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.Construction and Building Materials 16 2002 4963

Numerical analysis of time-dependent behavior ofpre-cast pre-stressed concrete girder bridges

Hyo-Gyoung KwakU, Young-Jae Seo

Department of Ci il Engineering, Korea Adanced Institute of Science and Technology, 373-1 Kusong-dong, Yusong-gu, Taejon 305-701,

South Korea

Received 21 July 2000; received in revised form 25 August 2001; accepted 25 October 2001

Abstract

This paper deals with the analysis of uncertainties associated with the behavior of bridges constructed with pre-castpre-stressed girders and a continuous deck. To analyze the long-term behavior of bridges, an analytical model which can simulatethe effects of creep, the shrinkage of concrete, and the cracking of concrete slabs in the negative moment regions is introduced.Based on the equilibrium of forces and the compatibility of strains with time, the prediction of stresses and strains in theconstitutive materials at any time is achieved. To consider the different material properties across the sectional depth, the layerapproach in which a section is divided into imaginary concrete and steel layers is adopted. An element stiffness matrix isconstructed according to the assumed displacement field formulation, and the creep and shrinkage effects of concrete are

considered in accordance with the first-order recursive algorithm based on the expansion of the compliance function. Correlationstudies between analytical and experimental results are conducted with the objective of establishing the validity of the proposedmodel. Also, many parameters related to the continuity of spans are analyzed to minimize deck cracking at the interior supports. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Continuous pre-cast pre-stressed concrete bridges; Long-term behavior; Analytical model; Creep; Shrinkage; Continuity of spans

1. Introduction

Among the various types of highway bridges beingconstructed, the pre-cast pre-stressed girder bridge hasbeen one of the most popular and economical typesadopted. While many inherit advantages due to themechanical connection between the cast-in-place con-crete slab and the pre-cast pre-stressed concrete girderexist, the complications caused by composite action,such as the variations of stresses with time in theconstitutive materials due to the effects of shrinkage

UCorresponding author. Tel.: q82-42-869-3621; fax: q82-42-869-

3610. . E-mail address: [email protected] H.-G. Kwak .

and creep of concrete, must also be considered. Fur-thermore, the differences in material properties andage at loading between the girder and the cast-in-placeconcrete slab lead to time-dependent differential

strains, and that difference may significantly affect thestress redistribution along the span length. Furthercomplications arise in the case of pre-cast pre-stressedgirder bridges. The girders behave as simple-spans fordead loads, while, with the continuous deck providing anegative moment connection at interior supports, thebridge girders may behave as a continuous structure forloads applied after casting the deck and diaphragms.The degree of continuity also depends on time-depen-dent material behavior.

Several analytical methods based on the creep-trans-formed section approach have been proposed to ana-

0950-0618r02r$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. .PII: S 0 9 5 0 - 0 6 1 8 0 1 0 0 0 2 7 - 7


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( )H.-G. Kwak, Y.-J. Seo r Construction and Building Materials 16 2002 496350

lyze the cracked pre-stressed and reinforced concretesections with the assumption of perfect bonds betweenthe constitutive materials, while a few analytical studiesfor the cracked composite sections have been ad-dressed. In addition, in spite of many numerical andexperimental studies of simply supported composite

members under service loads, relatively little researchhas been published on the time-dependent behavior ofcontinuous composite beams subjected to sustainedservice loads. In the case of continuous pre-cast pre-stressed bridge girders, more rigorous analyses arestrongly required because the structural behavior variessignificantly according to the construction sequencesand material in use in different states. Besides, the lack

w xof guidance in the design specifications 1,2 causes awidespread difference in the design approach used forthe connections at interior supports.

In the present study, an analytical model to predictthe time-dependent behavior of pre-cast pre-stressedgirder bridges is introduced. Concrete creep is evalu-ated by the first-order recursive algorithm based on the

w xexpansion of the compliance function 3 , and the varia-tion of concrete properties according to age is included.Moreover, to simulate the different material propertiesacross the sectional depth, the layer approach isadopted, and the material non-linearity due to thecracking of concrete decks in negative moment regionsis considered. The deflections predicted by the pro-posed analytical model are compared to the experimen-tal results to demonstrate the validity and applicabilityof the proposed model. In addition, several numerical

examples are given to investigate the effects of creepand shrinkage of concrete, and their contribution tostructural behavior.

2. Modeling of material properties

In order to formulate the constitutive relationshipsin a layer of composite beam elements, the following

.simplified assumptions have been made: 1 the ele-ment is divided into imaginary layers to describe the

.different material properties; 2 plane sections remainplanes to represent the linearity in the strain distribu-

.tion on any section at any time; 3 a perfect bondbetween the deck and the girder is assumed at service

.loads; and 4 the constitutive materials are assumed tocarry uniaxial stress only.

2.1. Concrete

Based on the principle of superposition, the total .uniaxial concrete strain t at any time t is assumedc

m .to be composed of the mechanical strain t causedcby short-term service loads and the non-mechanical

nm . cr .strain t consists of creep strain t , and shrink-c csh .age strain t .c

. m . nm . m . cr . sh . . t s t q t s t q t q t 1c c c c c c

Shrinkage strain can be evaluated directly by utiliz-

w xing the shrinkage model proposed in design codes 4,5since it is defined as the volume change which occursindependently of imposed stresses, whereas creep isdefined as the increase in strain under sustained stress.In this paper, the first-order recursive algorithm basedon expansion of creep compliance, proposed by Kabir

w xand Scodelis 7 , has been adopted because that modelcan simulate the stress history effectively in spite of itssimplicity in application.

The increment of creep strain from time t to timeny1t for the uniaxial stress state can be expressed asn

w xfollows 6 :

mc y g ti n . s A 1 y e ,n iny 1

is1 .2y i tn .A sA e qa i i i nn ny 1

where c is the increment of creep strain; is then i .inverse retardation times; a are the constants de-i

pending on the age at loading ; m is the number oftime steps; A is the hidden state variable by which theieffects of past time steps are considered. Initial values

.are A sa t at ns 1.i i 1 11Before starting the calculation of creep strain by Eq.

. .2 , parameters such as m, a and must bei idetermined. Since the use of compliance function . . ..J t, s1rE t qC t, in the form of Dirichlet se-ries induces some numerical difficulties caused by notconsidering a separate term to represent instantaneous

..deformation, creep compliance C t, is used directly .in this study, as shown in Eq. 2 . Accordingly m s 4 is

taken, and the assumed corresponding retardation times .are 5.0, 50.0, 500.0 and 5000.0. Also the values of a i

are determined by the method of least squares usingw xKabirs Dirichlet series creep compliance 7 .

m

y ty.i . . w x .C t , s a 1 y e 3 iis1

The concrete stress at each layer corresponding tomechanical strain, can be calculated by using thestressstrain relation of concrete. In describing theuniaxial stress strain behavior of concrete, the model

w xproposed by Hognestad 8 is used after some modifi- ..cations in this paper see Fig. 1 a . These modifica-

tions were introduced in order to increase the compu-tational efficiency of the model in view of the fact thatthe bending behavior of a concrete structure is muchmore affected by the tensile than by the compressive


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( )H.-G. Kwak, Y.-J. Seo r Construction and Building Materials 16 2002 4963 51

behavior of concrete. Moreover, the magnitudes of themaximum compressive stress f and the initial modu-c

w xlus E increase with time as the concrete ages 2,4,5 .cTo simulate the cracking of concrete and to describethe tension stiffening effect, the strain softening intension is included, and the maximum tensile strain 0

.in Fig. 1 a can be determined by s 2 Gt 0 f . . ln 3rb rf 3 yb proposed by Kwak and Filippout

w x9 , where b is the element width and G is the fracturefenergy dissipated in the formation of a crack of unitlength per unit thickness and the value of 87.61 Nrm isused in this paper based on the experimental study by

w xWelch and Haisman 10 . When the mechanical strainm . . t in Eq. 1 exceeds , a crack forms in a corre-c t 0

sponding layer. The normal stress and the modulus ofelasticity become zero to account for the effect of

w xcracking 9 .

2.2. Reinforcing steel

Steel is assumed to be a linear elastic-perfect plastic .material whose yield stress is as shown in Fig. 1 b .y

Thermal strain is the only non-mechanical strain ex-pected for steel. However, that is not considered in thisstudy so that the mechanical strain can be directly

. m ..calculated from the total strain of steel t s t .s s

2.3. Pre-stressing steel

Besides a large difference in the magnitude of thetensile strength, the stress strain curve of pre-stressingsteel is different from that of reinforcing steel in thatthere is no definite yield plateau for pre-stressing steel.To accommodate this different shape, a multilinearstressstrain curve, which is broadly adopted in numer-

w xical analyses of pre-stressed concrete structures 11 , isused. Since pre-stressing steel is never subjected tocompressive stress, the compressive stress strain curveis not considered. The pre-stressing steel placed in theconcrete with an initial tendon profile is approximatedby a series of straight pre-stressing steel segmentsmaintaining a constant force and sectional area, asshown in Fig. 2. The contribution of the pre-stressing

Fig. 1. Stressstrain relations of constitutive materials.


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Fig. 2. Pre-stressing steel segment.

steel to the element stiffness is computed as though thepre-stressing steel segment is a steel layer located at

.the distance of e qe r2 from the neutral axis.i jIn addition, thermal strain is not considered, while

relaxation with time is taken into consideration basedw xon the following equation by Magura et al. 12 .

f f f log ts si si .s 1 y y 0.55 , G0.55 4 /f 10 f fsi y ywhere f is the stress at time t; f is the initial stresss siimmediately after stressing; f is the 0.1% offset yieldystress; t is the time in hours after stressing.

.Since Eq. 4 is valid only for the condition in whichthe strain is kept constant and f is the only appliedsi

.stress; the total stress relaxation f at time t isrn n .obtained by summing up all the relaxation f atrk

. neach time interval t y t , that is f s f ,k ky1 rn ks1 rkaccording to the procedure suggested by Hernandez

w xand Gamble 13 . Once the pre-stressing steel force isknown, its contribution to the element end forces isobtained easily by transforming the pre-stressing force,

P, into equivalent loads and moments at both endnodes. More details such as friction coefficients andnumerical implementation of pre-stressing steel can be

w xfound in Ref. 11 .

3. Neutral axis in a cross-section

Composite members, consisting of a steel or pre-castconcrete girder and a cast-in-place concrete deck, aresensitive to creep and shrinkage so that the influenceof these phenomena on stress redistribution in compos-ite sections must be considered. When shrinkage ofconcrete is considered, in consequence of the restrainton the axial deformation by a girder or a reinforcementembedded in the concrete deck, an internal axial force

has occurred. Accordingly, the neutral axis of a sectioncan not be calculated directly by the equilibrium condi-

w xtion of normal force only. As mentioned by Ghali 14 ,the neutral axis depends upon the eccentricity of thesectional moment over the axial force.

To determine the neutral axis while consideringbending effects only, the mechanical strains of con-

m . m t .crete and steel s need to be partitionedc s s m t . minto an axial strain , and a bending strain ,ca sa cb

t . as represented in Fig. 3. Since the axial strain issb

Fig. 3. Strain components at a section.


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constant across the section and the bending strain iszero at the neutral axis, the bending strains of concreteand steel at any layer can be expressed by:

m t nm m m t t t . s y y , s s y 5cb c c ca sb b s sa

Based on the assumed neutral axis, the correspond-ing stress to the mechanical bending strain can becalculated from the stress strain curves of the constitu-tive materials, and the iterations using the bisectionmethod are repeated until an error for the axial forcecalculated by the bending strain is within the giventolerance.

In addition, the time rate in a stress increment has .been considered. If a stress increment t is intro-c 0

duced at time t and sustained without a change in0magnitude, the time variation of the strain in concretefollows the continuous line ABC in Fig. 4, and the totalstrain at time t, instantaneous plus creep, can be repre-sented by:

. tc 0cr . . . t s 1 q 6c .E tc 0

.where s t, t , the creep coefficient.0 .On the other hand, when a stress increment t, tc 0

is introduced gradually between t and t, the strain0 variation with time can be represented by the dashedline in Fig. 4. The total strain produced during theperiod t to t can be obtained by:0

. t ,tc 0cr . . . t ,t s 1 q 7c 0 .E tc 0

where is the concrete aging coefficient.

The concrete aging coefficient accounts for the effectof aging on the ultimate value of creep for stressincrements or decrements occurring gradually after theapplication of the original load. An average value ofs 0.82 can be used for most practical problems wherethe creep coefficient lies between 1.5 and 3.0 and t is0

greater than 5 days. An approximate value of 0.8 hasbeen frequently used for and the same value ofs 0.82 is adopted in this study.

4. Equilibrium equation for a composite beam element

Based on the assumed displacement field formula-w xtion 11 , the nodal displacement vectors of a two-di-

mensional beam element in its local coordinate system .can be expressed by u, v and see Fig. 5 . Assuming

that the independent axial displacement u varies lin-i

early withx

and the small rotation

at each node canibe calculated by derivation of the vertical displacement with respect to x.i

Then by adopting the plane section hypothesis andapplying the virtual work principle, the equilibriumequations can be rewritten in a differential form.

j nm .dR sdR q dR s K dr 8

where

K s BTE B dV;H TV

dRnm s BTE B dnm dV;H TV

Fig. 4. Time variation of strain caused by a stress increment .c


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dR j is the increment of nodal force vector; B is thestraindisplacement matrix; E is the tangent modulusTof the constitutive material.

Because of the uncoupled characteristic between theaxial force action and bending action, the elementstiffness can be divided into axial and bending terms.

TBaT w xK s B E B dVs E B ,yyB dVH HT T a bTyyBV V bK Kaa ab .s 9K Kba bb

where

1w xB s y1,1a L

2 3 3 . . .B s y1 q 2p , 1 y2p , y2 q 3p ,b L L L

.y1 q 3p

Moreover, each term of the stiffness matrix can berepresented by:

LT T .K s B E B dVs E d A B B dx 10H H Haa a T a T a b

V A 0

K sK T s yy BTE B dVHab ba a T bV

LT .sy E y d A B B dx 11H HT a b

A 0

L2 T 2 TK s y B E B dVs E y d A B B dxH H Hbb b T b T b bV A 0

.12

Since tangent elastic modulus E varies not onlyT

along the beam length but also across the sectionaldepth, each term of the stiffness matrix defined in Eqs.

. .10 12 can be calculated by using three-point Gauss-ian integration. In addition, with the layer approachwhere a typical section is divided into imaginary layers,the section integration on the right side of Eqs. . .10 12 , can be expressed by the following equations.

n n3 c s

E d A sEAs E A q E A w H T c c s s ki i j j /A ks1 is1 js1 .13

y E yd A sESH TA

n n3 c s

sy E y A q E y A w c c c s s s ki i i j j j /ks1 is1 js1

.14

E y 2d A sEIH TA

n n3 c s2 2s E y A q E y A w c c c s s s ki i i j j j /

ks1 is1 js1

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where p is the kth Gauss integration points; w is thek k

kth Gauss integration weight factors; n , n are thec smean numbers of the concrete and steel layers

More details for the finite element formulation canw xbe found elsewhere 15 .

Because the horizontal shear reinforcements betweenthe deck concrete and the girder are designed on the

w xbasis of the bridge design code 1 in a real structure, asufficient resistance to the applied horizontal shear isensured under service load conditions. In addition, theresearch aims are not on the prediction of the ultimateresistance of the structure but concentrated on theshrinkage cracking at interior supports during construc-

tion when only dead loads are applied. Accordingly, aprefect bond is assumed in this study.

Fig. 5. Components of a beam element.


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5. Solution algorithm

Every non-linear analysis algorithm consists of fourbasic steps: the formation of the current stiffness ma-trix, the solution of the equilibrium equations for thedisplacement increments, the state determination of all

elements in the model, and the convergence check.These steps are presented in some detail in the flowdiagram of Fig. 6. Since the global stiffness matrix ofthe structure depends on the displacement increments,the solution of the equilibrium equations is typicallyaccomplished with an iterative method through theconvergence check. The non-linear solution schemeselected in this study uses the tangent stiffness matrixat the beginning of the load step in combination with aconstant stiffness matrix during the subsequent correc-tion phase, that is, the incremental iterative method.

The criteria for measuring the convergence of theiterative solution are generally based on the accuracyof satisfying the global equilibrium equations or on theaccuracy of determining the total displacements. Theaccuracy of satisfying the global equilibrium equationsis controlled by the magnitude of the unbalanced nodal

forces. The convergence criteria for the unbalancednodal forces are used in this study, and these can be

w xexpressed as 11 :

< max < < max < .F F Tol , M F Tol 16unbal. F unbal. m

< max


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Table 1Material properties used in beam B1 and B2

f Creep model E Ec y cr sh c s

y6 4 527.44 MPa 274.40MPa ACI 1.68 520= 10 2.45= 10 MPa 2.00= 10 MPa

and Tol is the specified tolerance corresponding tomthe bending moment

6. Experimental verification

To verify the proposed analytical model, two two-spancontinuous composite beams have been investigated.These beams are B1 and B2 tested by Gilbert and

w xBradford 16 . The geometry and cross-section dimen-sions of the adopted beams are represented in Fig. 7,and the material properties of concrete and steel aresummarized in Table 1. Especially, the material proper-ties of concrete including the time-dependent deforma-tion properties were measured on companion speci-

w xmens, and used in a previous analytical study 17 . Thestructure is modeled by dividing each span into 20elements. The two beams are identical except for theload level. The first beam, B1, was subjected only to itsown self-weight of w s1.92 kNrm, while the second1beam B2 carried an additional superimposed sustainedload of w s4.75 kNrm.2

Fig. 8 compares the analytical results with the mea-

sured mid-span deflections for both beams. Very satis-

factory agreement between analysis and experiment isobserved. But, in the case of not considering the con-

.crete aging coefficient s 1.0 , overestimated de-flections at the initial time steps are obtained. Theresults are also compared with those of an earlier study

w xby Gilbert and Bradford 17 , who used the simplifiedtransformed cross-section approach based on the equi-librium equations and the compatibility conditions. Asshown in Fig. 8, there is, however, still some discrep-ancy between experiment and analysis. Those differ-ences seem to be caused by not considering the bond-slip effect between the concrete deck and the steel

girder.In addition, simply supported pre-cast pre-stressed

.girder bridges BEAMS A & B tested by Rao andw xDilger 18 have been investigated. The geometry and

cross-section dimensions are represented in Fig. 9, andthe material properties are summarized in Table 2. Thegirders were pre-stressed by a force of 196 kN at aget s 7 days, and a reinforced concrete deck was cast at0age 41 days while the girder was shored. The formwork

. was removed 7 days later age t s48 days and an1additional load of 25.77 kN at the third points of the

.span was applied to one of the girders Beam B at age

t s 53 days.2

Fig. 7. Two-span continuous beam.


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.Fig. 8. Mid-span deflections of beams B1 and B2 1 cms 0.01 m .

Table 2Material properties used in beams A and B

f w c c sh, cr,

3 y 6Slab 29.02 MPa 24.7 kNrm 810 = 10 2.453 y 6Girder 42.48 MPa 24.1 kNrm 811 = 10 2.18

Fig. 10 compares the analytical results with the mea-sured mid-span deflections for both beams, and thisfigure shows that the observed and predicted values ofcamber agree within reasonable limits of accuracy.Moreover, the response of the concrete to graduallychanging stress is best calculated by the age-adjustedeffective modulus method which was first introduced by

w x w xTrost 19 and further developed by Bazant 20 . Thisexample was previously analyzed by using the creep

w xtransformed section approach by Dilger 21 , and theobtained results are marked with squares in Fig. 10.That method gives a reasonable result for the un-

Fig. 9. Simply supported pre-cast pre-stressed girder bridge.

cracked simply supported concrete member, while thereis some restriction in application to the cracked contin-uous concrete members.

Through these correlation studies between analyticaland experimental results, we found that the layer ap-proach can be used effectively, and the concrete agingcoefficient must be taken into consideration in simu-lating the time-dependent behavior of structures.

7. Numerical verification

In a well-known example being affected by the time-dependent behavior of concrete, a double cantilever with an open joint at the center is selected for thenumerical verification of the introduced model. Thegeometry and the cross-section are shown in Fig. 11and the material properties are summarized in Table 3.The joint is assumed to be connected at 7 days and thisexample structure is subjected only to its own self-

.Fig. 10. Cambers of beams A and B 1 cms 0.01 m .


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Table 3Material properties used in numerical verification

Creep model E E f cr s y c c

5 42.5 ACI 2.01= 10 MPa 196 MPa 2.45= 10 MPa 27.44 MPa

Fig. 11. Configuration of example structure.

weight of w s 2.94 kNrm. Each cantilever is modeledwith ten elements and only creep deformation is takeninto consideration.

Fig. 12 represents the variations of bending moments with time along the span. From this figure, it can beseen that the bending moments converge to any limit-ing boundary values. Theoretically, the moment dis-tribution along the span must be converged to that ofthe fixed end beam with a span length of lls 10.1 m astime approaches infinity. This illustrates the fact thatmoment redistribution due to creep following a changein the static system tends to approach the moment

distribution that relates to the static system obtainedafter the change.In Table 4, the obtained fixed end moment at 365

w xdays is compared with that by the PTI method 22 .Since the shrinkage effect is not considered, it is alsoexcluded from the equation of PTI. As shown in thistable, the introduced model yields a very satisfactoryagreement with the PTI results.

8. Application to pre-cast pre-stressed girder bridges

8.1. Effect of positi e moment continuity reinforcement

Current bridge construction practices include a num-

Fig. 12. Moment distribution for cantilever system 1 t m s 9.8 kN.m .

ber of aspects such as the varying amounts of positivemoment reinforcement at supports, and the varyinggirder age at the start of bridge construction. To evalu-

ate the time-dependent behavior of pre-cast concretegirder bridges according to girder age under serviceload conditions, a sample structure, shown in Fig. 13,

Table 4Comparison of end moment

Method Fixed end beam PTI This study

2 22 2q ll q llq ll q ll1 1 y . .Formula Ms y Ms y q y q 1 y e 12 2 12 2

Moment y24.99 kN m y26.66 kN m y26.56 kN m

Note: the total span length lls 10.1 m, the cantilever span length ll s 5 m and at 365 days s 1.93.1


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Table 5 .Material properties used in concrete beam bridge I

f Creep model E E w Reinforcementc y cr sh c s c

y 6 4 5 3 2 .27.44 MPa 274.40 MPa CEB-FIP 78 2.5 200 = 10 2.15= 10 MPa 1.96= 10 MPa 23 kNrm A s 800, 400, 800, 1600 mms1,2,3,4

was analyzed. The assumed material properties aresummarized in Table 5, and the live loads of HS-20

w xlane loading 1 are assumed to be gradually applied at350 days after continuity by casting the deck and di-aphragm simultaneously. An initial pre-stressing forceof P s 3153.05 kN is assumed constant across thelength, and the applied force P is transformed into thedistributed upward load of w s 37.53 kNrm using thep

w xequivalent load concept 23 . None of the pre-stressinglosses due to friction, creep, shrinkage, and relaxationare taken into consideration because the primary pur-pose of this study is concentrated on the evaluation ofstructural behavior according to girder age.

Moreover, the composite member deforms along theneutral axis of the transformed section, while the rela-tive rotation is caused from the bearing located on the

bottom face of the girder. For the given end rotation ofthe beam, , therefore, the additional horizontal dis-placement of hr2 occurs at the slab. To simulatethis offset effect, the rigid members with the lengthequivalent to half of the beam height are imposed atthe support nodes.

The central support moments for continuity at thegirder age of 7 days are shown in Fig. 14. A largeproportion of girder pre-stress creep occurs after cont-inuity, causing positive restraint moments to develop.Also, negative restraint moments caused by differentialshrinkage between cast-in-place concrete decks andpre-cast girders are lessened because a relatively smallamount of girder shrinkage has occurred before thedeck is cast. Time-dependent restraint moments at thecentral support of point B are shown on the left side of

.Fig. 13. Two-span continuous pre-cast pre-stressed girder bridge I .


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Fig. 14, and the right side of this figure indicatesmoments upon incremental application of live loads at350 days. Especially, with a small amount of positivereinforcement, positive restraint moments may be neg-ligible, that is, the girder end rotation causes cracks todevelop in the bottom of the diaphragm. The flexural

stiffness of the diaphragm is, therefore, reduced andrestraint against girder end rotation is decreased. Incontrast, when a substantial amount of reinforcementis provided, girder end rotation is more effectivelyrestrained, and the moment redistribution according tothe time-dependent behavior has occurred along thespan length.

Fig. 14 also shows the decrease of support momentin four increments according to the application of liveloads at 350 days. The behavior of the reinforced andunreinforced sections differs. While the central supportmoment has been decreased linearly up to 100% of thelive load for the reinforced diaphragm, only a slight

decrease in the central support moment has beencaused because of the low rotational stiffness of thecracked diaphragm. Consequently, the positive momentcrack at the bottom of the diaphragm must be closedprior to inducing a negative moment at the continuityconnection.

On the other hand, some different structural behav-

Fig. 14. Central support moments for continuity at girder age of 7 days.


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iors are expected when the age difference between thedeck and the girder is large. Fig. 15 shows a time-dependent restraint moment at the center support inthe case of continuity at the girder age of 1000 days.Regardless of the amount of positive reinforcement atsupport, high negative restraint moment develops due

to differential shrinkage between the cast-in-place con-crete deck and the pre-cast concrete girder, becausealmost all girder concrete shrinkage had occurred al-ready.

As time passes, the restraint moment due to creep isalso increased because the structural system has beenchanged from the simply supported beam to the contin-uous beam. However, the total restraint moment due to

the consideration of both creep and shrinkage is notequal to the moment obtained by the summing of themoments due to the separate consideration of creepand shrinkage. It may be caused by the fact that creepreduces the large negative restraint moment due toshrinkage. Similar conclusions with those obtained in

this study for the pre-cast pre-stressed concrete girderw xbridge were also reached in a previous study 24 .

Different amounts of positive reinforcing steel atsupport have a negligible effect on resultant moments.Because the positive reinforcing steel is in the com-pression zone of the section under negative moment,varying amounts of reinforcing steel have a small effecton bending stiffness. Consequently, the pre-cast pre-

.Fig. 15. Central support moments for continuity at girder age of 1000 days 1 t m s 9.8 kN m .


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stressed concrete girder behaves almost identically withthe reinforced concrete girder as the girder age isincreased.

9. Conclusions

An analytical model to simulate the time-dependentbehavior of pre-cast pre-stressed girder bridges is pre-sented in this paper. The effects of creep, shrinkage ofconcrete, relaxation and losses of pre-stressing steel,and material non-linearity caused by concrete crackingwere taken into consideration, and the related numeri-cal implementations are described. In addition, to con-sider the different material properties across thesectional depth and the increase of the numerical ef-ficiency, the layer approach was adopted. The devel-oped model has been verified by comparison betweenexperimental results and numerical examples.

The representative pre-cast pre-stressed girderbridges were analyzed with the purpose of investigatingthe relative effects of creep and shrinkage of concreteon structural behavior, and the following conclusions

. were obtained: 1 the concrete aging coefficient mustbe taken into consideration to simulate the gradually

.increased time-dependent deformation; 2 the positivereinforcing steels at supports have a negligible effect

.on the resulting moment; 3 when the structural sys-tem is not changed during construction, the shrinkage

effect is dominant, while the creep effect is negligible; .4 when the structural system has been changed duringconstruction, the creep effect is dominant, while theshrinkage effect depends on the construction time of

.the deck slab and the diaphragm at support; 5 theeffect of shrinkage on structural behavior increases inproportion to the increase of the difference in con-

.struction time between the deck and the girder; 6 thetransverse cracking at interior supports are deeply re-lated to the shrinkage strain of the slab.

To increase serviceability and to remove the inherentstructural defects, therefore, effective construction con-

trol including concrete curing at an early age is re-quired. In addition, the parametric studies of compos-ite girder bridges reflecting the construction sequencemust be conducted, and the developed model can beused effectively whenever it is needed.

Acknowledgements

The research reported in this paper was made possi-ble by the financial supports from the Samsung Engi-

neering and Construction and BK21 project funded bythe Ministry of Education of Korea. The authors wouldlike to express their gratitude to both organizations fortheir support.

References

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w x2 British Standards Instituiton. Structural Use of Concrete. Part1. Code of Practice for Design and Construction BS 8110:Part

.1:1985 . London, 1985.

w x3 ASCE. State-of-art Report on Finite Element Analysis of Rein-forced Concrete, New York, 1982.

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