s = -4050 1.631 + 0.83 x 1.63

= -2805 ft-kipg (-3804KN/M).

The effect of creep is to induce a negative moment atsupport equal to about 69 percent of that obtained forthe continuous system that is, whole structure constructedin one operation.

In a similar way, the induced negative moment at support would be 78,62, and 53 percent of that obtained forthe continuous system if coupling time tea equals to 10,90, and 1oOq days respectively.

A C K N O W L E D G E M E N T S

Acknowledgement is given to the members of the Sub-committee II chaired by D.E. Branson, that prepared theprevious ACI-209-11 Report.”

Sub-Committee II would like to thank W.H. Dilger,W. Haas, A. Hillerborg, H. Hilsdorf, I.J. Jordaan, D.Jungwirth, K.S. Pister, H. Rusch, H. Trost and K. Willamfor their valuable comments on the draft of this report.However, it has been impossible for Sub-Committee II toincorporate all the comments without substantially af-fecting the intended scope of this report.

In the balloting of the nine members of Sub-Com-mittee II, AC1 Committee 209, all nine voted affirma-tively. In the balloting of the entire Committee 209consisting of twenty voting members, fifteen returnedtheir ballot, of whom fifteen voted affirmatively.

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56. Evans, R.H., and Bennett, E.W., PmStreswd Con-crete, lhcory amiLks&, John Wiley and Sons, Inc., NewYork, 1958,294 pp.

57. Sinno, R., “The Tie-Lcpendent Deflections ofPrestressed Concrete Bridge Girders,” Dissertation, TexasA&M University, 1968.

58. “Design of Continuous Highway Bridges with Pre-cast, Prestressed Concrete Girders,” But&tin EBO14.OlEPortland Cement Association, Aug. 1969, pp. 1-18.

59. Branson, D.E., and Chen, C.I., “Design Proceduresfor Predicting and Evaluating the Tie-Dependent De-formation of Reinforced, Partially Prestressed and FullyPrestressed Structures of Different Weight Concrete,Research Report, Civil Engineering Department, Univer-sity of Iowa, Aug. 1972.

60. Keeton, J.R., “Creep and Shrinkage of ReinforcedThin-Shell Concrete,” Naval Civil Engineering Labor-atory, Technical Report R704, Port Hueneme, CaIifomia,Nov. 1970, pp. l-58.

61. Comite Europeen Du Beton-Federation Inter-nationale de la Precontrainte, “International Recom-mendations for the Design and Construction of ConcreteStructures,” Cement and Concrete Association, London,June 1970, pp. l-80.

62. Branson, D.E., and Kripanarayanan, K.M., “Loss ofPrestress, Camber and Deflection of Noncomposite andComposite Prestressed Concrete Structures,” PClJoumal,V. 16, No. 5, Sept.-&. 1971, pp. 22-52.

63. Branson, D.E., “‘The Deformation of Noncompositeand Composite Prestressed Concrete Members,” AC1Publication SP 43-17 Deflections of Concrete Structures,1974, pp. 83-127.

64. Rao, V.J., and Diiger, W.H., “Time-DependentDeflections of Composite Prestressed Concrete Beams,”AC1 Publication SP 43-17, Deflections of Concrete Struotures, 1974, pp. 421442.

65. Bazant, Z.P., “Prediction of Concrete Creep Ef-fects Using Age-Adjusted Effective Modulus Method,”AC1 JOURNAL, Proceeding V. 69, No. 4, April, 1972, pp.212-217.

66. Bazant, Z.P., and Najjar, L.J., “Comparison of Ap-proximate Linear Methods for Concrete Creep,” Jounalof Struct. Div., Proceedings ASCE, V. 99, ST9, Sept. 1973,pp. 1851-1874.

67. Trost, H. “Implications of the SuperpositionPrinciple in Creep and Relaxation Problems for Concreteand Prestressed Concrete,” Beton und Stahlbetonbau(West Berlin), V. 62, 1967, pp. 230238, 261-269.

68. Neville, A.M., in collaboration with W. DiIger,Cnep of Concnte, Plain, Reinforced, and Prestresed,North Holland Publ. Co., Amsterdam, 1970.

69. Bazant, Z.P., “Lecture Notes for Course 720 D-28,Concrete Inelasticity,” Northwestern University, Evans-

ton, Ihinois, 1970, see aiso reference 80.70. h&Henry, D., “A New Aspect of Creep in Cott-

crete and its Application for Design,” he&b@ ASTM,v. 43,1943,1069-1086.

71. Ross, AD., “Creep of Concrete Under VariableStress,” ACI JOURNAL Proceedings V. 54, No. 9, Mar.1958, pp. 739-758.

72. AC1 Committee 435, Subcommittee 7, “Deflectionsof Continuous Beams,” AC1 JOURNAL, tie&&s V. 70,No. 12, Dec. 1973, pp. 781-787.

73. DiIger, W.H. and Abele, G., “Initial and Tiie-Dependent Shear Deflection of Reinforced Concrete T-Beams,” Deflection of Concrete Structures, AC1 SpecialPublication SP-43, American Concrete Institute, Detroit,1974, pp. 487-513.

74. AC1 Committee 435, Subcommittee 5, “State-of-the-Art Report, Deflection of Two Way Reinforced Con-crete Floor Systems,” Deflections of Concrete Structures,AC1 Special Publication SP-43, American Concrete Insti-tute, Detroit, 1974, pp. 55-81.

75. Niison, AH. and Walters, D.B. Jr., “Deflections ofTwo-Way Fioor Systems by the Equivalent FrameMethod,” AC1 JOURNAL, Pmceedbp V. 72, No. 5, May1975, pp. 210-218.

76. Browne, R.D. “Thermal Movement of Concrete,”Concrete, Z’Jre Journal of the Concrete Society, London, V.6, No. 11, Nov. 1972, pp. 51-53.

77. Kripanarayanan, K.M. and Branson, D.E., “SomeExperimental Studies of Thne-Dependent Deflections ofNon-Composite and Composite Reinforced ConcreteBeams.” Deflection of Concrete Structures, AC1 SpecialPublication SP-43, American Concrete Institute, Detroit,1974, pp. 409419.

78. PC1 Committee on Prestress Losses, “Recommen-dations for Estimating Prestress Losses.” Journal of thePrestmvsed Concrete Institute, V. 20, No. 4, July/Aug.1975, pp. 44-75.

79. England, G.L., “Steady-State Stress in ConcreteStructures Subjected to Sustained Temperatures andLose Nuclear figinecring and L&?&n, V. 3, No. 1, Jan.1966. North-Holland Publishing Comp. Amsterdam, pp.54-65.

80. Bazant, Z.P., “Theory of Creep, and Shrinkage inConcrete Structures: A Precis of Recent Developments,”Mechanics Today, V. 2, ed. by S. Nemat-Nasser, Perg-amon Press, New York, 1975, pp. l-92.

81. Bazant, Z.P., Caneira, D.J., Walser, A., “Creepand Shrinkage in Reactor Containment Shells,” JournalStructural Div., Prrxee&gs ASCE, V. 101, Oct. 1975, pp.2117-2131.

82. Bruegger, J.P., “Methods of Analysis of the Effectsof Creep in Concrete Structures,” Thesis at the Universityof Toronto, Dept. of Civil Engineering, 1974.

83. Branson, D.E., Deformation of Concrete Structures,McGraw-Hill Book Company, 1977.

84. Geymayer, “The Effect of Temperature on Creepof Concrete: A Literature Review” Miscellaneous PaperC-70-1 U.S. Army Engineer Waterways Experiment Sta-

tion, Corps. of Engineers, Vicksburg, Mississippi, Jan.1970.

85. Bazant, Z.P., and Wu, S.T., “Creep of Concrete atElevated Temperatures,” ASCE Annual and National En-vironmental Engineering Meeting, Oct. 20-NOV. 1,1973,New York, New York.

86. Bazant, Z.P., “Double Power Law for Basic Creepof Concrete,” Materiab d Structures, V. 9, Jan.lFeb. 1976.

87. Concrete So&g Technical Papr No. 101, TheCreep of Structural Concrete,” Report of a WorkingParty of the Materials Technology Divisional Committee,The Concrete Society, London, Jan. 1973.

88. Comite Europeen du Beton, “Effects Structurauxdu Fhtage et des Deformations Differees du Beton,Bulletin d’lnfonnation No. 94, Paris, 1973.

89. Comite Europeen du Beton, “Time Dependent Be-haviour of Concrete (Creep and Shrinkage), State of ArtReport, 1973,” Bulletin d’lnfomtation No. 97, Paris, 1973.

90. Klieger, P., “Effect of Miring and Curing Tem-perature on Concrete Strength,” AC1 JOURNAL, Pro-ceedings V. 54, June 1958, pp. 1063-1081.

91. Pauw, A., “Tiie-Dependent Deformations of Con-crete,” Study Prepared for Missouri State Highway De-partment, Department of Civil Engineering, University ofMissouri, Columbia, Missouri, Sept. 1971.

92. Rusch, H., Jungwirth, D., Hilsdorf, H., Remarks onthe First Draft (March 19, 1976) of AC1 Committee209-B Report, “Prediction of Creep, Shrinkage, andTemperature Effects in Concrete Structures,” PrivateCommunication to Subcommittee II, Munich, May 5,1976.

93. Bazant, Z.P., Osman, E., “On the Choice of CreepFunction for Standard Recommendations on PracticalAnalysis of Structures,” Cement and Concrete Research, V.5, 1975, pp. 129-137; Disc. V. 5, 1975, pp. 631641; andV. 6, 1976, pp. 149-155.

94. Bazant, Z.P., Osman, E., ‘Ihonguthai, W., “Prac-tical Prediction of Shrinkage and Creep of Concrete,”Mater-U and Structures (RILEM), V. 7, Nov.-Dec. 1976.

95. Bazant, Z.P., Thonguthai, W., “Optimization Checkof Certain Practical Formulations for Concrete Creep,”Materials & Structures (Paris), V. 9, Mar.-Apr. 1976.

96. AC1 Committee 209-H (Subcommittee II chairedby D.E. Branson) “Prediction of Creep, Shrinkage andTemperature Effects in Concrete Structures,” ACI-SP 27,“Designing for the Effects of Creep, Shrinkage andTemperature,” Detroit, pp. 51-93, 1971.

97. Illston, J.M., “Components of Creep in MatureConcrete.” AC1 JOURNAL, Proceedings V. 65, Mar. 1968,pp. 219-227.

98. Rusch, H., Jungwirth, D., Hilsdorf, H.K., “CriticalAssessment of the Methods of Allowing for the Effectsof Creep and Shrinkage of Concrete on the Behaviour ofStructure,” (in German), Beton and Stahlbeton, Nos. 3,4, and 6, pp. 49-60, 76-86, and 152-158, 1973.

99. Illston, J.M., and Constantinescu, D.R., andJordaan, I.J., Discussion of the Paper, “OptimizationCheck of Certain Practical Formulations for Concrete

Creep,” by Z.P. Bazant and W. Thonguthai (Reference95 in the Report) and Reply by Bazant, Z.P., and‘Ihonguthai, W., Materials and Structures (Paris), V. 10,No. 55, Jan.-Feb. 1977.

100. Rusch, H., Jungwirth, D., and Hilsdorf, H.K.,First and Second Discussions of the Paper, “On theChoice of Creep Function for Standard Recommen-dations on Practical Analysis of Structures,” by Z.P.Bazant and E. Osman (Reference 93 in this Report) andReplies by Z.P. Baxant and E. Osman, Cement and Con-crete Research, V. 5, 1975, pp. 631642 and V. 7, 1977,No. 1, pp. 119-130.

101. Haas, W., “Comparison of Stress-Strain Laws forthe Tie-Dependent Behavior of Concrete.” RILEM andCISM Symposium on Test and Observations on Modelsand Structures and Their Behavior Versus Tiie,UDINE, 18-20, Sept. 1974.

102. Agryris, J.H., Pister, K.S., Szimmat, J., andWilliam, K.J., “Unified Concepts of Constitutive Model-ling and Numerical Solution Methods for Concrete CreepProblems,” ZSD-Report No. 185, Stuttgart, 1976.

103. Tadros, M.K., Ghali, A., and Dilger, W.M.,‘Time-Dependent Prestress Loss and Deflection of Pre-stressed Concrete Members,” PCZ Journal, V. 20, Nov. 3,1975.

104. Jordaan, I.J., England, G.L., and Khalifa, M.M.A.,“Creep of Concrete a Consistent Engineering Approach,”Journal Struct. Div., Proceedings ASCE V. 103, Mar.1977, pp. 475-491.

105. Freudenthal, A.M., and Roll, F., “Creep andCreep-Recovery of Concrete Under High CompressiveStress,” AC1 JOURNAL, Proceedings V. 54, No. 12, June1958, pp. 1111-1142.

106. Roll, F., “Long-Time Creep-Recovery of HighlyStressed Concrete Cylinders,” AC1 Publication SP-9,Creep of Concrete, 1964, pp. 95-114.

107. AC1 Committee 517, “Low Pressure SteamCuring,” AC1 Report Title No. 60-48, American ConcreteInstitute, Detroit.

108. Bazant, A.P. and Kim, S.S., “Approximate Relax-ation Function for Concrete,” Jounrnal of the Struct.Div., Proceedings ASCE, V. 105, No. ST12, Dec. 1979.

109. AC1 Committee 444, “Models of ConcreteStructures, State-Of-The-Art,” Report No. AC1 444-79,Concrete International V. 1, No. 1, Jan. 1979, pp. 77-95.

NOTATION

1

2

A*4

4

= subscript denoting cast-in-place slab of acomposite beam or the effect of the slabdue to slab dead load

= subscript denoting precast beam= area of gross section, neglecting the steel= area of tension steel in reinforced members

and area of prestressed steel in prestressedmembers

= area of compression steel in reinforced

A,a

(ai)D

(41 )I$

aDS

au

as

‘i

ta.sh)s

atc

=PDdEET

members and area of nontensional steel onprestressed members

= area of transformed section= deflection in general. Also used as distance

from end of beam to the nearest of 2 sym-metrical diaphrams, or as the distance fromend to harped point in 2-point harping

= initial deflection under slab dead load= initial deflection due to diaphragm dead

load= initial deflection under precast beam dead

load= initial dead load deflection

= initial camber due to the initial prestressforce, F,

= live load deflection= ultimate (in time) deflection, camber

deflection‘ due to differential shrinkageshrinkage deflectionshear deflection due to shrinkagetotal deflection, camber, at any timesubscript denoting composite section. Alsoused to denote concrete, as E,; cement con-tent and initial curingsubscript denoting creep or curing periodsubscript denoting dead loadeffective depth of sectionage-adjusted effective modulusmodulus of elasticity of concrete at the timeof initial load, such as at transfer of pre-stress, etc., or of a sudden enforced defor-mation at ttme t(,modulus of elasticity of concrete at the timeof slab castingmodulus of elasticity of concrete at any timetmodulus of elasticity of steeleccentricity, also eccentricity of steeleccentricity of steel at center of beam. Alsoused, as indicated, to denote eccentricity ofsteel in composite sectioneccentricity of steel at end of beamthermal coefficient of expansion of concreteprestress force after lossesinitial prestress forceloss of prestress due to time dependent ef-fects only such as creep, shrinkage, steel re-laxation. The elastic loss is deducted fromthe tensioning force, Fi, to obtain F,prestress force at transfer, after elastic losstotal loss of prestress at slab casting minusthe initial elastic loss that occurred at thetime of prestressingtotal loss of prestress at any time minus theinitial elastic losstotal ultimate (in time) loss of prestressminus the initial elastic loss

= concrete stress such as at steel c.g.s. due toprestress and precast beam dead load in theprestress loss equations

= modulus of rupture of concrete= concrete stress at steel c.g.s. due to differ-

ential shrinkage= concrete stress at the time of initial loading,

such as at transfer of prestress= concrete stress at steel c.g.s. due to slab

dead load plus diaphragm, etc., dead loadwhen applicable

= compressive strength of concrete at 7 days;similarly, for subscript 2 for the avg. of 1 to3 days, subscript 28, for 28 days, etc

= compressive strength of concrete at anytime t

= ultimate (in time) compressive strength ofconcrete

= stress in prestressing steel at transfer, afterelastic loss

= ultimate strength of prestressing steel= steel stress at 0.1 percent strain= modulus of rupture of concrete= initial or tensioning stress in prestressing

steel= stress loss due to steel relaxation under

constant strain at any time or intrinsic relax-ation

= stress loss due to steel relaxation in pre-stressed members at any time

= ultimate (in time) stress loss due to steel re-laxation on prestressed members

= tensile strength of concrete= yield strength of steel, defined herein as

0.1 percent offset= average thickness of the part of the member

under consideration. Also, overall thicknessof the section

= moment of inertia of slab= moment of inertia of precast beam= moment of inertia of composite section with

transformed slab. The slab is transformedinto equivalent precast beam concrete bydividing the slab width by Ec21Ec,

= moment of inertia of cracked transformedsection

= effective moment of inertia= average effective moment of inertia= effective moment of inertia for the positive

zone of a beam= weighted (average) effective moment of in-

ertia51~ 42 = I, for each one of the negative moment end

zones of a beam

I = moment of inertia of gross section, neglect-ing the steel

I = moment of inertia of reinforcing steelI = moment of inertia of transformed section,

ie

4,LM

4

MID

MSJli

such as an untracked prestressed concretesection

= subscript denoting initial value= span length in general and longer span for

rectangular slabs= subscript denoting loading age= subscript to denote live load= total moment. Also bending moment, used

as the numerical maximum bending mo-ment, for prismatic beams uniformly loaded

= bending moment due to dead load= maximum bending moment under slab dead

load for composite beams= maximum bending moment under precast

beam dead load= bending moment between symmetrically

placed diaphragms= bending moment due to slab or slab plus

diaphragm, etc., dead loadM,,,M, = end bending moments

= modular ratio of the precast beam concreteat the time of additional sustained load ap-plication EJE, (e.g. at the time of slabcasting). Also subscript to denote mid-span

= modular ratio, EJE,, at the time of loading,such as at release of prestress for prestress-ed concrete members. Also usually used asEJE, for reinforced members

= modular ratio due to creep, defmed asE+%

m

n

“r

Q

sS

sht

U

W

YCS

Yf

zb

4

Y

= dtfferential shrinkage force= uniformly distriiuted load= IJAs= subscript denoting time of slab casting re-

ferred to the precast beam concrete. Alsoused to denote steel, slump and spacing ofstirrups

= subscript denoting shrinkage= time in general, time in hours in the steel

relaxation equations, and time in days inother equations herein. Also subscript todenote time-dependent

= temperature difference across the overallthickness

= subscript to denote temperature= age of concrete at fiit load application in

days= subscript denoting ultimate value in time= unit weight of concrete in pcf or Kg/m3= distance from centroid of composite section

to centroid of slab= distance from centroid of gross section to

extreme fiber in tension= Section modulus with respect to the bottom

fiber of a cross section= Section modulus with respect to the top

fiber of a cross section= shrinkage or creep correction factor, also

used as the product of all applicable correc-tion factors

6 = differential shrinkage strain, also subscriptdenoting differential strain or differentialstress

kh), = shrinkage strain in in/m. or mm/mm at anytime

hl)lL = ultimate (in time) shrinkage strain in inc./m.or mm/mm

A = relative humidity in percentAl l = prestress loss due to elastic shortening in

percent of initial tensioning stress or force4 = prestress loss due to steel relaxation in per-

cent4 = total prestress loss in percent at any time4 = ultimate (in time) prestress loss in percentvs = creep coefficient of precast beam concrete

at time of slab castingYl = creep coefficient at any timeVI1 = creep coefficient of the composite beam

under slab dead load, also creep coefficientat time t,

%2 = creep coefficient due to precast beam deadload

vll = ultimate (in time) creep coefficient%a = ultimate (in time) creep coefficient

us of the precast beam concrete correspond-ing to the age when the slab is cast for com-posite beams

i= deflection coefficient

t= deflection coefficient for flat plates= reduction factor to take into account the

effect of compression steel movement ofneutral axis, and progressive cracking inreinforced flexural members

f, = cross section shape coefficient

;:= deflection coefficient for two-way slabs= deflection coefficient for warping w due to

shrinkage or temperature changeP = reinforcement ratio, A,/bd for cracked

members, and As/As for untracked mem-bers. Used in percent in shrinkage warpingequations

7 = multiplier for additional long time deflec-tions due to creep and shrinkage

4 = curvatureX = aging coefficient

Table

2.2.1

Values of the Constants a, 8 and

a/B

and

the Time Ratio

From Eqs. (2-l)

and

(2-2).

Time

Ratio

W,)

,/(f

’,) *

*Eq.

(2-l)

Wc)

,/W

c),,

Eq.

(2-2)

Type

Cement

Constants

ofType

a,B

and

Curi

nga/B

Moist

Cured

Steam

Cured

Moist

Cured

Concrete

Age

Days

IYe

3 1

7

1 14

1

21

1 28

1

56

1 91

j

1

.46

1 .7

0

1 .8

8 1

.96

Il.0

Il

.08

11.i

-[;;

.59

.80

.92

.97

1.0

1.04

1.

06

1.08

.78

.91

.98

1.

0 1.

0 1.

03

1.04

1.

05

.82

.93

.97

.99

1.0

1.0

1.01

1.

01

.39

1 .6

0 i

.75

1 .8

2 1

.86

1 .9

2

1 .9

5 1

.99

1.0

I 1.

0.5

4 1

.74

1 .8

5 1

.89

1 .9

2 1

.96

1 .9

7 1

.99

.74

1 .8

7 1

.93

1 .9

5 1

.96

1 .9

8

1 .9

9 Il

.0.8

1 1

.91

1 .9

5 1

.97

1 .9

7 1

.99

1

.99

Il.0

1.09

I1.

09

1.05

1.05

---I--

1.02

1.02

1.0

1.0

--t

1.0

1.0

1.0

1.0

Table

2.2.2

Factors

Affecting

Concrete Creep

and

Shrinkage

and

Variables

Considered

in

the

Recommended

Prediction

Method.

Factors

Variables

Considered

Standard

Conditions

Concrete

(Creep

84Shrinkage)

Memb

erGeometry &

Environment

(Creep &

Shrinkage)

Loading

(Only

Creep)

Concrete

Composition

Cement

Paste

Content

Water-Cement

Ratio

Mix

Proportions

Aggregate

Characteristics

Degree

of

Compaction

Type of cement

Type I

and

IIISl

pAiyContent

2 / in,

(10

mn)

2.6 percent

Fine

Aggregate

Percentage

50

percent

Cement

Content

470

I52 lb/

(279?o

446

kg;i3j

d

Length of Initial

Curing

-Moist Cure

!d7

days

Steam

Curt%-

l-3

days

Initial

Curing

Temperature

Moist

Cured

-Steam Curt

?dI73.4+

4°F

(23+2"C)

_I

-Cu

ring

I(212"F,

(LlOOOC)

Curing

Humidity

F_

.-.lelative

Humidity

I _ ^-

~.1 LYS percent

I

Concrete

Temperature

Concrete

Water

Content

Geom

etry

Size and

Shape

Concrete T

c?mperature

73.4+4"F, (23+2"C)

Ambient

Relative

Humidity

40%

Volume-Surface

Ratio,

(v/s)

v/s

= 1.

5 in

MinimurThickness

(v/s = 38 mm)

6i

n (

l&n

mn\

Loading

Hist

ory

Concrete Age

at Load

Moist

Cured

Application

Steam

Cured

Duration

of

Loading

Period

1 Sustained

Load

Duration of Unloading Period I

7 days

l-3

days

Sustained

Load

Number of Load Cycles

IType of Stress and

Stress

Distribution

Across

the

Compressive

Stress

Axial

Compression

Conditions

Section

Stress/Strength

Ratio

Stress/Strength

Ratio

LO.50

Tabl

e 2.

4.1

Time

-Rat

io

Valu

es

for

Cree

p an

d Sh

rink

age

Cree

p an

d Sh

rink

age

Time

Time

Ra

tios

28 d

3 mt

h6

mth

1 yr

2 yr

5 yr

1oyr

2o

yr3O

yr

v,/

vu , Eq.

(2-8

)0.

420.

600.

690.

780.

840;

900.

930.

950.

96

(I2s

h)t/

( E&,

s Eq.

(2-g

)0.

440.

720.

840.

910.

950.

980.

991.

001.

00

(Es

&t/( E&,

9 Eq.

(2-1

0)

0.34

0.62

0.77

0.87

0.93

0.97

0.99

0.99

1.00

/

Tabl

e 2.

5.1

Corr

ecti

on

Fact

ors

for

Load

ing

Age,

fr

omEq

s. (2

-11)

an

d (2

-12)

Load

ing

Cree

pCr

eep

Age,

days

YRa

YQa

mois

t cu

red

stea

m cu

red

7 :: 50" 90

1.00

0.94

0.95

0.90

0.87

0.85

0.84

0.83

0.77

0.76

0.74

0.74

Table 2.5.3 Shrinkage Correction Factorsfor Initial Moist Curing

Moist curing duration,days

Shrinkage Ycp

: :::7

:"8:::3

90 i:E

Table 2.5.4 Correction FRelative Hwr

RelativeHumidity,percent

Creep Shrinkage

yx Y4

’ 1.001.000.940.870.800.730.670.60

actors fordity, from(2-15), and

’ 1.001.00

EEi0:700.600.300.00

Table 2.5.5.1 Correction Factors for AverageThickness of Members, fromEqs. (2-17) to (Z-20)

I Average CreepThickness

Shrinkage

of Member* 'h 'h

in. mn L 1 yr.ult. ult.

value (1 yr. valueP 51

1;: 127

1.17 1.30 1.30 1.17 1.35 1.25 1.25 1.35

ii 1.11 1.04 1.11 1.04 1.17 1.08 1.17 1.08

Eqs. (2-17) (2-18) (2-19) (2-20

6 152 1.00 1.00 1.00 1.008 203 0.96 0.96 0.93 O-94ii 254 305 0.91 0.86 0.93 0.90 0.77 0.85 0.82 0.88

15 381 0.80 0.85 0.66 0.74

*This method is recommended for averagethicknesses (part being considered) up toabout 12" to 15", (305 to 38 mm).

Table 2.5.5.2 Correction Factorsfor Volume-SurfaceRatios, from Eqs.(2-21) and (2-22)

Volume- Creep ShrinkageSurfaceRatio y v/s b/s

in. mn (2-21) (2-22)

:*i 2 5

i i!

1.00 1.09 1.00 1.06

0.92 0.94

: 1;; 0.75 0.81 0.84 0.742 127 152 0.70 0.72 0.58 0.66

1: 203 254 0.67 0.68 0.36 0.46

Examples:

For a rectangular section6"~ 12" (150 x 35Omm), v/s =2.0" (51 mn). For theStandard ASSHTO I-Beams,v/s varies from 3.0" to 4.7",(76 to 12Omn).

Table 2.7.1 Correction Factors Used in Example 2.7

Conditions

t&a = 28 daysA= 70%h = 8 in (200 mm)= 2.5 in (63 mm)

; = 60%C = 752 lbs/cu yd

(446kg/m3)(3 = 7%

Factors' product

Creep

Eq. Factor

(2-11) 0.840.800.960.991.02

%= 0.71

Shrinkage

Eq. I Factor

(2-28) 1.02(2-30) 1.01

Y sh = 0.68

Table 2.7.2 Creep Factors and Shrinkage Strains inExample 2.7

Concrete age, days 56 118 208 393

Time after initialcuring, days 49 111 201 386

Time after loadapplication, days 28 90 180 365

“ t , Eq. (2-8) 0.72 1.02 1.18 1.32

( Esh)t x lo+ Eq. (2.9) 309 403 451 486

( ’ & x 10%for tga = 56 days 0 93 142 176

Table 2.9.1 Suggested Values for the Degree of Saturation

Concrete Member Environmental Degree of emcConditions Saturation lo-~/OF lo-6/oc

Imnersed structures, high humidityconditions. Saturated

Mass concrete pours, thick walls, Between partiallybeams, columns and slabs, saturated andparticularly where surface is sealed. saturated

0 0

0.72 1.3

External slabs, walls, beams, 0.83 1.5columns, and roofs allowed to Partially saturateddry out or internal walls, columns decreasing with to toslabs, not sealed (e.g. by mosaic or time to the dryertiling) and where underfloor heating conditionsor central heating exists. 1.11 2.0

Table 2.9.2 Average ThermalCoefficient ofExpansion of Aggregate

Rock Group 1~

ChertQuartziteQuartzSandstoneMarbleSiliceous

limestoneGraniteDoleriteBasaltLimestone

6.65.76.2

2::

4.6

3::

3:;

11.810.311.19.38.3

i*;6:86.45.5

Table 2.9.3 Range of the Concrete Thermal Coefficientof Expansion

Aggregate, ea Concrete, ethRock Group lo-6/oF lo-6/x lo-6/v lo-6/w

Chert 4.1-7.2 7.4-13.0 6.3-6.8 11.4-12.2Quartzite 3.9-7.3 7.0-13.2 6.5-8.1 11.7-14.6Quartz - - - - 5.0-7.3 9.0-13.2Sandstone 2.4-6.7 4.3-12.1 5.1-7.4 9.2-13.3Marble 1.20-8.9 2.2-16.0 2.4-4.1 4.4-7.4Siliceous

limestone 2.0-5.4 3.6-9.7 4.5-6.1 8.1-11.0Granite 1.0-6.6 1.8-11.9 4.5-5.7 8.1-10.3Dolerite 2.5-4.7 4.5-8.5 - - -Basalt 2.2-5.4 4.0-9.7 414-5.8 7.9-10.4Limestone 1.0-6.5 1.8-11.7 2.0-5.7 4.3-10.3

*Test data for the concrete does not necessarilycorrespond to test data for the aggregate in Table2.9.2. These ranges are limited to the research workcompiled in Reference 76.

Table

3.5.1

Reduction

Factors

cr,

(3-8

), and

(3-9).

Sr v

u and

Sr mu

from Eqs.

(307

),

Eq.(

3-7)

'fro;"

Eq.(

3-8)

';,',"

Eq.(

3-9)

% %J

AZ/A

sfo

rcr

yp2.0

crTU

P2.0

5rTu

=2.0

00.

851.

71.

02.

01.

02.

0

0.5

0.62

51.

250.

701.

40.

667

1.3

1.0

0.40

0.8

0.40

0.8

0.5

1.0

&

Table

3.7.1

Intrinsic

Relaxation

Stress

Loss

(Steel

Relaxation

Under

Constant

Strain).

Wire

or

St

rand

(fsi

r)t

for fs

i/fp

y 0.60

(fsi

r)u/

fsi

at t=

105

hours

fPY

and

tl=

1 ho

urat

0.

1% st

rain

[.

Stre

ss0.

1 fs

i$-;

-0.55

logI

O(t/

tI)

1(0

.025

to

0.1

75)

reli

eved

fpy

= 0.85 f

pu

Stee

l'St

abil

ized

$b-;a

tion)

o*

o222

fs

if& fp

y-0

.55

1log1

0(t/

t1) (0.0055

to 0.39)

fPY

= 0.90 fpu

Table 3.7.2 Relaxation Reduction Factor

fsiifpy 0.50 0.55 0.60 0.65 0.70 0.75 0.80

0.00 1.000 1.000 1.000 1,000 1.000 1.0000.05 0.000 0.547 0.729 0.798 0.835 0.857 0.8720.10 0.000 0.289 0.516 0.627 0.689 0.729 0.756

w 0.15 0.000 0.172 0.361 0.486 0.564 0.615 0.6520.20 0.000 0.099 0.262 0.375 0.458 0.516 0.5580.30 0.000 0.013 0.150 0.238 0.305 0.361 0.4060.40 0.000 0.000 0.077 0.159 0.216 0.262 0.3000.50 0.000 0.000 0.029 0.102 0.157 0.197 0.230

Table 3.7.3 Minimum Time Intervalsto Compute Steel Relaxation

;tep BeginningTime, tl End time, t

Pretensioned:anchorage ofprestressing Age at pre-

1 steel. stressing ofPost-tensioned: concreteend of curingof concrete.

Age = 30 days,or time when a

2 End of Step 1 member is sub-jected to loadin addition toits own weight

3 End of Step 2 Age = 1 year

4 End of Step 3 End of servicelife

Table 4.2.1 Values of M, E and SW for Beams of Uniform Setand Uniform Load

Boundary Conditions M 5

Cantilever beam -q a212 l/4

Simple beam +q a218 5148

Hinged-fixed beam (one end continuous) -q a218 81185,Fixed-fixed beam (both ends continuous) -q a2112 l/32

.ion

l/2

l/B

111128

l/16

Table 4.4.1.1 Suggested Modular Ratios for Prestressed Beams

Sand-Modular Type of Concrete Normal light All-lightRatio Weight Weight Weight

145 120 1 0 0w in pcf, (kg/m3) (2323) (1922) (1602)

Curing M.C. S.C. M.C. S.C. M.C. S.C.

n At release of prestress 7.3 7.3 9.8 9.8 12.9 12.9

For the time between pre-stressing and slab casting

m = 3 weeks, 6.1 6.2 8.1 8.3 10.7 10.91 month, 6.0 6.2 8 2 10.5 10.72 months, 5.9 7.9 ii*; 8’2 10.3 10.63 months, 5.8 6.0 7:7 8:0 10.2 10.5

The above average modular ratios are based on f& = 4000 to 4500 psi (27.6

to 31.0 MPa) for both moist cured and steam cured concrete and type I

cement; up to 3-mths f; = 6360 to 7150 psi (43.9 to 49.3 MPa), using Eq.

(2-l) for moist cured, and 3-mths f; = 6050 to 6800 psi (41.7 to 46.9 MPA),

using Eq. (2-l) for steam cured concrete. Es = 27 x lo6 psi (18.62X104 MPa)

for ASTM A-416 Grade 250 (1725 Mpa) strands and Es = 28 x lo6 psi (19.3 x

lo4 MPa) for Grade 270 (1860 MPa) prestressing strands.M.C. = Moist Cured, S.C. = Steam Cured

Table

4.4.1.2

Typical

Loss

of

Prestress

Ratios

for

Different

Concretes

Type of Concrete

Norm

alI

Sand

-lig

htI

All-

ligh

twe

ight

co

ncre

tewe

ight

co

ncre

teweight

concrete

145

I12

0I

100

(232

3)(1

922)

(160

2)w

in p

cf,

(kg/

m3)

F /F for

3 weeks

to 1

month

batw

gen

prestressing

and

sust

aine

d lo

ad

appl

icat

ion,

incl

udin

g co

mpos

ite

slab

.

0.10

0.12

0.14

F IF

--

b&w@en

for

2 to

3

mont

hspr

estr

essi

ngan

dsu

stai

ned

load

ap

plic

atio

n,in

clud

ing

comp

osit

e sl

ab.

0.14

0.16

0.18

II

I

Fu’F

oI

0.18

I0.

21I

0.23

Table

4.4.1.3

Values of (f

sr)t

and

(fsr

)u for

Wires

and

Strands

(fsr

)t(f

sr)u

Wire

or

St

rand

for

f ./f

S' PY

from 0.65 to .8

0at

t =

lo5

hours

Stre

ss

Reli

eved

Stee

lo’0

15

fsi

(lOglOt)

0.07

5 fs

i

Stab

iliz

ed

(Low

rela

xati

on)

O*O

o5 fsi

(lOglOt)

0.02

5 fs

i

Table 4.4.2.1 Comnon Cases of Prestress MomentDiagrams and Equations forComrwtina Camber

Prestress Beam

Table 4.6.1 Material and Section Properties, Parameters andConditions for Example 4.6, (U.S. Customery Units)

UNSHORED COMPOSITE GIRDER

Y 9 2 ’ Y

. . . . . . . . . . .L 26” J

AASHTO IV

Material Properties:

Steam Cured Normal Weight Concrete

fPu

= 270 ksif'. = 4000 psi, fl = 5000 psi

D:;k f;: = 4000 psi

EGirder/ESlab = E2/El = 3.89/3.64 = 1.07

E.= 3.64 x 106psi, E = 3.89 x 10 psi&ion Properties andcLoading from

Pa* ”andbook

Girder:

SIMPLE SPAN

Calculated Section Properties and Loading -- Composite Section:

'iodified Slab Area = 7 x 92/1.07 = 602 in*, yb = (789 x 24.73 + 602 xi7.50)/(789 + 602) = 38.91”, Ig = 260,740 + 789(38.91 - 24.73)* + 92 x 73‘(12 x 1.07) + 602(57.50 - 38.91)* = 629,890 in4

It = 629,890/(61.00 - 38.91) = 28,510 in3, zb = 629,890/38.91 = 16,190 in3Including l/2" w.s., Slab D.L. = ws = (7.5 x 92)(150/144) = 719 lb/ftIS 20-44 AASHTO Loading, Impact = 50/(80 + 125) = 0.25\ssume Deck Slab Cast 2 Months After Prestressing

\rea of One l/2" Strand = 0.153 in* (Fig. 11.3.3, PC1 Handbook)

\ps = (32) (0.153) = 4.90 in*, fsi = (0.70)(270) = 189 ksi

'D = wDL*/8 = (822)(80)*/8 = 657,600 ft-lb, MS Di = (719)(80)*/8 + 50,000: 625,200 ft-lb, MD+MS,Di = 1,282,800 ft-lb, Inierior Girder ML + I = (1165 --

MASHTO Table) (l/2 -- Single Wheels)(1.25 -- L + 1)(7.67/5.5 -- AASHTO, S/5.5)

: 1,015,400 ft-lb

\ssume 60% Ambient Humidity

Bther Parameters: n = 7.3, m = 6.1 (Table 4.4.1.1), vu = 1.64, vs = vt,

yRa = (0.54) (1.64) (0.78) = 0.69, where vt/ vu = 0.54 (Eq. 2-8)

lnd creepYta= 0.78 (Eq. 2-12), ( &Sh)u = 487 x 10B6 in/in,(m/m)

p, = 0.14, F,/F, = 0.18 (Table 4.4.1.2),

uid (1 + n ~6~) = 1.25 (Design simplification)

Table 4.6.2 Material and Section Properties, Parameters andConditions for Example 4.6 (SI Units)

UNSHORED COMPOSITE GIRDER Material Properties:

Y 2.34m Y Steam Cured Normal Weight Concrete

178mm t fPu

= 1862 MPa

f . = 27.6 MPa, f; = 34.5 MPaCOMPO_SED N.A.

li:-1

- -

iDZk f; = 27.6 MPa

EGIRD_ER Se N . A .

iEGirder/ESlab = E2/El = 2.68/2.51 = 1.07

Inz E ci = 2.51x104MPa, Ec = 2.68x104MPa0-l b)c\I(D . . . . . . . . . .. . . . . . . . . . . Section Properties and Loading from

. . . . . . . . . . .k6Omm A

PC1 HandbookGirder:

A A S H T O I VSpacing = 2.34m, Ag = 0.509 m2

32 - 12.7mm S T R A N D S . 2 P T . D E P .

:: ; ;;~;;S;;;~;:~8<1 2KN,m

eO 9 CS I M P L E S P A N

Calculated Section Properties and Loading -- Composite Section:

Yodified Slab Area = 0.178x2.34/107 = 0.389m2, yb=(0.509x0.628+0.389x

1.46)/(0.509+0.389)=988m,=O.1O85+O.5O9(O.988-O.623)2+2.34xO.1783I/(12x1.07)+0.389(1.46-0.988) =0.2622m492

$=0.2622/(1.549-0.988)=0.4674m3, Zb=0.2622/.988=0.2653m3Including 12.7mm w.s., Slab D.L.=ws=(0.191x2.34)x2.4x9.807=10.5KN/m

iS20-44 AASHTO Loading, Impact=50/(80+125)=0.254ssume Deck Slab Cast 2 Months After Prestressing

4rea of One 12.7mn Strand=g.87x105m2 (Fig. 11.3.3, PC1 Handbook)lps=(32)(9.87x10~5)=3.16x10~3m2, f,i=(0.70)1862-1303MPa4D=WdL2/8=12(24.38)2/8=891.6Kllm, fsi=(0.70)1862-1303MPaq 847.gKNm, MD+MS,Di =1739.5KNm, Interior Girder ML+I=(1579.5--lASHTO Table) (l/2--Single Wheels)(l.Z5--L+I)(2.34x3.28/5.5--AASHTO, S/5.5)=13.76.7KNmlssume 60% Ambient Relative Humidity

1ther Parameters: n=7.3,m=6.1 [Table 4.4.1.1), vu=1.64,vs =vt,Ye =(0.54)(1.64)(0.78)=0.69, where vt/vu=0.54 (Eq. 2-8)

and creepYga =0.78 (Eq. 2-12), ( Esh)u’487XloB6in/in, (m/m)

-,/F,=O.14, F,/F, = 0.18 (Table 4.4.1.2),

and (l+noc s) = 1.25 (Design simplification)

Table

4.6.3

Term by Term Loss of Prestress

and

Ultimate (in

time)

Midspan

Loss for

Example

in 4.6, Composite

AASHTO Type IV Girder,

Normal

Weight

Concrete

Units

ksi

MPa %

Cree

pBe

fore

Slab

Cast

11.61

80.0

5

6.14

Losses

Shrink-

age

10.9

1

75.2

2

5.77

Relax-

ation

t

14.18

97.77

7.50

I

(ml) 49.0

Term by Term Camber,

Deflection

for

Example

in Composite

AASHTO

Weight

Concrete

and

Ultimate Midspan Values

Type IV Girder,

Normal

Initial

Creep

Cree

pCr

eep

Creep

Elastic

Defl.

Camber Camber Defl.

Defl.

Defl.

Due

ToUp To

After

Up To

After

Due

ToBe

amSl

abSlab

Slab

Slab

Slab

D.L.

Cast

Cast

Cast

Cast

Cast

-0.8

01.

220.

50-0

.71

-0.2

5-0

.74

-20.

331

.012

.7-1

8.0

-6.3

-18.

8

Cree

pDefl.

Due

ToSlab

D.L.

-0.39

-0.9

Live Load Plus Impact Deflection = -

0.50",

(-12.7mm)

Defl.

Due

ToDiff.

Shrink-

age

and

~ Creep

Total

-0.63

0.13

-16.

0 3.

3I

1 Gains

II

Elas-

tic,

Due

toSlab

Creep,

Due

toSlab

Diff.

Shrink-

age

and

Cree

p

-3.7

3-1

.58

-3.4

4

-25.

72-1

0.89

-23.

72

-1.9

7-0

.84

-1.8

2

Total

Loss L45.98

317.02

24.3

Table

4.6.4

Initial

Camber

Due

ToUnits

Prestr.

Inch

1.93

Table 4.7.1 Elastic Deflection Coefficients cfp and ctws (1~10~~) forTn+nrinr Dams1I,ICC1 I”, Cca11SI.

We Interior Panel Support E/S

1.0 1 . 1 1.2 m 1.4 1.5Cf

(Flat glates)Zero edge beam 0.0

c/a O.l*581 487 428 387 358 337'stiffness 441 372 320 283 260 243

Elastically support- Relativelyed edges. The appro- flexible 380 330 290 260 240 230priate coefficient is edge beamsin between the case (total depth 2:: 250" 21: 1;: 1;: 1::

5of zero edge beams about 2t)*

tws(Two-way

stiffness (flate plate) Relativelyslabs) and infinitely stiff stiff edge 290 260 230 210 190 180

edge beams (rigid sup- beams (totalports). depth 1:: 1:: 1;: 1:; ii; k:

about 3t)*Rigids supports. Built-in edges oninfinitely stiff edge beams 126 102 83 67 54 43

*approximate values c/a = column/span ratioa/s = longer span/shorter span ratio

Table 5.1.1 Aging Coefficient

t-Q).&

days

lo1

lo2

lo3

VU

0.5

12:;3.5

0.5

:::3.5

0.5

i-i:3:5

tll in days

.525 .804

.720 .826

.774 .842

.806 .856

.505 .888

.739 .919

.804 .935

.839 .946

.811

.825

.837

.848

.916

.932

.943

.951

.809

.820

.830

.839

.915

.928

.938

.946

.511 .912 .973 .981

.732 .943 .981 .985

.795 .956 .985 .988

.830 .964 .987 .990

.501 .899 .976 .994

.717 .934 .983 .995

.781 .949 .986 .996

.818 .958 .989 .997