a probabilistic comparison of prestress loss methods in

A Probabilistic Comparison ofPrestress Loss Methods inPrestressed Concrete Beams

Loss of prestress is a characteristicof all prestressed concrete members wherein the level of pre

stress force first applied to the member is reduced over time due to short-and long-term conditions.

Many methods have been used toestimate the prestress losses in prestressed concrete members. Thesemethods produce discrete values representing the expected losses, and vary

considerably in both length and complexity of calculations. The methodsconsider many factors in their respective calculations but do not considerthe effect of variability due to the parameters such as concrete strength,strand stress, strand area, dimensionalproperties, and environmental conditions.

The purpose of this study was to investigate the effect of input variation

Christopher G. Gilbertson, RE.Ph.D. StudentMichigan Technological UniversityCivil & Environmental EngineeringDepartmentHoughton, Michigan

Camber and long-term deflections are directly linked to the loss ofprestressing force, a characteristic of all prestressed concretemembers. In this investigation, the effects of the inherent variability ofthe parameters used to estimate prestress loss was studied for twotypical bridge beam cases using several prestress loss predictivemethods at final service conditions. A parametric study wasconducted to assess the general effect of single parameter variationon the calculation of prestress loss. Monte Carlo simulations wereused to assess the distribution of prestress loss considering thevariability of all parameters simultaneously. Results of the MonteCarlo simulations were also used to evaluate the impact of lossvariability on deflection and cracking moment calculations. Resultsshow that the variability of parameters has a notable effect onprestress loss variation and that this variation significantly influencesthe estimated deflection and cracking moment of prestressedconcrete beams. Indeed, the variation in camber between beams caston the same prestressing bed can be partially explained through thevariability of materials and predictive methods used.

Theresa M. (Tess) Ahiborn,Ph.D., P.E.Associate Professor of Civil EngineeringMichigan Technological UniversityCivil & Environmental EngineeringDepartmentHoughton, Michigan

52 PCI JOURNAL

on the computed losses from six common methods used to estimate prestress losses. To assess these effects,two composite bridge systems [a 70in. (1780 mm) I-beam and a 21 in.(535 mm) spread box beam] used insimple span configurations were considered. A parametric study was conducted to independently assess the influence of each parameter on the totalprestress loss.

A second assessment was performedusing a Monte Carlo simulation thatrandomly selected values for each parameter used by the loss calculationmethods based on the known or estimated variability of each parameter.The result was a more inclusive lookat the variability of total prestresslosses. The resulting variability wasthen used to assess variability in expected deflection and cracking moment.

BACKGROUND

An accurate prediction of the prestress loss in a prestressed concretemember is important in the designphase to assess the expected behaviorof the member over its life. Calculatedlosses are used to predict service conditions such as expected concretestress levels, camber, deflection, andcracking loads. However, an accurateprediction of prestress loss is difficultbecause of the complex interaction between the various sources of lossesand the inherent non-homogeneity ofconcrete members.

In general, short-term losses includeelastic shortening of the member at thetime of prestress application and anysteel relaxation that occurs prior totransfer of the prestress force. Long-term prestress losses include creep andshrinkage of concrete and any additional steel relaxation that occurs afterprestress transfer and during the life ofthe member.

There are numerous methods usedto estimate prestress losses. Somemethods require more knowledgeabout the member properties and loading conditions than other methods. Itis important to realize that all suchmethods are approximations.

Steinberg1 studied the effects of statistical variability in the parameters

used to calculate prestress losses.Using three common precast shapes —

a rectangular beam (12RB16), a double tee (1OLDT32), and an inverted Tbeam — he showed that prestress lossesare consistently calculated lower (byabout 33 percent) when using deterministic or nominal parameters thanwhen including statistical variability.Based on the analysis using the PC[General Method,2 Steinberg recom -

mended that nominal losses computedby this method be increased by 25 percent when checking calculated stressesat final conditions due to full load pluseffective prestress against allowablestresses. Further work was recommended to determine the effect ofvariability on prestress losses for additional member types.

The primary objective of the research presented herein was to compare the variability of total prestresslosses computed by several methodswhile accounting for parameter variability through a probabilistic assessment. Two bridge types — an I-beamsuperstructure and a spread box beamsuperstructure — were utilized forprestress loss comparisons and estimating the range of deflections andloads to cause cracking based on van -

ability of total prestress losses.The methods used to estimate pre

stress losses can be divided into thefollowing three primary categories:

1. Lump sum of total losses2. Lump sum of individual losses3. Time-dependent cumulative lossesThe lump sum of total losses

method (Category 1) is considered tobe the least accurate due to an oversimplification of the parameters involved in calculating prestress losses.The lump sum of individual losses(Category 2) is the most common category for prestress loss methods used

in design. These methods account formore variables than the Category I approach and are relatively straightforward in computation. Category 3, thetime-step method, is the most accurate, but is also fairly lengthy and involves knowledge of member loadingover its life duration.

Recommended calculation methodspublished by AASHTO, PCI, ACI,and others2t were considered. Eachmethod and the respective categoryare listed in Table 1.

The AASHTO LRFD4 prestressloss method is based on that providedin the AASHTO Standard Specifications,3 although steel relaxation is presented differently in the estimate ofelastic shortening. While additionalmethods can be found (e.g., CEB-FIPModel 9010), the study was limited tothe six methods in Table I. Thereader is directed to each respectivereference for details in calculatinglosses according to AASHTO LRFD4and Standard Specifications.3The Appendix contains details on the timestep method,5 PCI General2 and Simplified6methods, and the Zia method.8

The losses computed in this studyare long-term, and include the effectsof steel relaxation prior to transfer andelastic shortening. Anchorage losses atthe jack prior to seating of the strandsare not considered, as this value isknown to the precaster and taken intoconsideration during the initial stressing operation.

Prestressed ConcreteBridge Sections

Prestress losses were computed forcomposite interior beams in twobridge cross sections that representsections typically used by the Michigan Department of Transportation

Table 1. Summary of methods used to estimate prestress losses.

Lump sum Lump sum Time

- Calculation method total -- individual dependent

AASHTO Standard Specifications3—— L

— AAS1-ITO LRFD Specifications4 j X —

Time-Step Method5 X

PCI General Method2 X

- PCI Simplified Method6 —- X

________

ACI 3 l8-99 and Zia4 X

September-October 2004 53

Fig. 1. Crosssections of beams

considered showingdimensions and

reinforcing details.

(MDOT) in new construction of simple-span bridges. Case I studied a 70in. (1780 mm) I-beam system used fora composite bridge in Mt. Pleasant,Michigan (MDOT Bridge #37021).The 9 in. (230 mm) composite deck issupported by nine I-beams spaced at7.5 ft (2.3 m) on center.

Each beam was prestressed withfifty ‘2 in. (12.7 mm) diameter low-relaxation prestressing strands and hadan overall length of 126.5 ft (38.6 m).The beams spanned 125.5 ft (38.3 m)center-to-center of bearings. Fig. 1shows the dimensions and reinforcingdetails of the non-composite I-beamcross section.

Case II studied a 21 in. (535 mm)box beam used for a bridge in South-field, Michigan (MDOT Bridge#63 101). The 87 ft (26.5 m) widebridge required ten box beams spacedat 9 ft (2.7 m) on center, achieving aspread box system.

Each 34 ft (10.4 m) beam spanned33 ft (10 m) between bearing centers,supported a 9 in. (230 mm) compositedeck, and was prestressed with ten‘2 in. (12.7 mm) diameter strands in

the bottom flange. The dimensions

and reinforcing details of the non-composite box beam are shown in Fig.1 (right).

Statistical Descriptionof Parameters

The nominal (design) properties ofthe two prestressed concrete bridgesystems were taken primarily from thebridge plans. Table 2 contains a summary of the nominal beam propertiesused to estimate prestress losses. Values for steel and concrete strength,jacking stress, sectional area, perimeter, and moment of inertia came directly from the bridge plans.

The unit weight of concrete was estimated as 150 lb per cu ft (2400kg/rn3), and relative humidity wasbased on a site location in LowerMichigan using the AASHTO LRFDSpecifications4Fig. 5.4.2.3.3-1. Deadload moments from the girder self-weight and the deck dead load werecalculated using bridge plan details.The water content of the concrete wasestimated from relative humiditydata’1 and the time between initialstressing of the strand and release of

the member was determined from discussions with PCI certified plant managers that had produced the beams.’2

To proceed with a probabilistic assessment and include the variability ofeach parameter associated with prestress losses, statistical data were gathered from many sources. Data presented by several references weretraced back to the original source andare presented in detail by Gilbertson.’3Table 3 lists the referenced statisticaldata used in this study.

Several random variables were usedfor the prestress loss calculations forwhich no statistical information couldbe found. Most of the missing datawere specific to the bridge systemsand beams used in this study. Table 4summarizes each non-referenced statistical variable, the nominal value, estimated high-to-low values, and theestimated statistical values (mean, coefficient of variation, and type of distribution).

Non-referenced statistical propertieswere estimated by applying allowabletolerances or ranges to the knownnominal value. This produced highand low values with a reasonable

D0L3, TYP__\\\<5,9W)

D13 TYP

rYOno (B)-9Oon (35)

535’,,

-

(a) I-Beam

_— 6860

FUR POSITIURDOWELS

—l3on (O.5)

16 SPA. 510,, C6) = 816,,.,515.,,., LENSIR 33

12,7.,, PrestressInO Stronds 0/0 Bond Br.,,kersX 12,7,,,, Pr.,otreo0In Strondo Debond.,d 920,,,, Pro,, 0006 ond

(b) Box Beam

54 PCI JOURNAL

range. (For example, the allowablePCI design tolerances6were applied tothe cross sections studied and newsection properties were calculated todetermine the high and low values.)

The mean, ,u, represents the valueexpected from an actual measurement,taken here as the average betweenhigh (maximum) and low (minimum).The mean may be higher or lower thanthe nominal value depending on theskew of the range for the maximumand minimum estimated values. Statistical z-tables were then used to determine the standard deviation, u, andcoefficient of variation, V, as follows.

In a standard normal distribution,the mean, is zero and the standarddeviation, o, is one. A standard normal curve produces a bell shape centered at zero, with an area under thecurve of one. The z-value is on thebottom scale of this curve and is a random variable. The z-tables indicate theintegral of the normal curve fromminus infinity (_co) to z; thus, when z

0, 50 percent of the data fall to theleft of zero.

This property can be applied to anynormal curve by Eq. (1), which transforms a normal random variable toallow z-tables to be used on the data.Eq. (1) can be used to solve for a standard deviation given a mean and arange of values that encompass a determined amount of the data represented by the integral from —oo to z.

in which

Z=Xm_1L (1)

z = random variable for standardnormal distribution

Xmax = maximum value of x, calculated by multiplying percentdifference allowed by nominal value of x

= mean value of x determinedfrom Xm afld x5

= standard deviation of xThe z-value used in Eq. (1) may be

obtained from a statistical z-tablefound in most statistics and probabilitytextbooks.’7For this study, it was assumed that the probability of a randomvariable falling between the values ofx,, and Xmjfl was 90 percent; thus, a zvalue of 1.65, which corresponds to an

area under the normal curve of 0.95,was selected.

The o of the random variable wascalculated by rearranging Eq. (1) andapplying the mean and maximum val

ues from Table 4. The (coefficientof variation) was calculated by takingthe ratio of C.r,, to This process,along with the PCI Design Handbook6tolerances, was used to approximate

Table 2. Nominal design values used to estimate prestress losses.

Case I

70 in. 1-beamBeam properties

Ultimate strand strength,f (ksi) 270

Yield strength of strand,f,,4(ksi) 243

Jacking strand stress.J,j (ksi) 202.6

Concrete strength at release.f,1(psi) 5,800

Concrete strength at 28 days,f (psi) 7,000

Unit weight of concrete, ‘. (pcf) 150

Relative humidity, RH (percent) 75

Modulus of elasticity of strand, E, (ksi) 28.500

Gross area of cross section, A4 (sq in.) 774

Perimeter of cross section (in.)

Case II

21 in. box beam

270

243

202.6

3,046

5,076

150

75

28,500

467

183.3

1.53

24,600

8.50

794

2,385

315

36

Area of prestressing steel, A, (sq in.)

Gross moment of inertia, I (in.4)

Strand eccentricfty,e (in.)

Moment due to girder self-weight. M4 (in.-kips)

Moment due to superimposed dead load, Ma, (in.=kips)

Water content (lb/yd3)

Time, i (hours)

229.1

7.65

511,000

28.71

19,346

23,216

315

36

Note: I ksi = 6.89 MPa; t pcf= 16.02 kg/rn3; tin. = 25.4 mm; 1 in.-kip = 0.113 kN-m; 1 lb/yd3 = 0.593 kg/m3.

Table 3. Referenced statistical data.

Nominal Coefficient Distribution Reference

Variable value Mean of variation type number

/ 270 ksi 28l ksi 0.025 Normal 14

J 240 ksi I .027f,,.,, 0.022 Normal IS

,, y : 0.030 Normal 5

E,. 28400 ksi 0.020 Normal - 14 —

A 0.153 sq in. 0.1548 sq in. 0.0(25 Normal 5

e 0 to + /(, in. 0.04 to 0.68 Normal 15

Dead load DL, 1.05(DL,) 0.1 Normal 15

Note: 1 ksi=6.89MPa; 1 pcf= 16.02kg/rn3;I in.=25.4mm; 1 in.-kip=0.ll3kN-m; t tb/yd=0.593kg/rn3.

Table 4. Estimated statistical data.

Nominal High/Low Coefficient Distribution

Variable value (x,,,,./x,,,,,,) Mean, p, - of variation, Vr type

202.6 ksi 212.7/192.5 202.6 ksi 0.030 Normal

— l.lOf,, 0.174 Normal

— l.l0f14 0.200 Normal

RI-I 75 percent 89/60 75 percent 0.118 Normal

A, -1 774 in. 8 12.0/745.4 779 sq in. 0.026 Normal

A,. -II 467 in. 481.3/452.8 467 sq in. 0.018 Normal

Perimeter -1 229.1 in. 23 I .2/227.9 229.5 in. 0.004 Normal

Perimeter -11 183.3 in. 184.3/182.3 l83.3 in. 0.003 Norma4

_-I5ll,000in.’ 528392/493854 ..!..l23 in.4 0.020 Normal

I-ll 24.600 in.4 25,804/23,396 24,600 in.4 0.030 Normal

Water content 3I 5lb/yd3 325.0/305.0 j5 lb/yd3 0.014 _4qmal

r, time 36 hrs 48/24 36 hrs 0.202 Normal

Note: variable notation t and II refer to girder cross sections t) t-beam and II) box beam.Note: 1 ksi = 6.89 MPa; 1 pcf= 16.02 kg/rn3; tin. = 25.4 mm; 1 in-up = 0.113 kN-m; 1 lb/yd3 = 0.593 kg/rn3.


Fig. 2. Parametric study resu’ts using AASHTO

o, and V for the jacking stress,f, relative humidity, RH, gross section area,Ag, perimeter, moment of inertia, ‘g’water content, and time from jackingto release of strands, t.

The statistical parameters for concrete strength (V and distributiontype) were selected from Ellingwood16because the published values fit withinthe range presented by many others,the definitions of the nominal andmean values were clear, and thesource is used in many other studies.However, the mean value used to represent concrete strength was modifiedfrom the referenced value because theoriginal data were for reinforced cast-in-place concrete beams.

When constructing cast-in-placebeams, it is important to meet a required strength once, usually specifiedat 28 days. Thus, the actual concretestrengths are relatively close to the required compressive strengths, as reflected by a 1 percent increase fromthe nominal to the mean values givenby Ellingwood. Similar data for concrete strengths of precast, prestressedmembers remain unpublished.

The production of precast, prestressed concrete sections often requires two strengths, one at release(f), and the other usually at 28 days(fe’). Prestressed beams are typicallycast with a higher target compressivestrength to ensure that the required release strength is reached early and thebeam can be removed from the castingbed.

Discussions with PCI certified plantmanagers provided an estimation ofthe actual concrete strength being 20percent higher than the nominalstrength at both the time of release andat 28 days. A 10 percent increase fromnominal to mean strength (see Table4) was used as a conservative estimatebecause actual test/field data were unavailable. The V, value was referencedfrom Ellingwood’6for the 28-daycompressive strength, but was increased slightly for release strengthsto reflect more variation at earlystages of concrete curing.

It was found that correlation amongthe variables exists. For example, thearea of a precast section is directlylinked to the cross-sectional perimeterand the moment of inertia. Initially, allvariables were assumed mutually independent.

A test was performed to assess theeffects of correlating several of thevariables using two of the six prestressloss methods for four cases: (1) mutually independent variables, (2) mutually dependent correlation betweenarea, perimeter, and moment of inertia, (3) release concrete strength as afunction of 28-day strength, and (4) acombination of the second and thirdcases.

The analysis showed that the Vchanged slightly among the four casesbut showed little difference in themean total prestress losses, and general trends remained. Therefore, forsimplicity, all variables were assumed

independent of each other for comparison of total losses from the six prestress loss estimation methods.

PARAMETREC STUDYThe influence of each loss input pa

rameter (each variable) was investigated through a parametric study todetermine the individual effect on thetotal prestress loss for each method.Final losses were computed for eachof the methods (listed in Table 1) froma common set of input data representing nominal (design) values (see Table2) for the base case.

Not all of the data shown in Table 2were used by each of the predictiveloss methods. The PCI SimplifiedMethod,6 for example, does not require many of the parameters used bythe more complex time-step5 and individual lump sum methods.2’3’4’7Individual loss components were calculated for each base case as well as thetotal loss of prestressing steel stress inthe member at final service conditions.

Final prestress losses (or losses estimated at the end of service life) werealso computed plus or minus one standard deviation from the nominal basecase. A program was written to iteratethrough each variable, calculating theloss using the full range of the variable. The individual component lossesand total losses were stored for eachchanging variable.

The variable was then returned to itsbase case value, and the next variablewas changed to calculate losses forplus or minus one standard deviation.The process was repeated through allof the parameters used in the variousprestress loss methods.

The results calculated were prestresslosses due to elastic shortening, steelrelaxation, creep of concrete, concreteshrinkage, and total final loss for eachof the methods using individuallyvarying input parameters. A graphicalresult is shown in Fig. 2 for totallosses using the AASHTO LRFDmethod with the I-beam example.Similar results were found for otherprestress loss estimation methods andwhen using the box beam example. 13

From Fig. 2, the base case (horizontal line) represents using only nominal(design) values and does not varyfrom parameter to parameter. In this

S

C

C

C

‘I

28.50% - —

28.00%

2750%1 II — i.27.00%- — — — — — — — — — — —

Input Parameters

LRFD method for I-beam example.

56 PCI JOURNAL

Table 5. Prestress losses from Monte Carlo Simulation.

Note: I ksi = 6.89 MPa.

example, the base prestress loss of27.1 corresponds to the percent lossfrom initial jacking, or 54.9 ksi (380MPa) as listed in Table 5 for theAASHTO LRFD Method.4 (WhileTable 5 lists the results of the MonteCarlo simulation, the nominal lossesusing MCS match those computed inthe parametric study for the nominalbase case.)

The upper and lower bounds of thehistogram in Fig. 2 represent plus andminus one standard deviation aboveand below the nominal respective parameter values. A variation of zeropercent indicates that the parameter isnot used in the loss estimation methodpresented. Mean values, representingfield data values, were not used by theparametric study because the purposeof the parametric study was to showthe significance of variation of the individual design parameters.

The total prestress loss is affectedby the parameters mentioned above.However, it is important to note thatthe individual parameters may have astrong influence on the individual losscomponents, and may not have as significant of an effect on the total loss.For example, if the low value of a parameter causes one form of loss to in-

crease but another to decrease, the resuit of summing these effects may bereduced. Parameters f (concretestrength at strand release), RH (relative humidity) and e (eccentricity ofstrand) were found to have a high V(coefficient of variation) and thus agreater impact on the variability of thetotal loss. The jacking stress, didnot necessarily have a high V but hada strong influence because the finalstress is directly affected by this parameter.

Changes in parameters such as f,f, RH, and e caused very largechanges in the total prestress loss.However, these changes are smallcompared to some of the effects theyhad on the individual loss componentssummed to achieve the total loss. Theremaining parameters influenced totalloss to a lesser degree.

MONTE CARLOSIMULATION

All analytical systems contain a certain degree of variability. When thesesystems are formed by a combinationof random variables, the resultingvariability of the system generallycannot be found in a closed form ap

proach. An alternative approach thatallows the estimation of variability ina system given the variability of itscomponents is Monte Carlo Simulation (MCS).’7 MCS was used in thisstudy to determine the total variabilityof prestress loss estimated by eachmethod (and its components) considering all variables simultaneously, asopposed to the parametric study,which considered the prestress loss influence of only one parameter at atime.

The general process of an MCS involves selecting a random numberand using it to generate a randomvariable that falls within the statisticallimitations of each of the parametersor variables involved. The randomvalues produced for each parameter inthe loss calculation are then used tocalculate a prestress loss based on agiven estimation method. Repeatingthe process many times produces amatrix of values that can be used tocalculate the mean (u), coefficient ofvariation (Vs), distribution type, orother desired statistical parameters ofthe output.

In this study, 10,000 simulationswere used to estimate the prestressloss at final service conditions for

AASHTO AASHTO

Parameter Standard Specifications3 LRFD4 Time-Step5 PCI General2 PCI Simpifi. . ACI 3l8-99

Case I (1-beam) / Case H (box beam)

__________

Elastic Shortening (ksi)

_________________

Nominal

_________________

184/11)3 184/103 L03 4LLQ I — JTh±L92liMean — 18.1/10.2

______

18.1/10.2 , 18.1/10.2 18.1/10.2 — 18.1/9.63

CoefficientofVariation 0.119/0.124 0.119/0.123 L 0.118/0.123 0.117/0.100—

0.135/0.133

Shrinkage (ksi)

______

Nominal 575/575 575/575101/l1O 991/1274 — 466/495

Mean 5.74/5.76 5.74/5.76 10.1 / 11.0 10.1 / 12.3 — 4.64/4.96..

.

CoefficientofVariation 0.232/0.229 O.229/0.231j0.144/0.144 0.047/0.083 — O.358I0.357

Creep (ksi)

Nominal 26.8 / 8.68 26.7 / 8.66 13.4 / 7.92 16.3 / 5.69 —

Mean 26.9 / 8.71

Coefficient of Variation 0.084 / 0.1(19

Steel Relaxation (ksi)

Non3inal

Mean

Coefficient of Variation

Total Losses (ksi)

26.8 / 8.69 3.0 / 7.72 16.3 / 6.20 -

0.085 /0.109 0.128 /0.148 0.105 /0.253

.54/3.25

1.56/3.26

0.182/0.049

4.09 / 6.14

3.94 / 5.97

0(192 / (1.054

19.1 /4.20

Non1inal

Mean

Coefficient of Variation

3.97/4.77

3.61/4.39

0.188/0.169

18.1/3.82

0.172/0.359

52.4 / 27.9

52.3 / 27.9

0.069 / 0.074

3.95 / 4.91)

3.57 / 4.48

0.183/0.166

54.9 / 30.8

54.7 / 30.6

0,067 / 0.07)

3.3) / 4.24

3.37 / 4.26

0.602 / (1.027

45.8 / 34.1)

44.7 /33.3

0.094 / 0.093

48.5/ 33.6

48.1/33.1

0.071 /(1.1)91

34.6

63.3 / 34.8

0.054 / 1)036

45.5 /23.) -

44.2 / 22.7

(1.110/0.119


Nmnl: 18.4 Mean: 18.1Skew: 0.595 SDev: 2.15

Kurt:3.87

i

Iii.:

each of the methods considered. Sam-pies were conducted with 100,000simulations; however, the results weresimilar to samples with 10,000 simulations. Therefore, 10,000 simulationswere performed to conserve computation time in this study.

The MCS program also calculatedlosses using the nominal (design) values so as to establish a base case. Basecase values for nominal loss calculated by all of the methods for boththe I-beam and box beam exampleswere consistent between the MCS program and the parametric study, thusproviding a check balance against programming errors.

The loss of prestress at final serviceconditions due to individual components, as well as the total losses, arepresented in Table 5 for all six predictive methods for both the I-beam andbox beam examples. Nominal stresseswere computed using the design parameters. The mean and coefficient ofvariation computed from the MCS arealso listed for each simulated distribution.

Fig. 3 graphically depicts a histogram of each individual loss component using the AASHTO LRFDMethod4 for predicting losses in the I-beam example.

Values given on each distribution

include ‘Nmnl’ (using nominal parameters), ‘Skew’ (coefficient of skewness, a measure of whether a tail exitsto the right or left of the distribution,where a value of 0.0 describes a symmetrical distribution about the mean),‘Mean’ (the calculated mean of thedistribution), ‘SDev’ (standard deviation), and ‘Kurt’ (coefficient of kurtosis, a measure of the peakedness of thedistribution, where a value of 3.0 describes a normal distribution).

Fig. 3a shows that the predicted elastic shortening losses can vary fromabout 12 to 28 ksi (83 to 193 MPa) forthe I-beam example. Fig. 3e shows thedistribution of the combined stress lossin ksi (MPa). A mean of 54.6 ksi (375MPa) corresponds to the mean totalloss listed in Table 5 for this case. Fig.3f displays the percent loss of prestressrelative to the initial jacking force.Similar distributions were found for individual component losses when usingthe other predictive methods)3

Fig. 4 shows the resulting histograms of total losses computed foreach method using the 1-beam example. Fig. 5 shows total losses for thebox beam example. Total losses are asummation of the individual loss components and can vary greatly betweenthe predictive methods.

The scale on the loss axis of eachgraph is equal, thus allowing an easycomparison between the general distributions. However, the graphs do notnecessarily begin at the same value;thus, some losses are distributed similarly even though the distributions areabout a different mean.

Results from the two AASHTOmethods3’4are nearly the same for theI-beam example, while the time-stepmethod,5 PCI General Method,2 andACT 31 8997,t are similar to each other.The PCI Simplified Method6 producedthe highest estimation of total loss, followed by the AASHTO methods.

The highest standard deviation occurred using ACT 3 18-99, though allstandard deviations, cr, were between3.40 and 4.87 ksi (24 to 34 MPa). Thesmallest standard deviation occurredin the PCI Simplified Method. However, the standard deviation aloneshould not be used to determine theresolution of one method over theother methods.

a. Elastic Shortening Loss400

300

200000

100

b. Shrinkage Loss

Mean: 5.76SDev: 1.32Kurt: 2.94Nmnl: 5.75Skew: —0.00356

3001)C)C‘1)

p200C)00

o 100z

10 20 30 0 10 20Loss (ksi) Loss (ksi)

c. Creep of Concrete Loss d. Steel Relaxation Loss

Nmnl: 26.7Skew: 0.15

Mean: 26.8SDev: 2.28Kurt: 3.1

30

300

0

0C 200

C)

100

Mean: 3.94SDev: 0.364Kurt: 3.19Nmnl: 4.09Skew: —0.0964

3000ci)C)C

200C)C)0‘ 1000z

20 30Loss (ksi)

e. Total Loss

40 10 20Loss (ksi)

f. Total Percentage Loss

30

300‘cici)C)C

200zC)C)001000a


Kurt: 2.97

A .

50 60Loss (ksi)

20 30 40Loss (%)

Fig. 3. Monte Carlo Simulation output distributions using AASHTO LRFD individualloss components for I-beam example.

58 PCI JOURNAL

The PCI Simplified Method required far less input than the othermethods, and if input random variables do not vary significantly, theoutput will not vary appreciably, asshown here. In addition, all distributions are approximately normal basedon the coefficients of skewness (aboutzero) and kurtosis (about 3), except forthe PCI General Method.

The latter method is most likely notnormally distributed. A normal distribution would be expected if all parameters were summed to achieve the result. The complexity and nonlinearityof the loss equations allow for the possibility of other distribution types asfound in the PCI General Method.

The results of the 21 in. (535 mm)box beam example were somewhatdifferent than the 70 in. (1780 mm) I-beam example. For the smaller boxbeam cross section, the smallest predicted mean strand stress loss wascomputed by the ACI 318-99 method,which was approximately 30 percentsmaller than the typical value ofaround 33 ksi (228 MPa). The nextsmallest mean loss was computedusing the AASHTO Standard Methodfollowed by AASHTO LRFD.

The time-step and PCI GeneralMethod results were nearly the same,both in mean values and standard deviation. The largest mean loss wascomputed using the PCI SimplifiedMethod, though it was not much largerthan the time-step method or PCI General Method [34.8 ksi versus 33.3 ksi(240 MPa versus 230 MPa)}. Thelargest standard deviation was foundin the time-step and PCI GeneralMethods. The range of values for standard deviations fell between 1.25 and3.11 ksi (8.6 and 21 MPa).

Careful observation shows that theorder of prestress losses, from least tolargest mean loss is not consistent between the two cross sections used inthis study. Figs. 6 and 7 show thespread of the distributions of loss calculation methods on the same graphfor the two examples considered. Notethat these figures are merely a combined representation of Figs. 4 and 5,respectively.

The time-step and PCI GeneralMethods are near the lower end of therange on the I-beam, but near the upper

range for the box beam. This change oforder is a direct result of the cross-sectional properties and concrete strengthdifferences in the two examples.

A second observation from Figs. 6and 7 is that depending upon the variability present and method used, thetotal mean loss can vary from 30 to 75ksi (207 to 517 MPa) for the I-beamexample, and 15 to 42 ksi (103 to 290MPa) for the box beam example.When considering the percent prestress loss for each case, results fromthe Monte Carlo Simulation methodindicate total losses in the I-beamrange from 14.8 percent to 37.0 percent and in the box beam from 7.4 to20.7 percent.

The above is a very large spreadconsidering the fact that a designer hassome flexibility in selecting a methodfor prestress loss determination butmay not understand the impact of eachmethod or the corresponding variability. As a result, deflection and cracking moment estimations will vary anddepend on the prestress loss methodselected, as well as the geometric, material, and environmental conditionsrelated to the beam.

DEFLECTIONS ANDCRACKING MOMENT

The results of the probabilistic comparison of prestress loss methods have

a. AASHTO LRFD Total Loss

Nmnl: 54.9Skew:0.136

.1Mean: 54.6SDev: 3.67Kurt: 3.02

b. AASHTO Standard Total Loss

300

8 20000

100

Nmnl: 52.4Skew: 0.157


40C

300Cd)a)C)C1)

200C)C)

0

100

40 50 60 70 40 50 60Loss (ksi) Loss (ksi)

c. Time—Step Total Loss d. PCI General Total Loss


Kurt: 3.21

Nmnl: 48.5 Mean: 48.1

400 Skew: 0.518 Id. SDev: 3.4Caa)

Kurt: 4.76C)C

300C)C)

o 200‘15aZ 100

300a)C)Ca)

200C)C)00

o 100z

30 40 50 60Loss (ksi)

e. PCI Simplified Total Loss4(11)

Nmnl: 63.3 Mean: 63.7Skew: 0.118 . SDev: 3.43

Kurt: 2.96

040 6050

Loss (ksi)

f. Zia Total Loss

300a)C)C

200C)C)00

o 100z

Nmnl: 45.5Skew: 0.346

Mean: 44.2SDev: 4.87Kurt: 3.26300

Cd)

Ca)200

C)00

100

030

050 6o 70

Loss (ksi)40 50Loss (ksi)

Fig. 4. Monte Carlo Simulation output distributions showing total prestress losses for

all methods (I-beam example).


a significant impact on the expecteddeflection and cracking moment of thebeams. The variation in prestresslosses contributes to the variation inbeam deflection both immediately andover time. The higher the prestressloss, the lower the level of compressive stress in the bottom fiber at themidspan, and hence a greater net deflection.

Also, the additional applied momentneeded to cause beam cracking (overand above the self-weight and deckweight) is affected by prestress lossvariation. A lower compressive stateof stress in the bottom of the membermeans the member is closer to the tensile cracking load, thereby requiring a

lower additional moment to cause themember to crack.

It is important to realize that like prestress losses, deflection and camber values are not deterministic. They do, infact, differ with the inherent variationsin material and geometric properties,and environmental conditions. Variations in deflection and cracking moment for both bridge examples wereconsidered using the simulated prestressloss distributions from the AASHTOLRFD4and time-step5methods.

The AASHTO LRFD method iscommonly used in estimating deflection and cracking moment during thebridge design process, and the time-step method has been considered a

more accurate method when comparedto measured losses.18 Table 6 presentsupper and lower bounds for the simulated total loss distributions based on a95 percent confidence level for thetwo loss estimation methods considered. In addition, nominal designproperties were used with theAASHTO LRFD method to representdesign estimates for comparison.

These prestress loss ranges werethen used to estimate expected rangesfor deflection and additional appliedmoment to cause cracking. The otherdesign parameters used in the calculation of the deflection and crackingmoment were taken as the nominalvalues and were not treated as variables. This was done to produce aparametric study indicating the degreein which estimated losses affect deflection and cracking moment.

The final deflection in the beam reflects the deflection due to all prestress losses occurring in the prestressing steel and was calculated based onthe PCI Design Handbook6equivalentload estimations (with a sign convention of positive for camber and negative for deflection). The variation ofestimated deflection around the nominal deflection has the potential to belarge. For example, the I-beam has anestimated nominal long-term camberof 1.84 in. (47 mm), but due to thevariability of estimated prestresslosses using the AASHTO LRFDmethod, the camber could fall between1.59 to 2.14 in. (40 to 54mm).

When using the time-step method toestimate prestress losses, the predictedlong-term cambers are even largerthan nominal predictions. For the boxbeam example, the nominal designpredictions for final deflection indicate a negative camber of —0.112 in.(—2.8 mm). Based on a 95 percent confidence level with simulated lossesusing the two methods, the final deflection could be expected to fall between —0.101 to —0.133 in. (—2.5 to—3.4 mm).

The range of estimated deflectiondue to variability appears somewhatlarge for final conditions calculatedfor these two examples, and similar results can be expected for short-term orinitial camber. Indeed, precast manufacturers have seen large ranges in mi-

a. AASHTO LRFD Total Loss b. AASHTO Standard Total Loss


300 Kurt: 3.13

a

g 20000

100


Kurt: 3.04a300 iiaCi

a200

C)00

a 100z

020 4030

Loss (ksi)

c. Time—Step Total Loss

020 30

Loss (ksi)40

Nmnl: 34 J Mean: 33.3Skew: 0.208 . ii SDev: 3.11

Kurt: 3.15

d. PCI General Total Loss

300C)Ca

20000

100

NmnI: 33.6Skew: 1.06


400

C

300C)C)

2 2000

a100

20 30 40Loss (ksi)

e. PCI Simplified Total Loss

20 30 40Loss (ksi)

t. Zia Total Loss


300U,

C200

AKurt:

3.05

C)008 100az

300U)

C200

000‘S 100az

0

NmnI: 23.1Skew: 0.197

10


3020 30 40Loss (ksi)

50 20Loss (ksi)

Fig. 5. Monte Carlo Simulation output distributions showing total prestress losses for

all methods (box beam example).

60 PCI JOURNAL

tial cambers and short-term deflections in the precast plant for beamscast at the same time on the same bed.if a more accurate estimation of deflection is wananted, then a more in-depth approach may be needed beyondthe discrete value estimation that istypically used today with nominal design properties.

The additional moment applied tocause cracking over and above theself-weight and deck weight was alsodetermined for each bridge exampleusing prestress losses based on the 95percent confidence level as above, andare listed in Table 6. Values for the I-beam example indicate that a beammay crack from an applied moment11.5 percent lower than nominally estimated using AASHTO LRFD “high”losses, or may be able to withstand 31percent more moment using the timestep “low” losses.

Because of the variability in prestress losses, the estimated variation ofadditional moment to cause crackingimplies that load limits imposed on abridge based on nominal design valuesmay not be accurate. In fact, a muchsmaller moment may cause cracking ifthe prestress losses are near the distribution’s upper limit of the 95 percentconfidence level.

DISCUSSION OF RESULTS

The purpose of this study was toshow the variability found in severalof the current methods used to estimate prestress loss in prestressed concrete members. Two practical bridgeexamples were used to estimate lossesby six methods at final service conditions. These methods predict prestresslosses and, it must be emphasized, arenot exact procedures.

Prestress losses are typically calculated as discrete values in time whenin reality the short-term and long-termlosses are highly variable due to material, geometric, and environmentalvariations. However, the designer usesdiscrete nominal values to estimate deflections and safe loads which can beapplied to the beams. Variability ofthe loss methods was investigated intwo ways, first through a parametricstudy and then through a Monte Carlosimulation technique.

A parametric study varying singleinput parameters for prestress loss estimation showed that there is a degreeof uncertainty in the calculation ofprestress loss for all methods. Thevariation of each parameter had an impact on the individual components ofprestress loss. Those parameters having the greatest effect on total losseswere initial strand stress, initial concrete strength at release, relative hu

midity, and strand eccentricity. However, the summation of the loss components resulted in a lower impact onthe total loss.

Other parameters may have had ahigh positive effect on one componentand a high negative effect on another,thus causing the sum to be small andthe result on total loss to be minor.The parametric study was limited inthat the combined effect due to the

Fig. 6. Total prestress losses for all methods (I-beam example).

loss (ksi)

Fig. 7. Total prestress losses for all methods (box beam example).


variability of each parameter could notbe considered.

The Monte Carlo Simulation a!lowed for the interaction of variationbetween loss prediction parameters tobe studied, thus providing a more precise description of the variability thatshould be expected in total prestressloss calculations. The Monte CarloSimulation provided distribution descriptors including mean, lix, and standard deviation, o, for the prestressloss output representing the combinedvariability effects of the input parameters.

The results showed that a largerange of the predicted total loss couldbe expected for both bridge examplesconsidered. The outcome, thoughmeaningful in itself, has a greater impact on the designer when put intoterms of expected deflection andcracking moment.

While the variability of prestresslosses in themselves may seem like aminor detail, the implications reachbeyond theory and into reality. Calculations of long-term deflections andcracking moments based on a 95 percent confidence interval about thesimulated mean prestress loss showlarge differences from the nominalpredictions that would be estimated inthe design stage of these structures.The variation in deflection is important for constructibility and long-termserviceability.

During construction, the deck andwearing surface are placed such thatthe existing beam camber and the deflection due to additional dead loads

in the form of deck, wearing surface,diaphragms, traffic barriers, and additional permanent structures produces asmooth vertical profile on the bridge,thus reducing or eliminating “humps”in the vertical profile. Over time, long-term loss of prestress can cause these“humps” to form. In this case, theseoccur above the piers and abutmentsand can be unfavorable to the serviceability of the bridge.

An accurate approximation of thelosses allows the designer to predictlong-term deflections that can be accounted for during construction tominimize the likelihood of the formation of “humps.” Cracking moment isalso very important in determining theallowable loading that can be placedsafely on a structure. If nominal prestress loss values are used to determinethe allowable load, and the actual lossvalues are higher, the beam will crackunder a lesser load application.

It is important to realize that anexact method for predicting prestresslosses has not been established. Butmore importantly, variability of themany factors used in predicting prestress losses plays a large role in thedesigner’s ability to accurately estimate losses, deflection, and crackingmoment.

While this study provides insightinto the variability that can be expected when predicting prestresslosses, additional simulation techniques could be implemented for deflections and cracking moment calculations to refine the results. Furtherstudy to more accurately define van-

ability of some of the parameterscould also refine the procedures. Forexample, the ultimate creep coefficientused was a discrete value and verylikely contributes to the system variability. Variation in the shrinkagefunction is also very likely.

Note that the modulus of elasticityof concrete was estimated from theconcrete compressive strength. In addition, the variability of concretestrengths of the actual plant cast members should be investigated. Furthermore, consideration needs to be placedon the variability of functions used toestimate the geometric properties foruse in estimating prestress losses.

While this study compared the variability of losses using loss estimationmethods, it did not consider the actualfield measurements. Those reportingtest results or field measurements withcomparisons to predicted values needto be aware of the impact of materialand geometric properties, and environmental conditions on the reported results. Future design specificationsneed to include such variability so thatdesigners and owners can better understand the actual expected behaviorof prestressed concrete structures.

As the precast industry continues toward more efficient use of prestressedconcrete members through higher concrete strengths, smaller sections andlarger ratios of prestressing steel inmembers, deflections and crackingmoments may become more criticalsuch that variability will play a largerrole in these future designs. Severalmore elements of the Monte CarloSimulation can be refined. However,the information presented in this papergives a more reasonable understandingas to the variability involved with predictions for several prestress lossmethods, as well as deflections andcracking moments.

CONCWDING REMARKSPrestress losses, as well as camber

and deflection, are affected by severalparameters with inherent variability.This study compares predictive methods for prestress losses and the impactof losses on predicted camber and deflection due to the variability of parameters used for such predictions.

Table 6. Deflection and cracking moment summary.

Nominal

(AASHTO

Parameter LRFD4) AASHTO LRFD4 Time step5Low* High* Low’5 High*

Case I: 70-in. I-girder

___________

Total losses 27.1 percent 23.3 percent 30.5 percent 18.1 percent 26.2 percent

Deflection 1.84 in. — 1.59 in. 2.14 in. 1.92 in. 2.56 in.

Additional moment 2564 ft-kips 2896 ft-kip 2268 ft-kips 3363 ft-kips 2652 ft-kips

Case II: 21-in. Box Beam

___-

I

Total losses 15.2 percent 13.0 percent 17.2 percent 13.4 percent 19.4 percent

Deflection —0.112 in. —0.101 in._TZö122in. —0.104ij —0.133 in. —

Additional moment 327.9 ft-kips 346.5 ft-kips 310.8 ft-kips 342.8 ft-kips 292.1 ft-kips

Note: 1 in. = 25.4 mm; 1 ft-Lip = 1.36 kN-m.* Low/High Prestress Losses based on 95 percent confidence interval about the Monte Carlo Simulation mean lossfor each predictive method.

62 PCI JOURNAL

Using referenced and estimatedvariability in conjunction with MonteCarlo simulations, it is confirmed thatthe variability of material, geometric,and environmental factors leads tolarge variations in predicted losses andsubsequent camber and deflection predictions using common predictivemethods. Such variations have beennoted in practice for similar prestressed concrete members, includingmembers cast on the same bed.

For the two typical bridge systemsstudied herein, the primary influencingparameters were jacking stress (J),compressive strength of concrete atstrand release (J), relative humidity(RH), and eccentricity of strand (e).The inherent variability of these andall prestress loss parameters, compounded by the complex interactionsof prestress loss predictions, helps tojustify camber and deflection variations often seen in the field.

Lastly, the authors hope that thisstudy will provide designers withsome insight into the relative importance of the various factors that influence the determination of prestresslosses.

ACKNOWLEDGMENTThe authors of this paper would like

to express their appreciation to themembers of the PCI JOURNAL review team who provided many comments and recommendations whichhave been implemented in the finalversion of this paper. Also, formergraduate student Zhiqiang Chen is acknowledged for his contributions.

REFERENCES1. Steinberg, Eric P., “Probabilistic As

sessment of Prestress Loss in Pretensioned Prestressed Concrete,” PCIJOURNAL, V. 40, No. 6, November-December 1995, pp. 76-85.

2. PCI Committee on Prestress Losses,“Recommendations for EstimatingPrestress Losses,” PCI JOURNAL, V.20, No. 4, July-August 1975, pp. 44-75.

3. AASHTO, Standard Specifications forHighway Bridges, Sixteenth Edition,American Association of State Highway and Transportation Officials,Washington, DC, 1999, and 2001 Interim.

4. AASHTO, LRFD Bridge Design Specifications, Second Edition, AmericanAssociation of State Highway andTransportation Officials, Washington,DC, 2000.

5. Naaman, Antoine F., Prestressed Concrete Analysis and Design — Fundamentals, McGraw-Hill, New York,NY, 1982, pp. 275-3 15.

6. PC! Design Handbook, Fifth Edition,Precast/Prestressed Concrete Institute,Chicago, IL, 1999, pp. 2-5, 4-63 — 4-67.

7. ACI Committee 318, “Building CodeRequirements for Structural Concrete(3 18-99) and Commentary (318R-99),” American Concrete Institute,Farmington Hills, MI, 1999.

8. Zia, Paul, et al., “Estimating PrestressLosses,” Concrete international, V. 1,No. 3, June 1979, pp. 32-38.

9. PCI Bridge Design Manual, First Edition, Precast/Prestressed Concrete Institute, Chicago, IL, 2000.

10. CEB-FIP, CEB-FIP Model Code 1990,Design Code, Comité Euro-International du Béton, Lausanne, Switzerland, 1993.

11. Kosmatka. Steven H., and Panarese,William C., Design and Control ofConcrete Mixtures, Thirteenth Edition,Portland Cement Association, Skokie,IL, 1994, pp. 80-8 1.

12. Grumbine, Tom, Phone Interview,Grand River Infrastructure, PREMARC, Corp.. 2701 Chicago Drive,SW., Grand Rapids, MI 49509-1604,September 20, 2001.

13. Gilbertson, Christopher G., “Probabilistic Assessment of Several Prestress Loss Methods,” MSCE Thesis,Michigan Technological University,Houghton, MI, 2001.

14. Mirza, Sher Au, Kikuchi, Dennis K.,and MacGregor, James G., “FlexuralStrength Reduction Factor for BondedPrestressed Concrete Beams,” ACIJournal, V. 77, No. 4, July-August1980, pp. 237-246.

15. Siriaksorn, A., and Naaman, A., “Reliability of Partially Prestressed Beamsat Serviceability Limit States,” Thesis,Department of Materials Engineering,University of Illinois at Chicago Circle, Chicago, IL, June 1980,

16. Ellingwood, Bruce, “Reliability Basisof Load and Resistance Factors for Reinforced Concrete Design,” NBSBuilding Science Series 110, NationalBureau of Standards, Washington, DC,February 1978.

17. Ayyub, Bilal M., and McCuen,Richard H.. Probability, Statistics, &Reliability for Engineers, CRC Press,Boca Raton, FL, 1997.

18. Ahlborn, T. M., French, C. E., andShield, C. K., “High Strength ConcretePrestressed Bridge Girders: Long-Term and Flexural Behavior,” Minnesota Department of Transportation,Report # MN/RC—2000—32, 390 pp.,2000.


APPENDIX A — METHODS FOR ESTIMATING PRESTRESS LOSSES

Steel Relaxation:Stress Relieved Strand:

The methods for estimating prestress loss containedwithin this appendix are those which may not be readilyavailable to most engineers. The time-step method, PCIGeneral and Simplified methods, and the Zia method aresummarized herein. The AASHTO methods should be available to most engineers and, therefore, have not been reproduced in this article.

Time-Step Method [AC! 209 (1982)J

The total prestress loss at final service conditions underthis method is computed by the following equation:

Total Loss:

fpr=4fpEsfpR+1-1fps+’-1fpc (Al)

whereAfPT = total loss of prestress in pretensioned tendons, ksi

4fpE = loss due to elastic shortening of concrete at transfer, ksi

4f1,R = loss due to relaxation of prestressed tendons before and after transfer, ksi

fs = loss due to shrinkage of concrete, ksi

4fc = loss due to creep of concrete after transfer, ksi

Elastic Shortening:

fEs = [](fcsFi÷G)) (A2)

whereEpc = modulus of elasticity of prestressing tendons,

ksi= modulus of elasticity of concrete at time pre

stress is applied, ksi

fcgs (Fi + G) = stress in concrete at centroid of prestressingsteel due to initial prestressing force F1 andself-weight of member, ksi

f +Me0

rs (Fi+G)A i — i

whereF, = APS(fJ,j-4fPES-REI)

= gross area of concrete section considered, sq in.e0 = eccentricity of center of gravity of tendons with re

spect to center of gravity of concrete at sectionconsidered, in.

I = moment of inertia of gross concrete section, in.4MG bending moment due to dead weight of member

and any other permanent loads in place at time oftransfer, in.-kips

= area of prestressing strands, sq in.RE, = loss due to steel relaxation prior to transfer, ksi

4fpR

= [.tt,— 0.55J lo[J (A4)

where= time at beginning of time step, hours= time at end of time step, hours= stress in tendon at beginning of time-step, ksi

stress resulting in yielding of tendon, ksi

In Eq. (A4),f,,ç/fp must not be less than 0.55.

Low-Relaxation Steel:Use a denominator of 45 instead of 10 in Eq. (A4).Note that the time-step method should be used over sev

eral intervals, and the cumulative results used to estimateprestress loss. One time-step, from the beginning to the endof the member life will result in a poor estimate of total loss.

Shrinkage of Concrete:Composition of concrete mix, characteristics of the aggre

gates, relative humidity, and curing history affect the lossesdue to shrinkage between any number of time steps.

The shrinkage strain is assumed to be uniform throughoutthe member:

b(t. —t.)4tP = EPSESUKSHKSS

(b + t,)(b ± t)(AS)

where= ultimate shrinkage strain of concrete material

KSH = corrective factor which depends on the averagerelative humidity of the environment where structure is built (Table Al)

Kss corrective factor which depends on size and shapeof member (Table A2)

b = 35 for moist-cured concrete and 55 for steam-cured concrete

(A3)Esu = [2

+ ----

(w — 220)]l0 (A6)

wherew = water content, lb per cu yd

In place of Table A2, the following equation may be usedto find Kcs and Kss:

K5 = =l.l4_0.09!) (A7)

whereV = volume of the memberS = surface area of the member

64 PCI JOURNAL

Table Al. KSH values, type of curing influence. Table A3. C values.

Compressive strength (psi)_________-

C —

3000 3.1

4000 2.9

5000 2.65

6000 2.4

7000 2.2

8000 2

Table A2. K55 and Kcs values, volume-to-surface ratioinfluence.

Note that V/S is commonly taken as the cross-sectionalarea divided by the perimeter.

Creep of Concrete:

zif,,, (t1, t) = n1,CCUKCHKCA (t )[g(t) — g(t1)]

where

(A8)

n,,, = EPIEC (n1, can replace ni,, in the early stages, whereni,, = E1,5/E)

= ultimate creep coefficient (Table A3)KCH = correction factor depending on average relative

humidity of environment where structure is built

KCA = age at loading factorK5 = shape and size factor (Table A2)

f5 (t,)= (1+2.— MDeOS

A 1 r2)

where fcgs (ti) is taken at the beginning of a time-step, not atthe end.

The time function suggested by ACT Committee 209 is:

g(t)=10060

(AlO)

where t is the time, measured in days.By using the equations for fcgs(tj), the values of the stress

in the prestressing steel at time t, f(t) depend on the totallosses including creep, over all time intervals preceding theinterval considered, such that:

Moist-cured concrete Steam-cured concrete

40 percent<H<80 percent t7 days t 1 to3 days

KSH= l.40-0.01H K5ff= l.40-0.01H

80percent<H<loopercent t7days t 1 to3 days

KSH=3—0.03H KSH=3—0.03H

V/S ratio (in.) Kcs (Creep) J K,,5 (Shrinkage)

.05 1.04

2 0.96 0.96

3 0.87 0.86

4 0.77 0.77

5 0.68 0.69

6 0.68 0.6

PCI COMMITTEE ON PRESTRESS LOSSES:GENERAL METHOD (1 975)2

The total loss is calculated by the following procedure:Total Loss:

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

(A12)

whereTL total loss of prestress, psiANC loss of prestress due to anchorage of prestressing

steel, psiDEF = loss of prestress due to deflecting device in pre

tensioned construction, psiES = loss of prestress due to elastic shortening, psiCR = loss of prestress due to creep of concrete over

time interval t1 to t, psiSH = loss of prestress due to shrinkage of concrete

over time interval t1 to t, psiRET = loss of prestress due to steel relaxation over time

interval t1 to t, psi

Elastic Shortening:

ES—_--f (A13)E

where

(A9) E5 = modulus of elasticity of steel, psi= modulus of elasticity of concrete at time of initial

prestress, psi

fer = concrete stress at center of gravity of prestressingforce immediately after prestress force is appliedminus stress due to all dead loads acting at thattime, psi:

fcr=1liJ (A14)

whereP =A5(f-ES-RE1)f = stress at which tendons are anchored in prestress

ing bed, psiloss due to steel relaxation prior to transfer, psigross area of concrete section, sq in.eccentricity of center of gravity of tendons with respect to center of gravity of concrete at the sectionconsidered, in.

f,(t1)=f1

RE1 =

(All) =


Table A4. Minimum time increments. Table A5. SCF values (size and shape factor for creep).

r2 = I/Ar, sq in.M0 = bending moment due to dead weight of member

and any other permanent loads in place at transfer,in.-kips

I = moment of inertia of gross concrete section, in.4A, = area of prestressing strands, sq in.

Time-Dependent Losses:Note that Table A4 reflects the minimum time increments

recommended by the original reference. The authors of thisarticle feel that even more time steps are required.

Creep of Concrete (loss over each time step):

CR = (uCR)(sCF)(MCF)(PCR)(f) (A15)

ultimate loss of prestress due to creep of concrete, psi per psi of compressive stress in concrete

SCF factor that accounts for effect of size and shapeof a member on creep of concrete (Table A5)

MCF factor that accounts for the effect of age at prestress and length of moist cure on creep of concrete. Note: this should not be included for accelerated cured concrete (Table A6)

PCR = amount of creep over time interval t1 to t

f = concrete compressive stress at center of gravityof prestressing steel at time t1, taking into account loss of prestress force occurring over preceding time interval, psi

Normal weight concrete:For moist cure not exceeding seven days:

UCR=95__Cll (Al6)106

Age of prestress Period of

transfer (days) cure (days) MCF

3 3 1.14

5 5 .07

7 7 1.00

10 7 0.96

20 7 0.84

30 7 0.72

40 7 0.60

Table A7. AUCvalues (ultimate creep factor).

Tune after transfer (days) AUC

I 0.08

2 0.15

5 0.18

7 0.23

10 0.24

20 0.30

30 0.35

60 0.45

90 0.51

80 0.61

365 0.74

End of service 1.00

whereA UC = portion of ultimate creep at time after prestress

transfer (Table A7)= time at beginning of time interval, days

t = time at end of time interval, days

Shrinkage of Concrete (loss over each time step):

where

SH = (USH)(SSF)(PSH) (A 19)

USH = ultimate loss of prestress due to shrinkage ofconcrete, psi

SSF = factor that accounts for the effect of size and

Step Beginning time, t1 End time,t

1 Pretensioned Age at prestressing

anchorage of of concrete

_______________

prestressing steel

2 End of Step I Age = 30 days. or time

when a member is

subjected to load in

addition to its own

_______ _________

weight

________

3 End of Step 2 Age = 1 year

4 End of Step 3 End of service lifeTable A6. MCF values (moist cure factor for creep).

whereUCR

PCR = (AUC)- (AUC) (A 18)

For accelerated cure:

UCR=63_?P-l1 (A17)

where E is the modulus of elasticity of concrete, psi.

66 PCI JOURNAL

Table A8. 5SF values (size and shape factor for shrinkage).

USH = 27000— 30OOE 12000106

PSH = (A Us)1—

(AUS)1

Steel Relaxation:Low-relaxation strand:

RET=f(log24t_log24tLf055

St 45 } f

Stress-relieved strand:

ZIA METHOD (1 979)8

Elastic shortening:Members with bonded tendons:

ES= KesEs!S

V/S 5SF

1 1.04

2 0.96

3 0.86

4 0.77

5 0.69

6 0.60

Table A9. AUS values (ultimate shrinkage factor).

Time after end of curing (days) AUS

I 0.08

3 0.15

5 0.20

7 0.22

10 0.27

20 0.36

30 0.42

60 0.55

90 0.62

18(1 0.68

365 0.86

End ot service I

__________________ __________________

PCI COMMITTEE ON PRESTRESS LOSSES:SIMPLIFIED METHOD (1 975)6

The PCI Simplified Method (1975) requires the calculation of concrete stress at transfer and the concrete stress afterthe application of additional permanent dead loads. Concretestress at transfer is calculated from the stress in the steel attransfer, which is estimated by reducing the jacking stress by

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a reduction factor. These values are then entered into anequation based on concrete weight, strand type, and type oftensioning. The result is an estimate for total prestress loss

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in which

fcr = concrete stress existing immediately after prestresshas been applied to concrete, psi

fcds = stress due to all permanent (dead) loads not used incomputing fer’ pS

fAJ11Af,e2M’e (A24)A I

where= 0.90f1for stress-relieved steel= 0.9251; for low-relaxation steel

f = stress in tendon at critical location immediately

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after prestressing force has been applied to concrete, psi

shape of a member on concrete shrinkage (Table ft stress at which tendons are anchored in pretensioning bed, psi

PSH = amount of shrinkage over time interval to M’ = moment due to loads including weight of member

Normal weight concrete:at time prestressing is applied to concrete, in.-lb.

= cross-sectional area of prestressing tendons, sq in.

A = gross cross-sectional area of the non-composite(A20) concrete member, sq in.

e = tendon eccentricity measured from center of gravityof concrete section to center of gravity of tendons,

A8)

in.(A2 1) = moment of inertia of gross cross section of concrete

member, in.4

where A US is the portion of ultimate shrinkage at time after M eend of curing (Table A9). fd

= (A25)

RET = f[lot 24t — log 24t1

< — O.5510 )

where

fcds = concrete stress at center of gravity of tendons dueto all superimposed permanent dead loads that are

(A22) applied to the member after it has been prestressed,psi

Md = moment due to superimposed permanent deadloads and sustained loads applied after prestressing, in.-Ib

(A23) Adjustments: Equations are based on V/S = 2.0.See Tables AlO and All.

where

f = stress in prestressing steel at time t1, psi

f = stress at 1 percent elongation of prestressing steel,psi

f guaranteed ultimate tensile strength of prestressingsteel, psi (A26)


Table AlO. PCI Committee, Simplified Method equations.

Note: The above equations are only valid whenf., >fa.

whereKcjr =

Concrete weight

Normal Light

xx

xx

xx

xx

xx

Type of tendon

Stress Relieved Low Relaxation

xx

xx

xx

xx

Bar

xx

Tensioning JPre Post Equations

X TL=33.0+ 13.8fc—4.5fas

X TL=31.2+ 16.8f5,—3.8fa

X TL 19.8+i6.3J,,—5.4faX TL= l75+204f4fnis

X TL=29.3+5.1f,.—3.0f,1,

X TL= 27.1 + 1O.[f,—49fas

X TL= I2.5±7.Of4.lfr,a,

X TL=l1.9+11.1f,—6.2fa5

X TL=l2.8±6.9f,.r4.Ofcd,

X TL = 12.5 + I O.9j.,. —

Table Al 1. Adjustments to the Simplified Method.

V/S 1 2 3 4

Percent adjustment +3.2 0 —3.8 —7.6

Table Al 2. Kre and I values.

Type of tendon J270 Grade stress-relieved strand or wire 20000 0.15

250 Grade stress-relieved strand or wire I8500 (1.14

240 or 235 Grade stress-relieved wire 17600 0.13

270 Grade low-relaxation strand 5000 0.04

250 Grade low-relaxation wire 4630 0M37

240 or 235 Grade low-relaxation wire 4400 0.035

145 or 160 Grade stress-relieved bar 6000 0.05

Table Al 3. C-values.

Stress-relieved Stress-relieved bar or low

-- — strand or Wire C relaxation strand or Wire C

0.80 — 1.28

0.79 — 1.22

0.78 — 1.16

0.77 — 1.11

0.76 — 1.05

0.75 1.45 1.00

0.74 1.36 0.95

0.73 1.27 0.90

0.72 1.18 0.85

0.71 1.09 0.80

0.70 1.00 0.75

0.69 0.94 0.70

0.68 0.89 0.66

0.67 0.83 0.61

0.66 0.78 0.57

0.65 0.73 0.53

0.64 0.68 0.49

0.63 0.63 0.45

0.62 0.58 0.41

0.61 0.53 0.37

0.60 0.49 0.33

whereKes. = 1.0 for pretensioned membersEç = modulus of elasticity of prestressing tendons, usu

ally 28 x 106 psiE1 = modulus of elasticity if concrete at time prestress is

applied, psi

fcir = net compressive stress in concrete at center ofgravity of tendons immediately after prestress hasbeen applied to concrete, psi

(P. Pe2 Mefir —-—--p-- (A27)

C

‘c

0.9 for pretensioned members, adjustment factorbecause does not include losses for elasticshortening or steel relaxation prior to transfer

P,, prestressing force in tendons at critical location onspan after reduction for losses due to friction andseating loss at anchorages but before reduction forES, CR, SH and RE, psi

= area of gross concrete section at cross section considered, sq in.

e = eccentricity of center of gravity of tendons with respect to center of gravity of concrete at cross section considered, in.

= moment of inertia of gross concrete section at crosssection considered, in.4

MG = bending moment due to dead weight of memberbeing prestressed and to any other permanent loadsin place at time of prestressing, in.-lb.

Creep of concrete:For members with bonded tendons:

CR = K (f — fCd5) (A28)

where2.0 for pretensioned members. For members madeof sand lightweight concrete, the foregoing value

68 PCI JOURNAL

of Kcr should be reduced by 20 percent.

E = modulus of elasticity of concrete at 28 days, psi

f = stress in concrete at center of gravity of tendonsdue to all superimposed permanent dead loads thatare applied to member after it has been prestressed,

where

= M.de (A29)

Md = moment due to superimposed permanent deadloads and sustained loads applied after prestressing, in.-lb

Shrinkage of concrete:

SH = 8.2 x l0KShES(l —

o.o6Y(ioo— RH) (A30)

whereKSh = 1.0 for pretensioned membersV/S = volume-to-surface ratio. Usually taken as gross

cross-sectional area of concrete member divided byits perimeter, in.

RH = average relative humidity surrounding the concretemember, percent

Steel relaxation:

RE=[Kre —J(SH+CR+ES)]C (A31)

where Kre, J, and C are taken from Tables A12 and A13.

psi


a probabilistic comparison of prestress loss methods in

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