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Engineering Structures 33 (2011) 118126

Contents lists available at ScienceDirect

Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Accurate time-dependent analysis of concrete bridges considering concretecreep, concrete shrinkage and cable relaxation

Francis T.K. Au , X.T. SiDepartment of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong

a r t i c l e i n f o

Article history:

Received 12 January 2010

Received in revised form

20 September 2010

Accepted 26 September 2010

Available online 25 October 2010

Keywords:

Concrete bridges

Creep

Relaxation

Shrinkage

Stress relaxation model

Time-dependent behaviour

Time integration method

a b s t r a c t

This paper proposes a new relaxation model for steel tendons based on the equivalent creep coefficientto enable the accurate estimation of losses of cable forces. The equivalent creep coefficient worksnot only in the case of intrinsic relaxation but also under various boundary conditions. Based on theproposed relaxation model, an accurate finite element analysis of the time-dependent behavior ofconcrete bridges considering concrete creep, concrete shrinkage and cable relaxation is devised basedon the time integration method. Concrete members are modeled by beam elements while tendons aremodeled by truss elements with nodes connected to the beam axis by perpendicular rigid arms. Then theindividual and combined effects of concrete creep, concrete shrinkage and cable relaxation on the long-term performance of concrete structures are investigated. It is found that the proposed relaxation modeland time integration method can provide a reliable method for time-dependent analysis. The numericalresults obtained indicate that the interactions among these factors should be considered carefully inanalyzing the long-term performance of concrete bridges.

2010 Elsevier Ltd. All rights reserved.

1. Introduction

A long time has elapsed since the first observation of concreteshrinkage and the discovery of concrete creep [1]. Since then, en-gineers and researchers have been aware of the time-dependentbehavior of concrete structures [24], and steelconcrete com-posite structures [5,6] because the creep and shrinkage ofconcrete and the relaxation of cables interact with one anotherduring the construction and service stages, resulting in additionaldeflections, cracking, reduction of prestress and redistribution ofinternal forces, which in turn affect the long-term structural per-formance. Therefore the time-dependent effects are significant

considerations for serviceability limit states under which deflec-tions, stresses and crack widths should be limited [710].

A reliable method for time-dependent analysis of concretestructures is the finite element method combined with timeintegration [11]. In the method, the concrete members are usuallymodeled by frame elements and the tendons are treated astruss elements connected to the structural nodes with rigid arms[1215]. In the time-dependent analysis of prestressed concretestructures, the interactions among concrete creep, concreteshrinkage and cable relaxation are often considered approximatelyby introducing relaxation reduction coefficients taken from chartsor tables [1618]. It is therefore desirable to develop better

Corresponding author. Tel.: +852 2859 2650; fax: +852 2559 5337.E-mail address: [email protected] (F.T.K. Au).

methods for modeling the interaction between cable relaxationand other sources of time-dependent deformations.

This paper first introduces a convenient technique to modelcurved tendons using rigid arms. Then a new equivalent stressrelaxation model for steel tendons is developed. An accurate finiteelement method is then devised for accurate time-dependentanalysis of concrete structures with time integration taking intoaccount all three time-varying factors. Numerical investigationsare then carried out to demonstrate the versatility of the proposedmethod.

2. Modeling of tendons using rigid arms connected to a beam

The common assumptions for analysis of concrete girderbridges are made. The concrete, steel reinforcement and steelcables are modeled separately, ignoring their interaction withinthe element. It is also assumed that plane sections remain planeafter bending. The present analysis is mainly for the time-dependent behavior under working conditions before any cracksare formed. The tendons are assumed to be perfectly bonded tothe structuralconcrete members. Each tendon of an arbitraryshapeis modeled by a series of truss elements with nodes connected tothe beam axis by perpendicular rigid arms [12,14]. However, toavoid an unduly large number of degrees of freedom (DOFs) forcurved tendons, the rigid arms are modeled by the masterslavetransformation technique rather than using physical nodes.

Fig. 1 shows a spatial beam elements 12 of length Lc alignedwith the beam axis and a spatial tendon element ab of true

0141-0296/$ see front matter 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engstruct.2010.09.024

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F.T.K. Au, X.T. Si / Engineering Structures 33 (2011) 118126 119

Fig. 1. Beam element with tendon element.

length ls and projected length Lc on the beam. Note that the beamelement and tendon element need not be in the same plane. A

right-handed local xyz-coordinate axis system is chosen with the

x-axis coincident with the member axis having the origin at node

1. Each node of the beam has three translational DOFs u, v and w

along the local x-, y- and z-axes respectively, and three rotationalDOFs x, y and z about the local x-, y- and z-axes respectively. In

theglobal matrices, these DOFs areappendedby thenode numbers

as appropriate. Similarly each end node of tendon element ab hastranslational DOFs u, v andw along thelocalx-,y-andz-axes of the

beam element. Nodes a and b of the tendon element are treated

as slave nodes connected to master nodes 1 and 2 of the beam

element by rigid links a-1 and b-2 perpendicular to the x-axis ofthe beam.

The nodal displacement vectors {ab} of the tendon element ab

and {12} of the beam element 12 can be expressed respectivelyas Eqs. (1) and (2) are given in Box I.

The displacement vector {ab} is related to {12} by the transfer

matrix [H] [19] by

{ab} =

[A 00 B

]{12} = [H] {12} (3)

where the sub-matrices [A] and [B] are given explicitly by

[A] =

1 0 0 0 za z1 ya + y10 1 0 za + z1 0 00 0 1 ya y1 0 0

(4)

[B] =

1 0 0 0 zb z2 yb + y20 1 0 zb + z2 0 00 0 1 yb y2 0 0

(5)

in terms of the coordinates (x1,y1,z1), (x2,y2,z2), (xa,ya,za) and

(xb,yb,zb) of nodes 1, 2, a and b respectively in accordance withthe local x-, y- and z-axes of the beam element.

The stiffness matrix [k]t of the tendon element with respect tothe DOFs of the beam element can therefore be expressed as

[k]t = [H]T [T]T [k]s [T] [H] (6)

in terms of the local stiffness matrix [k] s of the tendon and the

transformation matrix [T] which comprises the direction cosines

of element ab in the local xyz-coordinate axis system. The load

vector {F}t of the tendon element with respect to the DOFs of thebeam elementcan be similarly expressed in terms of the local load

vector {F}s as

{F}t = [H]T [T]T {F}s. (7)

By adopting the masterslave technique, the total number of DOFswill be kept to a reasonable minimum.

3. A relaxation model for steel tendon

3.1. Intrinsic stress relaxation functions

When a tendon is stretched and fixed at two ends at a constant

distance apart while keeping the temperature unchanged, the lossof thetension in thetendon is referred to as theintrinsic relaxation.For convenience in analysis of prestressing losses, it is desirable to

express the stress relaxation pr as a function of time t (hours)since stressing and initial stress p0. An equation to evaluate thestress relaxation for stress-relieved strands as proposed by Maguraet al. [20] is

pr

p0=

log(t)

10

p0

fpy 0.55

(8)

where fpy is taken as the yield stress, which is defined arbitrarilyas the stress at a strain of 0.01. For low relaxation strands, theequation commonly used in North America [21] is

pr

p0=

log(24t)

40

p0

fpy 0.55

. (9)

Note that both Eqs. (8) and (9) are on the assumption thatrelaxation is negligible for initial prestressing ratio p0/fpy notexceeding 55%.

In a prestressed concrete member, creep and shrinkage tend

to reduce the member length thereby reducing the tendon stressand relaxation. To deal with it, most engineers have resorted toapproximate methods such as relaxation factors of BS8110 [22]andrelaxation reduction coefficients [11]. It is therefore desirable to

develop a more rational method to analyze the time-dependentbehavior of concrete structures considering the interactions

among concrete creep, concrete shrinkage and cable relaxation.

3.2. Equivalent creep coefficients of steel tendons

When a tendon is stretched under a constant stress s(t0)applied at time t0, the total strain s(t) at time t, including theinstantaneous andcreepstrain,is expressedin terms of theYoungs

modulus of elasticity Es and creep coefficient s(t, t0) as

s(t) =s(t0)

Es[1 + s(t, t0)] . (10)

The creep coefficient s(t, t0) also depends on the initial pre-stressing ratio s(t0)/fpy strictly speaking, but it is omitted for

convenience. With the applied stress changing with time, the to-tal strain of steel tendon s(t) is obtained by summation of re-

sponses of ds() applied at time based on the principle ofsuperposition

s(t) =s(t0)

Es[1 + s(t, t0)] +

s(t)0

1 + s(t, )

Esds () (11)

where s(t) is the stress increment from time t0 to t. If the creep

behavior of tendon is taken to be independent of age, which isa reasonable assumption, one can introduce the creep coefficients(t t0) = s(t, t0), so that Eq. (11) can be rewritten as

s(t) =s(t0)

Es[1 + s(t t0)]

+

s(t)0

1 + s(t )

Esds () . (12)

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120 F.T.K. Au, X.T. Si / Engineering Structures 33 (2011) 118126

{ab} =

ab

=

ua va wa ub vb wbT

(1)

{12} = 12 = u1 v1 w1 x1 y1 z1 u2 v2 w2 x2 y2 z2

T(2)

Box I.

Using Eq. (12) to write down s(t + t) explicitly and getting the

difference, the incremental strain s(t) can be expressed in termsof the incrementalstresss(t), mean modulus of elasticity Es(t)and incremental creep strain s(t) as

s(t) =s(t)

Es(t)+ s(t) (13)

where

s(t) = s(t + t) s(t)

s(t) = s(t + t) s(t)

Es(t) =Es

1 + s(t)/2

s =s(t0)

Es[s(t + t t0) s(t t0)]

+

s(t)s(t0)

s(t + t ) s(t )

Es

ds()

dd .

For the case of intrinsic relaxation at constant strain, there isno change in strain with time, i.e. s(t) = 0. Let t0 denote the

time of initial stressing and t be the time interval to define thesubsequent instants t1 = t0 + t, . . . , tn = t0 + nt. Applying

Eq. (13) to time t = t0 and noting that s(t) = 0, one gets upon

rearrangement

s(t) =s(t0)

s(t0) + s(t0)/2. (14)

Repeating the same procedure, one can derive thecreep coefficient

s [(n + 1)t] at time tn in terms of the previous values of creep

coefficient as

s [(n + 1)t] =s(t0)s(nt)

s(t0) + s(t0)/2

s(t0 + nt) [1 + s(t)/2] s(t0)s [(n 1)t] /2

s(t0) + s(t0)/2

ni=2

s [t0+(i1)t]

2{s [(n i + 2)t] s [(n i)t]}

s(t0) + s(t0)/2. (15)

In other words, the equivalent creep coefficient can be computedbased on the intrinsic stress relaxation function.

4. Time-dependent analysis of prestressed concrete bridges

In order to predict the time-dependent behavior of prestressed

concrete bridges, the proposed stress relaxation model for tendonstogether with the creep and shrinkage models for concrete can be

implemented by the finite element method using time integration.

The concrete members are modeled by beam elements whilethe tendons are modeled by tendon elements connected to the

beam axis by perpendicular rigid arms. The method also provides

benchmark solutions for other simplified methods.

To facilitate subsequent formulation for finite element analysisusing time integration, Eq. (13) for the tendon element for the timeinterval from t to (t + t) is rewritten to give the incrementalstress s(t) in terms of the mean modulus of elasticity Es(t),the incremental steel strains(t) and incremental creep strain ofthe tendon s(t) as

s(t) = Es(t)s(t) s(t)

. (16)

The strain vector {}s of the tendon element can be related tothe local displacement vector {u}s by the strain matrix [B]s as

{}s =

du

dx

s

= [B]s {u}s =

1/ls 1/ls u1

u2

s

(17)

where ls is the length of tendon element. Following the con-ventional formulation of finite element method and neglect-ing body forces, the incremental nodal force vector {qe}s =f1 f2

Ts

of the tendon element can be obtained by integrat-ing over the volume Ve and expressed asqe

s=

Ve

[B]Ts {}s dV

=

Ve

[B]Ts

Es(t)

[B]s dV

{u}s

Ve

[B]T

sEs(t)sdV

qe

s= [k]s {u}s + {f}s (18)

in terms of the stiffness matrix [k]s, incremental displacementvector {u}s and incremental load vector due to tendon creep{f}s. Note that the variable t has been omitted for brevity

hereafter. The stiffness matrix [k]s is given by

[k]s =Es(t)As

ls

[1 1

1 1

](19)

where As is cross sectional area. The incremental load vector dueto cable relaxation {f}s is given by

{f}s =Es(t)

Es

[s(t + t t0) s(t t0)]

Ns(t0)

Ns(t0)

+

tt0

[s(t + t ) s(t )]

Ns( )

Ns()

d

(20)

where Ns(t0) is the axial force of the element at time t0 andNs()

is the derivative of the axial force with respect to the dummy timevariable .

Similarly neglecting body forces, the incremental load vector of

a concrete beam element {qe}c =f1 f2 f12

Tc

canbe obtained as

qec = [k]c {}c + {f}c + {f}cs (21)in terms of the beam stiffness matrix [k]c, incremental displace-ment vector {}c, incremental load vector due to concrete creep{f}c and incremental load vector due to concrete shrinkage

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[k]c =

[[k11]c [k12]c[k21]c [k22]c

](22a)

[k11]c =

AcEc

Lc0 0 0 0 0

012EcIcz

L3c0 0 0

6EcIcz

L2c

0 012EcIcy

L3c0

6EcIcy

L2c0

0 0 0GcJc

Lc0 0

0 0 6EcIcy

L2c0

4EcIcy

Lc0

06EcIcz

L2c0 0 0

4EcIcz

Lc

(22b)

[k21]c = [k12]Tc =

AcEc

Lc0 0 0 0 0

0 12EcIcz

L3c0 0 0

6EcIcz

L2c

0 0 12EcIcy

L3c0

6EcIcy

L2c0

0 0 0 GcJc

Lc0 0

0 0 6EcIcy

L2c0

2EcIcy

Lc0

0

6EcIcz

L2c0 0 0

2EcIcz

Lc

(22c)

[k22]c =

AcEc

Lc0 0 0 0 0

012EcIcz

L3c0 0 0

6EcIcz

L2c

0 012EcIcy

L3c0

6EcIcy

L2c0

0 0 0GcJc

Lc0 0

0 06EcIcy

L2c0

4EcIcy

Lc0

0 6EcIcz

L2c0 0 0

4EcIcz

Lc

(22d)

Box II.

{f}cs. The stiffness matrix [k]c is given in Box II, where Ec is the

mean modulus of elasticity of concrete in the time step concerned

given as

Ec(t) =[Ec(t) + Ec(t + t)] /2

1 + c(t + t, t)/2. (23)

Gc is the mean shear modulus given in terms of the Poissons ratio

c as

Gc(t) =Ec(t)

2(1 + c). (24)

Lc is the length of element, Ac is the cross sectional area, Icyand Icz are the second moments of area about the y- and z-axesrespectively, and Jc is the torsional constant. In accordance withthe approach adopted by Au et al. [15], the axial force and bendingmoments in a member with member loads are decomposedinto time-dependent components (governed by overall time-dependent deformations) and time-independent components(governed by member loading). The incremental load vector due

to creep {f}c is given by

{f}c =

{f1}c{f2}c

(25a)

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{f1}c = Ec(t)

[c(t + t, t0) c(t, t0)

Ec(t0)

]

Nc(t0)

Mc1(t0) + Mc2(t0)

/Lc

Mc3(t0) +

Mc4(t0)

/LcTc(t0)

Mc3(t0)

Mc1(t0)

+

10

Nc0(x)6(2x 1)Mc0(x)/Lc6(2x 1)Mc0(x)/Lc

Tc0(x)2(3x 2)Mc0(x)/Lc2(3x 2)Mc0(x)/Lc

dx

Ec(t)

tt0

c(t + t, ) c(t, )

Ec( )

Nc() Mc1() +

Mc2()/Lc

Mc3() +Mc4()

/Lc

Tc()

Mc3()

Mc1()

d (25b)

{f2}c = Ec(t)

[c(t + t, t0) c(t, t0)

Ec(t0)

]

Nc(t0)Mc1(t0) Mc2(t0)

/Lc

Mc3(t0) Mc4(t0) /LcTc(t0)Mc4(t0)

Mc2(t0)

+

10

Nc0(x)6(1 2x)Mc0(x)/Lc6(1 2x)Mc0(x)/Lc

Tc0(x)2(3x 1)Mc0(x)/Lc2(3x 1)Mc0(x)/Lc

dx

Ec(t)

tt0

c(t + t, ) c(t, )

Ec( )

Nc()Mc1() Mc2()

/Lc

Mc3() Mc4()

/Lc

Tc()Mc4( )Mc2( )

d (25c)

where the location of a point in the element is defined by x =x/Lc, Nc(t0) is the axial force at time t0, Tc(t0) is the torsionalmoment at time t0, Mc1(t0) and Mc2(t0) are the bending momentsabout the z-axis of element at time t0 (t0 t) at the ends x = 0and x = 1 respectively, and Mc3(t0) and Mc4(t0) are the correspon-ding end moments about the y-axis, and the over-dot denotesdifferentiation with respect to the dummy time variable . The

time-independent forces Nc0(x) and Tc0(x) can be calculated fromapplied loads assuming both ends of the element to be fixed,while the time-independent moment Mc0(x) can be calculatedfrom applied loads assuming the element to be simply supported.

If all loads act at the nodes only, the integral in Eq. (25) involvingthe time-independent terms will disappear. The incremental loadvector due to concrete shrinkage is given by

{f}cs = Ec(t)Ac [cs(t + t, t0) cs(t, t0)]

1 0 0 0 0 0 1 0 0 0 0 0T

(26)

where cs is the concrete shrinkage strain from time t0 to thespecified time.

In each time interval, the incremental load vector for a beam

element with a tendon {qe}bt =f1 f2 f12

Tbt

canbe written asqe

bt

=

[k]c + [k]t

{}c + {f}c

+ {f}s + [N]T [H]T [T]T {f}s (27)

where [k]t is the stiffness matrix of the tendon element based onthe mean elastic modulus Es(t), namely

[k]t = [H]T [T]T [k]s [T] [H] . (28)

Before assembling these local stiffness matrices and loadvectors into the corresponding global matrices and vectors forsolution,they must be transformed to theglobal coordinate systemin the usual manner. Sufficiently fine time steps have to be usedin the time integration in order to obtain accurate results. Elasticanalysis fortime t0 is first performedto getthe initial deformationsand internal forces. The procedures for the subsequent time-dependent analysis using time integration for each of the timeintervals from t0 to the time of interest are summarized below.

(a) Estimate or update the mean moduli of the concrete andtendon elements.

(b) Based on the updated mean moduli, calculatethe local stiffnessmatrices and incremental load vector due to concrete creep,

concrete shrinkage and cable relaxation.(c) Transform and assemble the local stiffness matrices and the lo-

cal incremental load vectors to form the global stiffness matrixK

and global incremental load vector {Q} respectively.

(d) Solve the equation

K

{} = {Q} to obtain the incrementaldisplacement vector {}.

(e) From the incremental element load vectors estimated from theincremental displacement vector {}, calculate the total loadvector and nodal displacements.

(f) Repeat Steps (a)(e) until the time of interest.

5. Case studies

5.1. Simulation of creep coefficients of steel tendons

Define the initial prestressing ratio for a prestressed tendon interms of initial prestress s(t0) and yield stress of the tendon fpyas

R = s(t0)/fpy. (29)

Consider a prestressing tendon of unit length fixed at two ends.Based on the stress relaxation functions shown in Eqs. (8) and (9),the creep coefficients and stress relaxation of a stress-relieved anda low relaxation tendon under various initial prestressing ratios Rare simulated respectively for 10 years. The time interval adoptedis 8.76 h, which is one thousandth of one year, while the yieldstress is 1670 MPa.

It is observed that the variations of the creep coefficient shown

in Fig. 2(a) are largely similar to the variations of percentagestress relaxation shown in Fig. 2(b). Both the creep coefficientand cable relaxation are higher for higher initial prestressingratios. Both of them increase rapidly at first and tend to stabilize

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(a) Equ ivalent creep c oefficients. (b) Simu lated stress relaxation (markers) and assumed stress

relaxation (curves).

Fig. 2. Computed properties of a typical stress relieved tendon.

Fig. 3. Comparison of computed properties of stress relieved and low relaxation

tendons with R = 0.8.

ls

Fig. 4. A prestressing tendon connected to a perfectly elastic spring.

after a few years. To verify that the equivalent creep coefficientworks, the time-dependent behavior of the tendon of unit lengthfixed at two ends is solved by time integration based on theequivalent creep coefficients. There is perfect agreement between

the simulated percentage stress relaxation shown as markers inFig. 2(b) and the assumed relaxation functions shown as curves.The equivalent creep coefficients and percentage stress relaxationfor low relaxation tendons and stress-relieved tendons for initialprestressing ratio R = 0.8 are similarly worked out and shownin Fig. 3. In particular, the simulated stress relaxation results areshown as markers while the assumed stress relaxation functionsare shown as curves. Apart from verifying again the validity of theequivalent creep coefficient, the superiority of the low relaxationtendons is clearly seen.

5.2. A prestressing tendon connected to a perfectly elastic spring

In order to evaluate whether a time-dependent structural

model works in structural analysis, a common test problem isa model with partial fixity [23]. Fig. 4 shows a stress-relievedprestressing tendon of length ls fixed at the left end and connectedto a perfectly elastic spring of stiffness K at the right end. In

particular, the tendon-spring assembly is fixed at the left end first

and then tensioned at the right end. When the initial prestressingratio reaches R = 0.7, the right end of the spring is fixed. Therelevant data of the tendon include: sectional area As = 924 106 m2, Youngs modulus Es = 210 GPa, yield stress fpy =1670 MPa and length ls = 1 m. The time span investigated is 10years and the time interval for analysis is 8.76 h. The stiffness ratioRK is defined as the ratio of spring stiffness K to that of the tendonKt as

RK =K

Kt=

Kls

EsAs. (30)

The cases studied cover various values of stiffness ratio,i.e. RK = 0,0.5, 1, 10, 100 and .

Fig. 5(a) shows the percentage increase in tendon strain while

Fig. 5(b) shows that percentage stress relaxation in the tendon. ForRK = 0 which corresponds to an infinitely soft spring, the curve inFig. 5(a) is identical to the creep coefficient for the case R = 0.7shown in Fig. 2(a), while there is no stress relaxation as shownin Fig. 5(b). Fig. 5(b) also shows that the stress relaxation in thetendon generally increases with the spring stiffness. For RK = which means there is no change in strain, the curve in Fig. 5(b) isidentical to the stress relaxation for the case R = 0.7 as shown inFig. 2(b). The results show that this relaxation model works undervarious boundary conditions.

5.3. A simply supported prestressed concrete beam with a parabolic

tendon

Fig. 6 shows a prestressed concrete beam of negligible weighthaving a section of breadth b = 0.6 m and depth h = 1.2 m,whichis simply supported over a span of 40 m. The characteristiccompressive strength of concrete is fck = 32 MPa. Wet curing iscarried out until Ts = 3 days after which shrinkage begins. Therelative humidityis taken as 80%throughout.At theage ofTc0 = 28days, the beam is post-tensioned to an initial prestressing forceof P0 = 1108 kN with a stress-relieved prestressing tendon ofcross sectional area As = 924 10

6 m2 and Youngs modulusEs = 200 GPa. The initial prestressing ratio is R = 0.8, which isslightly on the high side for the study of relaxation. The tendonprofile is parabolic with a maximum eccentricity of 0.3 m at mid-span and zero end eccentricities. The parameters of CEB-FIP ModelCode 1990 [24] areadopted in thecalculation. Thebeam is modeled

by 20 identical beam elements with tendon elements connectedto the beam axis using the masterslave technique. Analysis ofthe time-dependent behavior is carried out by time integrationtaking into account the interaction among concrete creep,concrete

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(a) Percentage increase in tendon strain. (b) Percentage stress relaxation.

Fig. 5. Time-dependent behavior of a tendon connected to a perfectly elastic spring.

Fig. 6. A simply supported beam with a parabolic tendon (not to scale).

Fig. 7. Time-dependent deformation of simply supported beam.

shrinkage and cable relaxation for the period from t0 = 28 days tot = 365daysby 2000 equal time steps.The self-weight of thebeamis ignored in the present analysis.

To investigate the effects of various factors on the time-dependent behavior, hypothetical estimates of the axialshorteningalong the beam axis and the upward deflection mid-span areworked out for different scenarios when selected parameters areset to zero. For convenience in the present study, it is assumedthat the losses of prestress due to friction and anchorage draw-in are negligible. Fig. 7 shows the hypothetical estimates of axialshortening and vertical deflection due to concrete creep, concreteshrinkage with and without cable relaxation. It is observed fromthe results after one year that cable relaxation reduces theaxial shortening and vertical deflection due to concrete creepand shrinkage by 8.0% and 17.5% respectively. The hypotheticalestimates of loss of prestress at mid-span due to cable relaxationwith andwithout concrete creep andshrinkageare workedout andshown in Fig. 8. Also shown are hypothetical estimates of stressat bottom fibre at mid-span due to concrete creep and shrinkagewith and without cable relaxation. Concrete creep and shrinkagetherefore increase the loss of prestress due to cable relaxation by27.2% while cable relaxation reduces the stress at bottom fibre by8.9%. So the interaction among concrete creep, concrete shrinkage

and cable relaxation should be considered carefully for accuratetime-dependent analysis.

For estimation of the total long-term loss of prestress psdue to concrete creep, concrete shrinkage and steel relaxation for

Fig. 8. Loss of prestressand stressat bottomfibreat mid-span of simplysupported

beam.

sections having a single layer of prestressing steel without non-prestressed steel,CEB-FIP Model Code 1990 [24] gives the equation

ps =psc (t, t0)fcgp + Escs + 0.8pr

1 + psAsAc

1 +

Acy2

ps

Ic

[1 + c (t, t0)]

(31)

where ps = Es/Ec(t0) is the ratio of the modulus of elasticity ofprestressing steel to that of concrete, fcgp is the concrete stress atcentroid of prestressing steel at transfer, As is the cross sectionalarea of prestressing steel, yps is the y-coordinate of prestressingsteel measured from centroidal axis of beam section,Ac is the crosssectional areaof beam, is theageing coefficient that may be takenas 0.8 [21], and the other variables have been defined before. Thisformula for the evaluation of loss of prestress is one of the mostcomprehensive compared with various codes currently in use.Consider a concrete beam with a much larger cross sectional areacompared with the tendon (i.e. As/Ac 0) made with concretewith negligible creep (i.e. c(t, t0) 0) and shrinkage (i.e. cs 0). Eq. (31) gives ps = 0.8pr rather than the intrinsic stressrelaxation pr, which means that Eq. (31) cannot be degeneratedto this extreme case and it tends to under-estimate the loss ofprestress in this case.

Predictions of the loss of prestress at mid-span of the beamcalculated using Eq. (31) from day 28 to day 365 due to creep,shrinkage with and without relaxation are compared to thoseobtained by the proposed method in Fig. 9. It shows that the CEB-FIP Model Code 1990 [24] is conservative in predicting loss ofprestress, with substantial over-estimationin the presence of cablerelaxation.

5.4. A concrete cantilever with a stay cable

The long-term behavior of the hypothetical concrete cantileverwith a stay cable [11] as shown in Fig. 10 is investigated witha more complete set of parameters using the proposed time

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Fig. 9. Loss of prestress at mid-span of simply supported beam calculated by

various methods based on different assumptions.

Fig. 10. A concrete cantilever with a stay cable (not to scale).

integration method. The cantilever is 10 m in length with a squarecross section of 1 m 1 m and weight density of 25 kN/m3. Thecharacteristic compressive strength of concrete is fck = 36 MPa.Wet curing is carried out until Ts = 3 days after which shrinkagebegins. The relative humidity is taken as 80% throughout. The stay

cable is a stress-relieved tendon with a cross sectional area As =250 mm2, Youngs modulus Es = 200 GPa and initial tension P0 =200 kN applied at time t0 = 28 days with reference to the age ofcantilever. The initial prestressing ratio is R = 0.8. For simplicity,the beam is assumed to be supported on props along the lengthinitially. As the stay cable is tensioned, the props are graduallylowered to transfer part of the weight of beam to the stay cable.The parameters of CEB-FIP Model Code 1990 [24] are adoptedin the calculation. The beam is modeled by four identical beamelements while thecable is modeled by a truss element. In the timeintegration method, the period from t0 = 28 days to t = 365 daysis divided into 2000 equal time steps.

The sag effect of the stay cable can be taken into account byreplacing the modulus of elasticity Es with the Ernst modulus

Eeq [25] given by

Eeq =Es

1 + 2L2Es/123s(32)

where is the weight per unit volume of cable, L is the horizontalprojected length of the cable, and s is the tensile stress in thecable. The Ernst modulus of the cable in this example is calculatedas 199.996 GPa assuming a bare steel cable, which indicates thatthe sag effect has negligible influence and can be neglected duringthe time-dependent analysis in this case. For longer stay cables inwhich the sag effect is more significant, the Ernst modulus can beupdated for relaxation in each time step for accurate analysis.

The total axial shortening along the centroidal axis of thebeam and the vertical deflection at point A are predicted for

different scenarios. Fig. 11 shows the hypothetical estimates ofaxial beam shortening and vertical deflection at A due to concretecreep and shrinkage with and without cable relaxation. Comparingthe hypothetical estimates with and without cable relaxation

Fig. 11. Time-dependent deformation at point A of concrete cantilever.

Fig. 12. Loss of prestress in cable-stayed beam.

shows that, one year after casting the cantilever, cable relaxationreduces the axial shortening by 1.4% but increases the verticaldeflection at A by 136%. Similarly Fig. 12 shows the hypothetical

estimates of loss of prestress in the cable due to cable relaxationwith and without concrete creep and shrinkage. Comparing thesehypothetical estimates shows that concrete creep and shrinkagetend to reduce the loss of prestress in cable slightly. It should bementioned that the rather high initial prestressing ratio ofR = 0.8is for illustrational purposes. The much lower stress levels adoptedin practical stay cables implies that relaxation should not be assignificantas compared to tendons in prestressedconcrete bridges.Nevertheless it shows that the interactions among concrete creep,concrete shrinkage and cable relaxation should be consideredcarefully.

6. Conclusions

Based on the equivalent creep coefficient, a new relaxation

model has been developed for steel tendons to enable the accurateestimation of losses of cable forces. The numerical investigationsshow that the equivalent creep coefficient works not only inthe case of intrinsic relaxation but also under various boundaryconditions. Accurate analysis of time-dependent behavior ofconcrete bridges considering concrete creep, concrete shrinkageand cable relaxation is made possible by the implementationof the finite element method using time integration with theproposed relaxation model for cables. Apart from demonstratingthe versatility of the proposed method, the case studies also provethe need for accounting for interactions among various factors inthe accurate assessment of time-dependent behavior.

Acknowledgement

The work described in this paper has been supported bythe Research Grants Council (RGC) of the Hong Kong SpecialAdministrative Region, China (RGC Project No. HKU 7102/08E).

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