Stress and Strain Analysisin Prestressed Concrete:

A Critical Reviewby

Amin GhaliProfessor of Civil EngineeringThe University of CalgaryCalgary, Alberta, Canada

R einforced concrete structures with-out prestressing generally crack and

in some cases have excessive deflection.If some of the reinforcement is pre-stressed steel, cracking and deflectioncan be controlled. The amount of pre-stressing may be sufficient to avoidcracking. This condition is referred to asfull prestressing, as opposed to partialprestressing where controlled crackingis allowed. Partial prestressing is widelyused and is often considered preferableto the two extremes of no prestressingand full prestressing. When full pre-stressing is adopted, nonprestressedsteel is commonly provided to controlcracking due to temperature and shrink-age.

Thus, in general, cross sections ofreinforced concrete members have twotypes of reinforcements: prestressed andnonprestressed.

In current practice, the stress and

strain in a section due to prestressing arecalculated by considering the effect of acompressive force on a plain concretesection. The loss in tension in the pre-stressed steel due to creep, shrinkageand relaxation is estimated; and thetime-dependent changes in stress andstrain are determined accordingly.Equations which give the loss in tensionin a prestressing tendon can be found inmany codes and design recommenda-tions (e.g., Refs. 1, 2 and 3).

This concept, which ignores the non-prestressed steel, results in errors inprediction of stresses and in larger er-rors in strains and deflections. It will beshown that the magnitudes of the errorsare large with the amount of nonpre-stressed steel common in fully pre-stressed or in partially prestressed sec-tions.

A much more accurate approach isadopted in the CPCI Design Manual.

80

An equation is given to calculate 1P,representing the change in the resultantcompression on the concrete taking intoaccount (approximately) the effect of thenonprestressed steel. The force, APB, isconsidered to act on a plain concretesection and the time-dependent changesin stress and strain are calculated ac-cordingly; the advantages and limita-tions of this concept will be discussed.

This paper suggests that empiricalequations which predict the loss in ten-sion in prestressed tendons are notneeded. Accuracy in stress and strainanalysis can be achieved by considera-tion of equilibrium and compatibilityand recognition that, in general, pre-stressed sections have three compo-nents: concrete, prestressed steel andnonprestressed steel. The approach isrelatively simple and is general, i.e., itapplies to reinforced and compositesections with or without prestressing.

The paper will also show that whenpartial prestressing is used, the amountof nonprestressed steel in the cross sec-tions significantly influences the loadlevel at which cracking occurs, thecurvatures or deflections after cracking,the changes in stresses in the reinforce-ment at cracking and the crack widths.

Where appropriate, numerical exam-ples show the various discrepancies incurrent design methods.

SIGN CONVENTIONTensile stress and the corresponding

strain are positive (see Fig. Al, Appen-dix A). A positive bending moment andthe corresponding curvature are posi-tive. The symbol y denotes a coordinatedefining position of any fiber with re-spect to a reference point 0; y is posi-tive for fibers below O. The symbol Aindicates a change in value; a positiveand a negative A mean respectively anincrease and a decrease in value. Therelaxation of prestressed steel, Ao- ,., iscommonly a reduction in tension; thus anegative value.

CONVENTIONAL ANDACCURATE ANALYSES

In, a conventional analysis of pre-stressed concrete sections, empiricalequations are used to calculate Per^^l,,the tensile force in the prestressing steelafter occurrence of creep, shrinkage andrelaxation; different empirical equationsgive different results. But the correctvalue of Peffecti„e should satisfy the equi-librium of forces and strains in concreteand steel.

In this paper the term "accurate"analysis refers to an analysis in whichthe nonprestressed steel is consideredand compatibility of strains in concreteand in the prestressed and the nonpre-stressed steel is ensured. Only basicmechanics equations are used, withoutrecourse to empirical equations.

The results of accurate and conven-

PCI JOURNAL/November-December 1989 81

tional analyses will be compared. Theterm "conventional" analysis will meanan analysis in which the nonprestressedsteel is ignored. For a meaningful com-parison, the use of an empirical equationfor' effective is also avoided in the con-ventional analysis by the use of basicmechanics ensuring equilibrium andcompatibility. Thus, in the special caseof a section without nonprestressedsteel the conventional and the accurateanalyses are the same.

EQUILIBRIUM ANDCOMPATIBILITYREQUIREMENTS

The accurate analysis can be executedin four simple steps which are presentedbriefly in a separate section; more de-tails and derivation of the method aregiven in Refs. 5 and 6. The presentpaper compares results of severalanalyses to show the errors in the con-ventional analysis in which the nonpre-stressed steel is ignored.

A correct analysis must satisfy thefollowing compatibility and equilibriumconditions:

1. The strains in steel and concrete atany fiber must be equal before and afterthe time-dependent changes. An excep-tion where this compatibility require-ment does not apply is at a post-ten-sioned tendon at the time of prestress-ing; the requirement applies only to thetime-dependent strains occurring afteranchorage of the tendons.

The time-dependent strain in a non-bonded tendon can differ slightly fromthe strain in the adjacent concrete, butthis difference is ignored here.

2. In a statically determinate struc-ture, the prestressing produces no reac-tions. At the time of prestressing, thetensile force in the tendon and the com-pressive forces in the concrete and inthe nonprestressed steel are three forcesin equilibrium. They thus have zero re-

sultant components. Also, the changesAPB , AP,, and APn, in the forces in thethree components due to creep, shrink-age and relaxation must not disturb theequilibrium. In other words, the sum ofthe three 0 values must be zero andtheir moments about any referencepoint 0 must vanish.

Prestressing of statically indeterm-inate structures produces reactions andinternal forces, which require structuralanalysis of the static indeterminacy,before the stresses and strains at anysection can be calculated. The forces inthe concrete, in the prestressed and inthe nonprestressed steel in any sectionof a statically indeterminate structuremust have resultants equal to the stati-cally indeterminate internal forcescalculated a priori.

The results of stress and strain analysesgiven in this paper are for sections ofstatically determinate simple beams.For these cases it can be verified thatthe forces in the concrete, the pre-stressed and the nonprestressed steelhave zero resultants, before and after thetime-dependent changes.

EXAMPLEA simplified structure will be

analyzed by the conventional and by theaccurate methods and the results com-pared. The structure shown in Fig. 1represents a 1 ft (305 mm) wide strip of apost-tensioned, simply supported solidslab.

At time to , the structure is subjected todead load q = 0.40 kip/ft (5.8 kN/m) andan initial prestressing force P = 290 kips(1300 kN), which is assumed constantover the length. At time t, after occur-rence of creep, shrinkage and relaxation,a uniform live load p = 1.00 kip/ft (14.6kN/m) is applied. The intensity, p, issufficient to produce cracking at mid-span. The tensile strength of concrete attime t is fit = 360 psi (2.5 MPa). Themodulus of elasticity of concrete, E,(t0),

82

PARABOLA

l5T 24 in. 15

50ft (15m)

T Ans Aps=1.5 in230 in.

^Ans 216 in.

b=12 in.

CROSS - SECTIONAT MID-SPAN

4)(t,to)=3.0

Ecs (t, to) 2 -300 x 10-6

OFpr = -9.3 ksi (-64 MPa)fct = 36O psi (2.5 MPa )

Fig. 1. Example of a post-tensioned slab. Note: 1 in. = 25.4 mm.

is 4350 ksi (30.0 GPa) and is assumedconstant with time. The modulus ofelasticity of the prestressed and thenonprestressed steel, E 8, is 2900 ksi (200GPa).

Nonprestressed steel of equal crosssection area, is provided at top andbottom. The steel ratio, p„3 = A„8lbh, isconsidered variable between zero and 1percent. Other data are given in Fig. 1.

The objective of the analyses is todetermine stresses, strains and deflec-tions at time to and at time t just beforeand after application of the live load.The same values of the creep coeffi-cient, 4(t,to ), the free shrinkage, E 8(t,t0),and the reduced relaxation, ^Q areused in the accurate and in the conven-tional analyses. The two analyses differonly in that the nonprestressed steel isignored in the conventional analysis.The parameters O, ec$ and &5. are de-fined in Appendix B.

For clarity of presentation and for easein verification of the results presentedherein, the simplest cross section is cho-sen. Also, the variation of the initial pre-stressing force P because of friction isignored and the difference in the cross

section area of the tendon and that of theprestressing duct is neglected. In addi-tion, the modulus of elasticity of con-crete is assumed constant with time.None of these simplifications is essen-tial for the method of analysis used.

The bending moment at midspan dueto q and p are, respectively, 1500 and3750 kip-in. (169 and 424 kN-m).

Stresses and DeflectionsBefore Cracking

Table 1 gives the concrete stresses atmidspan at time to immediately afterprestressing and at time t after occur-rence of creep, shrinkage and relaxation,but before application of the live load. Itcan be seen that the stress at the bottomfiber varies between –1026 and –502psi (-7.08 and –3.46 MPa) as the non-prestressed steel ratio, p, is increasedfrom zero to 1 percent. The stress differ-ence is greater than f t .

In other words, ignoring the nonpre-stressed steel substantially overesti-mates the compressive stress providedby prestressing to prevent or to controlcracking by subsequent live load; the

PCI JOURNAL/November-December 1989 83

Table 1. Stresses and deflections at midspan before cracking in Example (Fig. 1).

0.4 withreduced

Nonprestressed steel ratio p18 (percent) 0 0.2 0.4 0.6 0.8 1.0 0 & ens

Concrete stresses at atop –302 –276 –246 –215 –184 –155 –250time t (psi) obt –1026 –879 –759 –659 –574 –502 –969

Change of Concrete OPT 52 76 97 114 128 140 59force inthe three Prestressed OP„^ 0 –28 –52 –72 –88 –102 –28materials steelbetweent o and t Nonprestressed APp, –52 –48 –45 –42 –40 –38 –30

(kips) steel

Deflection at time tbefore application of –923 –794 –696 –621 –560 –510 –553the live load (10-3 in.)

Ratio of deflection at time t,before application of live load 2.56 2.32 2.13 2.00 1.88 1.78 1.69to the instantaneous deflection

Steel stresses o (bot) –36 –33 –30 –28 –26 –24 –20at time t,before liveload application 0-ps 159 161 163 165 167 168 173

(ksi)

Remarks: Negative deflection means camber.Conversion factors: I psi = 6.9 x IU -3 VIPa; 1 kip = 4.45 kN.

overestimation is of the same order ofmagnitude or greater than the tensilestrength of concrete. The compressivestress reserve, intended to counteractthe tension due to live load, is substan-tially eroded as a result of the presenceofnonprestressed steel.

Table 1 also gives the force changesA P ' A P A P, in the. concrete, theprestressed steel and the nonpre-stressed steel due to creep, shrinkageand relaxation. It can be seen that thechange in forces in the concrete and inthe prestressed steel are equal in abso-lute value only when p,,, = 0. In othercases, A PP is substantially larger than

I A Ppg 1; the difference between the twoquantities is equal to I A P, 1, which rep-resents the compressive force picked upby the nonprestressed steel.

The sign of A P. is positive indicating

a decrease of the initial compressiveforce in concrete. The negative A P. in-dicates an increase in compression.Also, the negative A P., indicates loss oftension in the prestressing tendon.

The nondimensional graphs in Fig. 2represent the variation of p /t8 versus

(IA1? Il4er)or(APcld,.ef); where Areris a reference force equal to A PP / I A P„gwhen p, = 0. The difference betweenthe ordinates of the two curves in Fig. 2is equal to I A P. I / D,.f ; it represents therelative increase in compressive force inthe nonprestressed steel.

In a conventional analysis, the com-pressive stress in concrete after creep,shrinkage and relaxation will be greatlyoverestimated when considered equalto Pe ,tti„e = P + A P; where P is theabsolute value of the initial prestressingforce. Accordingly, the absolute value of

84

A ps= 1.5in2 ; b= 12in.;

Ans 2 AL1 P ns h = 30 in. ; P ns

A ref h = bh

30 in. Aps3.0

Ans ^ 6 in.

D Pc2.0

refIAPnsI

A ref I

1.0

IAPps1

ref0 i0 .002 .004 .006 .008 .010 °ns

Fig. 2. Relative time-dependent change in forces in concrete, prestressed steeland nonprestressed steel.

the compressive stresses in concreteafter losses will be greatly overesti-mated.

The loss in tension in prestressingsteel, 0 P„3 , should not be used to cal-culate the loss in compression in theconcrete. The empirical equations givenin various codes and design recommen-dations for estimating 0 P„$ are of lim-ited use, if any. It would be better todelete such equations because they arenot accurate and the answer they give isfrequently used incorrectly as equal inabsolute value to A P.

Table 1 also gives the deflection at thecenter of span with varying p„$ . Thenegative sign indicates camber. It isclear that camber is overestimated ifnonprestressed steel is ignored. Also, itcan be seen that the deflection aftercreep, shrinkage and relaxation cannotbe accurately predicted by multiplyingthe instantaneous deflection by a speci-fied number, because such a numbermust vary with p„$ and with the creep,

shrinkage and relaxation parameters(varies between 1.69 and 2.56 in thisparticular example).

Stresses, Deflections AfterCracking and Crack Width

Table 2 gives the stresses and centraldeflection after application of live load p= 1 kip/ft, which is more than sufficientto produce cracking. The correspondingmoment at midspan is 3750 kip-in. Theload p,,. and the corresponding momentMer at midspan at which cracking firstoccurs arc also given in Table 2.

Values which are frequently of con-cern in design for serviceability whencracking is allowed is the crack widthand the change in steel stresses due tolive load. The former is of concern forcorrosion protection and the latter forsafety against fatigue. Table 2 gives thecrack width w divided by the crackspacing s; it also gives the stresschanges Ao-. and Ao-,. in the prestressed

PCI JOURNAL/November-December 1989 85

Table 2. Stresses, deflection and crack width at midspan after live load application inExample (Fig. 1).

0.4 withNonprestressed steel ratio reducedp,, (percent) 0 0.2 0.4 0.6 0.8 1.0 6 &€,,

Live load bending moment at which 2600 2400 2300 2200 2100 2000 2700cracking occurs (kip-in.)

Ratio of uniform load intensityP^, at which cracking occurs top 0.69 0.65 0.61 0.58 0.56 0.54 0.73

(p = 1 kip/ft)

Deflection after application ofp 1250 1229 1182 1128 1074 1022 976(10-3 in.)

Steel stresses after cr„$(hot) -7 -6 -5 -5 -4 -4 -2application ofp (ksi) cps 180 181 182 183 183 183 187

Stress changes in steel Au- (bot) 29 27 25 23 21 20 18caused by p (ksi) Ao-, 22 20 19 17 16 15 14

Ratio of crack width tocrack spacing (10-s ) 0.72 0.68 0.65 0.62 0.58 0.55 0.39

Conversion factors: 1 kip/ft = 14.6 kN/m; 1 ksi = 6.9 N1Pa.

and in the nonprestressed steel due to p= 1 kip/ft.

Based on the results in Table 2, thefollowing remarks can be made:

(a) Ignoring the nonprestressed steelwill cause the deflection to be overesti-mated.

(b) The level of loading at whichcracking occurs will be overestimatedwhen cracking is ignored.

(c) The steel stress increments, &jand 0v decrease with the increase inp,,. Thus, p„S can be selected to ensuresafety against fatigue.

(d) The crack width can be controlledby appropriate choice of p. The ratiow/s, crack width to crack spacing, is in-cluded in Table 2 rather than w, be-cause s depends not only onp,^$ , but alsoon how the nonprestressed steel is ar-ranged in the section. The crack spacings decreases with the increase of p,,.Thus, w decreases faster than the ratiow/s asp,..is increased.

(e) The stress in the bottom nonpre-stressed steel is in compression despitecracking.

STRESS AND STRAIN INNONCRACKED SECTIONSThe instantaneous and time-depen-

dent changes in stress and strain in non-cracked sections can be determined infour steps (Fig. 3):

Step 1 — Apply the initial prestress-ing force and the dead load bendingmoment which becomes effective at thetime of prestressing on a transformedsection composed of A, plus (a13Ap +an3 Ans ); where the subscripts ps and nsrefer to the prestressed and the nonpre-stressed steel, respectively; a is equal toE93 orE„g divided by E, (t0).

Here, the transformed section in-cludes only the prestressed and thenonprestressed steel bonded to the con-crete at prestress transfer. Thus, AP$should be included when pretensioningis used, but when all the prestressing ispost-tensioned in one stage, A and thearea of the duct should be excluded.

When the structure is statically inde-terminate, the indeterminate bendingmoment due to prestressing should be

86

7RESTRAINT

AM

ND

STEP 3 : ARTIFITIAL RESTRAINTOF CONCRETE DEFORMATION

STEP i : INSTANEOUS STRAIN STEP 2 : FREE SHRINKAGEAND CREEP

Fig. 3. Steps of analysis of time-dependent strain and stress.

included in the value of the moment.The values eo (to ) and 4i (to ) defining

the distribution of the instantaneousstrain can be determined by Eqs. (Al)[Appendix A] ; where eo (to ) is the valueof the instantaneous strain at an arbitraryreference point 0 and qr (to ) is thecurvature which is equal to the slopede/dy of the strain diagram (Fig. Al).Multiplication by E, (to ) gives Qo (t0)and y (t0 ), which represent respectivelythe stress at Point 0 and the slopedo- /dy of the stress diagram (Fig. Al).

Step 2 — Determine the hypotheticalchange in strain distribution due tocreep and shrinkage of concrete if theywere free to occur. The strain change atO is equal to [4)eo (to ) + € I and thechange in curvature is 4) i[r (to ); where 0is the creep coefficient and e,$ is the freeshrinkage (see Appendix B).

Step 3 — Calculate artificial stresswhich when gradually introduced on

the concrete during the period to to twill prevent occurrence of the straindetermined in Step 2. The restrainingstress at any fiber y is:

o-reStrnint = E, l [e (to ) + t/! (to ) 1f] + Ecsl

(1)

where E, is the age-adjusted modulus ofelasticity of concrete [Eq. (B6)] .

Step 4 — Determine by Eqs. (A3) aforce at 0 and a moment which are theresultants of ioTeOtT62ft . The strain inconcrete due to relaxation of the pre-stressed steel can be artificially pre-vented by the application, at the level ofthe prestressed steel, of a force equal to( Q8 A,, ).

This force is substituted by a forceof the same magnitude at 0 plus acouple. Summing up gives AN and OM,the restraining normal force and thecouple required to prevent artificially

PCI JOURNAUNovember-December 1989 87

the strain due to creep, shrinkage andrelaxation.

To eliminate the artificial restraint,apply 0 N and A M in reversed direc-tions on an age-adjusted transformedsection composed of A, plus (&,3 A„8 +

&,A) where a is equal to Ep$ or E,;$ di-vided byEE (t,to ).

The strain distribution at time t is thesum of the strains determined in Steps 1and 4, while the corresponding stress isthe sum of stresses calculated in Steps 1,3 and 4.

Commentary:1. Superposition of strains or stresses

can be done simply by summing up theincrements Deo and 44i in various stepsor &r0 and dy, using the same referencepoint 0 in all steps. For this reason, Eqs.(A3) and (A4) are used here rather thanthe more familiar Eqs. (A5) and (A6)which require that 0 be the sectioncentroid.

2. The four steps give the stress andstrain at time t, without preceding theanalysis by an estimate of the loss intension in the prestressed steel. Noempirical equation is involved for losscalculation.

3. The compatibility of strain in con-crete and steel is maintained at all thereinforcement layers.

4. The procedure of analysis is simi-lar to the well-known displacementmethod of structural analysis, where thedisplacements are prevented artificiallyby restraining forces which are subse-quently eliminated by application of thesame forces in reversed directions.

5. The same four steps apply to rein-forced concrete sections without pre-stressing simply by setting A78 = 0; theeffect of cracking will be discussed inthe following section.

6. Composite sections made of morethan one type of concrete, cast or pre-stressed in stages or sections made ofconcrete and structural steel, can beanalyzed by the same procedure.6

7. The deflection can be determinedfrom the curvature by numerical inte-gration. In simple or continuous beamsof constant or variable cross sections, thefollowing equation can be used to giveapproximately the deflection at mid-span:

D 96 ( + 10413 + 413 ) (2)

where 0, and 4 are the curvatures at theends and qr is the midspan curvature.This geometric relation is exact whenthe variation of 0 between the three val-ues is parabolic.

STRESS AND STRAIN INCRACKED SECTIONS

Consider a prestressed section forwhich the distribution of stress at anytime t due to prestressing and dead loadare defined by two known quantities:a-, (t) and y(t). Assume that due to liveload the section is subjected to addi-tional bending moment M and axialforce N at centroid 0. Consider the casewhen the magnitudes of N and M aresufficient to produce cracking. Todetermine the strain and stress changesdue to live load, partition M and N:

N=N1 +N2 ; M=M1 +M2 (3)

Note that Nl and M1 , referred to as thedecompression forces, represent theforces which are just sufficient to bringthe stresses in the concrete to zero. Eqs.(A3) can be used to calculate Nl and Mlas the stress resultants of a stress dis-tribution defined by – o (t) and – y (t).

Cracking is produced by N2 and M2,which can be considered to act on rein-forced concrete sections, without dis-tinction between the prestressed andnonprestressed steels (except for theirrespective E8 values). After decompres-sion, N2 and M2 act on a section with zeroinitial stress in the concrete.

The forces N2 and M2 are considered toact on a transformed section composedof the area of concrete in compression

88

plus (a A,,$ + a„, A) where a is E93 orE 3 divided by E W. The depth c of thecompression zone is determined by theequilibrium equation and by the as-sumption that plane cross sections re-main plane. For rectangular or T-sec-tions, the determination of c involvesthe solution of a cubic equation given invarious references (e.g., Ref. 6).

Once c is determined, and the areaproperties of the transformed crackedsection are calculated, the equations ofAppendix A can be used to determinethe stress and strain changes due to N2

and M2.The total change in strain due to N

and M is the sum of the change in thedecompression stage due to N l and Ml,plus the change due to N2 and M2 . Butthe change in stress due to N2 and Mzgives the total stress in concrete aftercracking. The changes in stresses in theprestressed and the nonprestressedsteels due to NZ and M2 are the values tobe used in calculating the crack width(see Ref. 8 or 6) or to assess the fatiguelife of the structure. It should be notedthat the stresses due to N2 and M2 repre-sent the total stress in the concrete aftercracking but not the total stresses in thereinforcements.

The above procedure gives the strainand stress in a section due to transientlive load producing cracking. If the loadproducing cracking is sustained and it isrequired to determine the changes instrain and stress between time t and alater time t due to creep, shrinkage andrelaxation, Steps 2 to 4 of the precedingsection can be applied for a cross sectionof depth c, which is the depth of thecompression zone at time t. This ignoresthe time-dependent change in c, and re-sults in negligible error.'

COMPUTER PROGRAMSThe steps discussed above for analysis

of the instantaneous and time-depen-dent stresses and strains and thechanges in these values after cracking

involve repetitious calculations of sec-tion properties, which can be easilyperformed by the use of programmablecalculators. The computer programCRACK S can be used to give thestresses, strains and displacements inreinforced concrete and compositemembers with or without prestressing.The program CRACK, which is usableon IBM microcomputers, does not per-form an analysis of statically indetermi-nate forces.

A computer program CPF, CrackedPlane Frames,'° performs the sameanalyses as CRACK and in addition de-termines the statically determinateinternal forces and changes in theseforces due to creep, shrinkage and relax-ation and due to cracking. The effect ofcracking involves nonlinear iterativeanalysis discussed in Ref. 11. ProgramCPF is suitable for composite structuresbuilt and prestressed in stages (e.g.,segmental bridges and multistorybuilding frames) and is usable on mi-crocomputers.

EFFECTS OFVARYING CREEP AND

SHRINKAGE PARAMETERSIt is sometimes argued that the effort

required for an accurate prediction ofstresses and strains in service conditionsis not justified because accurate valuesof the creep coefficient (A and the freeshrinkage EC, are not commonly avail-able. A more rational approach, for im-portant structures, is to perform accurateanalyses using upper and lower boundsofthe parameters 0 and €CB.

The analysis for the midspan sectionin the Example is repeated for the casep78 = 0.4 with ¢ = 1.5 and e,$ = –150 x10^. The results are given in the lastcolumn in Tables 1 and 2. It can be seenthat, for the case considered, varying 0and EC8 by a factor of two has some effect,but the effect is not as important as theeffect of ignoring nonprestressed steel.

PCI JOURNAUNovember-December 1989 89

LOSS OF COMPRESSION INCONCRETE

As mentioned above, the time-depen-dent loss in tension in a prestressed ten-don is not generally equal in absolutevalue to the change in compression inthe concrete. The following equationgives the time-dependent increment ofthe resultant force on concrete:

EA Pc — #

0Qcst(to) Ec(to) +Ecs Est x

Ast + Aff A9. } (4)

where11

1 + ASt ESt (1 + y8t ^1 -(̂5)Ac E, r^^

with A3 = A AP$ = total steel area; Egt

= modulus of elasticity of steel assumedthe same for the two steel types; Y.,t is they coordinate of the centroid ofA$t , mea-sured downward from the centroid ofAc ; ° cSt (to ) is the concrete stress at timet, at y equals yet ; r, = Ic 1A,, where I, isthe second moment of A, about an axisthrough its centroid.

The quantity A P ' commonly posi-tive, represents a loss in compression; itis equal in absolute value to the loss intension in A„$ only in the absence ofA, .

Eq. (4) is based on the assumption thatall the reinforcements, A,^, and A93 , areconcentrated at one layer at distance yot

below the centroid of A, . The com-patibility of strains at this layer at time tin the nonprestressed steel, the pre-stressed steel and the concrete requiresthat:

otrns __ n-PB — A& .

Est Egt

_ ^ocst(to) +E + APQE,(to ) E,

A + Ict / (6)

The first and second terms on the last

side of this equation are the creep andshrinkage, while the third term is thestrain recovery due to a gradually intro-duced force, A PP .

For equilibrium, the sum of thechange in forces in the three materialsmust be zero:

A P + An$ OQns + AP$ Ao-P$ = 0 (7)

The solution of Eqs. (6) and (7) for P,,gives Eq. (4).

When the value PP is determined, itcan be used to calculate the time-de-pendent change in strain in concrete at y= ye as the sum of the three terms on thelast side of Eq. (6). The correspond-ing change in stress is simply equal tothe last of the three terms multiplied byE. The time-dependent changes insteel stresses, Ao and Otr9$ , also followfrom Eq. (6). The change in curvaturebetween to and t is given by:

AJ (t, to) _ , (t0) + 0 P^ tyat (8)

where jr (to ) is the instantaneous curva-ture.

Eq. (4), which is adopted in the CPCIDesign Manual4 is strictly valid onlywhen A.$ and A93 are situated in onelayer. When this condition is satisfied,use of Eqs. (4) or the four steps in Fig.3 gives identical results. Eq. (4) alsoapplies when the centroids of A., A93and A,,, coincide and the section is sub-jected to concentric force withoutbending moment.

When Eqs. (4) and (6) are applied inother cases, an approximate value of A P,will be obtained; the force PP should beapplied on A, at the centroid ofAst . Thisapproximate analysis will not ensurecompatibility of strains in concrete andsteel at all reinforcement layers.

Prestressed concrete designers ac-customed to using an equation for theprestress loss may prefer Eq. (4) to thefour steps represented in Fig. 3. How-ever, it should be noted that Eq. (4)gives the change in compressive force

90

on the concrete and not the loss in ten-sion in the prestressed tendon. Also, theinaccuracies in prediction of curvatureand the extreme fiber stresses should berecognized.

ACI-ASCE COMMITTEE 423,PRESTRESSED CONCRETEFor a section without nonprestressed

steel, the time-dependent changes inthe forces in the concrete and in the pre-stressed steel are of equal and oppositesigns; A P, = –A P. Thus, Eq. (4), re-versed in sign, can be used to give theloss in tension in the prestressed steeldue to creep shrinkage and relaxation.An example will be analyzed belowusing Eq. (4) and the results comparedwith equations given by ACI-ASCECommittee 423. The equations and theexample are taken from Ref. 3.

Consider a double T pretensionedsection of gross area 615 in 2 (Fig. 4);distance from centroid to soffit = 21.98in.; moment of inertia of gross area aboutcentroidal axis = 59720 in. 4; cross sec-tion area of prestressed steel A,, = 1.836in.2 and its centroid is at 4.43 in. aboutthe soffit.

At time to , a prestressing force P = 372kips and a dead load bending = 5232kip-in. are introduced. It is required todetermine the instantaneous change instress in the prestressed steel and thechanges in the same parameter due tocreep, shrinkage and relaxation betweento and a later time t.

The modulus of elasticity of concreteat to is E, (to ) = 2500 ksi. Consider creepcoefficient, 4) = 1.6; free shrinkage, E,, =–184x10-s ; intrinsic relaxation of pre-stressed steel = –5 ksi. Note that thelast two quantities are not explicitlygiven in Ref. 3; the values used here hadto be derived from the given data. Also,for simplicity, the self weight and theweight of concrete topping are assumedto have been introduced at to , producingthe moment given above.

CENTROID OF GROSSCROSS - SECTION AREA

21.984.43 in.

A ps = 1.836 inTf

Fig. 4. Pretensioned section treated byaccurate analysis and by equations fromRef. 3.

At the time of prestressing, the trans-formed section composed of A, plusa,, A„8 has the following properties: A =634 in?; distance between centroid andsoffit = 21.46 in.; moment of inertiaabout centroidal axis = 65320 in 4 Dis-tance from centroid of transformed sec-tion to prestressed steel = 21.46 – 4.43= 17.03 in.

At the level of prestressed steel, theconcrete stress at t, immediately aftertransfer[Eq. (A6)] is:

Qest to) _ – 372 + 5232 – 372(17.03)(

634 65320

x (17.03)

_ – 0.874ksi

The instantaneous change in stress inthe prestressed steel is:

OO1$ (to) = E6 (to) o-cst (to

= 2800 (-0.874)

= –9.789 ksi

The symbols used here for the two in-stantaneous stress changes have thefollowing equivalents in Ref. 3:–nest (t0 ) and ES = -- tom (to ). The twoequations recommended in Ref. 3 forthe two parameters [Eqs. (4.5.3) and(4.5.2)1 give: J. = 0.734 psi and ES =8.221 ksi. The answers differ because ofthe approximation resulting from the

PCI JOURNAL/November-December 1989 91

use of an empirical coefficient, KCZT , in-cluded in Eq. (4.5.3) of Ref. 3.

Because there is no prestressedsteel, EE = E. and A = A„$ . Division ofEq. (4) by A 9 and reversal of sign of theright-hand side, gives the following ex-pressions for the changes in the stress inthe prestressed steel due to the threecauses:Creep:

(LtJ s )creep = N ` 0-cst (to) EnsE,, (t0)

= –11.888 ksi

Shrinkage:

(Qo ns )shrinkage = N Ecs Ens=- 3.910ksi

Relaxation:

(Arils )relaxation = N A(Tnr_ – 3.036 ksi

The symbols used here for the three

time-dependent stress changes have thefollowing equivalents in Ref. 13: CR =— (,1 Qps )creep ; SH = — (o 8 )shrinkage ; RE— ( OO'ns )relaxation. Eqs. (4.5.4), (4.5.6)and (4.5.7) of Ref. 3 give: CR = 13.153;SH = 5.157 and RE = 3.937. The an-swers differ mainly because of the ab-sence of the dimensionless multiplier /j= 0.759 from the equations of Ref 3.The value of/3, calculated by Eq. (5), isalways smaller than 1.

It should be noted that the equationsof Ref. 3 cannot be applied in the moregeneral case when nonprestressed steelis provided. The answers obtained bythe equations of the same reference donot satisfy exactly the compatibilityconditions, i.e., Eq. (6). It can be ver-ified that the answers evaluated by Eq.(4) satisfy the compatibility require-ments; also, the same answers can bechecked following the four steps pre-sented in Fig. 3.

CONCLUSIONSReinforced concrete members are

generally composed of three compo-nents: concrete, nonprestressed andprestressed steels. Ignoring the nonpre-stressed steel in prestressed concretemembers can result in large errors in pre-diction of stresses and deformations. A rel-atively simple procedure, applicable tononcracked and cracked sections, can beused for a more accurate analysis basedon equilibrium of forces and compatibil-ity of strains in the three components.Two available computer programs whichcan be used for the analysis are described.

The current practice of analyzing theeffect of prestressing by considering acompressive force on a plain concretesection can produce significant errors. Itis suggested that such an analysis shouldbe treated as preliminary, particularlywhen the deflections are of concern orwhen cracking is not allowed. Theexample presented shows that the errorin prediction of stresses in concrete aftercreep, shrinkage and relaxation is of the

same order of magnitude as the tensilestrength of concrete.

The loss in tension in the prestressedtendon due to creep, shrinkage and re-laxation should not be used to representthe loss of compressive force on the con-crete. Empirical equations included inmany codes of practice for the loss intension in the prestressed steel are notreally useful. It is more useful to includein the codes an equation for 0 p , re-presenting the change in the resultantforce on the concrete[e.g., Eq. (4)].

The limitations of the equation shouldbe mentioned. The basic requirementsof equilibrium and compatibility whichlead to more accurate and more generalanalysis should be stated. Code com-mentaries and recommended practicesshould give the steps of analysis whichsatisfy the requirements. Unnecessaryapproximations should be avoided andthe equations which include empiricalvalues not accurate in all cases shouldbe eliminated.

92

REFERENCES

1. AASHTO, Standard Specifications forHighway Bridges, American Associationof State Highway and Transportation Of-ficials, 13th Edition, 1983, Washington,D.C.

2. Ontario Highway Bridge Design Code,Ministry of Transportation and Com-munications, Ontario, 2nd Edition, 1983.

3. PCI Design Handbook -Precast andPrestressed Concrete, 3rd Edition, Pre-cast/Prestressed Concrete Institute,Chicago, Illinois, 1985, pp. 4-40 to 4-42.

4. Canadian Prestressed Concrete Institute,Metric Design Manual, 2nd Edition,1987, CPCA publication, Ottawa (see p.3-26).

5. Ghali, A., "A Unified Approach for Pre-stressed and Nonprestressed ConcreteStructures," PCI JOURNAL, V. 31, No. 2,March-April 1986, pp. 118-137. See alsodiscussion, PCI JOURNAL, V. 32, No. 1,1987, pp. 133-140.

6. Ghali, A., and Favre, R., ConcreteStructures: Stresses and Deformation,Chapman and Hall, London and NewYork, 1986, 352 pp.

7. Tadros, M. K., "Expedient ServiceabilityAnalysis of Cracked Prestressed Con-crete Beams," PCI JOURNAL, V. 27, No.6, November-December 1982, pp. 67-86.

8. Favre, R., Beeby, A. W., Falkner,H.,

Koprna, M. and Schiessl, P., Crackingand Deformations CEB Manual, pub-lished and distributed by the Swiss Fed-eral Institute of Technology, Lausanne,Switzerland, 1985.

9. Ghali, A., and Elbadry, M. M., Manual ofComputer Program Crack — ResearchReport No. CE85-1, Department of CivilEngineering, The University of Calgary.Program is available on diskettes for IBMmicrocomputers, 1985.

10. Elbadry, M. M., and Ghali, A., Manual ofComputer Program CPF: Cracked PlaneFrames in Prestressed Concrete. Re-search Report No. CE85-2, Departmentof Civil Engineering, The University ofCalgary, 1985.

11. Elbadry, M. M., and Ghali, A., "Service-ability Design of Continuous PrestressedConcrete Structures," PCI JOURNAL, V.34, No. 1, January-February 1989, pp.54-91.

12. Ghali, A., and Trevino, J., "Relaxation ofSteel in Prestressed Concrete," PCIJOURNAL, V. 30, No. 5, 1957, pp. 82-94.

13. Trost, H., "Auswirkungen des Super-positionspringzips auf Kriechen — andRelaxations — Problems bei Beton andSpannbeton," Beton-und Stahl-betonbau, V. 62, No. 10, pp. 230-238 andNo. 11, 1967, pp. 261-269.

PCI JOURNAL/November-December 1989 93

APPENDIX A - RELATIONSHIPS BETWEENFORCES, STRESSES AND STRAINS IN A SECTION

The equations presented below givethe strain and stress distributions in across section subjected to a bendingmoment M and a normal force N at anarbitrary reference point 0 (Fig. Al).The cross section properties A, B and Iare given, representing respectively thearea, its first and second moment aboutan axis through the reference point. Useof Hooke's law and the assumption thatcross sections remain plane and equilib-rium consideration give:

E = E° +4jy ; Q = E(EO + ipy) (Al)

N= JcrdA ; M= Jo . ydA (A2)

Substitution of Eq. (Al) into Eq. (A2)

gives:

N=Aar° +By ; M=B y-° +Iy (A3)

Substitution of a-0 = E E° and y = E Jiinto Eqs. (A3) and solution gives:

_ IN–BM –BN +AM

EO E(AI – B 2 ) E(AI–Bz)(A4)

When 0 is at the centroid, B = 0 andEqs. (A3) and (A4) take the more familiarforms:

N=Av° ; M=Iy (A5)

EO = EA = I(A6)

Fig. Al. Definition of symbols and convention for positive, y, N, M, E, ,p, a- and y.

94

APPENDIX B - RELAXATION,CREEP AND SHRINKAGE

The term intrinsic relaxation of pre-stressed steel, Lo-,,,., will be used tomean the time-dependent drop in stressin a tendon stretched between two fixedpoints. The value OQpr is almost zerowhen the initial stress o- is equal to 40percent of the characteristic tensilestrength fptk ; but the value OQ„r in-creases with (A - 0.4)2 , where x = o lfptk

In a concrete member, the distancebetween the anchored ends of a tendonshortens due to creep and shrinkage ofconcrete, causing an additional drop oftension. The reduction in tensioncaused by creep and shrinkage has aneffect on the amount of relaxation simi-lar to the effect of reduction in X. Forthis reason, in prestressed concreteanalysis, a reduced relaxation valueshould be used:

1O-nr = Xr10nr (B1)

where Xr is a relaxation reduction factorgiven by (see Ref. 12):

r =e- (s.7+5.3X).ti (B2)X

with

Ao-" — OQpr(B3)SZ =

Qpr

Note that Ao is the change in stressin the prestressed steel due to the com-bined effect of creep, shrinkage and re-laxation. This value is not known apriori. A value Xr -_= 0.8 can be assumedand used to determine Lo-, from whichan improved value of Xr is determinedby Eq. (B2). The assumed and calcu-lated value commonly agree closelyafter one iteration.

The continuous line ABC in Fig. B1represents the time variation of strain inconcrete due to a stress increment Ao-(to ) introduced at time to and sustained,without change in magnitude. The totalstrain at time t, instantaneous pluscreep, is given by:

De (t) _ Ao-, (to) (1 + ) (B4)E, (to )

C

/ CREEP

z/ A

< / Ec(to) E(t0)S (t +X0)

B INSTANTANEOUS

O-c

c

STRAIN= 4a'cA E(to)

to t

STRESS INCREMENT Acrc INTRODUCED AT toAND SUSTAINED

---- STRESS INCREMENT Aoc (t, to) GRADUALLYINTRODUCED BETWEEN t AND to

Fig. B1. Time variation of strain caused by a stress increment

PCI JOURNAL/November-December 1989 95

where 0 _ 0 (t,t,) is the creep coeffi-cient, representing the ratio of creep oc-curring between to and t to the instan-taneous strain.

When a stress increment Ao- (t,to ) isgradually introduced between to and t,the strain variation with time can be re-presented by the dashed line in Fig. B1.The total strain produced during theperiod to to t is:

Ae (t ,t(, _ Ao-C (t,to) (B5)E'c (t, to )

where E, (t,to ) is the age-adjusted mod-ulus of elasticity of concrete:13

(B6)

where X is the concrete aging coeffi-cient.

The symbol e 0 = ec$ (t,to ) will be usedto represent the shrinkage which occursbetween to and t, when the shortening isfree to occur. The values of 0 and e^$depend upon the shape of the concretemember, the relative humidity of the air,the ages of the concrete at t,, and t.Suggested values of 0, €CS and X aregiven in Refs. 6 and 8. An approximatevalue of 0.8 is frequently used for both XandX,..

96

APPENDIX C - NOTATIONA = areaB = first moment of areaE = modulus of elasticityE, = age-adjusted modulus of elasticity

of concreteI = second moment of area1I = bending momentN = normal forceP = absolute value of prestressing

forcet = time or age of concretey = coordinateo- = stressa = ratio of modulus of elasticity of

steel to that of concrete= ratio of_modulus of elasticity of

steel to E',

X = APOlfIlk

X = aging coefficient for concreteX,. = relaxation reduction coefficientA = incremente = normal strain0 = creep coefficientiji = curvature (slope of strain diagram

= de/dy)y = slope of stress diagram (= do-/dy)

Subscriptsc = concretecs = shrinkagens = nonprestressed steelO = reference pointo = time of prestressingps = prestressed steelpr = prestressed steel relaxation

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

PCI JOURNAUNovember-December 1989 97