recomendations for estimating prestress losses

7/30/2019 Recomendations for Estimating Prestress Losses

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Recommendationsfor EstimatingPrestress Losses

Prepared byPCI Committee on Prestress Losses

PCI JOURNAL/Ju y-August 19753


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Recommendationsfor Estimating

Prestress LossesPrepared by

PCI Committee on Prestress LossesH. KENT PRESTONChairman

JAMES M. BARKERHENRY C. BOECKER, JR.*R. G. DULLHARRY H. EDWARDSTI HUANGJAIME IRAGORRIR. O. KASTENt* Replaced by Mario G. Suarez.Replaced by R. G. Dull.$ Previous Chairmen.

HEINZ P. KORETZKYPAUL E. KRAEMER*DONALD D. MAGURA$F. R. PREECEMARIO G. SUAREZPAUL ZIA

This PCI Committee report summarizesdata on creep and shrinkage of concreteand steel relaxation, and presents botha general and a simplified designprocedure for using these data inestimating loss of prestress after any giventime period. A Commentary explains thedesign provisions. Detailed design examplesfor pretensioned and post-tensioned concretestructures explain the procedures.

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CONTENTSCommitteeStatement ..............................46Chapter 1General Aspects Related to

PrestressLosses ......................471.1Tensioning of Prestressing Steel1.2Anchorage1.3Transfer of Prestress1.4Effect of Members in Structures

Chapter 2General Method for ComputingPrestressLosses ...................... 482.1Scope2.2Total Loss2.3Loss Due to Elastic Shortening2.4Time-Dependent Losses (General)2.5Loss Due to Creep of Concrete2.6Loss Due to Shrinkage of Concrete2.7Loss Due to Steel Relaxation

Chapter 3Simplified Method for ComputingPrestressLosses ......................523.1Scope3.2Principles of Simplified Method3.3Equations for Simplified Method3.4Adjustment for Variations fromBasic Parameters

Commentary ...................................... 55Notation......................................... 62References.......................................64DesignExamples .................................. 66

Example 1Pretensioned Double TeeExample 2Simplified MethodExample 3Post-Tensioned Slab

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COMMITTEE STATEMENTThis recommended practice is intended to give thedesign engineer a comprehensive summary of researchdata applicable to estimating loss of prestress. Itpresents a general method whereby losses arecalculated as a function of time.This report contains information and procedures forestimating prestress losses in building applications. Thegeneral method is applicable to bridges, althoughthere are some differences between it and the AASHTOStandard Specifications for Highway Bridges withrespect to individual loss components.A precise determination of stress losses in prestressedconcrete members is a complicated problem becausethe rate of loss due to one factor, such as relaxation oftendons, is continually being altered by changes instress due to other factors, such as creep of concrete.Rate of creep in its turn is altered by change in tendonstress. It is extremely difficult to separate the netamount of loss due to each factor under differentconditions of stress, environment, loading, and otheruncertain factors.In addition to the foregoing uncertainties due tointeraction of shrinkage, creep, and relaxation, physicalconditions, such as variations in actual properties ofconcrete made to the same specified strength, can varythe total loss. As a result, the computed values forprestress loss are not necessarily exact, but theprocedures here presented will provide more accurateresults than by previous methods which gave noconsideration to the actual stress levels in concreteand tendons.An error in computing losses can affect serviceconditions such as camber, deflection, and cracking. Ithas no effect on the ultimate strength of a flexuralmember unless the tendons are unbonded or the finalstress after losses is less thanIt is not suggested that the information and proceduresin this report provide the only satisfactory solution tothis complicated problem. They do represent anup-to-date compromise by the committee of diverseopinions, experience and research results into relativelyeasy to follow design formulas, parameters, andcomputations.

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CHAPTER 1-GENERAL ASPECTS RELATED TOPRESTRESS LOSSES

1.1Tensioning of PrestressingSteel

1.1.1 Pretensioned constructionFor deflected prestressing steel, lossDEF, occurring at the deflecting de-vices, should be taken into account.1.1.2Friction in post-tensioned con-structionLoss due to friction in post-tensionedconstruction should be based uponwobble and curvature coefficientsgiven below, and verified duringstressing operations. The losses dueto friction between the prestressingsteel and the duct enclosure may beestimated by the following equation:

FR = T o I1 e-^xz,z +aa^](1)When (K l t , + ,a) is not greater than0.3, the following equation may bused:

FR = T 0 (K l t ,. +a) (2 )Table 1 gives a summary of frictioncoefficients for various post-tension-ing tendons.1.2AnchorageLoss ANC, due to movement of pre-stressing steel in the end anchorage,should be taken into account. Slip atthe anchorage will depend upon theparticular prestressing system uti-lized and will not be a function oftime. Realistic allowance should bemade for slip or take-up as recom-mended for a given system ofanchorage.

1.3Transfer of PrestressLoss due to elastic shortening maybe calculated according to the pro-visions in this recommended prac-tice. The concrete shortening shouldalso include that resulting from sub-sequent stressing of prestressingsteel.1.4Effect of Members in

StructuresLoss of prestress of a member maybe affected by connection to otherstructural elements or composite ac-tion with cast-in-place concrete.Change in prestress force due tothese factors should be taken into ac-count based on a rational procedurethat considers equilibrium of forcesand strain compatibility.

Table 1. Friction coefficients forpost-tensioning tendons.Wobble CurvatureType ofoefficient, coeffi-tendon, per foot cient,Tend ons in flexiblemetal sheathingW ire tendons.0010 - 0.0015 0.15 - 0.257-wire strand.0005 - 0.0020 0.15-0.25High strengthbars.0001 - 0.000 6 0.08 - 0.30Ten don s in rigidmetal duct

7-w i restrand.0 0 0 2 0.15-0.25Pre-greased tendonsW ire tendons and7-wire strand.0003 - 0.002 0 0.0 5 - 0.15Mastic-coatedtendonsW ire tendons and7-wire strand.0010 - 0.0020 0.05-0.15PCI JOURNAL/July-August 19757


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CHAPTER 2-GENERAL METHOD FOR COMPUTINGPRESTRESS LOSSES2.1Scope2.1.1Materials2.1.1.1Lightweight concreteLightweight aggregate concrete w itha unit weight between 90 and 125lb per cu ft where the unit weightvaries because of replacement oflightweight fines with normalweight sand.2.1.1.2 Normal weight concreteConcrete with an approximate unitweight of 145 lb per cu ft where allaggregates are normal weight con-crete aggregates.2.1.1.3Prestressing SteelHighstrength prestressing steel that hasbeen subjected to the stress-relievingprocess, or to processes resulting inlow relaxation characteristics.2.1.2 Prestressed unitsLinearly prestressed members only.Excluded are closed sections pre-stressed circumferentially.2.1.3Curing2.1.3.1 M o isture Impermeablemembrane curing or other methodsto prevent the loss of moisture fromthe concrete.2.1.3.2Accelerated cure Curing inwhich the temperature of the con-crete is elevated to not more than160F for a period of approximately18 hours, and steps are taken to re-tain moisture.2.1.4EnvironmentPrestressed concrete subjected toseasonal fluctuations of temperature

and humidity in the open air or tonom inal room cond itions is covered.The values for UCR and USH arebased on an average ambient rela-tive hum idity of 70 percent.2.2Total Loss2.2.1Pretensioned construction

TL=ANC+DEF+ES+Z(CR+SH+RET)ti

2.2.2Post-tensioned constructionTL=FR+ANC+ESt

+ > (CR + SH + RET)t ,

2.3Loss Due to ElasticShortening (ES)

Loss of prestress due to elasticshortening of the concrete should becalculated based on the modulus ofelasticity of the concrete at the timethe prestress force is applied.

ES = fcr(E8/Ecz)5)2.3.1Pretensioned constructionIn calculating shortening, the loss ofprestress shall be based upon theconcrete stress at the centroid of theprestressing force at the section ofthe member under consideration.This stress, f^,., is the compressivestress due to the prestressing forcethat is acting immediately after theprestress force is applied minus thestress due to all dead load acting atthat time.

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Table 2. Minimum time intervals..3 .2Post - tensioned constructionThe average concrete stress betweenanchorages along each elementshall be used in calculating shorten-ing.2.4Time-Dependent Losses

(General)Prestress losses due to steel relaxa-tion and creep and shrinkage ofconcrete are inter-dependent andare time-dependent. To account forchanges of these effects with time, astep-by-step procedure can be usedwith the time interval increasingwith age of the concrete. Shrinkagefrom the time when curing isstopped until the time when the con-crete is prestressed should be de-ducted from the total calculatedshrinkage for post-tensioned con-struction. It is recommended that aminimum of four time intervals beused as show n in Table 2 .When significant changes in loadingare expected, time intervals otherthan those recommended should beused. AIso, it is neither necessary,nor always desirable, to assume thatthe design live load is continuallypresent. The four time intervalsabove are recommended for mini-mum non-computerized calculations.2.5Loss Due to Creep of

Concrete (CR)2.5.1 Lo ss over each stepLoss ov er each time interval is givenby

CR = (UCR)(SC F)(M CF) x(PCR )(f0)6)where f is the net concrete compres-sive stress at the center of gravityof the prestressing force at time ti,

Beginning time,Step t, End t ime, tPretensionedanchorage ofprestressing

steel Age at prestressing1 of concretePost-tensioned:end of curingof concreteA ge = 30. days, ort ime wh en a mem-2 End of Step 1 ber is subjected toload in addition toits own weight

3 E nd of Step 2 Age =1 year4 E nd of Step 3 End of service life

taking into account the loss of pre-stress force occurring over the pre-ceding time interval.The concrete stress f , at the time tlshall also include change in appliedIoad during the preceding time in-terval. Do not include the factorM CF for accelerated cured concrete.2.5.2Ultimate creep loss2.5.2.1 Normal weight concrete(UCR)Moist cure not exceeding 7 days:

UCR =95 20E 0 /106 _- 11 (7)Accelerated cure:

UCR = 6 3 2 0 E 0 /106 -11 (8)2.5.2 .2 L ightweight concrete (UCR)Moist cure not exceeding 7 days:

UCR =76 20E 0 /106 11 (9)Accelerated cure:

UCR =63 20E 0 /106 11 (10)

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Table 3. Creep factors forvarious volume to surface ratios.Volume to surfacratio, in. Creep factorSCF

1 1.052 0.963 0.874 0.775 0.68>5 0.68

Table 4. Creep factors for variousages of prestress and periodsof cure.Age of prestress Period of Creeptransfer, days cure, days factor, MCF3 3 1.145 5 1.077 7 1.0010 7 0.9620 7 0.8430 7 0.7240 7 0.60

Table 5. Variation of creep withtime after prestress transfer.Time after Portion ofprestress ultimatetransfer, days creep, AUC

1 0.082 0.165 0.187 0.2310 0.2420 0.3030 0.3560 0.4590 0.61180 0.61365 0.74End ofservice life 1.00

Table 6. Shrinkage factors forvarious volume to surface ratios.Volume to surface Shrinkage factorratio, In. SSF

1 1.042 0.963 0.864 0.775 0.696 0.60

2.5.3-Effect of size and shape ofmember (SCF)To account for the effect of izeand shape of the prestressed mem-bers, the value of SCF in Eq. (6) isgiven in Table 3.2.5.4-Effect of age at prestress andlength of cure (MC F)To account for effects due to theage at prestress of moist cured con-crete and the length of the moistcure, the value of MCF in Eq. (6) isgiven in Table 4. The factors in thistable do not apply to acceleratedcured concretes nor are they appli-cable as shrinkage factors.2.5.5-Variation of creep with time(AUC)The variation of creep with timeshall be estimated by the valuesgiven in Table 5. Linear interpola-tion shall be used between thevalues listed.2.5.6-Amountf creep over eachstep (PCR )The portion of ultimate creep overthe time interval ti to t, PCR in Eq.(6), is given by the following equa-tion:

PCR = (AUC)t-(AUC)tl11)2.6-Loss Due to Shrinkage of

Concrete (SH)2.6.1-Loss over each stepLoss ove r each time interval is givenby

SH= (USH)(SSF)(PSH)12)2.6.2-Ultimate loss . due to shrink-age of concreteThe following equations apply to50


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both moist cured and acceleratedcured concretes.2.6.2.1Normal weight concrete(USH)

U S H = 27 ,0 0 0 30 0 0 E 0 /10613)but not less than 12,000 psi.2.6.2.2 Lightweightoncrete(USH)

U S H = 41 ,0 0 0 10 ,0 0 0 E 0 /10614 )but not less than 12,000 psi.

2.6.3 Effect of . size and shape ofmember (SSF )To account for effects due to thesize and shape of the prestressedmem ber, the value of SSF in Eq. (12)is given in Table 6.

Table 7. Shrinkage coefficientsfor various curing times.Time afterend ofcuring, days

Portion ofultimateshrinkage, AUS1 0.083 0.165 0.207 0.2210 0.2720 0.3630 0.4260 0.5590 0.62180 0.68365 0.86End ofservice life 1.00

steel shall be taken as 1/24 of aday so that log tlat this time equalszero.)2.7.1Stress-relieved steel

2.6.4Variation of shrinkage with RET = ft { [log 2 4t log 2 4t 1 ] /10} Xtime (AU S)ft /f" 0.55]16)The variation of shrinkage with timeshall be estimated by the values giv- whereen in Table 7. Linear interpolationt/ fr 0.55 0.05shall be used between the values0.85 feulisted. 2.7.2Low-relaxationsteel2.6.5Amount of shrinkage overeach step (PSH )The portion of ultimate shrinkageover the time interval t1o t, PSHin Eq. (12 ), is given by the followingequation:

PSH = (AUS)t (AUS)t , ,15)2.7Loss Due to Steel

Relaxation (RET)Loss of prestress d ue to steel relaxa-tion over the time interval tlo tmay be estimated using the follow-ing equations. (For mathematicalcorrectness, the value for tl at thetime of anchorage of the p restressing

The following equation applies toprestressing steel given its low relax-ation properties by simultaneousheating and stretching op erations.RET = ft ([log 2 4t log 2 4t 1 ] /45} x

[f8t/fl , 0.55]17)where

0.550.05fl,= 0.90 f,.02.7.3O ther p restressing steelRelaxation of other types of pre-stressing steel shall be based uponmanufacturer's recommendationssupported by test data.

PCI JOURNAL/July-August 1975 1


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CHAPTER 3SIMPLIFIED METHOD FOR COMPUTINGPRESTRESS LOSSES

3.1ScopeComputations of stress losses in ac-cordance with the General Methodcan be laborious for a designer whodoes not have the procedure set upon a com puter program. The Simpli-fied Me thod is based on a large num-ber of design examples in which theparameters were varied to show theeffect of different levels of concretestress, dead load stress, and otherfactors. These examples followedthe General Method and the proce-dures given in the Design Examples.3.2Principles of the SimplifiedMethod3.2.1Concrete stress at the criticallocationCompute ft,,.and f8at the criticallocation on the span. The critical lo-

cation is the point along the spanwhere the concrete stress under fulllive load is either in maximum ten-sion or in minimum compression. Iff od8 exceeds for the simplified methodis not applicable.for and fidsare the stresses in theconcrete at the level of the center ofgravity of the tendons at the criticallocation. f ,.s the net stress due tothe prestressing force plus the w eightof the prestressed member and anyother permanent loads on the mem-ber at the time the prestressing forceis applied. The prestressing forceused in computing f O ,.s the forceexisting immediately after the pre-stress 'has been applied to the con-crete. f7dgis the stress due to all per-manent ((dead) loads not used incomputing for.

Table 8. Simplified method eq uations for computing total prestress loss (TL).Equationnumber Concreteweight Type of tendon Tensioning Equations

W LW SR LR BAR PRE PON-SR-PRE-70 X X X TL = 33.0 + 13.8f 4.5f,d,L-SR-PRE -70 X X X TL = 31.2 + 16.8fa,, 3.8f0.N-LR-PRE-75 X X X TL 19.8 + 16.3fo, L-LR-PRE-75 X X X TL =17.5 + 20.4fo, 4.8t,d,N-SR-POST-68.5 X X X TL = 29.3 -{- 5.1 fcr 3.0fca.L-SR-POST -68.5 X X X TL = 27.1 + 10.1 for 4.9fca,N-LR-POST-68.6 X X X TL = 12.5 + 7.01cr 4.1foa,L-LR-POST-68.5 X X X TL 11.9 + 11.1 fcr 6.2fea: 0.50HJLi0:0.25

1 000^AGE at LOADING , daysFig. 4. Relative ultimate creep versus loading age.0

0

0

C 90 100 150TIME ,-daysFig. 5. Percentage of ultimate creep versus time.PCI JOURNAL/July-August 1975 59


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1500vNNt_C0'E 1000wzN500w

t

pLOFJ

x. 0.0A.0.0MODULUS of ELASTICITY, 106si

Fig. 6. Ultimate shrinkage versus modulus of elasticity.

(.5

w 1.2zw I.02wI-I--J 0.5wI- 0.2wa :

O

5

05

0

5

20000RELATIVE HUMIDITY, percent

Fig. 7. Relative ultimate shrinkage versus relative humidity.60


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13 4 5 6 7 8I .50Ido 1.25Yz 1.00U)Wh - 0.75I.-JMW0.50W0.25e: VOLUME / SURFACE, inchesFig. 8. Relative ultimate shrinkage versus volume to surface ratio.225

200U)

U)N7 5w4-NJwLLI 150

1251 000000000000000TIME, hoursFig. 9. P restressing steel stress versus tim e for stress-relieved stee l.PCI JOURNAL/July-August 19751


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2.6 Comments on loss due to shrink-age are similar to those for creep, ex-cept that concrete stress is not a fac-tor. Ultimate shrinkage strain fromstandard tests is shown in Fig. 6(References 1, 2, 14, 15, 21, 24).These data were modified to 70 per-cent relative humidity using infor-mation illustrated in Fig. 7 (Refer-ences 3, 7, 8, 11, 17), and multipliedby the steel modulus of elasticity toobtain Eqs. (13) and (14). USH is theloss in steel stress due to shrinkageshortening.USH is influenced by the size andshape of the prestressed member.Shrinkage volume/surface data arepresented in Fig. 8 (References 7, 1117, 19). The size factor SSF is givenin Table 6. By. multiplying USH bySSF, standard test data can be ap-plied to actual prestressed members.The variation of shrinkage with timewas generalized using data in Ref-erences 2, 15, and 24. This informa-tion is given in Table 7 as the por-tion of ultimate shrinkage AUS for aspecific time after the end of curing.

The amount of shrinkage PSH oc-curring over a specific time intervalis the difference between the amountof shrinkage at the beginning of thetime interval and that at the end ofthe time interval.The loss of prestress over one timeinterval due to shrinkage of concreteis stated in Eq. (12).2.7Eqs. (16) and (17) (References13, 32) give the loss of prestress duesteel relaxation. The time tl inStep 1, listed in Section 2.4, is at an-chorage of the prestressing steel. Inpretensioned construction, whereelevated temperatures are used incuring, losses during curing can bestudied more closely using Refer-ences 5, 22, and 26.Fig. 9 shows typical steel relaxationwith time under constant strain. It isseen that losses are less for lowerinitial stresses. This illustrates thatby taking into account concreteshortening and steel relaxation overa previous time period, subsequentlosses are less than that under as-sumed constant strain.

NOTATIONA,gross cross-sectional area ofconcrete member, sq in.A .,cross-sectional area of pre-stressing tendons, sq in.ANC = loss of prestress due to anchor-age of prestressing steel, psiAUC = portion of ultimate creep at

time after prestress transferAUS = portion of ultimate shrinkageat time after end of curingCR= loss of prestress due to creepof concrete over time intervaltl to t, psi

DEF loss of prestress due to deflect-

ing device in pretensioned con-struction, psie = tendon eccentricity measuredfrom center of gravity of con-crete section to center of gravi-ty of tendons, in.

E,, = modulus of elasticity of con-cretet8ay sakens33w3/2 V1,,',psi= modulus of elasticity of con-cretetimef initialre -stress, psiE, = modulus of elasticity of steel,psi

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ES = loss of prestress due to elasticshortening, psifedR = concrete compressive stress atcenter of gravity of prestress-ing force due to all permanent(dead) loads not used in com-puting f c r ,psif ,concrete compressive stress atcenter of gravity of prestress-ing steel, psi= compressive strength of con-crete at 28 days, psi

L = initial concrete compressivestrength at transfer, psif rconcrete stress at center ofgravity of prestressing forceimmediately after transfer, psi

fPzG =guaranteed ultimate tensilestrength of prestressing steel,psi

fstress at 1 percent elongationof prestressing steel, psi

ice= effective stress in prestressingsteel under dead load afterlossesfs4= stress in tendon at critical loca-tion immediately after pre-stressing force has been ap-plied to concrete

fst = stress in prestressing steel attime t 1 , psiftstress at which tendons are an-chored in pretensioning bed,psiFR= friction loss at section underconsideration, psi1moment of inertia of grosscross section of concrete m em-ber, in.4Kfriction wobble coefficient perfoot of prestressing steell tlength of prestressing steelfrom jacking end to point x, ft

MCF=actor that accounts for theeffect of age at prestress andIength of moist cure on creepof concrete

Md4 = moment due to dead weightadded after member is pre-stressed

M= moment due to loads, includ-ing weight of member, at timeprestress is applied to concretePfinal prestress force in memberafter lossesP oinitial prestress force in mem-berPCR = amount of creep over time in-terval t l to tPSH = amount of shrinkage over timeinterval t l to tRE= total loss of prestress due torelaxation of prestressing steelin pretensioned construction,psi

REP = total loss of prestress due torelaxation of prestressing steelin post-tensioned construction,psiRET = loss of prestress due to steelrelaxation over time intervaltl to t, psiSCF = factor that accounts for the ef-fect of size and shape of amember on creep of concreteSH = loss of prestress due to shrink-age of concrete over time in-terval tl to t, psiSSF = factor that accounts for theeffect of size and shape of amember on concrete shrinkagettime at end of time interval,daystl= time at beginning of time in-terval, daysT o = steel stress at jacking end ofpost-tensioning tendon, psiT 7 = steel stress at any point x, psiTL= total prestress loss, psi

UCR=ultimate loss of prestress dueto creep of concrete, psi perpsi of compressive stress in theconcreteUSH =, ultimate loss of prestress dueto shrinkage of concrete, psiwweight of concrete, lb per cu ftatotal angular change of post-tensioning tendon profile fromjacking end to point x, radianstkfriction curvature coefficientPCI JOURNAL/July-August 19753


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REFERENCES

I. Chubbuck, Edwin R., "Final Re-port on Research Program for theExpanded Shale Institute," ProjectNo. 238, Engineering ExperimentSection, Kansas State College,Manhattan, K ansas, July, 195 6.2. Shideler, J. J., "Lightweight Aggre-gate Concrete for Structural Use,"Development Department BulletinD-17, Portland Cement Associa-tion; see also ACI Journal, V. 54 ,No. 4, October, 1957, pp. 29 9-32 8.3. Troxell, G. E., Raphael, J. M., andDavis, R. E., "Long Time Creepand Shrinkage Tests of Plain andReinforced Concrete," ASTM Pro-ceedings, Vol. 58, 195 8.4. Ross, A. D., "Creep of ConcreteUnder Variable Stress," ACI Jour-nal, V. 29, No. 9, March, 1958,pp. 739-758 .5. Preston, H. Kent, "Effect of Tem-perature Drop on Strand Stressesin a Casting Bed," PCI JOURNAL,V. 4, No. 1, June, 1959, pp. 54-57.6. Freyermuth, C. L., "Design of"Continuous Highway Bridges withPrecast, Prestressed Concrete Gird-ers," PCI JOURNAL, V. 14, No. 2,April, 1969 , pp. 14 -39.7. Jones, T. R., Hirsch, T. J., andStephenson, H. K., "The PhysicalProperties of Structural QualityLightweight Aggregate Concrete,"Texas T ransportation Institute, Tex-as A & M University, College Sta-tion, August, 1959.8. Lyse, I., "Shrinkage and Creep ofConcrete," ACI Journal, V. 31, No.8, February, 1960, pp. 775-782.9. Corley, W. G., Sozen, M. A., andSiess, C. P., "Time-Dependent De-flections of Prestressed ConcreteBeams," Highway Research BoardBulletin No. 307, National Acad-

emy of SciencesNational Re-search Council Publication No.937, 1961.10. Mattock, A. H., "Precast-Pre-stressed Concrete Bridges-5.Creep and Shrinkage Studies," De-velopment Department Bulletin D-46, Portland Cement Association;see also Journal of the PCA Re-search and Development Labora-tories, M a y , 1961.

11. Bugg, S. L., "Long-Time Creep ofPrestressed Concrete I-Beams,"Technical Report R-212, U.S. Na-val Civil Engineering Laboratory,Port Hueneme, California, October2 , 1962.12. ACI Committee 435, Subcommit-tee 5, "Deflections of PrestressedConcrete Members," ACI Journal,V. 60, No. 12, December, 1963,pp. 1697-1728 .13. Magura, Donald D., Sozen, M. A.,and Siess, C. P., "A Study of StressRelaxation in Prestressing Rein-forcement," PCI JOURNAL, V. 9,No. 2 , April, 1964 , pp. 13-57.14. Reichard, T. W., "Creep and Dry-ing Shrinkage of Lightweight andNormal-Weight Concretes," Na-tional Bureau of Standards Mono-graph 74, U.S. Department ofCommerce, March 4, 1964 .15. Hanson, J. A., "Prestress Loss asAffected by Type of Curing," De-velopment Department Bulletin D-75, Portland Cement Association;see also PCI JOURNAL, V. 9, No.2 , April, 1964 , pp. 69-93.

16. Zia, P., and Stevenson, J. F.,"Creep of Concrete Under Non-Uniform Stress Distribution and ItsEffect on Camber of PrestressedConcrete Beams," Project ERD-100-R, Engineering Research De-partment, North Carolina State

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University, Raleigh, North Caro-lina, June, 1964 .17 Keeton, J. R., "Study of Creep inConcrete, Phases 1-5," TechnicalReport Nos. R-333-I, -II, -III, U.S.Naval Civil Engineering Labora-tory, Port Hueneme, California,1965 .18. Selected Climatic Maps of heUnited S tates, Office of Data Infor-mation, Environmental ScienceService Administration, U.S. De-partment of Commerce, 1966.19. Hansen, T. C., and Mattock, A. H.,"Influence of Size and Shape ofMember on the Shrinkage andCreep of Concrete," DevelopmentDepartment Bulletin D-103, Port-land Cement Association; see alsoACI Journal, V. 63, No. 2, Feb-ruary, 1966, pp. 2 67-29 0.20. ACI Committee 435, "Deflectionsof Reinforced Concrete FlexuralMembers," ACI Journal, V. 63, No.6, June, 1966, pp. 637-674.21 Furr, H. L., and Sinno, R., "Creepin Prestressed Lightweight Con-crete," Texas Transportation Insti-tute, Texas A & M University, Col-lege Station, Texas, Octobe r, 1967.

22. Navaratnarajah, V., "An Analysisof Stresses During Steam Curing ofPretensioned Concrete," Construc-tional Review, December, 1967.23 Hickey, K. B., "Creep of ConcretePredicted from the Elastic Modulus

Tests," Report No. C-1242, Con-crete and Structural Branch, Divi-sion of Research, Bureau of Recla-mation, Denver, Colorado, January,1968.24. Pfeifer, D. W., "Sand Replacementin Structural Lightweight ConcreteCreep and Shrinkage Studies,"Development Department BulletinD-128, Portland Cement Associ-ation; see also ACI Journal, V. 65 ,No. 2, February, 1968, p. 131.25. Rokhsar, A., and Huang, T., "Com-parative Study of Several Con-

PCI JOURNAL/July-August 1975

cretes Regarding Their Potentialsfor Contributing to PrestressLosses," Fritz Engineering Labora-tory Report No. 339.1, Lehigh Uni-versity, Bethlehem, Pennsylvania,May, 1968.

26. Papsdorf, W., and Schwier, F.,"Creep and Relaxation of SteelWire, Particularly at Highly Ele-vated Temperatures," Stahl u.Eisen, July, 1968; Library Trans-lation No. 84, Cement and Con-crete Association, London, July,1969.27. Schultchen, E., and Huang, T.,"Relaxation Losses in %s in. Diam-eter Special Grade PrestressingStrands," Fritz Engineering Labo-ratory Report No. 339.4, LehighUniversity, Bethlehem, PennsyI-vania, July, 1969.28. Huang, T., and Frederickson,D. C., "Concrete Strains in Pre-Tensioned Concrete StructuralMembersPreliminary Report,"Fritz Engineering Laboratory Re-port No. 339.3, Lehigh University,Bethlehem, Pennsylvania, June,1969.29. Branson, D. E., Meyers, B. L., andKrinanarayanan, K. M., "Time-De-pendent Deformation of Non-Com-posite and Composite Sand-Light-weight Prestressed ConcreteStructures," Report No. 69-1, De-partment of Civil Engineering, Uni-versity of Iowa, Iowa City, Feb-ruary, 1969 .30. ACI Committee 318, "BuildingCode Requirements for ReinforcedConcrete (ACI 318-71)" and"Commentary on Building CodeRequirements for Reinforced Con-crete (ACI 318-71)," AmericanConcrete Institute, Detroit, Michi-gan, 1971.31. Branson, D. E., and Kripanarayan-an, K. M., "Loss of Prestress andCamber of Non-Composite andComposite Prestressed Concrete

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Structures,'" Report No, 70-3, De-partment of Civil Engineering, Uni-versity of Iowa, Iowa City, Iowa,June, 1970.32. Glodowski, R. J., and Lorenzetti,J. J., "A Method for PredictingPrestress Losses in a PrestressedConcrete Structure," PCI JOUR-NAL, V. 17, No. 2, March-April,1972, pp. 17-31.33. Design and Control of ConcreteMixtures, Portland Cement Asso-ciation, Old Orchard Road, Skokie,Illinois 60076.

34. Recommendations for an Interna-tional Code of Practices for Rein-forced Concrete, published by theAmerican Concrete Institute andthe Cement and Concrete Associa-tion.35. PCI Design Handbook Precastand Prestressed Concrete, Pre-stressed Concrete Institute, Chica-go, Illinois, 1971.

36. Interim Specifications Bridges1975, American Association ofState Highway and TransportationOfficials, W ashington, D.C., 1975 .

DESIGN EXAMPLESThe following three design exampleswere prepared solely to illustrate theapplication of the preceding recom-mended methods. They do not neces-sarily represent the real condition ofany real structure.Design aids to assist in calculatingprestress Iosses are included in thePCI Design Handbook (see Refer-ence 35). The aids will reduce thecalculations required. However, de-tailed study of losses and time-de-pendent behavior will follow thesteps outlined in the design exam-ples.The first example applies the generalmethod to a pretensioned double-teeand the second example uses thesimplified method for the samemember. The third example problemillustrates the general method for apost-tensioned structure.In these examples it is assumed thatthe member geometry, load condi-tions, and other parameters havebeen defined. Consequently, the de-tailed moment and stress calcula-tions are omitted.

DESIGN EXAMPLE 1Pretensioned Double TeeReference: PCI Design Handbook, p.3-33.Data: Double-tee section 1OLD T32 + 2 .Strand pattern 128-D1.Steam cured, ligthweight double-tee(115 lb per cu ft) with 2-in, topping ofnormal weight concrete (150 lb per cuft).The beam is designed to carry a liveload of 40 psf over a 70-ft span.Required: Calculate the losses at thecritical section, taken as 0.4 span inthe PCI Design Handbook. f', t i = 3500psi, f =5000 psiSection properties:Non-compositeA =615in.2I = 59,720 in.4Yb = 21.98 in.?It = 10.02 in.Z b = 2717 in.3Z t = 5960 in.3Weight: 491 lb per ft

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51 -0"

Design Example 1. Cross section of double-tee beam.

Composite UCR =63 2 0(E,/100)I8 3,001 in.4 (but not less than 11)y b ^ = 2 5.40 in. = 63 20(2.88) = 11ytc =8.60 in. USH = 41,000 10,000(E0/106)Z, = 3268 in. 3 (but not less than 12 ,000)Z t = 9651 in. 3 = 41,000 10,000(2.88 )Weight: 741 lb per ft = 12,200 psiThe beam is prestressed by twelve 1/-in. At critical sectiondiameter 270-grade strands, initially e = 12.98 + 0.8(18.73 12.92)tensioned to 0.70 f,..17.58 in.Eccentricity of strands:UCR)(SCF) =10.87

At ends = 12.98 in.USH)(SSF) =12 ,017 psiAt center = 18.73 in.tage 1: Tensioning of steel to transferfPy = 2 30 ksi

Transfer at 18 hours after tensioning tl = 1/24 daystrand, topping cast at age 30 days. t = 18/24 dayfst =189,000 psiLosse3B asic data

= 3500 psiE C Z = 1151 . 5 (33/3500)=2.41 x 10psi

= 5000 psiE, = 1151 . 5 (33V5000)= 2.88 x 106 psiVolume to surface ratio = 615/364

= 1.69SSF = 0.985SCF = 0.988

RET = fst [(log 24t log 24 t 1 )/ 10] X[fst/f,, 0.55]= 189,000 [,(log 18)/10][189/230 0.55]= 6450 psi

Dead load moment at 0.4 spanM DL = w (x/2 )(L x)_ (491/1000)(28/2)(70 28)= 289 ft-kips

Stress at center of gravity of steel dueto MDLPC! JOURNAL/July-August 19757


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f_289,000(12)/59,720] 17.58= 1020 psi (tension)

Assume ES 13 ksi, thenf,52 =189.0-6.45- 13.0= 169.55 ksiPo = 169.55(12)(0.153)= 311.3 kipsStress at center of gravity of steel dueto P0:f e = 311,300/615 +

311,300 (17.582/59,720= 2117 psi (compression)

f . = 2117 - 1020 = 1097 psiES = ferr(Es/Ee)

= 1097(28.0/2.41)= 12,750 psi - 13 ksi (ok)SH=CR=0Total losses in Stage 1 =

6450 + 12,750 = 19,200 psiStage 2: Transfer to placement of top-ping after 30 daystl18/24 dayt30 daysPCR=0.35PSH 0.42fv7.= 189,000 - 19,200= 169,800 psiRET = 169,800 [(log 720 - log 18)/10] x [169.8/230 - 0.55]= 5119 psiCR = 10.87(0.35)(1097) = 4173 psiSH = 12,017(0.42) = 5047 psiTotal losses in Stage 2

5119 + 4173 + 5047 = 14,339 psiMoment due to weight of topping

250(28/2)(70 - 28) = 147,000 ft-lbStress at center of gravity of steel dueto weight of topping147,000(12)(17.58)/59,720 = 519 psiIncrease in strand stress due to topping

519(28.0/2.88) = 5048 psiStrand stress at end of Stage 2169,800 - 14,339 + 5048 = 160,509psi

Stage 3: Topping placement to end ofone yeart =0 dayst1 year = 365 daysPCR = 0.74 - 0.35 = 0.39PSH 0.86-0.42=0.44f ,, t =160,509 psiRET = 160,509 [(log 8760 - log 720/

log 720/10] X[160.5/230-0.55]= 2577 psi

= 2117(160,509/169,550) -1020 - 519

=465 psiCR = 10.87(0.39)(465) = 1971 psiSH = 12,017(0.44) = 5287 psiTotal losses in Stage 3

2577 + 1971 + 5287 = 9835 psi

Summary of steel stresses atvarious stages (Design Example 1)

SteelStress level attress,various stagessi PercentStrand stress aftertensioning anddeflection (0.70fp0) .. 189.0 100.0Losses:

Elasticshortening12.75 6.7Relaxation: 6.45+5.12 + 2.58 +2.54 = 16.69 8.8

Creep: 4.17 +1.97 + 0.97 = 7.11 3.8

Shrinkage: 5.05 +5.29 + 1.68 = 12.02 6.4

Total losses, TL ....8.57 25.7Increase of stressdue to topping ......05 2.7Final strand stressunder total deadload (f0).........45.48 77.068


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Stage 4: One year to end of service lifet l = 1 yeartend of service life (say 40 years)PCR = 1 - 0.74 = 0.26PSH = I - 0.86 = 0.14f," f160,509 - 9835 = 150,674 psiRET = 150,674 [(log 350,400 -log 8760)/10] x [(150.7/230) - 0.55]

= 2537 psi= 2117(150,674/169,550) -

1020 - 519= 343 psi

CR= 10.87(0.26)(343) = 969 psiS H = 12,017(0.14) = 1682 psiTotal losses in Stage 4 =

2537 + 969 + 1682 = 5188 psi

DESIGN EXAMPLE 2Application of SimplifiedProcedure to Design ExampleCompute f 1 d ,fcd,v = eMd,/I= 17.58(147)(12)/59,720

= 0.519 ksiCompute f 1 1f, A,,f,, /A 1 + AJM e2 /IC + M' e/1f.4ti = 0.9O f.f = 0.90(189) = 170.1 ksif c r = 1.84(170.1)/615 +1.84(170.1)(17.58) 2/59,720 -2 89(12)(17.58) /59, 72 0

= 0.509 + 1.620 - 1.021= 1.108 ksi

Equation L-SR-PRE-70 from Table 8 isTL =31.2+ 16.8fcr-3.8fcd3

= 31.2 + 16.8(1.108) - 3.8(0.519)=31.2+18.61-1.97= 47.84 ksi

Adjustment for volume to surface ratio= 1.69Use a straight-line interpolation be-tween adjustment values for V/S = 2.0and V/S = 1.0Adjustment = (0.31)(3.2) = + 0.99%

Net TL =1.0099(4 7.84) = 48 .31 ksiIn Design Example 1, TL = 48.57 ksiDifference = 0.26 ksi

Compute f 8 1To find f, 1n accordance with discus-sion under Section 3.32, and stress intendons due to dead load applied aftermember was prestressed.This stress is equal tofcds(Es/E c) = 0.519(28/2.88) = 5.05 ksif91 = 189 - 48.31 + 5.05 = 145.74 ksiNote that f3e can also be computed fromthe equations shown in Table 9.Equation L-SR-PRE-70 from Table 9 isf 3 P =ft-31.2+ 16.8f - 13.5f,4 )=189-(31.2+16.8 X1.108-13.5 X0.519)= 189 - (31.2 + 18.61 - 7.01)

= 189 - (42.8)An adjustment for variations in thebasic parameters should be applied tothe quantity in parentheses. In thiscase, adjust for a V/S of 1.69. The ad-justment is +0.99 percent. The ad-justed quantity becomes1.0099(42 .8) = 43.2 2f s ^ = 189 - 43.22 = 145.78 ksiChecking the assumed valueof 9 z :In the application of the simplifiedmethod to Design Example 1, the valueof f 3 L was assumed to be 170.1 ksi.The following procedure can be used tocheck the accuracy of this assumedvalue.For this example the exact value off , 4 2 is the initial stress of 189 ksi re-duced by strand relaxation from ten-sioning to release and by loss due toelastic shortening of the concrete as theprestressing force is applied.From Section 2.7.1, the relaxation lossin a stress-relieved strand isRET = fst [(log 24t - log 24t 1 )/10] X

[fst/fpu - 0.55]



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For stress-relieved strand41 = 0.85(270) = 229.5 ksiBy definition in Section 2.7.1, whentime is measured from zero, log 24t, = 0RET = 189 [ (1.2 55 0) /10][189/229.5 0.55]

= 6.49 ksiStress loss due to elastic shortening ofconcreteES = (E,,/Ec)fer=28/0.24)1.108

= 12.93 ksiThen, f st i = 189 6.49 12 .93 =169.58 ksiand 0.90 f t = 170.10 ksiTherefore, there is a 170.10 169.58 =0.52 ksi stress error in fsi.Consequently, in this particular casethere is no need for a second trial.As an example, assume a large error inthe estimated fS 2 ,say 10 ksi, and checkits effect. The strand relaxation will notchange.Therefore, the change in ES will be

AES = (10/170.1)12.93 = 0.76 ksiIf desired, the original estimate of f 4;can be adjusted by 10 ksi and f,.,. canbe recalculated. One such cycle shouldalways give an adequate accuracy.

DESIGN EXAMPLE 3Post-Tensioned Unbonded SlabsThe following is a procedure for calcu-lating the prestress losses in the longi-tudinal tendons which extend from endto end of the slab (see sketch showingfloor plan and tendon profiles).Dataw = 150 lb per cu ftf,' (28 days) = 4000 psiPrestressed at age 4 daysf3000 psiMoist cured 7 days.

Loads7 1/a-in. slab = 94 psfSuperimposed load = 63 psfThe tendon profile shown is designedto balance 85 psf.Friction Loss (FR)The slab is prestressed by 270-grade,1/z-in, diameter strand, pregreased andpaper wrapped.Coefficient of friction, p. = 0.08Wobble coefficient, K = 0.0015fry = 230 ksi.Angu'ar changes along tendon will be:OA /1= 2(2.5)/ [12(12)]= 0.0347 radiansOLU = OFF = OFG _ OKL =2(4.0)/ [12(9.6)]

= 0.0694 radianscn = ODE = 0GH = OHK2(1.0)/ [12(2.4)]

= 0.0694 radiansAngular change between A and Lcr = 0.0347 -{- 4(0.0694 ) T 4 (0.0699)

= 0.59 radiansFR at L (middle of length of slab)

=TO [1-e-(KL+a)]= T. [1-e{(o.00r5)(6o)+(o.os)(o.5s)}]= T. [1- e (0.000+0.047)]= 0.12 8 To

The distribution of frictional loss is notuniform, but nearly proportional to(K + p.a/L). However, the variation ofstrand stress before anchoring is ap-proximately as shown on p. 72.Anchorage loss (ANC )Anchorage set in a single strand anchor= 1/8 in.

= Shaded area in diagram X (1/Es)Area = (1/8)29,000 = 3625 ksi-ft= 302 ksi-in.The maximum strand stress after seat-ing of anchorage occurs x ft from end,

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9 4 ' 94' 9a' 9 4 9 4 '.i. , r.RABOLAS BETWEEN PIS.

ADE FHK L21.6'9.2'9.2'9.2'1.6'2.4.4.4.4ACTUAL LONGITUDINAL TENDON PROFILE(TENDON STRESSED FROM BOTH ENDS)DIMENSIONS FROM SLAB FACE TO TENDON CGS

4.9 K / FT. 1/4"/4"I/4 " I/q".9 K / FT.n..3/41/4/4/4/4/43/414.7 K/FT.THEORETICAL TENDON PROFILE(REQUIRED FINAL PRESTRESS FORCE SHOWN)Design E xam ple 3. P lan and tendon profiles of post-tensioned unbo nded slab.EIIr



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n n n r C r_ u V0.0194 To0.0380

STRANDBEFORETo

0.0550To

STRANDAFTER

STRESSSEATING

STRESSSEATING

0.0923 To0109410

LI -

X

1 2 9.6' 4.8'

Approximate variation of strand stress.

and this stress, must not exceed ANC = 2(T0 - T o m ) =2 0.8 ksi0.70f = 189 ksi. T4 = 200 - 20.8 = 179.2 ksiT o - T, = 0.0380 T o + FR = 0.128 T o = 2 5.6 ksi(0.0550 - 0.0380)T0 (x - 21.6)/4.8 Tr = 200 - 25.6 = 174.4 ksi

.1280To

Area(To-T^)12+(0.9806 T p - T^)2 1.6 +(0.9620 T o - Tx ) (x - 12 )ThereforeT., = 0.9620T0 =0.0170T 0 (x - 21.6)/4.8

= 189 ksi(T0 - T^)12 + (0.9806T0 - T_)21.6 +(0.9620T 0 - T o m ) (x - 12)= 302 ksi-ftThese equations can be solved by trialand error.Approximate solution:x = 25.5 ftT o = 200 ksiT,=192.4 - 2.8 = 189.6 189 ksiArea = 124.8 + 140.8 + 37.8= 303.4 302 ksi-ft (ok)For initial end tension before anchorageT o = 200 ksi (=0.74 fpu)Maximum stress after anchorageT ,, = 189.6 ksi

Average stress after anchorage:T B= 183.2 ksi, T, = 186.9 ksiTav =T 0 /60)(0.5) [(0.896)(12) +(0.916)(21.6) + (0.935)(13.5) +

(0.94 8)(4.8) + (0.94 5)(2 0.1) +(0.908 )(24 .0) + (0.891)(14.4) +(0.872 )(9.6) ]

= 182.8 ksiElastic shortening loss (ES)In post-tensioned structural members,the loss caused by elastic shortening ofconcrete is only a fraction of the corre-sponding value in pretensioned mem-bers.The fraction varies from zero if alltendons are tensioned simultaneously to0.5 if infinitely many sequential stepsare used.In a slab, strands are spaced far apartand it is unlikely that the stretching ofone strand will affect stresses in strandsother than those immediately neighbor-ing.

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A factor of 0.25 will be used.ES = 0.25(E 8 E) ter

In a design such as this in which theprestress approximately balances thedead load, and the level of prestress islow, a sufficiently close estimate of f(. rcan be obtained by using the averageprestress P/ A .The design final prestress force is 14.7kips per ft for interior spans and 19.6kips per ft for end spans. Assuming along-term prestress loss of 15 percent,the initial prestress force will be 17.3kips per ft and 23.1 kips per ft, respec-tively. The average strand stress afteranchorage is 182.8 ksi.Therefore, the required area of steel forthe end spans isA, = 23.1/182.8 = 0.126 sq in. per ft

This required area is supplied by 1/z-in.diameter strands spaced at 14 in.A,.= 0.131sg in. per ft

Every fourth strand will be termi-nated in the first interior span, leavingan A;, of 0.098 sq in. per ft.The actual initial prestressing forcesare:End span

0.131(182.8) = 23.9 kips per ftInterior span0.098(182 .8) = 17.9 kips per ftThe average concrete stresses are 266and 199 psi, respectively.fr r (1/12 0) [(266)(48 ) + (199)(72)]= 226 psiE. = 33w'5V' 33(150)1.6V/3000= 3.32 x 10 6 psiES= 0.25(226)(29/3.32) = 494 psiAfter all the strands have been ten-sioned End anchored:

= 182 .8 0.494 = 182 .3 ksiAt midspan of the middle spanStrand stress (at L)

174.4 0.494 -- = 173.9 ksifir [(173.9)(0.098)] / [(12)(7.5)]0.189 ksiLong-term lossesThe calculation for long-term losseswill be for the midspan of the middlespan (at Section L).Stage 1: To 30 days after prestressingRelaxation:tl = 1/24 dayt = 30 daysfl, = 173.9 ksifst/f,, = 0.756RET = f, [(log 24 t log 2 4t 1 )/10] X

[(f,t/f,) 0.55]= 173,900(0.28 57)(0.2 06)= 10,230 psi

Creep:CR = (UCR)(SCF)`MCF)(PCR)(f,.)UCR = 95 (20E,/106)

but not less than 11 psiE 0 = 33(150)1.5/4000

= 3.83 X 10, psiUCR = 95 - 76.6 = 18.4 psiV/S ratio = 0.5(slab thickness)= 0.5(7.5)

= 3.75 in.SCF = 0.80MCF = 1.07 (estimated)(UCR)(SCF)(MCF) = 15.75fc=f^,,. 189psiPCR = 0.35CR = 15.75(0.35)(189) = 1042 psiShrinkage:SH = (USH)(SSF)(PSH)USH = 2 7,000 (3000 E, /10G)

but not less than 12,000

PCI JOURNAL/Juy-August 1975 73


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US H = 27,000 - 11,490 = 15,510 psiV/S = 3.75 in.SSF = 0.79(USH)(SSF) = 12,2 70 psiTime after end of curing = 27 daysPSH = 0.402SH = 12 ,270(0.402 ) = 4933 psiTotal losses in S tage 1:RET + CR + SH = 10,230 + 1042 +4933

= 16,205 psiTendon stress at end of Stage 1

173,900 - 16,205 = 157,695 psiConcrete fiber stress

189(157,695/173,900) = 171.4 psi

Stage 2: To 1 year after prestressingRelaxation:tl = 30 dayst = 1 year = 365 daysfst = 157,695 psif.c/fpy =157.7/230 = 0.671RET = 157,695 [(log 8760-log 30)/10]x [(0.671 - 0.55)]

= 2070 psiCreep:f = 171.4 psiPCR=0.74-0.35=0.39CR = 15.75(0.39)(171.4) = 1053 psiShrinkage:PSH = 0.86 - 0.402 = 0.458SH = 12,270(0.458) = 5620 psiTotal losses in S tage 2:

2070 + 1053 + 5620 = 8743 psiAt end of Stage 2, tendon stress at Sec-tion L

157,695 - 8743 = 148,952 psiConcrete fiber stress

189(148,952/173,900) = 161.9 psi

Stage 3: To end of service life (takenas 50 years)Relaxation:tl = 1 year = 365 dayst = 50 years = 18,250 dayslog 24t - log 24t 1 = 1.699fl ,= 148,952 psifst/frz, = 0.634RET = 148,952(0.1699)(0.084)

= 2126 psiCreep:f, = 161.9 psiPCR = 1 - 0.74 = 0.26CR = 15.75(0.26)(161.9) = 663 psiShrinkage:PSH = 1 - 0.86 = 0.14SH = 12,270(0.14) = 1718 psi

Sum mary of steel stressesat various stages(Design Example 3)SteelStress level attress,various stagessiercent

Tensioning stressatend...........00Average stressafter seating...... 182.8Middle sectionstress afterseating...........74.4 100.0Losses:Elasticshortening ...5 0.3Relaxation .... 14.4 8 .2Creep.........7 1.6Shrinkage.... 12.3 7.0Total lossesafter seating......9.9 17.1Final strand stress at mid-dle section withoutsuperimposed load .. 144.5 82 .9

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Total bog-term losses:RET = 10,230 + 2070 +

2126 = 14,426 psiCR=1042+663=053 + 2,758 psiSH =1933 + 5620 +1718 = 12,271 psiTotal losses29,355 psiThis example shows the detailed stepsin arriving at total losses. It is not im-plied that this effort or precision is re-quired in all design situations.The PCI Post-Tensioning Manual pro-vides a table for approximate prestressloss values which is satisfactory formost design solutions. The value rec-

omm ended for slabs with stress-relieved2 70-kip strand is 30,000 psi which com-pares with the calculated value of29,355 psi.Final tendon stress at Section L

173.9 = 29.4 = 144.5 ksiPercentage loss after anchorage(174.4 144.5)/174.4 = 17.1 percent

but greater than 15 percent (assumedinitially)Assum ing the same percentage loss pre-vails over the entire tendon length, theaverage prestressing forces after lossesare 19.8 and 14.8 kips per ft, respec-tively, which are adequate when com-pared with the design requirements.Therefore, revision is not needed.

Discussion of this report is invited.Please forward your discussion toPCI Headquarters by December 1, 1975.

recomendations for estimating prestress losses

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