Relaxation of Steel inPrestressed Concrete

Jose TrevinoResearch Assistant

Federal Institute of TechnologyLausanne, Switzerland

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

A steel tendon stretched between twofixed points gradually loses a part

of its tension due to creep. The loss intension under constant strain, as in a testin which the length of the tendon ismaintained constant after stretching,will be referred to as the intrinsic relax-ation.

The amount of intrinsic relaxation de-pends on the ratio:

= og (1)f wherefpx , = initial stress immediately after

stretching (at time tt)= strength of prestressed steel

When X is smaller than 0.5 the intrin-sic relaxation is negligible, but its valueincreases rapidly as A approaches 1.

A tendon in a prestressed concretemember loses a part of its initial tensiondue to the combined effect of creep andshrinkage of concrete and steel relax-ation; the loss in tension is associatedwith shortening of the tendon. The re-

duction in tension caused by creep andshrinkage has the same effect on themagnitude of the relaxation as if the ini-tial stress were smaller.

Thus, the value of relaxation to beused in predicting prestress loss and theassociated deformations of prestressedconcrete members should be smallerthan the intrinsic relaxation obtainedfrom a constant length relaxation test.Hence, one can write:

L = XrLr (2)

whereLr = intrinsic relaxation which oc-

curs in a constant Iength ten-don; units of L, are force/length2

L. = a reduced relaxation value to beused in predicting prestress lossand deformation of concretemembers

Xr = relaxation reduction coefficient,a dimensionless value smallerthan unity

The purpose of this paper is to derivean expression for the relaxation reduc-

82

tion coefficient X,. and demonstrate howit can be used in practice for calculatingthe loss of prestress and determining thestress and strain after loss in a prc-stressed concrete cross section.

Ghali et aIs used a step-by-step com-putation procedure to derive values forthe relaxation reduction coefficient. Thepresent paper offers a more accurateevaluation of X, based on an equation forthe intrinsic relaxation adopted from thrCEB-FIP Model Code for ConcreteStructures, 1978 (MC-78). 2 A graph andan equation are presented herein for thecoefficient Xr.

The relaxation reduction coefficient X ris intended for use in practice as a mul-tiplier to the intrinsic relaxation value,Lr . The latter value may be based on testresults, often reported by steel suppli-ers, or calculated by empirical equa-tion.r

Most codes recognize the fact that themagnitude of relaxation of a tendon in aprestressed concrete member increaseswith the increase in initial steel stressand decreases with the increase of lossdue to creep and shrinkage. Some codesgive empirical expressions for the steelrelaxation as a function of the abovementioned parameters.

The equation presented here for therelaxation reduction coefficient X, in-cludes all the necessary parameters andis reached by rational derivation; henceit is more accurate. The equation is sim-ple to use without complicating the de-sign calculations, A numerical exampleis included in the paper, while the der-ivation of the coefficient ,.is given in anappendix.

SIGN CONVENTIONTensile stress or tensile force in steel

or concrete is assumed positive. Thesymbol AP represents a force incrementin concrete or in steel. A positive A Pindicates an increase in tension or a re-duction in compression. Loss of tensionin prestressed tendons due to the corn-

SynopsisRelaxation of prestressed steel in a

concrete member is of a smaller mag-nitude than the intrinsic relaxationwhich occurs in a tendon stretchedbetween two fixed points. A reducedrelaxation value should be employedin the calculation of prestress loss andthe corresponding deformation.

In this paper a graph and an equa-tion are presented for a relaxation re-duction coefficient to be employed asa multiplier to the intrinsic relaxationfor use in prestressed concrete de-sign. A numerical example is includedto show how the method can be ap-plied.

bined effects of creep, shrinkage andrelaxation, A P, is generally a negativevalue.

The same effects produce a change,AP,., in the resultant of the stress onconcrete; ] P, generally represents a re-duction in compression, hence a posi-tive quantity. Similarly, relaxation inprestressed steel is a reduction in ten-sion, hence a negative value.

INTRINSIC RELAXATIONThe magnitude of intrinsic relaxation

depends on the value of A and also onthe quality of steel. MC-782 refers to twogroups of steel. The first group withhigher intrinsic relaxation includescold-drawn wires and strands. The sec-ond group, with low relaxation, includesquenched and tempered wires andcold-drawn wires and strands which aretreated (stabilized) to achieve low relax-ation.

In the absence of relaxation tests,MC-78 gives a table which may beused to determine the value of the in-trinsic relaxation as a function of A (Fig.

PCI JOURNAL'September-October 1985 83

LrCO-_ ABSOLUTE VALUE OF THE ULTIMATE

fpsi LraD INTRINSIC RELAXATION

fpsi = INITIAL STRESS

f psu = CHARACTERISTIC TENSILE STRENGTH

0.25

^_!®

s M=

GROUP

0. 12 GROUP 2

0.100 06

.06

00300.3 0.4 0.5 06 0.7 0.8

= tpsi

tpsu

Fig. 1. Very long-term intrinsic relaxation of prestressing steels according tothe CEB-FIP Code.2

1). The values in the graph correspondto the two steel groups mentioned aboveand to aconstant length relaxation over aperiod of 0.5 x 10" hours, After this pe-riod the ultimate relaxation is consid-ered to have been reached.

The following equations closely ap-proximate the MC-78 values for the in-trinsic relaxation:

fI. ns+1(-0.4? when0.41

p I

and (3)

L.T„= 0 when X<O.4

whereL,m = value of the intrinsic relaxation

at time infinity= 1.5 or ( 2 ) for steels of Groups 1

and 2, respectivelyWhen the value of the intrinsic relax-

ation is known for a particular value of X,

Eq. (3) may be used to derive a value ofrt ,, for the type of steel considered. Sub-sequent use of the same equation vary-ing x gives the intrinsic relaxation forany initial stress value.

The intrinsic relaxation at any instantr depends upon the length of the period(7 – ti ); where t, is the time at which theinitial tension is applied.

For any type of steel or initial tension,the intrinsic relaxation at any timer maybe expressed as a product of the ultimateintrinsic relaxation L,,, and a dimen-sionless function of the period (r – t 1 ) inhours:

Lr(r) =Lrm L

1 Inl r 10t. + 1 (4)

when 0u(r`–t,)--1000

r — t ^Lr(T) = Lr (5j)0,5 x 10)C.2j

when 1000<(r– t,)-- 0.5 x 108

84

^ w is the absolute value of change ofstress in the prestressed steel due to thecombined effect of creep, shrinkage andrelaxation. is the absolute value ofthe intrinsic relaxation.

The parameter X is the ratio of the ini-tial tension at transfer, f, to the ulti-mate strength, f. With pretensioning,f„ t is the stress after elastic shortening.The relaxation occurring during the pe-riod between jacking and transfer (givenby Eq. (4)] takes place without changein tendon length and thus should not besubject to any reduction. With post-ten-sioning, the value fp, i is the initial stressat any section after a reduction ofthe loss due to friction and anchorset.

The relaxation reduction coefficient Xr

calculated by Eq. (7) or read from Fig. 2applies for any type of prestressingsteel.

PRESTRESS LOSSFig. 3 represents a prestressed con-

crete cross section of a member withprestressed and nonprestressed rein-forcement. At time t i prestress is intro-duced simultaneously with the load dueto the self weight of the member. Theinstantaneous stress and strain distribu-tions at t t can be determined by conven-tional equations.

Due to creep and shrinkage of con-crete and relaxation of prestressed steel,concrete loses a part of its compressionand prestressed steel loses a part of itstension. The nonprestressed steel gen-erally picks up some compression. Forany period (t – t o ), where t > to , thesum of the force increments in the threematerials must be zero; thus:

APB +A p" +AP,,,=0 (9)

The symbol A P represents a force in-crement, when positive it indicates anincrease in tension (or reduction incompression). The subscripts c, ps andns refer to concrete, prestressed steeland nonprestressed steel, respectively.

and

L, (r) = Lr„ (6)

when (r– tt) >0.5x10'

Results of intrinsic relaxation tests areusually reported for a period (r – t f ) =1000 hours. Eqs. (4) to (6) may be ap-plied to derive the relaxation value cor-responding to (T – t,) = x or to any otherperiod of time. The equations are basedon experimental values reported in Ref-erence 4.

REDUCED RELAXATIONCompare two tendons of the same

steel quality: the first in a constantlength relaxation test and the secondused in prestressing a concrete member.Assume that the initial stress a-„m , at timet, in the two tendons is the same. Be-cause of creep and shrinkage of con-crete, the stress at any instant r issmaller in the second tendon comparedto the first. Thus, the relaxation in thesecond tendon should be smaller thanthe first.

This may be accounted for empiricallyby considering that the relaxation in thesecond tendon to be the same as the in-trinsic relaxation for a reduced initialtension equal to the actual tensionminus a fraction of the total loss of pre-stress due to the combined effect ofcreep, shrinkage and relaxation. Thisfraction is 0.3 according to MC-78.2

A more accurate approach is to multi-ply the intrinsic relaxation by a reduc-tion coefficient Xr to obtain a reducedrelaxation value for a tendon in a pre-stressed member [Eq. (2)]. For practicalapplication, the value X, may be readfrom Table 1 or the graph in Fig. 2 orcalculated by a closely fitting equation:

xr = + 5.35)n (7)

where

Il r jL(g)Jnet

PCI JOURNALJSeptember-October 1985 85

X,

'or

0E

0 E

)UCED RELAXATION

= Xr Lr

= INTRINISTICRELAXATION

'.81.751.701.65)60) 55

ITOTAL PRESTRESS LOSSL INTRINSIC RELAXATIONISTEEL STRESS IMMEDIATELY AFTER TRANSFER

STEEL STRESS IMMEDIATELY AFTER TRANSFER

ULTIMATE TENSILE STRENGTH

0 0.1 0.2 0.3 0.4 05 51

Fig. 2. Relaxation reduction coefficient xr.

In the common case discussed above and nonprestressed steel, A:AP, is positive while APB and AP,,, are

A„, + AA, (12)negative quantities. App =

The change in force on concrete, 0 PP,and

during the period (t – t 1 ) due to creep,shrinkage and relaxation may be calcu- ezlated by the equation (derived in Ref. 7): a = 1 + 2 (13)

r

'^ Pe = — f [C feet n. A., + s E, A x, + I+rAp, 1 in which

(10) e = eccentricity, i.e., the distancefrom Point 0, the centroid of

where concrete area (A„) to the centroidof A.,

= f 1 I + (1 + XC)

_ i

(1I) r2 = I IA square of radius of raJtionrofarea,A,

I, = moment of inertia of the crossin which the total steel area, Aa,, is the section area of concrete about ansum of the areas of prestressed steel,A,, axis through its centroid

86

In Eqs. (10) and (11) the following ad-ditional symbols are explained:

fce^ = instantaneous stress at y = e(Fig. 3) due to prestressing anddead load applied at time tt

E, = modulus of elasticity of the re-inforcement, assumed the samefor prestressed and nonpre-stressed steels

s = free (unrestrained) shrinkagen = E,/E,. (t i ), where Et (t 1 ) = modu-

lus of elasticity of concrete attime tj

C = creep coefficient = ratio of creepof concrete to instantaneousstrain due to a stress introducedat time t j and sustained, withoutchange in magnitude, up totime t

x = aging coefficient dependingupon the ages of concrete at t iand t; its value generally rangesbetween 0.6 and 0.9. Tables andgraphs for the value of x areavailable. 1.9.5 Note that X is usedas a multiplier to C, when astress increment is graduallyintroduced during the period t,to t.

Lb,. = reduced relaxation = intrinsic

relaxation multiplied by the re-laxation reduction coefficientXr [Eq. (2)1

Eq. (10) expresses the loss of com-pression in the concrete as the sum ofthree terms inside the square bracketswhich represent, respectively, the ef-fects of creep, shrinkage and relaxation.The dimensionless coefficient 63 ac-counts for the fact that the loss due toeach of the three causes depends uponcreep and the cross section areas and lo-cations of the prestressed and nonpre-stressed steels.

Post-tensioned and pretensionedmembers differ only in the calculationof f,. With post-tensioning, the area ofcross section to be employed in calcu-lating ff,t includes the cross sectionareas of the nonprestressed steel and ofconcrete, excluding the area of the pre-stressing duct. With pretensioning, thecross section to be used is composed ofthe areas of concrete and of prestressedand nonprestressed steels.

The value of the relaxation reductioncoefficient X r depends upon the totalloss, L,,, due to the combined effects ofcreep, shrinkage and relaxation. Sincethis is generally not known before Eq.

-4"Ans2

45TEEL

CENTROID CF A r AE0

A fY 0¢

p5

CENTROICAnstcei

lOF TOTAL

Ast=Apsns

STRESSED CROSS SECTION STRESS AT TIME CHANGE IN STRAINt, IN THE PERIOD t -ti

Fig. 3. Definition of symbols used in Eqs. (10) to (18).

PCI JOURNAL/September-October 1985 87

(10) is applied, a value for x r must firstbe assumed (for example 0.7) and latercorrected by iteration. A single iterationis sufficient in most cases (see Exam-ple).

The force P. acts on the concretecross section at eccentricity e; it pro-duces increments in normal strain atPoint 0 and in curvature. Adding theeffects of A P. to creep and shrinkagegives the total change in strain at Point0 during the period (t - t o ):

AEp = C e, (ta) + S + P' (14)E,, A,

Similarly, the change in curvatureover the same period is:

u = C0 (to)+ ] (15)E, I,

where E, is the age-adjusted elasticitymodulus of concrete:

k E, (t i )=

(16)1+xC

The age-adjusted modulus representsthe stress necessary to produce a totalstrain (instantaneous plus creep) ofmagnitude unity; the stress is here as-sumed to be gradually introduced overthe period t i to t.

The change in concrete stress at anyfiber is:

AP`e y (17)

where y is the coordinate of the fiberconsidered (measured downwards fromPoint 0).

The change in stress in the pre-stressed steel is:

L. =E( 0 (Aeo + ypr AO) + L,(18)

where y„g is the y coordinate of the pre-stressed steel.

The following numerical example il-

lustrates the application of the proposedmethod.

EXAMPLECalculate the changes in stress and in

strain which occur in a period (t - t i ) inthe concrete cross section shown in Fig.4a which is post-tensioned at time t,.

The following data are given.Ec (t i ) = 4500 ksi; E, = 29 x 10s ksi;

initial force in prestressed tendon = 315kips; bending moment due to dead loadintroduced at the same time as the pre-stress = 3500 kip-in.; free shrinkage s =-240 x 10-s ; creep coefficient C = 3;aging coefficient x = 0.8; intrinsic relax-ation L,. = -17 ksi, strength of pre-stressed steel, f ,, = 270 ksi.

Fig. 4b shows the strain and stressdistributions at time t 1 , immediatelyafter prestressing. These are calculatedby considering the initial prestress andthe dead load bending moment to beapplied on a transformed section ofmodulus of elasticity E, (t ! ) and com-posed of the area of concrete (less pre-stressing duct) plus n times the area ofnonprestressed steel; where n = 29 x103/4500 = 6.44.

The centroids of A. and A., are deter-mined (Fig. 4a) as well as the followinggeometric properties:

A, = 570.4 in.'; I, = 108000 in!; r 4 =189.6 in 2; a = 1.36; Aat = 5.6 in.'; A„8 =

1.7 in.'Assume a value for the relaxation re-

duction coefficient: Xr = 0.7.Thus, the reduced relaxation [Eq. (2)]

is:

L.r = 0.7(-17) _ -11.9 ksi

Eq. (11) gives:

f3-^1+1.36(6.44)(5.6) (1+0.8x3)}

570.4

= 0,7738

The change in force on concrete dur-ing the period considered [Eq. (10)] is:

88

iAns2'1.6 in22 ^Sto) 0,095ksi

-r

_4!23.9 =-4.04xI06

1 CENTROID

in. in-I

48 Iin. 0. OF Ac E0(to)

CENTROID e_g,3 =-118x106

OF A st aAp=1.7in2 fce1 =-0.681

ksi4 DUCT AREA=4.6 in2

2 Ansi=2.3 in 0.967 ksi

in

(a) CROSS-SECTION DIMENSIONS

(b) STRAIN AND STRESS AT ti,

IMMEDIATELY AFTER PRESTRESSING

-0009 ksi

LE 452x10-6 0.1850 =-

Sao _

-6-I-597x10i

0-382 ksi(c) CHANGES IN STRAIN AND IN STRESS DUE TO

CREEP SHRINKAGE AND RELAXATION

Fig. 4. Analysis of time-dependent strain and stress in a post-tensioned cross section.

A P, = -0.7738 [3(-0.681) (6.44) (5.6) AEa = 3(-118 x 10 ) - 240 x 10-g+ (-240 x 10-) (29000) (5.6

+102.8

+ (-11.9) (1.7)] = 102.8 kips 1324 (570.4)

The age-adjusted modulus of elastic- _ -458 x 101

ity [Eq. (16)] is: and the change in curvature [Eq. (15)]

4500 is:Ec _ = 1324 ksi

1+0.$x 102.8(8.3}^cb = 3(-4.04 x 10- 6 ) +

The change in axial strain [Eq. (14)] 1324 (108000)

is: _ -6.15 x 10-R in.-$

PCI JOURNAL1September-October 1985 89

The change in stress in prestressedsteel [Eq. (18)] is:

L^ = 290001-458 + 18.1 (-6.15)]x10-8 -11.9= –28.4 ksi

To calculate an improved value of thereduced relaxation, substitute fast _315/1.7 = 185.3 ksi in Eqs. (1) and (8):

X= 185.3 -0.7270

R 28.4-17 _0.06185.3

Table I or Fig. 3 gives X r – 0.85.Thus, a more accurate value of the re-

duced relaxation lEq. (2)] is:

L, = 0.85(-17) _ –14.4 ksi

Substitution of this value in Eq. (10)gives r3 P, = 106.1 kips. The corre-sponding changes in axial strain in cur-vature and in stress [Eqs. (14), (15) and(17)], are plotted in Fig. 4c. The changein stress in prestressed steel [Eq. (18)] isL„e = –30.7 ksi. Further iteration wouldchange these results only slightly.

It should be noted that the loss of ten-sion in the prestressed steel (30.7 x 1.7 =52.2 kips) is smaller in absolute valuethan the loss of compression in concrete(106.1 kips). The difference representsthe compressive force picked up by thenonprestressed reinforcement.

CONCLUSIONA prestressed tendon exhibits relax-

ation in a concrete member of smallermagnitude than the intrinsic relaxationwhich would occur if the length of thetendon was maintained constant. Thereduction in relaxation is caused by theshortening of the tendon due to shrink-age and creep of concrete. The relax-ation coefficient X,. can be used in pre-stressed concrete design as a multiplierto the intrinsic relaxation in the predic-tion of the prestress loss and the associ-ated deformation.

ACKNOWLEDGMENTThis paper is based on research con-

ducted at the Federal Institute ofTechnology, Lausanne, Switzerland,while the first author was a visitingprofessor for six months in 1983.

SI Conversion Factors

1 in. = 25.4 mm1 in.' = 645 mm'1 kip = 4.448 kN1 ksi = 6.894 MPaI kip-in. = 0.1130 kN-m

NOTE: Discussion of this paper is invited. Please submityour comments to PCI Headquarters by May 1, 1986.

90

REFERENCES1. Bazant, Z. P., "Prediction of Concrete

Creep Effects Using Age-Adjusted Effec-tive Modulus Method,"ACJ Journal, Pro-ceedings, V. 69, No. 4, April 1972, pp.212-217.

2. Comite Euro-Internationale du Beton(CEB) — Fed@ration Internationale de IaPricontrainte (FIP), Model Code for Con-crete Structures, 1978, (MC-78), CEB, 6rue Lauriston, F-75116 Paris.

3. Favre, R., Kopma, M., and Radojicic, Ef-fets Differes Ffssuration et Deformationsdes Structures en Beton, Ceorgi Sanit-Saphorin, VD, Switzerland, 1980.

4. Federation Internationale de la Prtcon-trainte, "Report on Prestressing Steel, Part1 — Types and Properties," FJP15,i3, Au-gust 1976, Published by Cement and Con-crete Association, Wexham Springs,SIough 513 6PL, England.

5. Chali, A., and Favre, R., Concrete Struc-tures. Stresses and Deformations, Chap-man and Hall, London, England, approx.400 pp. (in print).

6. Chali, A., Sisodiya, R. G., and Tadros,G. S., "Displacements and Losses inMulti-Stage Prestressed Members," Jour-nal of the Structural Division, ASCE,V. 100, STII, November 1974, pp. 2307-2322.

7. Chali, A., and Tadros, M. K., "PartiallyPrestressed Concrete Structures,"Journalof Structural Engineering, ASCE, V. 111,ST No. 8, August 1985, pp. 1846-1865.

8. Magura, D. D., Sozen, M. A., and Siess,C. P., "A Study of Stress Relaxation inPrestressing Reinforcement," StructuralResearch Series No. 237, University of Il-linois, Civil Engineering Studies, Urbana,September 1962, 102 pp.

APPENDIX A - DERIVATION OF EXPRESSION FORRELAXATION REDUCTION COEFFICIENT Xr

The reduction coefficient for the re-laxation of prestressed steel during anyperiod (t – t,) may be expressed as:

x(17cr = J a (1 –Sly)– 0.4 ] d^

(A1)

where ,~ is a dimensionless time func-tion defining the shape of the stress-time curve for a constant length tendon(Fig. Al).

The value t= equals zero when r = t,and equals 1 when r = t; r representsany instant between t i and t. Thus, theintrinsic relaxation at any instant r is:

Lr (7) = [Lr (t )] S (A2)

Eq. (Al) is derived assuming that theprestress loss due to the combined ef-fects of creep, shrinkage and relaxationvaries with time according to the sameshape function i•'. Thus:

Lna (r) _ [Ln+ (t)] f (A3)

At any instantr (Fig. Al) the tendon inthe prestressed concrete member ex-hibits relaxation as if its initial tensionwere:

.fw, (r) = Lfna, – (r) – Lr (r)j ] (A4)

The term in absolute value representsa reduction in tension caused by theshortening of the tendon; f^„ (r) is a re-duced initial tension at the instant r.Substitution of Eqs. (8), (A2) and (A3)into Eq. (A4) yields:

fmi (r) = ffaf (I — u S) (AS)

Eqs. (3), (4) ]or Eqs. (5) and (6)] andEq. (A2) may be combined to expressthe intrinsic relaxation. At any instant:

L,. (T) = —TllfAai (x — 0.4) Swhen A -- 0.4 (A&)

andLr (r) = 0 when x < 0.4 (A7)

PCI JOURNAL/5eptember-October 1985 91

STRESS CONSTANTi rr.lr_TU

T

Fig. Al. Stress versus strain in a constant length relaxation test. Definitionof the shape function e.

where af t = rq„ multiplied by the valuebetween square brackets in Eq. (4) or(5), with T = t. Note that when (t — t0)>0.5x106 , -qe= 71 «.

Employing Eq. (A6) or (A7), a differ-ential of the intrinsic relaxation may beexpressed as:

di., _ — mfg, (x — 0.4)2 d^

when X 0.4 (A8)

and

dL,,=0 whenX<0.4 (A9)

It can be seen from Eq. (A8) that thedifferential intrinsic relaxation dependsupon the initial tension value f,,, and A

(= fa,,,lfp,,,). For the tendon in pre-stressed concrete, the effective initialtension at any instant is reduced by thefactor (1 — SZg). Thus, Eq. (A8) may beused to express the differential reducedrelaxation by replacing f , by f andsubstituting for the latter by Eq. (A5):

d LT = rlr fvgi (1 — fl) x[X(1 — fl) — 0.4 j2df

92

Table Al. Relaxation reduction coefficient Xr.

Ii A=0.55 a-0.6() a =0.65 A=0.70 A=0.75 A=0.80

0.0 1.000 1.000 1.000 1.000 1.000 1.0000,1 0.6492 0.6978 0.7282 0.7490 0.7642 0.77570.2 0.4168 0.4820 0.5259 0.5573 0.5806 0.59870.3 0.2824 0.3393 0.3832 0.4166 0.4425 0.46300.4 0.2118 0.2546 0.2897 0.3188 0.3429 0.36270.5 0.1694 0.2037 0.2318 0.2551 0.2748 0.2917

when x(1-Sli)-0.4(A10)

or

d =0 when X(1-(f)<0.4(All)

Integration of each of Eqs. (A10) and(A8) and then division gives the relax-ation reduction coefficient Eq. (AI). Theequation applies when A(I -- Ili) -- 0.4.This restriction can be accounted for

simply by replacing the upper limit ofthe integral in Eq. (Al) by the smaller of1 and the value [(A - 0.4)/(X )l.

The values of the relaxation reductioncoefficient xr in Table Al or Fig. 2 arecalculated by evaluating the integral inEq. (Al) for various values of a and Sl. Aclosed form expression resulting fromintegration of Eq. (Al) is rather lengthy.Instead, Eq. (7) (obtained by curve fit-ting) may be used for the relaxation re-duction coefficient.

PCI JOURNAUSeptember-October 1985 93

APPENDIX B -- NOTATION

A = cross section areaC = creep coefficient = ratio of creep

which occurs during a period(t – t f) to the instantaneous straindue to a stress introduced at t f andsustained constant thereafter

E = modulus of elasticitye = eccentricity of the centroid of total

steel area measured downwardfrom centroid of concrete area

f = stressI = moment of inertiaL r = intrinsic relaxation of prestressed

steel = change in stress in a tendon_ stretched between two fixed pointsL r = reduced relaxation (for prestress-

ing tendons in concrete)n – E,1E, (to)

square of radius of gyrationof concrete area with respect to itscentroid

s = strain due to shrinkage when it isfree to occur without restrain

t = timea =1 + (e21a-2)A = an incremente = strain(A = curvatureX = aging coefficient for a specified

period tj tot. A value smaller thanunity used as a multiplier to thecreep coefficient when a stress in-crement is gradually introducedduring the period (t – t1)

x,. = relaxation reduction coefficientp = dimensionless coefficient defined

by Eq. (11)rt = dimensionless coefficient em-

ployed as multiplier in Eq. (3) forthe intrinsic relaxation. The valueof depends upon the steel qualityand on the length of the period ofrelaxation.

A = ratio of steel stress immediatelyafter transfer to ultimate tensilestrength

Q = stressT = any instant of time

= dimensionless function varyingbetween 0 and 1 (Fig. Al)

St = ratio of total prestress loss minussteel stress immediately aftertransfer to steel stress immediatelyafter transfer

Subscripts

c = concreteps = prestressed steelns = nonprestressed steeli = instant t i, the time of introduction

of prestressingst = total steel areau = ultimate strengthoo = very long period (exceeding 0.5 x

106 hours)

94