analysis of concrete cable-stayed bridges for creep, shrinkage

0045-7949(95)00131-x

Com,wer.s & Swticrurrs Vol. 58. No. 2. pp. 337-350. 1996 Copyright :I 1995 Elsewer Science Ltd

Prmted in Great Bntam All nghts reserved 0045.7949196 $9 50 + 0 00

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ANALYSIS OF CONCRETE CABLE-STAYED BRIDGES FOR CREEP, SHRINKAGE AND RELAXATION EFFECTS

N. C. Cluleyt and R. Shepherd1

TDepartment of Civil Engineering, California State Polytechnic University, Pomona, 3801 West Temple Ave, Pomona, CA 91768, U.S.A.

iDepartment of Civil Engineering, University of California, Irvine. CA 92717. U.S.A.

(Receiced 30 Jww 1994)

Abstract-The time-dependent effects of creep and shrinkage of concrete and relaxation of prestressing tendons on stresses and deflections in segmentally erected, cable-stayed, concrete bridges are investigated. Specifically, these effects should be considered when determining the girder cross-sectional stress redistribution necessitated by shear lag. A special purpose, three-dimensional finite element code is developed to analyze these effects. Time-independent effects considered are large displacements in girders and pylons, sag effects in cable stays and anchorage slip loss.

NOTATION (GJ”

cross-sectional area of post-tensioning elements, also :a

the surface area of shell elements beam cross-sectional area equivalent joint loads creep compliance coefficients

L*

“” P 0

strain displacement matrix for shell elements stress-strain relationship for shell elements

50

joint displacement vector modulus of elasticity for cable-stays effective modulus of elasticity for concrete at the beginning of a time interval loss of force in chorage seating 28 dav strength

post-tensioned elements due to an-

of concrete prestr&s at the beginning of a time interval yield strength of steel change of prestress over a time interval moment of inertia for beams end of the previous time interval end of the current time interval creep compliance function time interval prior to i structure stiffness matrix element stiffness matrix horizontal projected length of cable-stays length of post-tensioning elements axial force in beams moment in beams vector of equivalent joint loads time vector of combined joint loads applied to the structure retardation times which govern the shape of the creep curve product of AC1 209 correction factors used in concrete shrinkage calculations axial strain in beams due to creep step function change in stress total strain creep strain instantaneous elastic strain shrinkage strain shrinkage strain in beams

ultimate shrinkage strain thermal dilatation sum of elastic, shrinkage, and thermal strain storage vector for past stress histories ultimate creep coefficient weight per unit volume of cable constant uniaxial stress age of concrete at loading

INTRODUCTION

The objective of the investigation described is to

determine the time-dependent effects of creep and shrinkage of concrete and relaxation of prestressing tendons on stresses and deflections in segmentally erected, cable-stayed, concrete bridges. The importance of these time dependent effects, when determining the girder cross-sectional stress redistribution necessitated by shear lag, is well established [I]. Time independent effects considered are large displacements in girders and pylons, sag in cable stays and anchorge slip loss. Such effects occur during both the construction phase and the service life of the structure. It is possible that allowable stresses may be exceeded during either or both periods. Consequently the designer needs to know, as a function of time, the maximum stresses and deflections due to construction and service loads.

To accomplish such complex analyses a three-dimensional finite element code is developed to simu- late the nonlinear effects together with construction and service loads as a function of time. The theory and methodology to develop and implement this type of analysis is presented in this paper along with the results of a simplified sample analysis. The nonlinear effects considered are defined in the following sections.

337

338 N. C. Cluley and R. Shepherd

Time dependent nonlinear effects MODELING OF CREEP, SHRINKAGE AND RELAXATION

(a) Creep strain, the time-dependent change in strain under sustained load.

(b) Shrinkage, the time-dependent change in strain under constant temperature. This study focuses on drying shrinkage.

The total strain in a uniaxially loaded concrete specimen can be found by the superposition of elastic, creep, shrinkage and thermal strains as follows:

(c) Relaxation, the loss of prestress in elements when subjected to constant strain.

c(f)=tE(f)+ff(f)+CS(f)+~7(t)r (1)

(d) Aging-as concrete ages the modulus of elas-

ticity increases, quickly at first and more slowly as curing slows down.

where t(t) is the total strain,+(t) is the instantaneous elastic strain, c,(t) is the creep strain, c,(t) is the shrinkage strain, ET(t) is the thermal dilatation.

Time independent nonlinear effects

The principal of superposition is considered valid if the following conditions are upheld

PI: (a) Large displacements, due to the inherent

flexibility of cable-stayed bridges whereby the stiffness characteristics may change significantly due to changes in geometry. Such changes may be accounted for using an incremental or iterative finite element solution procedure or a combination thereof.

(1) The stresses are less than about 45% of the concrete strength.

(2) Appreciable reductions in strain magnitude

due to unloading do not occur. (3) No significant change in moisture content dis-

tribution during creep occurs.

(b) Sag effects, in the cable stays, due to their weight. The tensile load resistance will vary with the

amount of sag.

(4) No large, sudden, stress increase long after the initial loading occurs.

(c) Anchorage slip loss in post-tensioned members due to the seating of wedges in the anchors when the jacking force is transferred to the anchorage.

SEGMENTAL BRIDGE CONSTRUCTION BY THE BALANCED CANTILEVER METHOD

In practice all of these conditions may be violated to some extent, but experience has shown that conformity with the first and third conditions,

which are the most important, is generally true under good design and construction practices. The second and fourth conditions, which are the least important, suffer more substantial violations but experience has shown that strain predictions are still acceptable.

For purposes of analysis during the construction phase, only the balanced cantilever method of construction is considered. This construction method proceeds as follows. Consider a cable stayed bridge which consists of two approach spans and a central span. First, the piers, towers and abutments are constructed. Next, the pier segments are sequentially added to each side of each tower by traveling cranes and temporarily secured at the tower to ensure stability during construction. After preliminary post- tensioning of segments, cable stays are attached and post-tensioned to prevent excessive deflection and overstress when additional segments are added. During this process the cranes are repositioned at the tower. Additional girder segments are then added using the cranes and a second set of cables are attached and post-tensioned. Upon completion of adding and post-tensioning of segments and cables, the spans are closed at the abutments and closure is achieved at the center with a cast-in- place closure pour. All cables are then post-tensioned to comply with design requirements for camber and initial force and moment distribution. Such cable adjustments are dependent upon deflections and the attendant stress redistributions caused by the nonlinear effects which are the subject of this investigation.

Traditionally, practical prediction of creep strain has been confined to the linear-elastic range (i.e. stresses less than or equal to 0.45f:) such that the principle of superposition applies as discussed above. Since cable-stayed bridges are designed with the intent that their stresses remain in the linear elastic range, the principle of superposition will be utilized in this investigation.

Creep strain is dependent on stress and may be graphically depicted by creep isochrones which are lines connecting the value of strain under a sustained stress during a given time interval. The plot of creep isochrones in Fig. 1 shows that for stresses up to about 50% of the concrete strength, the creep is approximately proportional to stress.

, , - T(, = I 11,111

‘.. ,: : ,,-r<,=ldav

Fig. I. Creep isochrones.

Analysis of concrete cable-stayed bridges 339

JO, .‘,,I ,,_/ ro =5 days

;. r,=50dayr

Fig. 2. Typical creep curves for loading ages TV

For constant uniaxial stress, 0, the strain may be written as

where t, is the total uniaxial strain and r~ is a constant uniaxial stress, J(t,, rO) is the compliance (creep) function which represents the strain at time t, due to a unit constant stress which has been acting since time 7,, Due to the proportionality of creep strain to stress the creep is fully represented by J(t,, TV), a plot of which is shown in Fig. 2. It can be seen from this plot that creep strains can be much larger than initial elastic strains, hence the importance for considering the effect of creep strains on deflection and stress redistribution.

Practical prediction of creep strain involves devel- oping a function which accurately predicts the com-

pliance function J(t,, r,,). The American Concrete Institute [3] recommends

the following expression:

1 J",-"i)=J,(r,, '+

[

(t, - Top"

IO + (t, - 70)o.6 ", I ’ (3)

where rg is the age at loading in days, t, is the current age in days, V, is the ultimate creep coefficient which is defined as the ratio of the assumed creep strain at infinite time to the initial strain at loading. It is a function of environmental humidity, loading age, minimum thickness of structural member, slump, cement content. percent fine aggregate and air content.

Since this investigation is concerned with the variations in stresses and deflections during both the construction phase and the service life of cable-stayed bridges and stresses and strains are continually changing during the bridge’s lifetime, due to both structure-modification and creep and shrinkage, the expressions presented above for creep, which are based on constant stress, must be adapted to situations of varying stress.

Since cable-stayed bridges are designed such that their stresses are not to exceed the elastic range, concrete may be treated as an aging, viscoelastic, material [3]. Using superposition, the strain, due to

any stress history o(t), may be obtained regarding the history as the sum of increments da(r,) applied at

increments of time, TV. Equation (2) may then be written as

s

‘i t(t) = J(t,> 7,,) W7,) +tO(t,). (4)

0

Equation (4) relates general histories of stress, (T

and strain, t. The integral in this equation, known as the Stieltjes integral, has the advantage that it is applicable even for discontinuous stress histories. The numerical implementation of the integral in eqn (4) can be accomplished by using a step-by-step time

integration [4] as follows:

&(i+ ])=+I +J(t;+,,T,)]

+c-- ’ d”O’)[J(t,+I.7,)--J(t,,7,)1” (5) ,=I EC/‘)

where At(i + 1) is the total strain increment for time interval i to (i + I), (i + 1) is the end of the current time interval, i is the end of the previous time interval, j represents time intervals prior to i.

The first term on the right side of eqn (5) accounts

for the instantaneous plus creep strain in the time interval i to (i + 1). The summation term accounts for the creep increment of strain over the time interval i

to (i + I) due to all previous increments of stress. To implement eqn (4) using a step-by-step numerical technique, it is required to store the entire stress history for all elements in the analysis and to retrieve all these stresses for each new time interval in which the strain increment is to be calculated. This would require an enormous amount of peripheral storage and significant retrieval times. Fortunately, the need for storing and retrieving the complete history of stresses can be eliminated if the integral-type creep law [eqn (4)] can be converted to a rate-type creep law (i.e. a creep law represented by a system of first-order differential equations). This can be done by approxi- mating the kernel, (/(t,, TV)), of eqn (4) by the so-called degenerate kernel which takes the following form [2, 51:

J($,7,)=C ( ) I-e- “’ a, 5o [ ,=,

(I( -r”j6], (6)

where t, is the time in days when the creep strain is desired, to is the time of loading, r, are retardation times which goveren the shape of the creep curve, ai are the creep compliance coefficients.

The method of evaluating coefficients [6], a,(to), is the same whether the creep strains are measured in a laboratory or computed using some formula for their estimation such as those of AC1 209 [3].

Drying shrinkage is the major cause of shrinkage and is largely dependent on the initial moisture


content, the level of environmental humidity, member size and the age of the concrete at which drying commences.

Shrinkage strain is usually about one half of that for creep strain and, hence, can contribute significantly to deflection and stress redistribution.

Practical prediction of shrinkage strain in accord- ance with AC1 209 can be accomplished using the

following equations. For concrete which has been moist cured for seven

days:

%h ct) = & (Ch)“.

For concrete which has been steam cured for one

to three days:

c,h ct) = & (Gh)“’

where (c,,), is the ultimate shrinkage strain, which

may be taken as 780y,,(10)-h where ysh is the product of correction factors as prescribed by AC1 209.

Stress relaxation in prestressed elements is the loss of stress under conditions of constant strain. The magnitude of this loss is dependent on both the duration of the sustained prestressing force and the ratio of the initial prestress to the yield strength of the steel.

The following expression [7] is commonly used to calculate relaxation and is adoped for this investigation:

A/R~=~~,[logI2lOlogfi]~-o.55] (9)

where A& is the change in the prestress over the time interval t, to t,. f,, is the prestress at the beginning of the time interval t,. t, is the beginning time interval in hours. t2 is the end of the time interval in hours. &, is the yield strength of steel in psi. For low relaxation tendons use 45 in place of 10 in the denominator of eqn (9). For&,/& < 0.55 relaxation is negligibly small.

MODELING OF TIME INDEPENDENT NONLINEAR EFFECTS

The time independent nonlinear effects considered in this investigation are as follows.

Large displacements

Large displacements of the towers and girders of cable-stayed bridges are due to their inherent flexibility. The resulting changes in geometry serve to alter the stiffness characteristics of the structure. To account for this alteration of stiffness a geometric nonlinear analysis must be performed.

There are basically two ways in which the stiffness matrix of a nonlinear system may affect displacement

variation. The first corresponds to a situation in which the stiffness of the structure decreases with increasing deformation. The girder and towers typify this response. The other corresponds to a situation in which the stiffness of the structure increases with increasing deformation. The cables typify this response.

Two procedures which are well suited for the above situations are the incremental approach and the iterative approach.

Incremental or stepwise approach. In this approach the total load is applied in increments and the structure is assumed to respond linearly within each increment. The joint displacements calculated for an increment are used to update the joint coordinates in preparation for analysis using the next load increment. At the beginning of the next load increment the structure stiffness matrix is reformulated using the updated set of joint coordinates. The general form of the stiffness equation which must be solved for each load increment, i, is

Wl,(AD),= {AW, (10)

where [IQ is the structure stiffness matrix correspond- ing to the deflected shape of the structure at the beginning of the ith load increment. {AD), are the joint displacement increments which occur due to the application of a load increment. (A W}, is the magni-

tude of each joint load increment. The final displacements and member forces are

found by adding the incremental displacements and member forces that correspond to all load increments.

Iterative approach. In this approach the total load is applied to the structure in a single increment. The initial analysis uses the tangent stiffness matrix of the undeformed structure to calculate the joint displacements. The joint coordinates are updated and the stifiness matrix is reformulated before member end loads are calculated. Since this final stiffness, used to determine the member end loads. is different from the

initial stiffness used to determine the joint displacements, equilibrium will not be satisfied and unbalanced loads will exist at the joints. These unbalanced joint loads are applied to the structure and the resulting displacements are used to update the joint cordinates, thereby acting as a correction factor to minimize diversion from the true solution. The stiffness matrix is reformulated once again and the member end loads due to the unbalanced loads are calculated. Since the most recent stiffness matrix, used to find the member end loads, is different from the stiffness matrix used to find the displacements due to the unbalanced loads, equilibrium will not be satisfied and unbalanced loads will exist at the joints. The above procedure is repeated until the unbalanced loads at the end of any load cycle are less than some


previously accepted tolerance. This itertive procedure is known as the Newton-Raphson method. The final

displacements and member end loads are found by

summing those found in each iteration.

Combined incrementul and iterative approach. In this approach the loads are applied in an incremental fashion and the Newton-Raphson method is used to iterate to the true solution for each load increment before the succeeding load increment is applied. This procedure has been shown to be the most accu- rate [20,21], although computationally time consum- ing, and is the method used in this investigation.

One of the most important large displacement effects in cable-stayed bridges is the P-delta effect in the girder and towers. This is due to the coupling of large lateral deflections with high compressive axial forces. This coupling reduces both the axial and rotational stiffnesses of the members involved.

For the bridge towers an efficient approach is to use the geometric stiffness matrix of each beam- column element to modify its elastic stiffness matrix. The resulting tangent stiffness matrix [8] of eqn (11) becomes the element stiffness matrix used in the

analysis.

VTIE = LGIE -t kilE3 (11)

where [kTIE is the element tangent stiffness matrix, [k& is the element elastic stiffness matrix, [kFIE is the element geometric stiffness matrix.

The bridge girder is modeled using a highly efficient triangular plate/shell element. It has been shown [9] that for a geometric nonlinear analysis by the combined incremental-iterative approach, the tangent stiffness matrix for such plate/shell elements can be represented acceptably by the conventional elastic stiffness matrix. This is especially true for prestressed box girders which are quite stiff. For these types of girder cross sections the contribution of local plate P-delta effects to the P-delta action of the full span of the girder are negligible. In the interest of compu- tational efficiency, the conventional elastic stiffness matrix is used in this investigation with the girder P-delta effects being accounted for through the combined incremental iterative analysis approach.

Sag eJk~/s in cable-.staJx. The sag in cable-stays is caused by their deadweight.

The relative axial movement of the ends of the cable-stay is the result of the following distinct ac- tions:

( I) The elastic strain in the material which is linear and governed by the modulus of elasticity, E.

(2) The change in sag of the cable which is a function of its geometry and tensile load. This change varies in a nonlinear fashion with the tensile load.

Since the variation of sag with the axial force in the cable is nonlinear, the axial stiffness of the cable will also vary in a nonlinear fashion. An effective and

convenient approach for considering this nonlinearity is to consider the length of the sagging cable to be

equal to the length of its chord, Lc and to represent

the nonlinearity through the use of an equivalent modulus of elasticity which accounts for sag and elastic strain. A widely used expression [lo] for this equivalent modulus is as follows:

E Eeqv =

p2L2E’ (12)

I+---- 12a3

where E is the effective modulus of elasticity, p is the

weight per unit volume of cable. L is the horizontal projected length of the cable. u is the tensile stress in the cable.

By using eqn (12) the cable can be modeled as a linear three-dimensional truss element in which L and cr are updated after each load increment is processed.

Anchorage slip loss. Anchorage slip loss in post- tensioned elements is due to the seating of anchor wedges when the jacking force is transferred to the anchor. The magnitude of this movement usually ranges between l/4 and 3/8 in. The magnitude of the loss of tensile force in the element due to this slip can be calculated as follows [7]:

AF=+EA. (13)

where AF is the loss of force in the post-tensioned element due to anchorage seating. A, is the magnitude of anchorage slip at transfer of load from jacking equipment to anchor wedges. E is the modulus of elasticity of the post-tensioned elements. I is the length of the post-tensioned element. A is the cross- sectional area of the post-tensioning element.

FINITE ELEMENT IMPLEMENTATION

The general theoretical foundation of this investigation is implemented in a special purpose finite element computer program, CSTAY [I 11. It is capable of performing a three-dimensional, geometrically nonlinear, time domain analysis of a prestressed, post-tensioned, box girder cable-stayed bridge. The structure may be modeled with a combination of beam, cable, prestressing and shell elements. The towers may be modeled using a combination of beam and prestressing elements. The box girder is modeled using a combination of prestressing and shell elements and is supported by cables connected to the tower. Each element has constant cross sectional properties over its length or surface. Nonprismatic portions of the structure may be modeled using a series of short prismatic elements with varying cross sectional properties.

The analysis is divided into two phases, the construction phase and the service life phase. During the


construction phase, successive segments of the box girder together with post-tensioning members and cables, are added to the existing bridge structure using the balanced cantilever construction method [12], as outlined at the beginning of this paper. Each of these successive additions to the bridge takes place during a specified time step. The analysis during the service life of the bridge takes place over a series of time steps. Time steps for both phases of the analysis are fully controlled by the analyst.

At the beginning of each time step of the construction phase, any element, concentrated joint force or

moment, prestressing loads, temperature changes and beam element concentrated and/or distributed loads may be added to the structure. Dead loads may also be automatically generated and combined with the above loads at the beginning of each time step. A full, geometric nonlinear, finite element analysis is performed, appropriate to these loads and the resulting

joint deflections are used to update the bridge geometry. At this point analysis for creep, shrinkage and aging commences over the specified time step. Creep and shrinkage are implemented using initial strain loads in a similar manner to that used for temperture loads. Aging is accounted for by modifying the modulus of elasticity as a function of time. This procedure is also a full geometric, nonlinear analysis followed by an update of the bridge geometry based on the resulting joint deflections. When the bridge construction phase is completed analysis for the service life phase commences. During this phase all of the above loads together with creep. shrinkage and aging may be implemented in the same fashion as during the construction phase. At the end of each time step during both the construction and service life phases the analyst has the option of adjusting the bridge’s camber by modifying cable and/or post-tensioning member axial loads. If this option is exercised a full geometric, nonlinear analysis is again performed followed by an update of the bridges geometry. This modification of bridge camber can be performed as many times as the analyst wishes at the end of each time step so as to fine tune the bridge profile before proceeding to analysis during the next time step.

The beam element used is the classical Bernoulli-Euler, three-dimensional, linear elastic formulation [13] with the stiffness matrix modified by the geometric stiffness matrix to account for the P-delta effect as shown in the following equation:

WTIR = PEIH + [kl, (14)

where [kTIS is the tangent stiffness matrix in local coordinates, [k& is the elastic stiffness in local coordinates and [liolR is the geometric stiffness in local coordinates.

This beam element may be subjected to concentrated or uniformly distributed member loads, a

temperature gradient and automatic generation of deadweight loads. In addition, the numerical implementation of creep strain in beams is accomplished with the application of the following general equations [2] during each time step:

Ait(t,- t, ,)= f c,*(t,_,)[l -_em”,-‘r 1’.“1] /= I

(154

c:(t,)=f,T(t, ,)e (‘,-‘, “‘.,+n,(t,~,)6a(t,~,)

(15b)

cf(to)=O, (15c)

where t, is the time in days at the end of the current time interval. 1, , is the time in days at the end of the previous time interval. At (t, - t,~ ,) is the creep strain increment over the current time interval. t:(t,_ ,) is

the “hidden” material value which stores the past stress history. r, is the “retardation time” which serves to adjust the creep curve so as to fit the AC1 committee 209 data or experimental values if available. u,(t,_ ,) is the pseudo-elastic modulus which is analogous to Young’s modulus. h(t, ,) is a step function change in stress at time t, ,

The specialization of eqn (15a-c) for beam axial loads takes the following form [14]:

(16a)

6: (to) = 0, (16~)

where AC,, is the axial strain due to creep. AN is the increment in axial load at the beginning of the current time interval. A, is the beam cross sectional

area. Equivalent joint loads due to the axial creep strain

in eqn (l6a) are found as follows [l4]:

AN’(c) = Acdt, - t, ,)A,E,(~, ,), (17)

where AN’ is the equivalent axial joint force. E, is the effective concrete modulus of elasticity at the beginning of the current time interval.

The specialization of eqn (I 5a-c) for beam moment loads takes the following form [14]:

h#~~,(t,-t[,_,)= 1 +z,(t, ,)[I -ee-“r ‘r I)‘,] ,=I

(18a)


+ai(L,)

4: (&I) = 0, (184

where A4M is the beam curvature due to creep. AM is the increment of moment at the beginning of the current time interval. I, is the moment of inertia of the beam.

Equivalent joint loads due to the axial creep strain in eqn (18a) are found as follows [14]:

where AM’ is the equivalent beam joint moment. The equivalent axial joint loads for shrinkage in

beams can be found in the same manner as those for temperature gradients. For adaptation to incremental time analysis, this requires a modification of eqns (7) and (8) as follows.

For concrete which has been moist cured for seven days:

A+,(t,) = & - -!!-%- , 35+t,_,

(E,,,)~. (20)

For concrete which has been steam cured for one to three days:

55+t, 55+t,_, (21)

The resulting equivalent axial joint loads are:

A ML,., = EA (k,,(fr)) (22)

A ML7.c = - EA(Ac,,(t,)). (23)

Equivalent joint loads at all remaining degrees of freedom are equal to zero.

The cable and prestressing elements used are based on the three-dimensional tangent stiffness formulation for the truss element. The response, under load, of a cable-stayed bridge is such that cable stiffness is affected by its sag, whereas prestressing elements do not experience sag. The basic tangent stifiness formulation, used for all three elements, is thus modified for cables to account for the effect of sag on cable stiffness by using the equivalent modulus of elasticity [lo] in the stiffness matrix.

The tangent stiffness matrix is based on the elastic stiffness matrix for a three-dimensional truss element [13] as modified by its geometric stiffness matrix [ 151 to account for the effect of large displacements as shown in eqn (24):

[kTl = IkEI + [&Ir (24)

CAS 58,2--H

where [kT] is the element tangent stiffness matrix in element coordinates. [kE] is the element elastic stiffness matrix in element coordinates. [kG] is the element geometric stiffness matrix in element coordinates.

This element may be loaded with axial preload and automatic generation of deadweight load. In addition a uniform temperature increase and anchorage slip loss may be accounted for.

The shell element is developed by superimposing a constant strain triangle [ 161 with the DKT [17-l 91 plate bending element. As such the membrane and bending stiffnesses are not coupled. The rotational stiffness about the normal to the surface of the element at each node is estimated as being equal to 1O-4 times the smallest bending stiffness at each respective node. This results in six degrees of freedom at each node and eliminates the possibility of singu- larities in the global stiffness matrix. Such singular- ities could occur in the event that all elements meeting at a node are coplanar, with the degree of freedom normal to their surfaces being parallel to one of the global coordinate axes. The element is explicitly integrated (i.e. numerical integration is not used) which gives exact results and, hence, is computationally efficient. Due to the element’s relative simplicity and proven accuracy, it serves as an excel- lent element for use in microcomputer finite element analyses. Loading options available for the element are dead weight and uniform temperature change. In addition, creep and shrinkage may be accounted for as follows.

The numerical implementation of creep strain in shell elements is accomplished by adapting equation (15ax) to the three-dimensional case of shell membrane stresses as follows:

+Q,(& - 1) [Cl Va(t, - , )I3 x , Wb)

[~:(kJ3 x I = 101, X I1 (25~)

where t, is the time in days at the end of the current time interval. t,_ , is the time in days at the end of the previous time interval. [Ac(t, - t,_ ,)I .is the matrix of creep strain increments over the current time interval. [c:(t,- ,)I is the matrix of “hidden” material values which stores the past stress history. r, is the “retardation” time which serves to adjust the creep curve SO as to fit the AC1 committee 209 creep data of experimental creep data if available. a,(t,_ ,) is the pseudo-elastic modulus which is analogous to Young’s modulus. [C] is the matrix shown below


commonly used in the strain-stress relationship of two-dimensional elements.

(26)

6a(r,_ ,) is the vector of normal stress increments in the element local coordinate x and J’ directions and the shear stress in the x-y plane.

Equivalent nodal loads [16] due to membrane creep strains in eqn (25a) are found from the following equation:

[PO16 x I = s [Bl~,,[El,.,[At(t,-t,. ,)l~.,dK Y

(27)

where [p,,] is the vector of equivalent joint loads. [B] is the strain displacement matrix. [E] is the matrix

relating stress and strain. For the constant strain triangle the terms in [B] are

all constant. For concrete, values for the variable v, which occurs in [E] vary between 0.15 and 0.25 and may be taken as 0.18 with negligible error. Also, the values for [AC] are constant. Consequently it is not necessary to integrate to find the equivalent joint loads and eqn (27) may be written as shown in eqn

(28):

l&16x I = [Bl~,z[El,.,[A~(~,--t,~,)l).,At, (28)

where A is the surface area of the element. t is the element thickness.

The equivalent nodal loads for shrinkage in shell elements can be found in the same manner as those for temperature changes.

SUMMARY

A finite element computer code, CSTAY [l I], is developed in ANSI FORTRAN 77 which will run on an IBM PC or 100% compatible with an 80386 or higher CPU. CSTAY consists of a main program and 54 subprograms and is capable of performing three- dimensional, time-dependent, geometrically nonlinear analyses of prestressed, box girder, cable-stayed

bridges. The finite element model may be constructed using

a combination of beam, cable, prestressing and shell elements. The beam, cable and prestressing elements are geometrically nonlinear in that large displacement effects are accounted for. The shell is a flat facet triangular element which is derived by combining a plate bending and membrane element. The membrane portion of the element is based on a constant strain formulation which requires a finer element mesh than would be required if a higher order element were used. This formulation was chosen in the interest of efficient microcomputer implementation.

It does not require the use of numerical integration to formulate its stiffness matrix which results in significant savings of CPU time. Also, since a fine mesh is required, the additional modifications of the element stiffness matrix which would normally be required to account for the P-delta effect, are not necessary. In essence, the P-delta effect is adequately accounted for by the combination of an incremental, iterative, geometrically nonlinear analysis and a fine mesh.

The general modeling approach is to use beam elements to represent the towers, cable elements to represent the cable-stays, shell elements to represent the box girders and prestressing elements as needed in both the towers and the girder. The entire structure, including all geometry, material properties, loads and necessary control parameters, is fully modeled before analysis is initiated. Each joint, joint restraint, element and load is associated with a time at which it becomes active and remains active for the rest of the analysis. These “activation times” correspond to times when new segments are added to the structure during the construction phase and when load changes are added during the service life phase. Once analysis commences the program automatically begins constructing the bridge as a balanced cantilever. Girder segments, along with the necessary cables and prestressing elements, are included or excluded dependidng on their desig- nated time of activation. The analyst has the option of interactively adjusting the cable-stays and post-tensioning elements at the end of each time so as to maintain a proper bridge profile (i.e. camber). The time-dependent effects of creep and shrinkage in the towers and girders and relaxation in the prestressing elements may be optionally

included.

SAMPLE ANALYSIS

The following sample analysis illustrates the effects of creep and shrinkage of concrete and relaxation of prestressing members on cross sectional stresses and deflections in a simplified representation of a segmentally erected, cable-stayed, concrete box girder bridge. In particular, the effects examined are girder deflections, forces in prestressing members and in-plane stresses in shell elements of the

girder.

GENERAL BRIDGE INFORMATION

The model is supported by six cables on each side which are attached to a centrally located tower as shown in Fig. 3a<

The model is symmetric about both the longitudinal centerline (plane X-J in Fig. 3c) and the transverse centerline (plane y-2 in Fig. 3a). This symmetry makes it possible to significantly reduce the size of the


finite element modei by using a quarter model bounded by the x-y longitudinal plane and the y-z

transverse plane. Symmetry boundary conditions are used at the node points located in these planes. Since the deflection of the tower has a negligible effect on the stress distribution due to shear lag in the girder cross section, the model was further simplified by eliminating the tower and attaching all cables at the top of the tower to a fixed point. Cables, prestressing and girder elements are modeled using the cable, prestressing and shell elements previously described.

Loads on each model consist of deadweight due to cable-stays, prestressing elements, the box girder superstructure, asphalt paving and guard rails. The live loads used are traffic lane loads. It is assumed that one third of this live load is applied as a sustained load over the life of the structure for purposes of creep analysis of the concrete and relaxation of the

prestressing elements. As mentioned above, this is a simplified analysis. The intent is not to analyze a full scale cable-stayed bridge, rather, it is to examine the effects of creep and shrinkage of concrete and relaxation of prestressing elements on the girder cross-sectional compressive stresses and deflections.

The following load cases are analyzed and com- pared so as to assess the effects of creep. shrinkage, and relaxation on the above mentioned stresses and

deflections:

(I) DL + SLL:

(2) DL+SLL+C+S+R;

where DL is dead load, SLL is the sustained live load (l/3 of total live load), C is creepage, S is shrinkage and R is the relaxation of prestressing and post-tensioning elements.

The analysis progresses through the construction phase, which is based on the balanced cantilever method, and the service life phase. The construction phase consists of progressively adding five equal length box girder segments to each side of the tower alternately so as to maintain balance. The first seg-

ment is placed 35 days after fabrication, which is one week after its 28 day curing-period has been reached. The addition of each segment is completed 14 days after the preceding segment has been positioned and all necessary cables and post-tensioning elements have been adjusted. It is assumed that the fabrication of each segment is completed 14 days after the fabrication of the segment preceding it and is com- pletely in place seven days after its 28 day curing- period has been reached. The time sequence for constructing the entire bridge is then 35, 49, 63, 77 and 91 days. The bridge is fully analyzed after the addition of each segment. The analysis accounts for all nonlinear and time-dependent effects of the load case being considered. During the construction phase the first load case consists of dead load due to the

superstructure and the second load case consists of

dead load due to the superstructure plus creep and shrinkage of the concrete and relaxation of the

prestressing elements. Once the service life phase has commenced, dead load due to asphalt and guard rails and the sustained portion of the live load are added to the above loads. Analysis continues from com- mencement of the service life phase at 91 days and accounts for all loads and nonlinear and time dependent effects for a total construction and service life phase of 1825 days (i.e. five years) at time periods of 105, 119, 133, 500 and 1825 days. These time steps were chosen so as to accurately step through the creep and shrinkage curves for concrete as shown in Fig. 4.

The model is configured for two traffic lanes and has the dimensions shown in Fig. 3a and b.

Post-tensioning elements are used at four points, A, B, C and D in Fig. 3b, where the webs intersect the flanges. In addition, prestressing elements are used in the top and bottom flanges, in both the transverse and longitudinal directions, so as to provide a uniform compressive stress of 6895 kPd.

This is about 55% of the maximum allowable stress of 12411 kPa which gives enough margin to prevent overstressing and, hence, violation of the 1241 ! kPa allowable stress and also enough margin to prevent reduction of the compressive stress to the point where the concrete would experience tensile loads and,

hence, cracking. Review of the output data for all time sequences

for each load case reveals that the most extensive effects of creep, shrinkage and relaxation exist upon completion of the last time step. The following sub- sections evaluate and compare the results from this last time step for longitudinal and transverse girder stresses, the forces in the longitudinal and transverse prestressing elements and the vertical deflections along the length of the bridge. Please note that in the following discussions with respect to stress distribution, comparisons between the analyses which do not consider the effects of creep. shrinkage and relaxation with those that do are made with respect to their variation from the design prestress value of

6895 kPa.

In -plane girder stresses in the longitudinal direction

In-plane longitudinal girder stresses are evaluated at locations where their values are both maximum and minimum, so as to provide an envelope of the stress levels in the bridge based on both load cases. The largest in-plane normal stresses were found to be in the top flange of the cross section at a distance of 23.47 m from the tower. The variation of stresses from the tip of the flange overhang (i.e. at origin of the “bridge half width” axis) to the centerline of the bridge at 4.57 m is shown in Fig. 5. The intersection of the web and flange is located at 1.2 m in this figure.

The maximum compressive longitudinal stresses are due to a combination of design prestress and positive bending of the girder which induces ad-


Y

17.37 m

[email protected] m=30.5m [email protected]=30.5m

(cl Y

(b)

230111"

460 mm

23Omm

/’ . . cables

Fig. 3. (a) Overall bridge dimensions; (b) girder cross sectional dimensions; (c) bridge cross section.

ditional compressive load in the flange. These maximum stresses, as shown in Fig. 5, indicate that for both load cases shear lag alters the stress distribution over the girder cross section. Shear lag is due to the variation of cross sectional stiffness between the tip of the overhanging flange and the bridge centerline. For the load case which does not account for creep and shrinkage of concrete and relaxation of prestressing elements (Fig. S), the stress at the tip of the flange overhang is reduced by 20%, at the intersection of the web and flange the stress is increased by 9% and at the centerline of the bridge the stress is reduced by 6%. For the load case which includes creep, shrinkage and relaxation (Fig. 5), there is a significant overall stress reduction of 45% at the tip of the overhanging flange, 7% at the intersection of the web and flange and 21% at the centerline of the bridge. Comparison of these two load cases shows that creep, shrinkage and relaxation has an overall positive effect by reducing maximum longitudinal girder stresses relative to the load case which does not consider creep, shrinkage and relaxation by an additional 25%, 16% and 17% at the tip of the overhanging flange, intersection of web and flange and bridge centerline, respectively.

The minimum in-plane normal stresses were found to be in the top flange of the cross section at a distance of 13.1 m from the tower. The variations of stresses from the tip of the flange overhang to the centerline of the bridge are shown for both load cases in Fig. 6.

The minimum compressive longitudinal stresses are due to a combination of design prestress and negative bending of the girder, which tends to induce a tensile load in the top flange. These minimum stresses, as shown in Fig. 6, indicate that for both load cases

- creep

shrinkage

r > 80 160 240 320 400 480 560 ."' 1825

Twae(days)

Fig. 4. Standard creep and shrinkage vs time curves.

Analysis of concrete cable-stayed bridges 34-l

10500

- - - Without creep, shrinkage & reiasation

- - With creep, shrinkage &

04

0 1219 2337 3454

Bridge Half Width (mm)

Fig. 5. Maximum longitudinal compressive stresses.

I

4572

shear lag alters the stress distribution over the girder cross section.

For the load case which does not account for creep and shrinkage of concrete and relaxation of prestressing elements (Fig. 6), the stress at the tip of the flange overhang is reduced by 20%, at the intersection of the web and flange the stress is reduced by 40% and at the centerline of the bridge the stress is reduced by 25%. For the load case which includes creep, shrinkage and relaxation (Fig. 6), the overall stress reduction is not much different from that for the case without creep, shrinkage and relaxation. Figure 6 shows a stress reduction of 15% at the tip of the overhanging flange, 38% at the intersection of the web and flange and 21% at the centerline of the

bridge. Comparison of these two load cases shows that creep, shrinkage and relaxation has a very small overall effect by increasing maximum longitudinal girder stresses relative to the load case which does not consider creep, shrinkage and relaxation by an additional 5, 2 and 4% at the tip of the overhanging flange, intersection of web and flange, and bridge centerline, respectively.

In-plane girder stresses in the transverse direction

Transverse stress variations tended to be very similar over the length of each segment supported by cable-stays. The stresses in segment no. 1, beginning at the tower and extending out 6.1 m, were typical of the stresses in the remaining segments and are shown

Ol

0 1219 2337

77” -. - - - -_- ” .- _y. =

- - - Without creep,

- - With creep,

3154 4572

Bridge Half Width (mm)

Fig. 6. Minimum longitudinal compressive stresses.


10500

0

c *.

c a . -*

c - -

_ Y.‘\ 0 . .“,“.._---**

_ . - - I \‘y

_*.---

r, \ . . .._----

-Design prestress I I - Without creep,

shrinkage & relaxation tl, - - With creep, shrinkage & relaxation

I

0 1016 2032 30-18 -106-I 5080 6096

Length of Segment No. 1 along bridge centerline (mm)

Fig. 7. Maximum transverse compressive stresses.

in Fig, 7. These stresses are in-plane stresses in the top flange along the horizontal centerline of the bridge.

As was done for the longitudinal stresses, the transverse stresses for both load cases are evaluated using the design prestress of 6895 kPa as a basis. For the load case which does not account for creep, shrinkage and relaxation (Fig. 7) the stress varies by an increase of 2% at the beginning of the segment to a maximum increase of 30% near the cross section where the cable is attached and then drops to an increase of 4% at the end of the segment. The 30% jump in stress is due to the fact that the cable load is inducing a large transverse bending stress at this point. For the load case which accounts for creep, shrinkage and relaxation (Fig. 7), the stress varies from a decrease of 17% at the beginning of the segment to a maximum increase of 19% near the

cross section where the cable is attached and then

drops to a decrease of 8% at the end of the segment. Comparison of these two load cases shows that creep, shrinkage and relaxation has an overall positive effect by reducing maximum transverse girder stresses relative to the load case which does not consider creep, shrinkage and relaxation by 19, 11 and 12% at the beginning, near the cable attachment and at the end of the segment, respectively.

Vertical dq4ections along the longitudinal centerline

The vertical deflections along the longitudinal centerline for both load cases are shown in Fig. 8. Note that cable-stays are attached at 0,6.1, 12.2 and 18.3 m along the outer edge of the top flange.

A comparison of the vertical deflections between

the load case which does not account for creep, shrinkage and relaxation and the load case which

0 4064 8128 17192 16256 20370 24384 28448

‘5

i _,:

-51 I

Bridge Length Along Centerline (mm)

Fig. 8. Vertical deflections along centerline.


does (Fig. 8) indicates that deflections for creep, shrinkage and relaxation considered are significantly larger with a maximum increase of about 66% at the

bridge tower.

Forces in the longitudinal pretensioning elements

Forces in the longitudinal pretensioning elements were applied so as to induce a uniform compressive stress in the concrete cross section of 6895 kPa. Comparisons between the two load cases shows that

for the load case which does not account for creep, shrinkage and relaxation the loss of pretension varies between 4 and I I % of the design pretension. For the load case which accounts for creep, shrinkage and

relaxation the loss of pretension varies between 15 and 29% of the design pretension. The largest percent loss in pretension for both load cases was at the tip of the overhanging flange and the centerline of the bridge, whereas the smallest percent loss was in the vicinity of the web to flange intersection. The pattern of reduction in pretension is consistent with the maximum longitudinal compressive stress distri-

bution which is smallest at the tip of the flange and centerline of the bridge cross section and largest at the intersection of the web to flange intersection.

Forces in transverse pretensioning elements

Forces in the transverse pretensioning elements were calculated so as to induce a uniform compressive stress in the concrete cross section of 6895 kPa. Comparisons between the two load cases show that for the load case which does not account for creep, shrinkage and relaxation the loss of pretension varies between 4 and I I % of the design pretension. For the load case which accounts for creep, shrinkage and relaxation the loss of pretension varies between I7 and 38% of the design pretension. The largest percent loss in pretension for both load cases was at the cross section where the cable-stays are attached whereas the smallest percent losses were at locations between

cable-stays. The cable-stays induce a large transverse compressive force in the top flange of the girder which increases elastic and creep deflections in this region resulting in a larger loss of pretension than at locations of the cross section away from the cable- stays.

Examination of the above mentioned figures shows that creep, shrinkage and relaxation tend to shift stresses back to the initial prestress levels which is desirable from a design perspective in that it gives a designer more control over the design process. It can also be concluded that flange thickness may be reduced in regions where bending is such that additional compressive stress is induced in the flange resulting in maximum stresses being reduced due to creep, shrinkage and relaxation. By the same token. flange thicknesses should be increased in regions where bending is such that compressive stress is reduced in the flange, resulting in minimum stresses

being increased due to creep, shrinkage and relax-

ation. It should be noted here that the viability of the

analysis presented in this paper is dependent on a multitude of assumptions made both by the authors of this paper and the developers of the various theoretical principles incorporated in the computer program, CSTAY. One area of improvement in the program would be the use of true catenary elements to represent the cables instead of truss elements with an equivalent modulus of elasticity. Another area of improvement lies in the development of more accu- rate creep models.

REFERENCES

1. V. Kristek and Z. P. Bazant, Shear lag effect and uncertainty in concrete box girder creep. J. struct. Eng ASCE 113, 557-574 (1987).

2. Z. P. Bazant (Ed.), Maihemai~caf Modefling of Creep and Shrinkage of Concrere. Wiley, New York (1988).

3. Prediction of creep, shrinkage and temperature effects in concrete struciures. Amirican Concrete Institute, AC1 Committee 209, Reoort no. AC1 209R-82 (1982).

4. M. K. Tadros, A. Ghali and W. H. Dilger, Time deoendent analvses of comoosite frames. J. sfrucl. Di&

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AiCE 103(ST4), 871-884 il977). Z. P. Bazant and S. T. Wu, Dirichlet series creep function for aging concrete. J. Engng Mech. Div. ASCE 99(EM2), 367 -387 (1973). M. A. Ketchum and A. C. Scordelis, Redistribution of stresses in segmentally erected prestressed concrete bridges. Report no. UCB/SESM-86/07, University of California, Berkeley, CA (1986). E. G. Nawy, Presstressed Concrete, a Fundamenial Aooroach. Prentice-Hall, Englewood Cliffs, NJ (1989). C‘.‘Oran. Tangent stiffness L space frames. J. ‘struck. Die. AXE 99(ST6), 987~1001 (1973). D. W. Murray and E. L. Wilson, Finite-element large deflection analysis of plates. J. Engng Mech. Die. ASCE 95(EMI), 143-165 (1969). M. S. Troitskv. Cable-Staved Bridpes. 2nd Edn. Van Nostrand Reinhold, New ?ork (1988). N. C. Cluley, The effect of time-dependent phenomena on force redistribution in segmentally erected, cable- stayed, concrete bridges. Ph.D. dissertation (unpub- lished). University of California, Irvine, CA (1993). F. Harris, Modem Canstrucrion Equipment and Method.?. Longman Scientific, and Technical, Harlow (Copublished in the U.S. with Wiley, New York (1989). W. Weaver and J. M. Gere, Marrh Analysis af Framed Strucrures, 2nd Edn. Van Nostrand Reinhold, New York (1980). Young-Jin Kang and A. C. Scordelis, Nonlinear analysis of prestressed concrete frames. J. Struct. Dir. ASCE 106(ST2), 445462 (1980). K. J. Bathe, Finife Elemenr Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ (1982). W. Weaver and P. R. Johnston, Finite Elements for Structural Analysis. Prentice-Hall, Englewood Cliffs, NJ (1984). J. L. Batoz, K. I. Bathe and L. W. Ho, A study of three-node triangular plate bending elements. In/. J. mtmer. Me&. En&g I& 1771-1812<1980). K. J. Bathe and L. W. Ho, A simple and effective element for analysis of general shell structures. Compur. Struct. 13, 673-681 (1981).


19. J. L. Batoz, An explicit formulation for an efficient solution schemes for nonlinear structures. Compu~. triangular plate-bending element. Int. J. numer. Meth. Struct. 9, 223-236 (1978). Engng 18, 1077T1089 (1982). 21. W. F. Chen and D. J. Han, Plasticity for Structural

20. D. P. Mondkar and G. H. Powell, Evaluation of Engineers. Springer, Berlin (1988).

analysis of concrete cable-stayed bridges for creep, shrinkage

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