A METHOD FORPREDICTING PRESTRESSLOSSES IN APRESTRESSED CONCRETESTRUCTURER. J. GlodowskiArmco Steel Corp.Middletown, Ohio

J. J. LorenzettiArmco Steel Co-p.Midd.e:own, 0

A method is presented for predicting actual prestresslosses in a prestressed concrete structure. It usesavailable information on concrete creep andshrinkage and steel stress relaxation, and includesthe effect of interaction of the various factorscontributing to prestress loss. The required iterativeprocedure and curve transfer concepts are explained.

In the past several years, there has beenconsiderable interest within the pre-stressed concrete industry concerningthe amount of prestressing force lostduring the lifetime of a structure due tostress relaxation in the steel tendons. Asubstantial amount of fairly reliablestress relaxation data has been devel-oped from testing stress-relieved wire(ASTM-A421) and strand (ASTM-A416). However, stress relaxation dataare usually obtained from tests whichdo not allow for external load andlength changes andtherefore do not in-clude the variables encountered in ac-tual use. It is also apparent that the ba-ss of the steel relaxation test (constant

strain and stress changing withtime) isinconsistent with the test basis used inconcrete creep evaluation (constantstress and strain changing with time).

Therefore, any attempt to estimateprestress losses by direct prestress losssources based on the initial conditions issubject to considerable error. Instead,an overall evaluation of the prestressedconcrete structure is needed. An under-standing of the effect of external loadand length changes and of the inter-action between prestress losses resultingfrom concrete creep and shrinkage andsteel stress relaxation is required forsuch an evaluation.

This paper explains a method that

PCI Journal / March-Apr? 1972 17

utilizes available information on con-crete strain-time relationships and steelrelaxation 'to predict the actual prestresslosses in a prestressed concrete struc-ture. Curve-transfer concepts used indeveloping the prestress loss estimationdue to steel relaxation are presented.Also included is a description of the re-quired iteration procedure which canbe readily adapted to computer calcula-tion.

Information

required for

prestress loss analysis

In developing an interaction approachfor estimating prestress losses from steelrelaxation and concrete creep andshrinkage, it is necessary to have para-meters available , describing the steeland concrete properties. Certain designand construction considerations are alsorequired. The required parameters are:

1. Stress relaxation of steel as a func-tion of time and stress level.

2. Average steel strength properties.3. Creep strain of concrete as a func-

tion of time and stress level.4. Shrinkage strain of concrete as a

function of time.5. Modulus of elasticity of the steel

and of the concrete in the loadrange.

6. Sequence of casting, curing,stressing, and prestress transferused in the construction proce-dure,

7. Stress levels expected in the steeland the concrete resulting fromexternally applied loads, initially,and during the life of structure.

Steel stress relaxation. The prestressed.concrete industry is presently using thestress relaxation test as the best indica-,tion of a steel tendon's ability to main-tain the major portion of the initial pre-,stressing force in a concrete structure..In this test, stress relaxation is defined

as the time dependent loss of stress in atendon held at constant strain. In gener-aI, stress relaxation increases with in-creasing stress, time, and temperature.

For prestress loss analysis, it is neces-sary that the stress relaxation be ex-pressed as a function of time and stresslevel, assuming a relatively constanttemperature. Many papers have beenpublished reporting stress relaxationtest results, but only one, that byMagura, Sozen and Siess( 1 ), has at-tempted to describe relaxation in amathematical equation including bothtime and stress level. The equationgiven is as follows:

% SR = [lif t (R — 0.55) X 100 (1)

for R > 0.55where % SR = '% stress relaxation

t = test time in hrs.R = ratio of initial stress to

yield stress

This description of stress relaxation is asimple, straight line semi-log curve with0% SR at 1.0 hr., and with the slopedependent upon the initial stress levelratio, R. An example of this curve isgiven in Fig. 1.

Our experience in testing for stressrelaxation properties has shown that thestraight line semi-log curve is not an ac-curate description for short periods (lessthan 1000 hrs.). This is understandablesince the equation was developed usinglong-time data.

A method has been developed in ourlaboratory for extrapolating relativelyshort-time stress relaxation data to pre-dict steel losses at` much longer times.Basically, it is a quadratic equation ofthe form:

%SR=A+B(lnt)+C (Int)2 (2)

where A, B and C are functions of thestress level ratio. This stress level ratiois defined as the initial steel stress di-vided by a measure of the steel

18

12

109

0I-

><

JWQ'

W

I-

JWWN

% S—R = [ LI? (R — .55) 1 x 1

% S—R = A+B !nt + C (It)2

0'10ҟ100ҟ1,000ҟ10,000

ҟ100,000 40 Yr.

Time (Hours)

Fig. 1. Time dependent steel stress relaxation at R = 0.735

strength, which is similar to the R-valuepresented by Magura, et al., except thatthe yield strength can be replaced byanother measure of steel strength, suchas the ultimate tensile strength. As inEq. 1, the higher the stress level ratiothe greater the expected stress relaxa-tion. The functions A, B and C can bedetermined for most steel strength mea-surements.

This quadratic expression for stressrelaxation has the advantage of beingquite accurate at short times as well asbeing reasonably consistent with othermethods of predicting long-time stressrelaxation losses. An example of a curvepredicted by this equation is given inFig. 1.Concrete creep and shrinkage. In eval-uating the concrete's contribution toprestress losses, creep and shrinkagestrain have been accepted by the indus-try as the proper indicators of the timedependent changes in the prestress

force. In the creep test, the concretetest specimen is subjected to a constantcompression load and the strain changeis noted as a function of time. An un-loaded duplicate specimen is left in thesame room in which the temperatureand the humidity are controlled. Thestrain change of the unloaded specimenis also noted as a function of time. Thestrain change of the unloaded specimenis defined as the shrinkage, and the dif-ferential strain between the unloadedand loaded specimen is defined as thecreep strain.

In predicting interaction prestresslosses, the creep strain of concrete mustbe known as a function of time andstress level. One common method of ex-pressing creep strain uses a value calledthe ultimate specific creep coefficient orunit creep strain( 2 , 3 1. This number spe-cifies the ultimate strain expected in thelifetime of the concrete per unit psi ofconcrete stress. Along with this number,

PCI Journal / March-April 1972 19

an equation which expresses the frac-tion or percent of the ultimate creep asa function of time must be included. Al-most any mathematical form can beutilized, such as straight line semi-log,segmented line semi-log, hyperbolicfunction, or even a polynomial describ-ing empirical data. Since the ultimatespecific creep coefficient is the ultimateexpected strain per psi of concretestress, total creep is a function of thatstress. This form of expressing creepthen satisfies the requirements stated, afunction of both time and stress level.

A method of describing the expectedshrinkage of the concrete is alsoneeded. Since, by definition, shrinkageis independent of applied stresses, totalshrinkage strain need only be known asa function of time. One method of ex-pressing the shrinkage is to state the ul-timate shrinkage strain expected in thelife of the structure, and then describethe rate of this shrinkage with a mathe-matical function similar to that used inthe prediction of creep strain.

When describing expected creep andshrinkage, the effect of various con-struction and environmental factorsshould be considered. Construction fac-tors that will change the expected creepand shrinkage of concrete, as deter-mined by laboratory testing, include thetype and length of curing, time fromcure to prestress transfer, surface to vol-ume ratios, the use of supplemental re-inforcing steel, and the presence ofbiaxial stresses. The type of environ-ment, specifically temperature and hu-midity, also should be evaluated as toits effect on expected concrete proper-ties.Elastic moduli. When the stress or strainin a prestressed concrete systemchanges due to non-elastic factors suchas creep, shrinkage and stress relaxa-tion, there is an elastic rebound effect.This is necessary to maintain an equilib-rium of forces in the system assuming

that strain differentials between thesteel and concrete cannot exist. Byknowing the elastic moduli of the steeland concrete, the elastic rebound orequalizing effect can be calculated us-ing force and strain balance equations.The solution to these balancing equa-tions is shown in Appendix I.Construction procedure. The interactionprogram was developed to apply toeither pretensioning or post-tensioningtypes of construction. Because of thedifferences between the two types ofconstruction, and differences betweenmethods used in the same type of con-struction, it is advantageous to knowthe sequence of events leading up tothe initial condition for prestress lossconsiderations when the full steel ten-sion is applied to the concrete. Concreteshrinkage occurs independent of thestressing situation, and stress relaxationwill occur during situations where thesteel tension is not applied to the con-crete but is held by an external anchor-ing system. These factors could affectsubsequent prestress losses and there-fore should be considered before esti-mations are made.Expected stress levers. The appliedstress levels in the prestressed concretesystem have a direct effect on pre-stress losses. When estimating initialstress levels in the structure, it is im-portant to consider stress changes dueto factors such as elastic shortening inpretensioned systems, or friction andanchorage slip in post-tensioned sys-tems. Even after an accurate estimateof initial stress levels is made, it may benecessary to consider changes in exter-nal loading during the life of the struc-ture. An example of this is a beam thatlies on the ground for several weeks af-ter manufacture; then it is placed in astructure where it must support its ownweight and other dead load weights.Any live or working load, if significant,should also be considered.

20

Interaction analysisfor prestress losses

As pointed out above, the rates of steelstress relaxation and concrete creepduring the life of a prestressed concretestructure are dependent upon load andtime. When the steel and the concreteare combined in a prestressed structure,there is a continuing change in thestress and strain of each component.The basis of this prestress loss methodof analysis is to divide the service lifeinto small time intervals during whichthe stress relaxation, creep strain, andshrinkage strain can be assumed to beindependent of each other. In eachsmall time interval prestress loss due tostress relaxation and strain changes dueto creep and shrinkage are calculated.At the end of each time interval, elasticexpansion and/or contraction of thecomponents is considered so that thecompression load on the concrete isequal to the tension load on the steeland the total lengths of each are equal.The stress levels at the end of each timeinterval are used in the next successivetime interval. This procedure is contin-ued until the required service life ofthe structure is reached.

The basic steps used in the prestressloss estimation method for one time in-terval are as follows:

1. Determine the load in the steel,P1 , and load in the concrete,P,,., and the length of the steel,L, and length of the concrete,L ie., at the start of the time inter-val.

P18 = P 1 (3a)L,. = L,, (3b)

2. Calculate the independent creepand shrinkage strains that occurduring the time increment Atbased on the load and time at thestart of that increment. Assume

the loads have not yet changedbut the calculated length of theconcrete, L26 is less than the orig-inal length of the steel.

Pis = P18 (4a)Lis > L28 (4b)

3. Calculate the elastic contractionof the steel and the elastic expan-sion of the concrete necessary toequalize lengths while maintain-ing equal loads. (See Appendix Ifor formulas.) New loads, P 28 andPlc, and lengths, L28 and L,now describe an equilibrium situ-ation.

P28 = P20 (5a)

L2, = L38 (5b)

4. Calculate the stress loss due tostress relaxation during the sametime interval, At. The curvetransfer procedure necessary forthis calculation will be explainedbelow. The lengths remain equalbut the new calculated tensionload on the steel, Pas, is less thanthe compression load on the con-crete, P20.

P38 < P20 (6a)

L23 = Lac (6b)

5. Calculate the elastic expansion ofthe steel and concrete necessary toincrease the steel tension load anddecrease the concrete compressionload until they are equal. Newloads, P48 and Pic, and newlengths Las and L40, define theload situation at the end of thetime interval considered.

P48 = P30 (7a)

Lss = L40 (7b)

6. The prestress loss due to creepand shrinkage of concrete in thistime interval is P1$—P28.

7. The prestress loss due to stress re-

PCI Journal / March-April 1972 21

laxation of the steel during thistime interval is P23—P48.

8. Load and length conditions arenow defined for the next time in-terval as stated in Step 1 and thesteps are repeated.

The length of the time intervalsshould be selected carefully. Theremust be a compromise between makingit as short as possible to simulate an in-stantaneous situation and long enoughto keep the number of computations ata manageable level. Both the concreteand the steel time dependent propertieshave a continuously decreasing ratewith time. Therefore, it is advantageousto have the time increments at the be-ginning of the structure's life smallerthan those at the end. This was accom-plished in the computer program by de-fining the first increment to be from 0 to1.1 hours, and then dividing each in-creasing logarithmic cycle into a givennumber of increments. Therefore, inthe 1 to 10 hour range, the incrementsof time are much smaller than each suc-ceeding log cycle-10 to 100 hrs., 100to 1,000 hrs., etc.—until the life of thestructure has been reached. The neces-sity of having a computer for these cal-culations is quite evident when usingsmall time intervals.

For a particular post-tensioning situ-ation studied, 50 increments per logcycle were adequate and even 20 in-crements per cycle gave reasonable pre-cision. In this situation creep and stressrelaxation were considered to start atthe same time. If a situation is consid-ered where this is not the case, a largernumber of increments per log cyclemay be necessary to keep the actualtime of the increment small during theinitial effects by any of the factors.

Curvetransfer procedureFor a prestress loss analysis where in-teraction between steel and concrete is

considered, it is necessary to establish aprocedure for determining steel stressrelaxation losses during conditions ofchanging strain. A common procedureinvolves transfer from one stress levelrelaxation curve to another for eachchange in strain. The problems are todefine the proper stress level curve touse after each strain change, and to de-fine the proper starting point on thatcurve.

The steel stress relaxation equationspreviously discussed will define stressrelaxation percentage vs. time curvesfor a given initial stress. To define theproper curve to use after a strainchange, the stress equivalent to the ini-tial stress for that situation must be de-termined. At least two methods are pos-sible for selecting this initial stress.

One method uses the actual stressafter the strain change. This actualstress is the algebraic sum of the initialstress and stress changes due to bothstrain changes and prior stress relaxa-tion. There is no attempt to separatethe contributions of each. Using theactual stress to determine a new initialstress level ratio implies that this ratioincludes the effect of prior stress relaxa-tion. However, the original equationswere derived with the stress level ratioas an independent variable. Any changein this variable should be independentof prior stress relaxation. For these rea-sons, using the actual stress level fordetermining the new stress level ratiois not a satisfactory method.

A second method for selecting theproper relaxation curve uses an initialstress that is independent of prior stressrelaxation. This is accomplished by in-troducing an effective initial stresswhich is defined as the algebraic sumof the initial applied stress and anylater changes in that stress due to fac-tors other than stress relaxation. Thesefactors include any elastic strainchanges in the steel. The effective ini-

22

tial stress is used in the stress levelratio, R, which is now the ratio of theeffective initial stress to the yield stress.Therefore, the R-ratio is based on ex-ternal load or strain changes and is nota function of the amount of stress lostdue to the relaxation process. Deter-mining the R-ratio in this manner solvesthe problem of defining the properstress level curve to use after eachstrain change in the prestress loss analy-sis. The effect of prior stress relaxationis accounted for in the curve transferprocedure as explained in the followingparagraphs.

To determine the proper startingpoint on the new stress level curve, it isnecessary to define the method of trans-ferring from the previous curve to thenew stress level curve. Since the R-ratiohas been defined, two variables—stressrelaxation percentage and time—are leftin the relaxation equation to determinea starting point on that curve. If actualtime is used to define the starting pointon a new stress relaxation percentagevs. time curve, the curve transfer isvertical from the previous curve to thenew curve. However, the effect of priorstress relaxation at the higher initialstress level is neglected in this proce-dure. If prior stress relaxation is used todefine the starting point, the curvetransfer should be horizontal from theprevious curve to the new curve. Theprevious stress loss due to relaxationwill then define the rate of relaxationafter a strain change. In other words,the R-ratio defines the curve and theprevious stress relaxation defines " thestarting point on that curve.

To verify the proper curve transferprocedure, stress relaxation experimentswere conducted using changing strainconditions on 0.250 in. dia. wire speci-mens. Since a strain change in the steelmust, according to Hooke's Law, be ac-companied by a proportional amount ofstress change, the effect of a strain

change is the same as changing the ap-plied stress. In both of the experiments,two wires were tested, one at a refer-ence stress level and another at higherinital stress levels for various periods oftime.

The curves in Fig. 2 compare stressrelaxation test data taken on a wire withno external stress change (Test A) vs.another wire with two external stresschanges (Test B). Test A was loaded to75% of the guaranteed ultimate tensilestrength (GUTS) and had an initial R-ratio of 0.797. This test was conductedover 600 hours. Test B was loaded to ahigher initial stress ratio, R = 0.860,then manually reduced in load after 9min. to R = 0.833, and reduced againafter 3 hr. 40 min. to R = 0.797, thesame as for Test A. The problem is todefine the proper transfer procedurethat will determine the position on theTest A curve for predicting the stressrelaxation response of Test B. It can beseen that using actual test time as thecriterion for determining the proper po-sition is the same as dropping verticallyto the Test A curve. Plotting the actualdata from that point shows a lower re-laxation rate than would be predictedby the reference curve (Test A). Theother alternative is a horizontal move toa point on the Test A curve, indicatingequivalent stress loss due to stress re-laxation. When the actual data areplotted in this manner, the predictedand actual relaxation rates are nearlyidentical. The slight initial offset is dueto the necessity of resetting the straincontrolling extensometer.

The curves in Fig. 3 represent a simi-lar situation but with a lower initialstress level and a longer time in thehigh stress situation. Two test wireswere used from the same coil, the first(Test C) originally loaded to 70%GUTS (R = 0.728) and continued to400 hours. The second (Test D) wasoriginally loaded to 76% GUTS for 24

PCI Journal / March-April 1972 23

0I-><JW

w

I-NJWWI-

i

/ //HORIZONTAL fir'TRANSFER //

/,

__________________________________

VERTICALTRANSFER

TEST B —f

—TEST A — ACTUAL DATA BEFORE CURVETRANSFER.

- - ACTUAL DATA FROM "TEST B" AFTERTRANSFER TO 'TEST A" CURVE.

10 100 1,000Time (Hours)

Fig. 2. Effect of curve transfer methods for Tests A and B

hr., then manually reduced by 6% ofGUTS to be equivalent to the 70%GUTS test (R = 0.728). Again, droppingvertically at the 24 hr. point from TestD to Test C and plotting the actualdata substantiates that a much lowerrate of relaxation actually occurredthan would be predicted from the origi-nal 70% GUTS curve. The horizontaltransfer for equivalent stress relaxationwas beyond the time range for Test C,but it appears to fit an extension of theTest C curve.

As the data presented in Figs. 2 and 3illustrate, the best choice of the two al-ternatives of horizontal and verticalcurve transfer is the horizontal methodwhich is based on equivalent stress re-laxation losses. This procedure was

used in developing the full interactionapproach to predicting prestressed con-crete stress losses.

It is possible that the concept of thehorizontal curve transfer procedure asdescribed for steel stress relaxation isalso applicable to creep strain in con-crete. This possibility should be verifiedby test. For situations in which there isa large variation in the concrete stressduring the life of the structure, anequivalent creep concept is neces-sary(4 ). In a structure where there is lit-tle or no external stress change in theconcrete, either curve transfer methodcould be used since concrete creep isconsidered to be directly proportional tothe total applied stress( 5 '6 ), and therforesmall changes are not as critical as they

24

ACTUAL DATA BEFORE CURVETRANSFER. HORIZONTAL—>/

TRANSFER /--- ACTUAL DATA FROM "TEST D" AFTER

/TRANSFER TO "TEST C" CURVE.

VERTICALTRANSFER

TEST D --►

TEST C

to IUU 1,0110

Time (Hours)

3.(

2.!

t 2.(z0I-5<JW l^

WH

JW 1•(

0.;

0.l

Fig. 3. Effect of curve transfer methods for Tests C and D

are in steel stress relaxation.

Application ofprestress loss analysis

A computer program was developedbased on the procedures described forinteraction analysis and curve transfer.The program was written so that allavailable pertinent information couldbe included in the prestress loss calcula-tions. Use of the computer allows alarge number of situations to be con-sidered using small time interval inter-actions.Results. Table 1 and 2 present a sum-mary of estimated prestress losses as-suming a particular post-tensioning sit-uation. The assumed initial conditions

are as follows:

Creep and shrinkage as proposed byPCI Committee on Prestress Loss-es.

Yield strength of steel = 220,000 psiModulus of elasticity of steel

29 x 106 psiOriginal stress on steel = 168,000

psiModulus of elasticity of concrete

4.2 x 10 6 psiOriginal stress on concrete = 1,500

psiTotal shrinkage strain = 500 x 10-6

in. /in.Time before prestress transfer = 30

daysAverage temperature = 68 F

Table 1 includes estimations using a

PCI Journal / March-April 1972 25

Table 1. Estimated load losses in a prestressed concrete structure using various steel and concrete grades and based on Eq. 2

Estimated percent load loss in 40 years

Steeltype

Ultimatespecific

creep coef.x 10-6 /psi

Due to steel relaxationDue to concrete creep

and shrinkage Total load loss

Nointeraction

Interactionwith concrete

NoҟInteractioninteractionҟ-ҟwith steel

Nointeraction Interaction

1 0.20.40.8

13.713.713.7

6.95.23.4

10.1ҟ9.215.3ҟ13.425.6ҟ20.9

23.829.039.3

16.118.624.3

2 0.2040.8

8.58.58.5

4.33.02.0

10.1ҟ9.315.3ҟ13.625.6ҟ21.2

18.623.834.1

13.616.623.2

3 0.20.8

3.93.9

2.1" 2.0

10.1ҟ9.415.3ҟ13.7

14.019.2

11.515.7

0.4 3.9 1.8 25.6ҟ21.2 29.5 23.0

ti

ti

Table 2. Estimated load losses in a prestressed concrete structure using various steel and concrete grades and based on Eq. 1

Estimated percent load loss in 40 years

Due to steel relaxationDue to concrete creep

and shrinkage Total load lossUltimatematpemat

Steel creep coef. No Interaction No Interaction Notype x 10-6 /psi interaction with concrete interaction with steel interaction Interaction

1 0.2 13.7 7.7 10.1 9.1 23.8 16.80.4 13.7 6.2 15.3 13.3 29.0 19.50.8 13.7ҟ0 4.8 25.6 20.7 39.3 25.5

2 0.2 7.4 5.1 10.1 9.3 17.5 14.40.4 7.4 4.1 15.3 13.5 22.7 17.60.8 7.4 3.1 25.6 21.0 33.0 24.1

3 0.2 2.5 1.7 10.1 9.4 12.6 11.10.4 2.5 1.3 15.3 13.8 17.8 15.10.8 2.5 0.9 25.6 21.4 28.1 22.3

21

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J ^12

WN }

9

J6

0h

3

AL.

ҟ

0 ■ҟI I I 1

0 ҟ.2ҟ.4ҟ.6ҟ.8ҟ1.0

Steel Stress Level Ratio (R)

(b) — Total Stress Relaxation at 40 Yrs. vs. Steel Stress Level Ratio

700

600

z500

a zI— O

' 400a —WW

c^ 300J aQ W

}

F Q 200

100

10 —6

0ҟ300ҟ600ҟ900ҟ1200ҟ1500

Concrete Stress Level (psi)

(a) — Total Creep Strain at 40 Yr. vs. Concrete Stress

Fig. 4. Response of concrete creep and steel relaxation to stress levels

28

stress relaxation equation of the typegiven as Eq. 2, and Table 2 gives esti-mations using the straight line semi-logEq. 1. In each of these tables, threesteel types are considered based ontheir basic "no interaction" stress re-laxation properties. Also, three concretesituations are considered by using dif-ferent unit creep strains. The lower "in-teraction" load loss due to stress relax-ation using Eq. 2 (Table 1) is due to thelower predicted stress relaxation rateduring the important initial time peri-ods (see Fig. 1).Discussion. As would be expected, inevery situation that was investigated,the interaction consideration yieldedlower total prestress losses than if theloss causes were treated independently.The decrease in expected losses due tothe steel relaxation was generally moresignificant than the decrease in lossesdue to concrete creep. The reason forthis is the variation in the response tostress levels as shown in Fig. 4. The ap-plied stresses in the concrete are usuallyless than one half the ultimate compres-sive strength and, in this range, thecreep is considered to be directly pro-portional to the applied stress (Fig. 4a).Significant steel stress relaxation occursonly at relatively high stresses startingabove R = 0.60 (Fig. 4b). A typicalinitial stress ratio in the steel would be0.75. Therefore, the expected steel re-laxation can change quite drasticallycompared to the expected creep strainfor an equivalent amount of loadchange.

Most prestressed concrete structuresare designed using a minimum prestressforce requirement during the expectedlife of the structure. Therefore, the ex-pected prestress losses are compensatedfor by using additional prestressingforce in the initial construction. Thisadditional prestressing force is providedby increasing the number of steel ten-dons since the stress on each tendon is

limited by construction codes and can-not be increased. For this reason, anaccurate estimate of prestress losses isdirectly related to the economics of theconstruction because an over-estimationof losses will result in unnecessary costs.

With recent emphasis in the industryon steel tendons specially processed toproduce low relaxation properties, it isinteresting to note the effect of theseproperties on total load loss when theinteraction approach is considered. Asshown in Table 1, on the basis of totalpercent load loss, the interaction effectfor each steel type in a relativelypoor concrete (creep coefficient = 0.8)indicates that there is very littleeconomic advantage to a low relaxationsteel (Type 3). The total decrease in ex-pected percent load losses based on thesame original load, replacing normalstress-relieved material (Type 1) withlow relaxation material (Type 3), is only1.3%. This improvement is significantlyless than the 9.8% decrease expectedwhen not considering the interactioneffects.

The effect of improved concreteproperties (changing the creep coeffi-cient from 0.8 to 0.2) on total percentload loss is not as significant when con-sidering interaction. For example, aload loss decrease of 15.5% is predictedusing no interaction with stress relievedsteel compared to 8.2% when consider-ing the interaction effect. Improvingthe concrete properties is more effectivewhen using a steel tendon with lowerrelaxation properties as shown by the11.5% decrease in interaction losses us-ing Type 3 steel.

Evaluation of the data in Tables 1and 2 points out that if the most eco-nomical method of construction is de-sired, it is necessary to have a good bal-ance of steel and concrete properties.The use of low relaxation steel has moreof an advantage when higher qualityconcrete is used, and vice-versa. The

PCI Journal / March-April 1972 29

decision to use higher priced materialsto decrease prestress losses should bebased on the best estimate of the totaleffect of these materials and the result-ing economic advantage.

The particular values stated in theprevious paragraphs are only examplesbased on one particular set of condi-tions and described properties. The ef-fect of all material properties and con-struction techniques must be consid-ered for a proper evaluation.

A specific example of constructiontechniques is the case of the preten-sioned I-beam. Normally the steel isstressed to 70% of the guaranteed ulti-mate tensile strength and anchored toabutments. The concrete is then castaround the steeltendons. A normal pro-cedure is to steam cure this concrete attemperatures around 150 F for at leastone night. After the designated con-crete strength is reached, the steel ten-dons are cut from the anchorages andthe stress is transferred to the concrete.Because of the expected high amount ofrelaxation at 150 F, and the subsequentloss of stress due-to elastic shortening ofthe beam, it is possible that the relaxa-tion of the steel after the prestresstransfer is insignificant. An accurate de-scription of the stress and temperaturelevels during the curing process and re-laxation test data under these condi-tions is necessary to substantiate thispossibility.

Program limitations. , The main limita-tion to the interaction program for esti-mating prestress losses is the quality ofthe available input data. The accuracyof the estimate based on interaction ef-fects can be no greater than the accura-cy of the data determined from "no in-.teraction' tests, it is important to theacceptance of this program that thebest information be made available tothose who wish to use it. This means'that''materials testing should.be contin-ued and intensified.

The major deficiency in most of thedata now available is that the effect ofenvironmental conditions has not beendetermined. Concrete creep and shrink-age strain are known to be affected bytemperature and humidity, and steel re-laxation is also known to be substantial-ly affected by temperature. Yet theseparameters are not well described formost situations actually encountered,such as the pretensioning situation de-scribed above.

Conclusions

1. The basis of the interaction meth-od of predicting prestress lossesdescribed in this paper is a stressand strain balancing techniquewhich is not new to the industry,although it has not been used ex-tensively.

2. The method of horizontal curvetransfer for predicting stress relax-ation rates after strain changes oc-cur is a new approach and has notbeen used before. It is an impor-tant part of the interaction meth-od.

3. The interaction of `concrete creepand shrinkage strains, and steelstress relaxation reduces the totalexpected prestress losses.

4. Both the stress losses due to steelstress relaxation and those due toconcrete creep and shrinkage aredecreased when the interactionapproach is used. The reduction isgenerally more significant for steelstress relaxation.

5. Improving the steel stress relaxa-tion properties alone by a givenamount will not reduce the totalexpected losses by an equivalentamount.

6. The accuracy of the' interactionmethod "-of predicting prestresslosses is dependent upon the ac-curacy of the material property

3ff

determinations and on the properdescription of actual stress levelsand the environment of the struc-ture during its lifetime.

References1. Magura, D. D., Sozen, M. A. and

Siess, C. P., "A Study of Stress Re-laxation in Prestressing Reinforce-ment", PCI Journal, April 1964, pp.13-57.

2, Lorman, W. R., "The Theory ofConcrete Creep", ASTM Proceed-ings, Vol. 40, 1940, P . 1082.

3. Corley, W. G., Sozen, M. A. andSiess, C. P., "Time-Dependent De-

flections of Prestressed ConcreteBeams", Highway Research BoardBulletin 307, 1961, pp. 1-25.

4. McHenry, D., "A New Aspect ofCreep in Concrete and Its Applica-tion to Design", ASTM Proceedings,Vol. 43, 1943, p. 1069.

5. Neville, A. M. and Meyers, B. L.,"Creep of Concrete: InfluencingFactors and Prediction", Symposiumon Creep of Concrete, ACI Publica-tion SP-9, 1964, pp. 1-30.

6. Petersen, P. H. and Watstein, D.,"Shrinkage and Creep in PrestressedConcrete", National Bureau of Stan-dards, Building Science Series 13,1968.

APPENDIX I

Known conditions at load equilibrium:Load in concrete = Load in steel

Po = P$If a length differential ALT exists, thento equalize that differential

ALT = AL3 + AL,or AET = AE8 + AE0

Therefore:

P0(original) = P3(original)

A0o 0, = A8

0-80

. .

AP, = AP,A, 0o0 = A8 0u-3

A, E6 (DE,) = As E8 (&Es)

DE, = DE$Ac Ec

DES (T,0 E8 A E,go-so Ec

AEC _ o- E8 (LET - AEc)Aso Ec

AEp 1 +(

(r00 E8 ) _ u-0D E3

AET 080 E0 o 8D E0

where A = cross-sectional area Therefore:

A0 _ (r8oDE — (rc° E3 AeAso Ec + a vo E$ T

But any change in loads must be equal QE$ = ACT — QED

Discussion of this paper is invited.Please forward your discussion to PCI Headquarters by July 1 topermit publication in the July-August 1972 issue of the PCI JOURNAL.

PCI Journal / March-April 1972 31