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Prestress Losses inContinuous Composite
Bridges
Mohamed H. SolimanResearch Assistant
Department of Civil EngineeringUniversity of Windsor
Windsor, Ontario, Canada
John B. KennedyProfessorDepartment of Civil EngineeringUniversity of WindsorWindsor, Ontario, Canada
D uring the last 30 years composite con-struction of concrete slab-on-steel gir-
ders has been widely used to form thesuperstructure of bridges. Generally, suchconstruction has been applied only to sim-ple spans or to the sagging moment regionsof continuous spans since transverse crackswill inevitably develop in the region of in-termediate supports.
Such transverse cracking, caused by thepresence of large negative (hogging) mo-ments, reduces significantly the stiffness ofthe bridge and leads to costly maintenance.Moreover, even in the case of simple spanbridges, extensive damage to the expansionjoints can occur due to seepage of deicingsalt-laden water. Fig. 1 shows an exampleof such damage in one of the many expan-sion joints on the Gardiner Expressway inToronto, Canada.
Several researchers 1-3 have proposed toprestress part of the concrete deck in the
vicinity of the intermediate supports of con-tinuous bridges as shown in Fig. 2(a). A re-cent study2 has shown that such prestressingincreases substantially the cracking load inaddition to the stiffness of such bridges.
For proper design it is essential to esti-mate accurately the anticipated losses in theprestressing force due to creep, shrinkageand steel relaxation.'`' The objective of thispaper is to propose a simple method to-gether with design aids by which these long-term losses can be reliably estimated. Theresults from this simple method are com-pared to those from an elaborate method ofsolution requiring the use of a computer.
In this paper it is assumed that prestress-ing of the concrete deck is carried out onthe composite sections (with the connectionbetween the concrete and steel beams re-alized prior to prestressing). Previousexperience' has shown that the alternativescheme of prestressing a free-to-slide con-
84
crete deck can lead to problems not only inconstruction but also in bridge mainte-nance.
ANALYTICAL PROCEDURES
Two analytical methods are presented toestimate the losses in the prestressing forcein continuous composite bridges. The firstmethod is an iterative approach requiringthe use of a computer, while the secondmethod is a one-step explicit solution withdesign aids suitable for design office use.
Both methods are based on well-estab-lished principles used by other inves-tigators3,4.6- s and are dependent on thefollowing assumptions: plane sections re-main plane after any change in the stresses;curvature and axial strain at any cross sec-tion of the bridge can be related to the ap-plied forces.
Iterative MethodIn this method, time is divided into in-
tervals the length of which is assumed toincrease with tine;"" each interval such ast is subdivided into t '2 , t and tj.f1/2, i.e.,
a beginning, middle and end of interval j,as shown in Fig. 3. Any stress variation isassumed to take place at the midpoint of theinterval, while the creep and shrinkagestrains as well as the relaxation stress of theprestressing steel are determined at the be-ginning and end of the interval.
During any interval j, the incrementalaxial strain, AE., and the incremental cur-vature, AK, , can be expressed, respectively,
.1as 4:
Deg = —a A
(1 + 4J *'12J) +
1 AN.
AE ̀ (^,+ v2.. – ; ii2.) + A€ ,,h . (1)
AK = .AM
(1 + ^^+u2,,) +
-1 AM
E .I z (4',+v2., – 4;- "2. ) (2)
SynopsisDesign aids are developed to esti-
mate the long-term losses in the pres-tressing force in continuous compositeconcrete-on-steel girder bridges. Theprestressed portions of the concretedeck are in the vicinity of the inter-mediate supports.
The use of the design aids and ac-counting for the reduced relaxation inthe prestressing steel are illustratedwith a numerical example. The resultsare compared to those from an itera-tive method of solution requiring acomputer.
in whichi, j = time interval number; when
used with time-dependentparameters they indicate timeat the middle of intervals i, j,respectively
Fig. 1. Damage at one of the expansionjoints in the Gardiner Expressway inToronto (Courtesy of A. Tork)
PCI JOURNAL/January-February 1986 85
AN,, = change in axial force on the I, = moment of inertia of concrete
concrete, N _ deck about its own neutral axisD& ,hj = incremental axial shrinkage •j+I/2., = creep coefficient at the end of
strain the jth interval for stress ap-A " = effective area of concrete deck plied at the middle of the ith
Transverse cracks
(a) Composite Concrete deck-on-steel girderBridge - Reinforced concrete deck
Non-prestressed Prestressed deckdeck Np ` Np
(b) Composite Concrete deck-on-steel girderBridge - Part of deck prestressed
Reinforcing steel Prestressing tendonyp = oCentroid of JConcrete deck
doys _ _ N.A. of Composite
Sectio
of steel girder
(c) Cross-section
Fig. 2. Geometry of a continuous two-span composite bridge showing a longitudinalsection with (a) cracked reinforced concrete deck, (b) uncracked prestressed concretedeck and its cross section in (c).
86
N
N
1-I/2 I 1+1/2 J-1/2 J +y2
Fig. 3. Definition of the time interval used in the iterative method of solution.
Time
intervalOM, = change in moment M carried
by the concrete deckE ,= modulus of elasticity of con-
creteUsing Eqs. (1) and (2) at any time j, six
equations can be generated relating the fol-lowing unknowns:" changes in the normalforce acting on the concrete deck and on thesteel girder, AN, and ANq, respectively;changes in the induced force in the pre-stressing steel and in the nonprestressedreinforcing steel, ON and ON,,, , respec-tively; and, changes in the bending mo-ments acting on the concrete deck and thesteel girder, OM ,. and OMs, respectively.
The six equations are derived from thefollowing conditions:
1. The suns of the changes in the forcesat any section is zero for equilibrium.
2. The change in the bending moment atany section is equal to that due to the changein the prestressing force N.
3. Compatibility of curvature and strainat the interface between the steel sectionand the concrete deck.
4. Compatibility of strains at the centerof gravity of the steel section in the concrete
deck.5. Compatibility of strains at the center
of gravity of the prestressing steel in theconcrete deck.
6. Compatibility of strains at the centerof gravity of the nonprestressed reinforcingsteel in the concrete deck.
These six equations were solved for eachtime interval up to time t = 10,000 days(considered to be infinity) yielding the long-term losses in the prestressing force. Acomputer program was written to treat two-span and three-span continuous compositebridges. Values for the concrete creepcoefficient, 4, and shrinkage strain, es,,,were based on the recommendations ofACI Committee 209" and CEB-FIB.18Furthermore, the reduction in the stress re-laxation in the prestressing steel was con-sidered based on the 1975 PCI recommen-dations. 12
The main steps in the computer programfor an analysis by the iterative method areas follows:
1. Read in the following data: Age of con-crete at the beginning, middle and end ofeach time interval; material properties ofsteels and concrete (including slump, rela-
PCI JOURNAL/January-February 1986 87
tive humidity, water content, curing pe-riod, etc.); section properties; initialprestressing force N„0 and long-term mo-ment due to sustained loading.
2. Initiate sectional forces, N, Ns, NP , N^,
M,, M at time of transfer.3. For each time interval:
(a) Determine the creep coefficient 4)as a function of age of concrete attime of loading and of period ofloading.
(b) Compute the shrinkage strain andconvert it to an equivalent normalforce N,,, acting on the concrete deckwhen considering the continuity ofthe structure.
(c) Compute the relaxation stress in theprestressing steel based on the 1975PCI recommendation 12 with the in-itial steel tendon stress f,, given by:
(N,,, — AN„)IA„
where AN,, = 0 at the first time interval (i= 1).
(d) Based on the six conditional equa-tions mentioned above formulatethe matrix equation with the sixunknowns AN, AN ,, AN,, , AN,,,,,AM and AM,.
(e) Solve the matrix equation to deter-mine the unknowns and in partic-ular AN .
4. Print the long-term loss in the pre-stressing steel for each time interval.
Results based on the iterative method us-ing the computer program were generated;they are discussed in connection with theillustrative bridge design example pre-sented later on.
One-Step Method
The total strain in the concrete of age Tand loaded at age t with a stress f , when T>> t, can be shown to be:'''fi
f,ET.r — E (1 + 4)T,) + Esh(Tl
r
in whichE , = concrete elastic modulus at
age t(, r = creep coefficient at time T
with load applied at concreteage t
64.O = concrete shrinkage strain attime T
If the stress is continuously varying withtime, then the total strain becomes:
E T.r = E (I+4),)+
Jr , (1 + 4r.r) afr dt' + e ti^^ (4)at'
The difficulty of continually varyingstress and concrete characteristics withtime is surmounted by the use of the creeprelaxation coefficient introduced firstby Trost'6 and then refined by Bazant. 6 Ineffect, this coefficient accounts for the re-duction in creep deformation at time T =oo due to continuous change in the appliedprestressing force caused by the long-termlosses.
Thus, at time T = , Eq. (4) can be writ-ten as:
Ex r = (1 + 4k,) +
Af' (1 + 'n 4r) + E.,h(:, (5)
Two different 9's are possible'° depend-ing on whether the strain is associated withflexure (q ,,, ) or with axial force (i , ). Valuesof q ,,, (or ii„) are obtained from Fig. 4 fordifferent values of 4) ,,, (or i„ ) given by:'6
and
= a,, 4)_.i
where
cx =1+E,I
EI
(3) and
88
08
Age at Loading = 10 daysConcrete Age = 10,000days
N
Age at Loading = 7 days0.6 Concrete Age =10,000days
0
0.5 F //0.4
0 3I, 1 I I 1 I i
0 0.5 1.5 2.5 3.5
0m,n
Fig. 4. Relationship between the flexural and thrust relaxation coefficients qand creep factor b.
1a = /
in whichA = area of steel girderE = modulus of elasticity of steel
girderI , = moment of inertia (about
its own centroid) of steelgirder
y s = distance from centroid of steelgirder to centroid of concretedeck
The coefficients a and a reflect the factthat creep is influenced by the restrainingaction of the steel girder, '° ignoring that ofthe nonprestressed steel in the deck whichwas shown to be relatively small. Dealingonly with long-term strains by omitting theelastic strain in Eq. (5), the changes inthe strain e and curvature K can be writtenas:
Nhex. = A E fix ' +
A E (1 + ^„ ^__,) + A E (6)
and
OK = E L
. + AM "„ (1 + ^,,, ^_.,) (7)
in whichN , = initial axial force on concrete
deckAN A d<Mer+
mnA, ml,
M^1 = moment at critical section (seeFig. 5) due to long-term loads
A, = area of composite sectionI, = moment of inertia of compos-
ite sectionm = modular ratiod = distance from neutral axis of
composite section to centroid
PC I JOURNAUJanuary-February 1986 89
N P N ,—WJt =D.L±0.3L.L.
o Po(a)
T !
(b)
(c)
(d)
ili
Fig 5. (a) Long-term (It) loading on bridge; bending moment due to(b) applied prestressing force; (c) w it ; and (d) combinedloading (b) and (c).
of concrete deckON- = change in N , at time T = 00
N ,,, = axial shrinkage force in con-crete deck assumed to deformfreely
M 0
= initial moment resisted byconcrete deck
AM = change in M,,„ at T = 00
Results from the iterative solution indi-cated that the effect of nonprestressed steelin the concrete deck is relatively small (lessthan 0.7 percent) and, therefore, was ig-nored in this analysis in order to arrive atan easy-to-use method of analysis and moretractable design aids. Thus, applying theequilibrium conditions to the changes in theforces and in the moments on the compositesection yields, respectively:
ON . +RNs + N = 0.0 (8)
AM , + OM + (AN , 0)y , (9)+ (AN ,,) y „ = OMer
in which
AN— = change in axial forces carriedby steel section
ONE,,, = change in axial forces carriedby prestressing steel
OM q„ = change in moment carried bysteel section
y„ = distance from center of gravityof prestressing steel to cen-troid of concrete deck
OMet = change in moment at criticalsection (see Fig. 5) due to long-term losses in prestressingforce, given by (ON ,, ) k, whereA is a continuity factor derivedin Appendix A.
Such a moment change was ignored inRefs. 7 and 19. It should be remarked thatresults from the iterative solution programshowed that the largest prestressing losseswere incurred at the cross sections coincid-ing with the ends of the prestressed deck(s)due to the presence of a sagging moment.
Invoking compatibility of curvature and
90
strains at the interface between the con-crete deck and the steel section, will leadto:
M + -
o (1 + i1^, 4 _.^) AM-
EJ(10)
N ANAE (t, .7 +
--°(1 + "1„4J) +
AM .o, + N N
.O, = As (11)E lls A,E, AsE,
A E fix•' + A E- (1 + 11
AM.- + N +N
Ell s ACE, A,,E,,
in whichA„ = area of prestressing steelE , = elastic modulus of prestressing
steelN „ = prestressing steel relaxation force
at time T =ooIt should be remarked that the above
analysis could have been dealt with in termsof only two unknowns; however, such amethod would perforce require the use ofone q only since it cannot account for bothr) and -q. This would lead to an unneces-sary approximation which was circum-vented by the use of the method adoptedherein.
A long-handed solution of Eqs. (8) to (12)yielded the following explicit expression forthe long-term losses in the prestressing forceat time T = oo:
AN -`i[(1-. - A ) +(I+A)-yp] N°° W(10)3
(13)
in which
N =N , +w (14)
where w = N ,51(fi x ) and A zlirN,.The values of z (without considering the
"reduced" relaxation of the prestressing steel)are obtained from Fig. B1 and that for
from Table 3. The expressions for the factorsiii, E, C, 'y, p and p., incorporating and,q ,, , are given in Appendix B.
The numerical values for the factors inEq. (13) have been determined for two- andthree-span continuous composite bridgeshaving various cross sections (see Tables 1to 3). For brevity, only a limited number ofsteel sections were considered. However,values for steel sections not listed in thetables can be determined by interpolation.The influence of the bending moment car-ried by the concrete deck (M,.,,) was foundto be relatively insignificant and thereforewas omitted from Eq. (13).
It should be noted that the results in Ta-bles 1 and 2 are based on a prestressed deckof length ranging from 0.21 to 0.351 whichis the normal range for this type of construc-tion, I being the length of one span. Forbrevity, only the most common concrete deckthicknesses of 8 and 9 in. have been consid-ered in Tables 1 to 3. However, these re-sults can be linearly extrapolated for thinnerdecks, say 7 in. thick, to give slightly over-estimated prestressing losses and, there-fore, be conservative.
To account for the "reduced" relaxationof the prestressing steel, the value of z, ob-tained from Fig. B1, must be reduced by afactor a ,. as suggested by Tadros et al` andDilger. 8 They have shown that a , is a func-tion of the ratio:
H _ Loss due to creep and shrinkageInitial prestressing force
_ AN,('PIA) (15)N,,,,
and the ratio:
Initial prestress f_ ^ 1= 6
Ultimate strength fN,,
Fig. 6 (taken from Ref. 8) gives a , as afunction of fl and (3. Thus, the "reduced" zto be used in Eq. (13) becomes:
z' = a ,z (17)
where z is determined from Fig. B1. This
4_) +
AE (12)
PCI JOURNAUJanuary-February 1986 91
CDN)
Table 1. Factors to estimate the long-term prestress losses for 8 in. concrete deck.
w x 10" µ
Two Span Three Span For (A ,/A,m) x 100 =Relative Section
0.1 percent 0.2 percent 0.3 percent 0.4 percent 0.5 percentHumidity No. p y (1, 1) (1, 1.21, 1)
36 x 300 5.709 1.629 16.009 -3.800 470880. 375732. -118.025 -60.174 -40.891 -31.249 -25.464
36 x 260 5.712 1.735 16.039 -4.411 435395. 346299. -139.457 -71.050 -48.247 -36.846 -30.005
36 x 230 5.798 1.851 16.069 -5.046 405336. 321457. -163.137 -83.052 -56.358 -43.010 -35.002
36 x 210 5.902 1.931 16.101 -5.837 384704. 303237. -187.576 -95.489 -64.794 -49.446 -40.238
36 x 182 5.966 2.099 16.079 -6.759 351480. 276407. -222.873 -113.363 -76.860 -58.609 -47.658
36 x 160 5.999 2.268 16.127 -7.782 322073. 252716. -261.372 -132.868 -90.033 -68.616 -55.766
36 x 135 5.984 2.519 16.150 -9.626 286479. 223890. -324.098 -164.656 -111.509 -84.935 -68.991
33 x 241 5.507 1.802 16.082 -5.121 417091. 330752. -155.180 -79.034 -53.652 -40.961 -33.347
50 percent 33 x 201 5.570 1.985 16.139 -6.249 373631. 295127. -194.372 -98.901 -67.078 -51.166 -41.619
33 x 152 5.748 2.350 16.262 -8.766 311199. 243973. -282.367 -143.498 -97.208 -74.063 -60.176
33 x 130 5.646 2.575 16.222 -10.558 279088. 218144. -337.543 -171.467 -116.108 -88.429 -71.821
33 x 118 5.684 2.760 16.292 -11.940 259688. 202589. -385.312 -195.655 -132.436 -100.826 -81.861
30 x 211 5.161 1.938 16.043 -6.518 385082. 304188. -185.016 -94.163 -63.879 -48.737 -39.651
30 x 173 5.198 2.159 16.260 -8.082 339014. 266855. -236.216 -120.120 -81.422 -62.073 -50.463
30 x 132 5.264 2.551 16.306 -11.447 282341. 220564. -337.703 -171.564 -116.184 -88.494 -71.880
30 x 116 5.287 2.786 16.407 -13.293 256927. 200352. -397.529 -201.856 -136.632 -104.019 -84.452
30 x 99 5.259 3.125 16.441 -16.269 227306. 176804. - 487.100 - 247.207 -167.243 - 127.260 - 103.271
Note: 1. For any section not listed above and falling between any two listed sections (such as WF36 >factors may be estimated by averaging (e.g., p = (5.709 + 5.712)/2 = 5.7105 for WF36 x 2
2. The values listed are valid only for prestressed deck of length (0.2 to 0.35)1.
Table 1 (cont.) Factors to estimate the long-term prestress losses for 8 in. concrete deck.
w x 10 µTwo Span Three Span For (k /Am) x 100 =
Relative Section0.1 percent 0.2 percent 0.3 percent 0.4 percent 0.5 percentHumidity No. p 5 y (1, 1) (1, 1.21, 1)
36 x 3(H) 5.599 1.692 14.302 -3.800 427244. 414107. -112.192 -57.133 -38.780 -29.603 -24.098
36 x 260 5.569 1.803 14.332 -4.411 395047. 381667. -132.414 -67.386 -45.711 -34.873 -28.370
36 x 230 5.706 1.938 14,362 -5.046 367774. 354289. - 156.403 -79.526 -53.901 -41.088 -33.400
36 x 210 5.746 2.015 14.393 -5.837 349054. 334208. -178.802 -90.920 -61.626 -46.979 -38.191
36 x 182 5.758 2.183 14.437 -6.759 318909. 304638. -211.763 -107.609 -72.891 -55.532 -45.116
36 x 160 5.721 2.350 14.486 -7.782 292227. 278527. -246.402 -125.157 -84.742 -64.535 -52.411
36 x 135 5.656 2.616 14.405 -9.626 259931. 246756. -303.512 -154.076 -104.264 -79.358 -64.414
70 percent 33 x 241 5.364 1.875 14.375 -5.121 378439. 364533. -147.556 -75.067 -50.904 -38.823 -31.574
33 x 201 5.418 2.073 14.432 -6.249 339007. 325269. -185.270 -94.166 -63.798 -48.614 -39.504
33 x 152 5.410 2.435 14.442 -8.766 282360. 268891. -263.002 -133.551 -90.401 -68.826 -55.881
33 x 130 5.336 2.675 14.477 -10.558 253225. 240425. -316.252 -160.525 -108.616 -82.661 -67.088
33 x 118 5.345 2.869 14.492 -11.940 235623. 223281. -359.635 -182.476 -123.423 -93.896 -76.180
30 x 211 5.062 2.035 14.467 -6.518 349397. 335256. -177.737 -90.359 -61.232 -46.669 -37.931
30 x 173 4.957 2.249 14.440 -8.082 307597. 294110. -221.946 -112.756 -76.359 -58.161 -47.242
30 x 132 4.995 2.649 14.617 -11.447 256176. 243091. -317.727 -161.287 -109.140 -83.067 -67.423
30 x 116 4.972 2.897 14.607 -13.293 233118. 220815. -371.219 -188.351 -127.394 -96.916 -78.630
30 x 99 4.981 :3.255 14.771 -16.269 206241. 194861. -458.917 -232.729 -157.333 -119.635 -97.016
Note: 1. For any section not listed above and falling between any two listed sections (such as WF36 x 280 falls between WF36 x 3(X) and WF36 x 260), the appropriatefactors may be estimated by averaging (e.g., p = (5.709 + 5.712)/2 = 5.7105 for WF36 x 280).
2. The values listed are valid only for prestressed deck of length (0.2 to 0.35)!.
cDw
Table 2. Factors to estimate the long-term prestress losses for 9 in. concrete deck.
w x 10" µ
Two Span Three Span For (A /A,m) x 100Relative Section0.1 percent 0.2 percent 0.3 percent 0.4 percent 0.5 percentHumidity No. p y (l, 1) (1, 1.21, 1)
36 x 3(X) 4.782 1.838 9.916 -3.885 553092. 482647. -79.408 -40.430 -27.437 -20.940 -17.04336 x 260 4.787 1.978 9.945 -4.511 507078. 441358. -94.050 -47.8,51 -32.451 -24.751 -20.13136 x 230 4.834 2.127 9.976 -5.161 468703. 407013. -109.857 -55.857 -37.857 -28.857 -23.457
36 x 210 4.920 2.233 10.007 -5.968 442565. 382201. - 126.716 -64.423 -43.658 -33.276 -'27.04736 x 182 4.896 2.439 9.971 -6.912 101226. 345928. -148.768 -75.586 -51.192 -38.995 -31.677:36 x 160 4.924 2.661 10.020 -7.959 :365159. 314326. -174.609 -88.667 -60.020 -45.697 -37.10336 x 135 4.874 2.966 10.108 -9.847 :322125. 276422. -215.517 -109.398 -74.025 -56.338 -45.726
33 x 241 4.594 2.062 9.989 -5.243 483608. 419840. -104.625 -53.217 -36.081 -27.513 -22.37250 percent
33 x 201 4.558 2.289 9.966 -6.399 428790. 371122. -129.161 -65.648 - 44.477 -33.891 - 27.540
33 x 152 4.648 2.746 10.088 -8.978 351931. 302775. -186.503 -94.696 -64.093 -48.792 -39.61133 x 130 4.605 3.038 10.180 -10.815 313277. 268938. -224.940 -114.170 -77.246 -58.785 -47.70833 x 118 4.636 3.277 10.250 -12.233 290194. 248779. -257.009 -130.399 -88.195 -67.093 -54.432
30 x 211 4.265 2.229 10.001 -6.686 443091. 383455. -124.378 -63.230 -42.847 -32.656 -26.54130 x 173 4.263 2.518 10.086 -8.293 385896, 333125. -157.754 -80.135 -54.262 -41.326 -33.564
30 x 132 4.300 3.007 10.264 -11.746 317124. 272125. -225.960 -114.696 -77.608 -59.064 -47.93730 x 116 4.285 3.311 10.285 -13.643 286893. 245924. -264.064 -133.976 -90.613 -68.932 -55.92330 x 99 4.210 3.716 10.370 -16.701 252079. 215750. -320.486 -162.550 -109.905 -83.582 -67.788
Note: 1. For any section not listed above and falling between any two listed sectionsFactors may be estimated by averaging (e.g., p = (5.709 + 5.712)/2 = 5.7101
2. The values listed are valid only for prestressed deck of length (0.2 to 0.35)/.
Table 2 (cont.). Factors to estimate the long-term prestress losses for 9 in. concrete deck.
w x 10:1 µ
Ts C) Span Three Span For (A /A,m) x 100 =Relative Section
0.1 percent 0.2 percent 0.3 percent 0. 1 percent 0.5 percent[lwnidity No. p ! ry (1, 1 (1, 1.21, 1)
36 x :300 4.582 1.904 8.822 -3.885 503939. 439754. -74.582 -37.936 -25.720 - 19.612 -15.948
:36 x 260 4.583 2.055 8.851 -4.511 462014. 402134. -88.473 -44.970 -30.469 -23.219 - 18.868
36 x 230 4.578 2.207 8.848 -5.161 427050. 370842. -102,589 -52.116 -35.292 -26.879 -21.832
36 x 210 4.633 2.313 8.879 -5.968 403234. 348235. -118.003 -59.944 -40.590 -30.914 -25.108
36 x 182 4.594 2.514 8.923 -6.912 365569. 315185. -138.512 -70.329 -47.601 -36.237 -29.419
36 x 160 4.629 2.765 8.903 -7.959 332708. 286392. -162.827 -82.621 -55.885 -42.518 -34.497
36 x 135 4.581 3.090 8.991 -9.847 293497. 251857. -201.310 -102.107 -69.040 -52.506 -42.585
(1 percent 33 x 24133 x 201
4.4274.328
2.1572.37:3
8.8618.918
-5.243-6.399
440630.390684.
382529.338141.
-98.984-121.306
-50.292-61.605
-34.062-41.705
-25.947-31.755
-21.078-25.785
33 x 152 4.369 2.856 8.972 -8.978 320655. 275868. -174.044 -88.301 -59.720 -45.430 -36.856
33 x 130 4.294 3.166 8.995 -10.815 285437. 245037. -208.582 -105.785 -71.520 -54.387 -44.107
33 x 118 4.318 3.417 9.066 -12.233 264405. 226670. -238.320 -120.825 -81.661 -62.078 -50.329
30 x 211 4.018 2.309 8.884 -6.686 403713. 349378. -115.885 -58.865 -39.858 -30.354 -24.652
30 x 173 4.011 2.615 8.970 -8.293 351601. 303520. -147.183 -74.706 -50.547 -38.467 -31.220
30 x 132 4.010 :3.132 9.079 - 11.746 288941. 247942. -209,604 -106.311 -71.880 -54.664 -44.335
30 x 116 3.980 :3.426 9.181 - 13.643 261397. 224069. -244.597 -124.022 -83.831 -63.735 -51.678
30 x 99 3.955 3.883 9.277 -16.701 229677. 196576. -300.329 -152.212 -102.840 -78.154 -63.342
Note: 1. For any section not listed above and falling between any two listed sections (such as WF36 x 280 falls between WF36 x 300 and WF36 x 260), the appropriatefactors may be estimated by averaging (e.g., p = (5.709 + 5.712)/2 = 5.7105 for WF36 x 280).
2. The values listed are valid only for prestressed deck of length (0.2 to 0.35) 1.
(D01
Table 3. Values for k, under different rel-ative humidity (RH) and slab thickness t.
DeckThickness,
Ill
==5 RH = 60 RH = 70t percent percent
8 in. 70.04 65.37 60.10
9 in. 54.02 50.86 46.61
procedure in addition to the use of Tables1 to 3 as design aids are illustrated in thefollowing example.
DESIGN EXAMPLEIt is required to estimate, at a relative
humidity of 50 percent, the prestress lossesin a continuous two-span composite con-crete-steel girder bridge with a portion ofits deck prestressed as shown in Fig. 7. Thegeometric and material properties as well as
the loading are also given in Fig. 7.The applied prestressing force, N = 880
kips per composite beam. The length of theprestressed deck is 0.251. The width of thebridge is 32 ft with five equally spaced WF33x 221 steel beams.
The total dead load = 1.45 kips per ft.The total sustained (long-term) load, w, , =dead load + (0.3) live load = 1.75 kips perft. From structural analysis the bending mo-ment, due to w,,, is shown in Fig. 5(c); andthat due to the prestressing force is shownin Fig. 5(b). By superposition, the netbending moment diagram is shown in Fig.5(d) where the largest positive moment M„= 301 kip-ft at sections coinciding with theends of the prestressed concrete deck.
The appropriate values for the various pa-rameters in Eq. (13) are determined fromTables 1 and 3. It should be noted that sincethe chosen section of WF33 X 221 is notlisted in Table 1, interpolation is used. Thus,
1.0
W0
0.8V
Z0 0.6I-VDQc 0.4
Z0I-X 0.2
WQ'
0 0.1 0.2 0.3 0.4 0.5JL
Fig. 6. Relaxation reduction factor a, as a function of parameter (1 fordifferent values of (3 (from Ref. 8).
96
\80
17.5No
17.5
Longitudinal Section of Bridge
4.. 8 .. 8 8
c NA-- - ys Non Prestressed
Y Steel J4-WF 33x221
Cross-Section
Prestressingtendons
33.93
LoadingsConcrete weight = 145 lb/ft3Surfacing weight = 60 lb/ft2Girder weight = 200 lb/ftU. D. L. L. =1000lb/ft
Material PropertiesEc =3.9x10 3 ksi; Es = 29x103 ksi ; m=7.44;fpy =230 ksi; fc = 8.0 ksi; fpu=270ksi
Geometric PropertiesAc =768in2 ; At =168.3in 2 ; Ap=4.65in2 ; dc=8.lin; yp=0;ys =20.98in; y=29.85in; S= 757in 3 ; I t =30885in4;OT vo = 2.95; Ap /mA t -0.37'6 ; Anp=6.00 in2
Fig. 7. Data for estimating the loss in prestressing force in a continuous two-spancomposite bridge cited in the design example.
for N in Eq. (14): for sections 33 X 241 and 33 x 201 in Table
A . = 768 in. 2; Nr„ = 880 kips; rn = 7.44; 1.
A, = 168.3 in.'; M,, = (301) (12) kip-in.; d^ Thus-= 8.1 in.; 1, = 30885 in. ;; (w x 10) for asteel section 33 x 221 and an 8 in. concrete (W 103) =
417091 + 373631x = 395361
deck is estimated by averaging the values 2
PCI JOURNAL/January-February 1986 97
from which w = 395.4 kips.Using the definition of N,,,, given by Eq.
(7):
A^Nu^ AdMN =--+mA, ml
_ (768)(880)(7.44) (168.3) +
(768) (8.1) (301) (12)(7.44) (30885)
= 637.8 kips
Thus, from Eq. (14):
N = N + w
= 637.8 + 395.4
= 1033.2 kips
Now A = z/(tiN,), in which z = 97 X 10'from Fig. B1; and = 70.036 from Table3. Thus, A = 1.345.
For Eq. (13): Averaging the values forsections 33 X 241 and 33 X 201 in Table 1yields: p = 5.539; = 1.89; = 16.11; -y_ - 5.685; and µ = -50.35.
Thus, using Eq. (13) and without ac-counting for the "reduced" relaxation in theprestressing steel:
70.036[(1 - 1.89 - 1.89x 1.345)16. 11 + (1 + 1.345)(5.539)(- 5.685)](1033.2)_^Nv (-50.35) (10:')
= 185.8 kips
Hence, the percentage long-term losses steel, respectively.in the prestressing force is (185.8/880)(100) To account for the "reduced" relaxation,or 21 percent. The corresponding percent- a reduced value of z and hence A must beage losses estimated by the iterative method used in Eq. (13). First, the loss in prestress-using the computer solution program were ing force due to creep and shrinkage only is20.7 and 20.0 percent without and with al- estimated from Eq. (13) by putting A = 0lowing for the nonprestressed reinforcing (since z will be zero). Thus:
70.036[(1-1.89)16.11 + (5.539)(-5.685)]1033.2_ (-50.35)(10)= 65.9 kips
Hence, from Eq. (15):65.9 Thus, with fl = 0.075 and 13 = 0.70, Fig.
H = 880 = 0.075 6 gives a, = 0.8, and hence z' = 0.8z and
and 13 from Eq. (16) is:the factor A in Eq. (13) will be reduced by
f (880/4.65)the same factor a = 0.8; using Eq. (13)
R = 270 = 0.70
fF 270again:
70.036[{1-1.89-1.89(1.345)(0.8)116.11+11+(1.345)(0.8)1(5.539)(-5.685)1(1033.2)_NNW (-50.35)(10)
= 161.4 kips
yielding a percentage of (161.4/880)(100) or duced" relaxation of the prestressing steel18.3 percent. the percentage long-term losses in the pre-
Thus, by taking into account the "re- stressing steel is reduced by approximately
98
20.0
15.0a
C 10.0a)a)a. 5.0
0.00
C-)C-0CXz
CvC
T<UC-Cv
ODWQ)
Losses estimated using the one steprmethod t'= ca
Notation RH A in 2 An in2----- 507. 4.65 0.00--- 50 7 4.65 6.00--- 70 4.65 0.00--- 5090 5.00 0.00
50' 4.65 0.00
1200 2400
3600 4800 6000
Excluding Steel Relaxation
Time in daysFig. 8. Percent losses in prestressing force with time under various conditions.
Table 4. Comparison of results for various composite bridges.
Percent Losses
Number of Span Length, 1, andM^^ and Npo by
Spans Steel Section Iterative One-StepMethod Method
f = 50 ft; WF36 M,, = 116.5 kip-ft 15.5 17.6x 182 N,0 = 490 kips
f = 60 ft; WF36 M, = 232.5 kip-ft 16.8 18.3Two Equal x 135 Noo = 700 kips
Spans e = 70 ft; WF33 M1t = 301 kip-ft 16.5 18.3X 221 N, = 880 kips 20.7* 21.1*
e = 80 ft; WF36 M 1, = 463 kip-ft 16.5 18.1x 300 N,,,= 1115 kips
f = 50 ft; WF33 M1, = 111 kip-ft 15.4 16.9Three Spans x 118 N, = 445 kips
(1, 1.21, l) C = 60 ft; WF36 M,, = 188 kip-ft 15.7 17.3x 160 N,, = 560 kips
C = 70 ft; WF36 M,, = 294 kip-ft 16.1 18.6x 210 N^ = 725 kips
C = 80 ft; WF36 M,, = 515 kip-ft 16.4 18.4x 260 Npo = 920 kips
Note: In all cases: t = 8 in.; w = 96 in.; u.d. Dead load = 1.45 kips per ft; u.d. Live load = 1kip per ft. *Losses without accounting for reduced relaxation of steel.
3 percent. The corresponding losses by theiterative method using the computer pro-gram are 16.5 and 15.9 percent without andwith allowing for the nonprestressed rein-forcing steel, respectively.
DISCUSSIONThe percentage loss in the prestressing
force versus time was studied under variousconditions by means of the iterative method.Fig. 8 shows the variations in these losseswith relative humidity, area of prestressingsteel, and area of nonprestressed steel inthe concrete deck, according to the ACICode." The reduction in the relaxation inthe prestressing steel was taken into ac-count in calculating these losses.
The comparison of results shows that thelosses become smaller as the relative hu-midity increases, as expected. Under thesame applied prestressing force, an increasein the area of the prestressed steel tends toreduce the prestress losses. The reason forthis is that the stress ratio of initial pre-
stressing stress to yield stress (f, ,,/f,,) de-creases, thus effecting a reduction in thelosses due to the relaxation of the prestress-ing steel (see Fig. B1).
Fig. 8 also shows that, unlike the situa-tion in concrete-concrete bridges, the influ-ence of nonprestressed steel in the concretedeck on the losses is relatively small here.This is due to the fact that the area of non-prestressed steel in the concrete deck isusually a small proportion of the steel girderarea which provides the major restrainingforce.
It can be also observed that the resultbased on the one-step method is in fairagreement with that estimated by the iter-ative method. Comparison of results onprestress losses based on the CEB Code,1efor brevity given elsewhere, 13 shows that suchlosses are only slightly higher than thosebased on the ACI Code."
Glodowski and Lorenzetti 1 ° indicated thatsteel relaxation has an important influenceon the prestress losses in prestressed con-crete structures. Fig. 8 also shows this in-
100
fluence for the composite concrete deck-on-steel girder structure used in the design ex-ample treated earlier. Here it is observedthat the losses due to steel relaxation are amajor component of the total losses.
To demonstrate the adequacy of the one-step method and use of the design aids inestimating the prestress losses, results basedon both methods were obtained for two- andthree-span continuous bridges with differ-ent spans and cross sections. From a com-parison of the results given in Table 4, it isevident that the one-step method and thedeveloped design aids can be reliably usedto estimate conservatively the prestress lossesin composite concrete-steel girder bridgeswith portions of the concrete deck pre-stressed in the vicinity of the piers.
From this study it can be concluded that:1. The long-term prestress losses in con-
tinuous composite concrete-steel girderbridges with prestressed decks can be read-
ily estimated by means of a simple easy-to-use method.
2. The long-term losses are significantlyinfluenced by the relaxation of the pre-stressing steel; smaller losses can be ex-pected where low relaxation prestressingsteel is used. Furthermore, it appears thatthese losses are not much influenced by thepresence of nonprestressed steel in the con-crete deck.
3. The use of the CEB charts and ACIrecommendations for the estimation of creepcoefficients and shrinkage strain leads to al-most identical values for the long-termprestressing losses.
ACKNOWLEDGMENTThis research work was supported by the
Natural Sciences and Engineering ResearchCouncil of Canada under Grant No. 1896.
NOTE: Discussion of this paper is invited. Please submit yourcomments to PCI Headquarters by September 1, 1986.
PCI JOURNAL/January-February 1986 101
REFERENCES
1. Johnson, R. P., and Buckby, R. J., Compos-ite Structures of Steel and Concrete, Vol. 2,Granada Publishing Ltd., 1979.
2. Kennedy, J.B., and Grace, N. F., "Pre-stressed Decks in Continuous CompositeBridges," Journal of the Structural Division.ASCE, V. 108, No. 11, 1982.
3. Roik, K., "Methods of Prestressing Contin-uous Composite Girders," Proceedings,Conference on Steel, June 1968, BSSA, Lon-don, 1969.
4. Tadros, M. K., Ghali, A., and Dilger, W.,"Time Dependent Prestress Losses and De-flection in Prestressed Concrete Members,"PCI JOURNAL, V. 20, No. 3, May-June 1975,pp. 86-98.
5. Dorton, R. A., Holowka, M., and King, J.P. C., "The Conestogo River Bridge — De-sign and Testing," Canadian Journal of CivilEngineering, V. 4, 1977.
6. Bazant, Z. P., "Prediction of Concrete CreepEffect Using Age-Adjusted Effective Modu-lus," ACI Journal, V. 69, No. 4, 1972.
7. Dilger, W., and Neville, A.M., "Effect ofCreep and Shrinkage in Composite Mem-bers," Proceedings, Second Australian Con-ference on Mechanics of Structures andMaterials, Adelaide, Australia, 1969.
8. Dilger, W., "Creep Analysis of PrestressedConcrete Structures Using Creep Trans-formed Section Properties," PCI JOURNAL,V. 27, No. 1, 1982, pp. 98-118.
9. Ghali, A., Sisodiya, R. G., and Tadros, G.S., "Displacements and Losses in MultistagePrestressed Members," Journal of the Struc-tural Division, ASCE, V. 100, No. ST11, No-vember 1974.
10. Knowles, P. R., Composite Steel ConcreteConstruction, John Wiley and Sons, NewYork, N.Y., 1973.
11. Neville, A. M., and Dilger, W., Creep ofConcrete: Plain, Reinforced and Prestressed,North Holland Publishing Co., 1970.
12. PCI Committee on Prestress Losses, "Rec-
ommendations for Estimating PrestressLosses," PCI JOURNAL, V. 20, No. 4, 1975,43-75; and discussion in V. 21, No, 2, 1976,pp. 108-126.
13. Soliman, Mohamed H., "Long-Term Lossesin Prestressed and Thermal Effect in Com-posite Steel Concrete Structures with Pre-stressed Concrete Decks," MASc Thesis,University of Windsor, Windsor, Ontario,Canada, 1985.
14. Tadros, M. K., Ghali, A., and Dilger, W.,"Time Dependent Analysis of CompositeFrames," Journal of the Structural Division,ASCE, ST4, April 1977.
15. Tadros, M. K., Ghali, A., and Dilger, W.,"Effect of Non-Prestressed Steel on PrestressLoss and Deflection," PCI JOURNAL, V. 22,No. 2, March-April 1977, pp. 50-63.
16. Trost, H., "Implications of the SuperpositionPrinciple and Creep and Relaxation Prob-lems for Concrete and Prestressed Concrete"(in German) Beton - and Stahlbetonbau,Berlin - Wilmersdorf, V. 62, No. 10, 1967.
17. ACI Subcommittee 209, "Prediction of Creep,Shrinkage and Temperature Effect, 2," ACIReport 209 R-82, Detroit, Michigan, October1982.
18. CEB-FIP, "International Recommendationsfor Design and Construction of ConcreteStructure — Principles and Recommenda-tions," Comit6 Europeen du Beton — Fed-eration International de la Pre-Contrainte, FIP6th Congress, Prague, June 1970, Cement &Concrete Association, London, England.
19. Ghali, A., and Tadros, M. K., "Partially Pre-stressed Concrete Structure," Journal ofStructural Engineering, ASCE, V. 111, No.8, August 1985.
20. Glodowski, R. J., and Lorenzetti, J. J., "AMethod for Predicting Prestress Losses in aPrestressed Concrete Structure," PCIJOURNAL, V. 17, No. 2, March-April 1972,pp. 17-31.
102
APPENDIX A - DETERMINATION OFCONTINUITY FACTOR
This continuity factor A is found usingthe unit load method. 13 Let AM be thechange in the bending moment with the in-termediate support(s) assumed removed andthe structure made simply supported; andAM, be the redundant bending moment withthe intermediate support(s) replaced by aunit force X,. Thus, with the structure as-sumed to he brought to its original contin-
uous condition, the final long-term bendingmoment may be expressed as:
AM,, = AMo + AM,X
At the ends of the prestressed deck (seeFig. 5) this becomes:
AM,, = AM o + AM,X, f3'
where
JAM"AM ' dl —(AN,0d„ . AM, dl
=
EI J EI
X J (AM`)2 dl f (AM ' )2 dlEI EI
and 13'1 is the length of the unprestressed deck in one span (see Fig. 5). Thus:
d OM,
AM, = AN , • d, — ON,,,, (OM )2 dl AM . R'
EI
This may be rewritten as:
d`AM' dlEl
,AM,, = AN„,, d J (AM,)' dl AM R']
El
Or:
AMet = AN,,,, - A
in which A is the continuity factor.
PCI JOURNAL/January-February 1986 103
APPENDIX B - FACTORS FOR DETERMININGEQ. (13)
The explicit expressions for determining 1 + (1 + I „ ^,, 7)
the factors in Eq. (13) are as follows: [Ej, ELI
= ^a,7
EA
_ EA + (1 + i„ 4..7)
(1 + Tl,, -,7)
ys
ry= EI,
Y, ( 1 + "1 ^x.7) A_P (1 + rl„-,7) I,
1 (1 + -,7) 1 (1 + I, ^x,7)
_ 1 + ii _ G A .= + E,A,, ) \E A,, + E ft,
LL EA 1 + TI,. 4 7
ErA
( 1(1 + ^1 4 7)l ( /.,
x \E SI.9 + ELI, / + \ESIa
J ' (1 + ^h ^m7) (E,,A,, + (1+(1 +” 1L 4)-,7) E,I x
E,1, + (1 + 11, x7)
EA,_
Note that the expression for the factor z is given in Fig, B1.
0.8
0.7
O6
Z _ 0_lfpo log t o ( fpo -0.551E p io \ f py /
t.= 240,000 hours
.2 4
6 8 10(x10")z
Fig. B1. Values for the coefficient z defining the factor A in Eq. (13).
104
APPENDIX C — NOTATION
A = area of sectionE = modulus of elasticityf = stressI = moment of intertiaM = moment acting on sectionN = normal force acting on sectionw = uniformly distributed loadz = factor defined in Fig. B1a = reduction factorA = incrementE = strain
= creep coefficient= thrust relaxation coefficient
X = continuity factorK = curvature of beam under loading
13 = ratio between initial and ulti-mate prestressing stress in thetendons
(3'l = length of unprestressed deck inone span
fl = ratio given in Refs. 4 and 8,, i;, ^, -y, p, p, are factors defined in Ap-
pendix B.
SubscriptsC = concretes = steelt = timet' = variable time t - t' Ti, j = time interval numbers, when
used with time dependent pa-
rameters they indicate time atmiddle of intervals i, j, re-spectively
i — V2 = beginning of interval i, ji + V2 = end of interval i, jm, n = flexural and axial, respectivelyr = steel relaxation reductionT, t = age of concrete and time of load-
ing, respectivelyp = prestressing steelIt = long termnp = nonprestressing steelsh = shrinkageo = initial00 = very long period
Metric (SI) ConversionFactors1 in. = 25.4 mm1 in.z = 645.2 mm21 in. 4 = 416231 mm41 kip = 4.448 kN1 psi = 6.895 kPa1 kip-ft = 1356 N.m1 kip/ft = 14.58 kN/m1 ft = 0.305 m
PCI JOURNAL/January-February 1986 105
