post-tensioned tendon losses in a spliced-girder bridge

Post-tensioned Tendon Losses ina Spliced-Girder Bridge,Part 2: Analysis of Losses

Chris P. Pantelides, Ph.D., P.E.ProfessorDepartment of Civiland Environmental EngineeringUniversity of UtahSalt Lake City, Utah

BrandtW. Saxey, M.Sc.-larris Saxey Consultants Inc.Salt Lake City, Utah

This paper presents analytical procedures for determining post-tensioning tendon losses in a typical spliced, precast concrete girderused in the 4500 South Bridge at Interstate 15 in Salt Lake City, Utah.The bridge is composed of eight post-tensioned, spliced, precast concrete girders that have three segments each to produce a single clearspan of 61.44 m (201.6 ft). The analysis procedure specified in thernAmerican Association of State Highway and Transportation Officials’AASHTO LRFD Bridge Design Specifications is used and comparedwith experimental measurements of losses in the post-tensioned,spliced, precast concrete girders monitored at the 4500 South Bridge,as described in the companion paper, “Post-tensioned Tendon Lossesin a Spliced-Girder Bridge, Part 1: Field Measurements,” in the May—June 2007 PCI Journal.The time-dependent method predicts measured post-tensioning lossesin the precast concrete girders with sufficient accuracy at the abutments and at midspan. The measured modulus of elasticity of the girderconcrete was found to be predicted more accurately by the equationproposed by AC! Committee 363. Concrete shrinkage and creep testsperformed on the girders were used to obtain the ultimate creep coefficient and ultimate shrinkage strain. These results were compared withvalues recommended by AC! Committee 209. Shrinkage and creeptests indicate an asymptotic achievement of ultimate shrinkage strainand ultimate creep coefficient in approximately eight months, a resultsupported by strain readings from vibrating wire strain gauges embedded in the concrete of the spliced girders.

A

PCI JOURNAL

As indicated in the PCI Journal companion paper,

“Post-tensioned Tendon Losses in a Spliced-Girder

Bridge, Pait 1: Field Measurements,” research on

monitored losses in post-tensioned, spliced, precast concrete

girders is limitecL’ In preparation for the 2002 Olympic Win

ter Games, which were held in Salt Lake City, Utah, exten

sive interstate highway reconstruction work was undertaken,

including construction of 81 prestressed concrete I-girder

bridges. The 4500 South Bridge at Interstate 15 (1-15) in Salt

Lake City, a typical spliced, precast concrete girder bridge,

was selected for monitoring of post-tensioning tendon losses.

This paper presents the analytical procedures used to deter

mine post-tensioning losses in the spliced, precast concrete

bridge girders. Shrinkage and creep test results are presented

and compared with recommended values.

LITERATURE REVIEW

The American Concrete Institute (ACT) and the American

Society of Civil Engineers (ASCE) Joint Committee 3232 rec

ommended lump-sum losses, for routine design purposes, in

prestressed concrete construction as early as 1958. For post-

tensioned members, the recommended value of lump-sum

losses was 172 MPa (25 ksi). Lump-sum losses include those

due to elastic shortening, shrinkage, creep, and relaxation but

exclude those due to friction and anchorage slip. Branson and

Kripanarayanan3presented equations for predicting loss of

prestress and camber in concrete members subjected to either

full or partial prestressing. Equations were presented to ac

count for the time-dependent effects of loss of prestress and

the results were compared with experimental data.Meyers et al.4 provided simple empirical equations for de

termining the long-term creep and shrinkage of concrete byusing experimental results. Using these equations, prediction

accuracy that previously required four months of testing could

be achieved with only 28 days of creep and shrinkage data.Modifications of the lump-sum method were subsequently

included in the 1975 American Association of State Highway

and Transportation Officials’ (AASHTO) Interim Specica

tions—Bridges.5The treatment of losses as lump-sum quantities, however, was recommended only for standard condi

tions. For members with unusual proportions, exceptionally

long spans, or lightweight concrete, AASHTO recommended

a separate estimate of the individual losses.In 1975, the PrecastlPrestressed Concrete Institute (PCI)

Committee on Prestress Losses6 issued “Recommendations

for Estimating Prestress Losses.” This document presented

both general and simplified methods for computing prestress

ing tendon losses. In the general method, the total loss for post-tensioned construction was separated into one-time losses and

time-dependent losses. The one-time losses included friction,anchorage, and elastic shortening, and the time-dependent

losses were creep, shrinkage, and losses due to steel relax

ation. Tadros et al.7 presented an analytical method to predict

time-dependent prestressing losses, axial strain, and curva

ture at a section of a prestressed concrete beam or frame. The

method was applicable for non-composite pretensioned and

post-tensioned structures.Tadros et al.8 presented a step-by-step computer method

for predicting the stress distribution due to creep and shrink

age of concrete and relaxation of prestressing steel. In a dif

ferent paper, Tadros et al.9 presented multipliers and design

aids for predicting time-dependent deflections. In that paper,

the design aids considered the influence of creep and shrink

age of concrete, relaxation of prestressing steel, and the presence of nonprestressed steel on time-dependent deflection of

prestressed concrete members.The PCI Design Handbook: Precast and Prestressed Con

crete’° provides simple equations for estimating losses of

prestress that enable designers to estimate the various types

of prestressing losses rather than using a lump-sum value.

Although a number of methods are available for determining

loss of prestress, all handbook methods must be modified for

use in post-tensioned concrete construction.Abdel-Karim and Tadros” reviewed the state of the art for

spliced-girder bridges. They presented a computer-based pro

cedure for analyzing composite precast concrete girder bridges

with cast-in-place concrete topping. The computer program

evaluates stresses in concrete and steel at any cross section in

a statically indeterminate composite beam or plane frame and

gives the deflection at various stages of construction.Oh and Yang12 presented a method for improving the long-

term prediction of prestressing force changes in prestressed

concrete structures. The method used short-term measure

ments in the field before opening the structure to traffic, and

the measurements were then used to update predictions of

prestress losses due to creep and shrinkage. The method was

supported by statistical data.Stallings et al.’3 recorded camber and strains from 31 gird

ers from the time of prestressing force transfer until construc

tion completion. An incremental time-step analysis was used

to predict girder strains, camber, and prestressing losses up

to the time of deck construction. The authors concluded that

current analytical techniques are adequate provided that the

correct time-dependent material properties are appropriately

integrated into the procedure.Wollmann et al.’4 used the age-adjusted concrete modu

lus approach in the time-dependent analysis of spliced, pre

cast concrete girder composite bridges. The approach allows

designers to take advantage of the greater concrete strength

at the time of post-tensioning and the resulting reduction in

creep and shrinkage strains when determining prestressing

losses for design.The authors use a time-dependent loss method described

in the AASHTO LRFD Bridge Design Specfications’5in this

paper to analyze the losses of a post-tensioned, spliced, precast

concrete girder at the anchorage zones and at midspan. They

compare the method with actual losses measured through load

cells at the anchorage zones and vibrating wire strain gauges

at midspan. They also compare predictions for the concrete

modulus of elasticity, the ultimate creep coefficient, and ulti

mate shrinkage strain to test results of the same concrete used

to construct the post-tensioned spliced girder.

CONCRETE PROPERTIES

The authors performed creep and shrinkage tests to evalu

ate the ultimate creep coefficient and ultimate shrinkage

July—August 2007 59

1.OE-03

Fig. 2. Actual and theoretical shrinkage strain at concrete age of 3 days. Note: 1 mm = 0.03937 in.

Table 1 shows the measured and predicted moduli of elasticity found using Eq. (1) and (2) and assuming w, = 2323kg/m3 (3917 lbIyd3). Equations (1) and (2) overestimate themeasured elastic moduli of the 3-day-, 30-day-, and 90-day-old concrete. The equation suggested by ACT Committee 363(Eq. [2]), however, calculated values much closer to the experimental results.

Ultimate Creep Coefficient

CTLGroup performed creep and shrinkage tests per ASTMC512.2°Tests were performed on 150 mm x 300 mm (6 in.x 12 in.) cylinders with an applied unit stress of about 40%of the concrete compressive strength. Laboratory relative humidity levels of 50% and temperatures of 23.0 °C ± 1.7 °C(73.4 °F ± 3.0 °F) were maintained for both the preloadedenvironment and the loaded environment.

The creep and shrinkage test for the 3-day-old concretecylinders began December 18, 1999, and continued for 895days, until May 31, 2002. For all tests, the authors tested twocylinders for creep and two for shrinkage. Figures 1 and 2,respectively, show the results of the 3-day-old concrete creepand shrinkage tests. At the time of the 3-day test, the cylinders had a compressive strength of 44 MPa (6.4 ksi).

Table 2 contains the values of ultimate creep coefficient v

that best satisfy the equation:

‘ 0.6t_t1)

— 0.610 + (t — t1)

t—t1 = the number of days that elapsed since concreteloading began

Equation (3) is a basic model used to predict creep as givenby Branson and Kripanarayanan and ACT Committee 209.21

The best fit value of v (measured) in Table 2 and used inFig. 1 was chosen by:

1. Estimating a value of v.

2. Using this initial trial of v to calculate the sum of thesquared deviations between the measured and predictedvalues of v.

3. Trying a range of values greater and lesser than the initial predicted value and calculating and recording thesum of the squared deviations for each trial value.

4. Finding an absolute minimum sum of squared deviations or best fit for V by plotting the trial values versusthe results and determining the point of zero slope.

The value shown in the column labeled ACI 209 in Table 2was determined using the adjustment factors found in the reportby ACT Committee 209. Table 2 shows the values used to compute the correction factors for creep and shrinkage. Adjustmentswere made for loading age y, relative humidity y, and minimum thickness using the volume-to-surface ratio method 2’vs

Figure 1 shows a plot of the measured and predicted creepversus age of concrete. The values of v shown in Table 2 forthe 3-day-old concrete were used in the best fit and the ACT209 methods, respectively. At eight months, the creep hadreached 87% of its ultimate value. Based on the tests performed with the concrete used in the 4500 South Bridge, therecommended value of the ultimate creep coefficient v forconcrete initially loaded at an age of 3 days is 2.16. From theACI 209 method, a value of 2.56 is obtained for the ultimatecreep coefficient for the same age concrete.

9.OE-04

8.OE-04

E7.OE-04

E6.OE-04

5.OE-04

4.OE-04

(1) 3.OE-04

2.OE-04

1.OE-04

O.OE+OO

0 100 200 300 400 50 600 700 800 900 1000

Age of Concrete (day)

wherev = creep at time

(3)


Table 2. Ultimate Creep Coefficient Values from CTLGroup Creep and Shrinkage Test Compared with ACI 209

Ultimate Creep Coefficient v, Ultimate Shrinkage Coefficient (e51J,

Best Fit ACI 209 Best Fit ACT 209

2.16 2.56 780 x 10.6 729 x 10.6

Loading Age— 1 098

Component y,

Humidity Component— 0.935 — 0.900

YA

Volume/Surface— 1.061 — 1.039

Component

Creep Correction —

1.089 — —

Factor y

Shrinkage Correction — —

— 0.935Factor 2’s,,

Note: For the ACI 209 method, v=(235)y, (e,) = [780(10j]yh, y=(y)(Ya)(yv), and Ysh =0’)(Yys).

Ultimate Shrinkage Strain

Table 2 shows the value of the ultimate shrinkage strain(sh) that satisfies the following equation.

where

(t — t)(ESh), =

(r,,), = shrinkage strain at time t

Branson and Kripanarayanan and ACI Committee 209 recommended Eq. (4) for calculating shrinkage strain. The basevalue for (ES6)u of 780 x 106 was adjusted for humidity andminimum thickness as described for creep in Eq. (3) with thevalues shown in Table 2.

Table 2 shows a best-fit value of (Eh)U using the most appropriate value found, as described previously for the ultimate creep coefficient. Figure 2 plots the shrinkage strainfrom the experimental, the best fit, and the ACI-209-deter-mined values of (eh). Values are corrected using the correction factors listed in Table 2. At eight months, the shrinkagestrain had reached 92% of the ultimate value. Based on thetests performed on concrete used in this bridge, the recommended value of the ultimate shrinkage strain for concreteinitially loaded at an age of 3 days is 780 x 106. From theACT 209 method, a value of 729 x 10.6 is obtained for theultimate shrinkage strain.

POST-TENSIONED SPLICED GIRDER LOSSES

Actual tendon losses, which may be larger or smaller thanthe estimated losses, affect service load behavior such as deflection or camber, cracking load, and crack width, as wellas deformation during construction. Overestimation of losses, which may be considered conservative, can actually beas detrimental as underestimation of losses. Overestimationmay lead the designer to specify a prestressing force that istoo large, resulting in excessive camber and shortening. It is

thus necessary to make the best possible estimate of losses.Loss of prestressing force is considered using a time-

dependent method based on the AASHTO LRFD Bridge Design Specifications. The AASHTO total-loss method is notsuitable for calculating losses in the present case because it

(4) does not take into account the concrete age at the time ofpost-tensioning and the presence of a precompressed decksupported on shored, post-tensioned, spliced girders. The effect of light pretensioning of the girder segments was ignoredin the calculation of losses. Pretensioning was applied main

ly for erection purposes and was limited to individual girdersegments. As such, the pretensioning strands’ contribution tolosses is small compared with the losses from post-tensioningof the spliced girder.

The precast concrete deck sections measured 2440 mm x2440 mm (96 in. x 96 in.) and were 90 mm (3.5 in.) thick,with a design compressive strength of 35 MPa (5100 psi).The thickness of the cast-in-place concrete deck was 125 mm(5 in.) with a design compressive strength equal to 35 MPa.Thus, the overall thickness of the deck was 215 rrun (8.5 in.).The girder spacing was 3.34 m (11 ft).

Time-dependent losses were calculated for a period ofthree years and eight months, corresponding to the monitoring period of the instrumented girder. The following time increments were used:

• At the time of stressing;• Every week for a period of three months after post-

tensioning; and• Every month for the remaining period until December

31, 2003 (three years and eight months).The following losses (which were added to give the total

loss) were considered:• Elastic shortening;• Creep;• Shrinkage;• Tendon relaxation;• Anchorage seating; and• Friction loss.

62 PCI JOURNAL

AASHTO LRFD TIME-DEPENDENTLOSS METHOD

Elastic Shortening Loss

The amount of elastic shortening loss for post-tensionedgirders depends on the sequence of jacking. In the presentcase, tendon 1 (at the top) was stressed first, followed by tendons 2, 3, and 4. In general, if N is the number of tendons thatare sequentially tensioned, the elastic shortening loss 4fEs iS

given as:22

where

j

I

LlfES=(--l\

)EiJ CS

= number of jacking operations, which in this case isequal to four

= modulus of elasticity of concrete at transfer calculatedusing Eq. (2) with f = 82 MPa (12 ksi)

E5 = modulus of elasticity of prestressing steel (193 GPa or28,000 ksi)

f5 = stress in the concrete at the tendon centroid

Following are the total losses due to elastic shortening:• Tendon 1 = 100.3 MPa (14.5 ksi);• Tendon 2 = 66.9 MPa (9.7 ksi);• Tendon 3 = 33.4 MPa (4.8 ksi); and• Tendon 4 = 0.0 MPa (0.0 ksi).Tendon 4 (at the bottom) was tensioned last and did

not suffer any losses due to elastic shortening. Tendon 1,which was tensioned first, experienced the greatest amountof loss.

Time-Dependent Creep Loss

In establishing creep loss, the time-dependent properties of concrete were not considered because thegirder was post-tensioned when the concrete was 134days old and it had a measured compressive strengthof 82 MPa (12 ksi) at 90 days. Creep loss occurred inthe girder after post-tensioning, which was performedafter the deck was cast. The effective area of concretewas included in the calculation of the creep coefficient.The volume-to-surface ratio V/S used was based onan average value of the girder section and the girder-plus-deck combined section.

The time-dependent creep coefficient J.’ according to theAASHTO LRFD Bridge Design Specifications is definedas:

wheret = time (day)

= age of concrete when load is initially applied (day)H = average annual ambient relative humidity, taken as 50%

= time-dependent factor for the effect of volume-to-surface ratio V/S, given as:

26e00142(WS) + t 1.80 + 1 .77eO213(W5)k,

= 2.587(mm)

45 + t

kf = factor for the effect of concrete compressive strengthgiven as:

where

62kf

= 42+f(MPa)

= 6+f(ksi)

f = specified concrete compressive strength at 28 days,measured as 75.8 MPa (11.0 ksi).

An average value of V/S equal to 100 mm (4 in.) was used,as described previously. The loss due to creep AfcR(t) can befound from the expression:22

where

E4fcR(t)= f(t,t1)—-f

ECCS

f. = the stress in the concrete at the level of the centroid ofthe tendon, given as:22

f =----1i4-2÷QCS r2,) 1

whereA = area of the girder section

= moment of inertia of the girder sectionr = radius of gyration of the girder sectione = the tendon eccentricityMD = the bending moment due to total superimposed dead

loadCreep recovery due to prestress loss was accounted for in

the calculation of creep loss in the stress at the center of gravity of the strands.

In calculating the axial load P, was reduced as the prestress loss was reduced, thus adjusting the creep loss as otherlosses (and prior creep losses) accumulated.

Time-Dependent Shrinkage Loss

According to the AASHTO LRFD Bridge Design Specifications, for steam-cured concrete, the strain due to shrinkage 6ch

at time t is given as:

k (in.)

(t—t.)

0.6H

= 3.5k.kf (1.58 120)ti

lO.0÷(t_ t1)0.6


Table 3. Time-Dependent Losses Compared with Measured Losses from Load Cells at Jacking End

Time, years Ti-C, % T2-C, % T3-C, % T4-C, % LC-i, % LC-2, % LC-3, % LC-4, %

½ 18.5 15.3 12.8 10.0 10.5 9.8 8.0 6.5

1 20.2 17.0 14.5 11.6 12.0 10.8 9.0 7.2

2 21.7 18.4 15.9 12.9 14.2 12.4 10.6 —

3.7 22.6 19.3 16.8 13.8 16.5 13.8 14.4 —

Note: Ti-C Calculated prestressing force in tendon i; LC-i = measured prestressing force from load cell i.

Table 4. Time-Dependent Losses Compared with Measured Losses from Load Cells at Anchored End

Time, years T5-C, % T6-C, % T7-C, % T8-C, % LC-5, % LC-6, % LC-7, % LC-8, %

½ 20.5 16.8 14.0 10.8 9.9 7.9 8.0 6.7

1 22.5 18.7 15.8 12.5 11.3 9.2 9.2 —

2 24.1 20.2 17.3 13.9 13.0 10.8 11.2 —

3.7 25.2 21.3 18.3 14.9 14.8 12.9 12.6 —

Note: Ti-C calculated prestressing force in tendon i; LC-i measured prestressing force from load cell i.

Table 5._Time-Dependent Losses_Compared_with Measured Losses from_Vibrating_Wire Strain Gauges at Midspan

Time, years TC-1, % TC-2, % TC-3, % TC-4, % VW-1, % VW-2, % VW-3, % VW-4, %

½ 17.9 14.6 11.9 11.4 12.3 12.0 12.1 11.4

1 19.8 16.4 13.7 13.0 13.2 12.8 12.9 12.7

2 21.3 17.8 15.1 14.4 13.9 13.6 13.7 13.5

3.7 22.3 18.8 16.1 15.2 14.8 14.4 14.5 14.3

Note: TC—x = calculated prestressing force in tendon x; VW—x — measured force from vibrating wire strain gauges for tendon x.

(e ) =—k k ( 056(10-3)h55o+t)

wherek5 = size factor, the value of which depends on the vol

ume-to-surface ratio V/S as:

resulting in a value of kh = 1.29The loss due to shrinkage zJf11 can be found from the ex

pression:22

zifSH (t) = (Esh)t.Eps

Time-Dependent Steel Stress Relaxation Loss

Relaxation losses of prestressing tendons after transfer AfRmay be defined according to the following expression fromthe PCI Committee on Prestress Losses general method:6k

— [ 26eOO142) +1064— 3.70 (v / s)]

(mm)

45 + t

k (in.)

whereV/S = 100 mm(4in.)kh = humidity factor based on the annual ambient relative

humidity HH =50%

zlfR (t) = f [[ 0.55]

where

f, = initial stress in the tendon at the end of stressing= 0.70f

f = specified tensile strength of prestressing steel= 1860 MPa (270 ksi)= yield strength of prestressing steel= 0.9ff,= time in hours over which relaxation is considered

It should be noted that the removal of the erection

PCI JOURNAL64

(shoring) towers affected the strain in the girders asshown in the companion paper,’ but shoring removal hadno effect on the relaxation losses because post-tensioning lifted the girders off the supports as post-tensioningwas applied.

Anchorage-Seating Loss

Losses due to anchorage seating zlfpA are calculated basedon an estimated slip of 10 mm (0.4 in.). The assumption ofa slip z14 equal to 10 mm, also referenced in AASHTO, resulted in an anchorage loss given as:

where

‘-1fPA =

L tendon lengthmodulus of elasticity of prestressing steel

The length of tendon affected by anchorage set was considerably less than half the girder span. Thus, this loss did notaffect the calculation of losses at midspan.

Friction Losses

Losses due to friction between the internal tendon and theduct wall AfPF were calculated according to the expression:23

4fF=fpj(l_e_(Ki))

where

f = stress in the prestressing steel at jackingx = length of a prestressing tendon from the jacking end

to any point under considerationK = wobble friction coefficient per unit length of tendon

= coefficient of frictiona sum of the absolute values of angular change of the

prestressing steel path from jacking end to the pointunder investigation

For example, a is calculated from the jacking end to mid-span approximately as:22

Lg

wherey’ total vertical change of tendon path, that is, the height

of the tendon at girder end minus the height of thetendon at the girder midspan

L5 = girder span

The measurements taken were not sequenced correctly, sothe coefficient of friction and the wobble friction coefficientK could not be found for evaluating friction losses. However,Wollmann et al. recently provided site-specific data for dU andK for a similar girder located at the 7200 South Bridge atthe 1-15 overpass in Salt Lake City, near the location of thebridge used in the present study. The 7200 South Bridge wasconstructed using the same girder cross section and with the

Fig. 3. Effective prestressing force at jacking end: time-dependent method (Ti -C, T2-C, T3-C, T4-C) and measurements from loadcells (LC-1, LC-2, LC-3, LC-4). Note: 1 kN = 0.2248 kip.

4400

4200

I—.—-LC-1 +--LC-2—-h---LC-3—•—LC-4 —a-—Ti-C •“>“T2-C —t--T3-C —.o—T4-CI

4000

3800

0

3600

3400

3200

3000

2800

Time (day)

0 200 400 600 800 1000 1200 1400


—*— LC-5 - - + - - LC-6 — -A- - LC-7 — — LC-8 —0--— T5-C “0 T6-C — T7-C — 0— T8-C

Fig. 4. Effective prestressing force at anchored end: time-dependent method (T5-C, T6-C, T7-C, T8-C) and measurements from

load cells (LC-5, LC-6, LC-7, LC-8). Note: 1 kN = 0.2248 kip.

same post-tensioning method. The testing reported by Wollmann et al.14 recommended a friction coefficient of 0.25 anda wobble coefficient of 7 x 107/mm (2 x 10-5Iin.), and thesevalues were used in the calculations in this paper.

COMPARISON OF AASHTO METHOD WITHEXPERIMENTAL MEASUREMENTS

Tables 3 through 5 show the total losses using the time-dependent method for the four tendons at the following times:

• Seating;• After ½ year;• Atlyear;• At 2 years; and• At 3.7 years.Table 3 compares the time-dependent losses with the mea

sured losses from load cells at the jacking end, Table 4 compares the time-dependent losses with the measured lossesfrom load cells at the anchored end, and Table 5 comparesthe time-dependent losses with the measured losses from vibrating wire strain gauges at midspan. The 3.7-year benchmark corresponds to the last day of experimental load cellmeasurements on December 31, 2003.

Table 3 shows that tendon T4 (bottom) incurred the smallest losses from the measurements at the jacking end (LC-4),confirming the time-dependent method prediction. The time-dependent method overestimates the losses obtained from

the load cells, except that from load cell 4 (LC-4), whichproduced erratic readings. At 3.7 years, this overestimation.ranges from 2.4% to 6.1% of the total losses.

Values in Table 4 also show that tendon T4 (bottom) hasthe smallest losses for measurements taken at the jacking end(LC-8), confirming the time-dependent method prediction.With regard to the jacking end, the time-dependent methodoverestimates the losses obtained from the load cells, exceptthat from load cell 8 (LC-8), which gave erratic readings. At3.7 years, this overestimation ranges from 5.7% to 10.4% ofthe total losses.

Table 5 indicates that tendon T4 (bottom) recorded thesmallest losses from the measurements at midspan (VW-4),confirming the time-dependent method prediction. At 3.7years, the time-dependent method overestimates the lossesobtained from the vibrating wire strain gauges from 0.9% to7.5% of the total losses.

Figure 3 plots the effective prestressing tendon forcesremaining after losses at the jacking end using the time-dependent method. Figure 3 also shows the actual measuredtendon forces from the load cell readings at the jacking end(north-abutment anchorage zone). The time-dependent method predicts the prestressing forces in a satisfactory manner,with the exception of readings from load cell 4 (LC-4), whichproduced erratic readings after 1.5 years.

Figure 4 identifies the tendon forces remaining after lossesat the anchored end using the time-dependent method. Figure4 also shows the actual measured tendon forces from load cell

L

4500

‘fUVU

3500

z

U00

G

0--

-A— ---•

9nn - . — - . —.- — - - -._ .

_________________________________________

4/27/2000

0 200

Jr400

Time (day)

600 800 1000 1200 1400

66 PCI JOURNAL

Fig. 5. Effective prestressing force at midspan: time-dependent method (TC-l, TC-2, TC-3, TC-4) and measurements from vibrating

wire strain gauges (VW-1, VW-2, VW-3, VW-4). Note: 1 kN 0.2248 kip.

readings at the anchored end (south-abutment anchorage zone).The prediction from the time-dependent method is less satisfactory than that for the jacking end, but it is still conservative,except for the erratic readings from load cell 8 (LC-8).

The tendon forces remaining after losses at midspan usingthe time-dependent method are plotted in Fig. 5. These values are compared with the tendon forces obtained from actualvibrating wire strain gauge readings at midspan. From Fig. 5,it can be seen that the time-dependent method conservativelypredicts the prestressing forces for all four tendons.

CONCWSIONS

Shrinkage and creep tests were performed on samples ofconcrete used in constructing the post-tensioned, spliced,precast concrete girders under investigation. From the testsperformed in this study, the recommended value of the ultimate shrinkage strain for 3-day-old concrete is 780 x 10.The recommended value of ultimate creep coefficient fromthis study for 3-day-old concrete is 2.16. The shrinkage andcreep tests indicate an asymptotic achievement of ultimateshrinkage and creep in approximately eight months. This result is supported by strain readings from vibrating wire straingauges embedded in the concrete of the spliced precast concrete girders being monitored. The concrete modulus of elasticity was overestimated by ACI 318 but was predicted in asatisfactory manner by the ACT 363 equation.

After 3.7 years, computations using the AASHTO LRFD

time-dependent method at the jacking and anchored ends ofthe anchorage zones and at midspan predict a larger remaining prestressing force at the bottom tendon compared withthe top tendon. This is confirmed by the measurements fromthe load cells at the anchorage zones and the vibrating wirestrain gauges at midspan.

The AASHTO LRFD time-dependent method predictedthe total prestress losses in the range of 14% to 23% at thejacking end and from 15% to 25% at the anchored end of theanchorage zones. In this study, the load cell loss measurements at the jacking end ranged from 14% to 17%, and valuesranged from 13% to 15% at the anchored end. The AASHTOLRFD time-dependent method predicted midspan losses inthe range of 15% to 22%, whereas the vibrating wire straingauge measurements recorded midspan losses in the range of14% to 15%. The AASHTO LRFD time-dependent methodpredicts the measured prestress losses with sufficient accu

racy, is slightly conservative for all locations, and comparesfavorably to the experimental measurements.

ACKNOWLEDGMENTS

The authors acknowledge financial support from the UtahDepartment of Transportation (UDOT) and thank DanielHsiao, Stan Bums, and Doug Anderson of UDOT for theirsupport and encouragement. In addition, the authors acknowl

edge the assistance of Professor Lawrence D. Reaveley of theUniversity of Utah and CTLGroup.

z

a01aaaaaa-

200 400 600 800 1000 1200 1400

Time (day)


APPENDIX

Notation

= area of concrete girder section

E = modulus of elasticity of concrete

E1 = modulus of elasticity of concrete at transfer= modulus of elasticity of prestressing steel

e = eccentricity of the tendon

f = concrete compressive strength

f = stress in the concrete at the level of the centroid ofthe prestressing tendons

= initial stress in prestressing steel

f = specified tensile strength of prestressing steelyield strength of prestressing steel

H = average annual ambient relative humiditymoment of inertia of the girder section

j = number of jacking operationsL = tendon lengthLg = girder spanMD = bending moment due to the total superimposed

dead loadN = number of identical tendonsP. = axial loadr = radius of gyration of the girder sectiont = time (day)

= age of concrete when load is initially applied (day)= time (hour)

V/S = volume-to-surface ratiow = concrete densityx = length of a prestressing tendon from the jacking

end to any point under considerationy’ = total vertical change of tendon patha = sum of the absolute values of angular change of

prestres sing steel path from jacking end to the pointunder investigation

= creep coefficient

Ysh = shrinkage coefficient

Yla = loading age adjustment factor= shrinkage (creep) correction factor for humidity

Yvs = shrinkage (creep) correction factor for volume-to-surface ratio

4fcR (t) = loss due to creep of concrete at concrete age t

4fEs = loss due to elastic shortening

4fR (t) = loss due to relaxation of steel after transfer

4fsH (t) = loss due to shrinkage

(ESh)r = time-dependent shrinkage strain

(s5h) = ultimate shrinkage strainK = wobble friction coefficient per unit length of

tendonk = creep factor for the effect of volume-surface ratiokf = creep factor for the effect of concrete strength

kh = shrinkage factor for the effect of humidityk5 = shrinkage factor for the effect of volume-to-surface

ratio= coefficient of friction= time-varying creep= ultimate creep coefficient= time-dependent creep coefficient

/4

Vt

Vu

(t,t1)


post-tensioned tendon losses in a spliced-girder bridge

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