Time-dependent analysis of composite beams with continuous shear connection based on a space-exact stiffness matrix

Engineering Structures 32 (2010) 2902–2911

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Time-dependent analysis of composite beams with continuous shear connection based on a space-exact stiffness matrix Quang-Huy Nguyen a,c , Mohammed Hjiaj a,∗ , Brian Uy b a

Structural Engineering Research Group, INSA de Rennes, 20 avenue des Buttes de Coësmes, 35043 Rennes Cedex, France

b

School of Engineering, University of Western Sydney, NSW 1797, Australia

c

Faculty of Engineering, University of Wollongong, NSW 2522, Australia

article

info

Article history: Received 20 August 2009 Received in revised form 11 May 2010 Accepted 17 May 2010 Available online 20 June 2010 Keywords: Steel–concrete composite beams Partial interaction Space-exact solution Creep Shrinkage Space-exact stiffness matrix Beam element

abstract In this article, the time-dependent behavior of continuous composite beams with partial interaction is investigated using a space-exact, time-discretized finite element formulation. The effects of creep and shrinkage taking place in a concrete slab are considered by using age-dependent linear viscoelastic models. The Euler–Bernoulli’s kinematical assumptions are considered for both the connected members and the shear connection is modeled through a continuous relationship between the interface shear flow and the corresponding slip. Based on above key assumptions and the time-discretized form of the constitutive relationships, the governing differential equations are derived in terms of the displacements at a generic instant. These equations are analytically solved and the corresponding space-exact stiffness matrix is deduced for a generic composite beam element. This stiffness matrix may be utilized in a classical finite element procedure for the time-dependent analysis of composite beams with partial interaction. The present finite element formulation requires a minimum number of elements depending on the support and loading conditions. Finally, a time-dependent analysis of two-span continuous composite beams is presented. The results compare favorably with experimental data as well as previous numerical studies. It can be seen that shrinkage and creep can have a significant influence on the beam’s deflection. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The behavior of composite beams is strongly influenced by time effects (creep and shrinkage) that take place in the concrete slab. Under a sustained load, the deflections due to concrete creep and shrinkage may be significantly larger than their short-term values. The actual behavior of the beam being dependent on the past history, the analysis of a composite beam needs to be carried out over the whole time interval of interest. To model the time-dependent behavior of uncracked concrete in the linear range, an appropriate mathematical formulation includes linear viscoelasticity. In what follows, the interaction between cracking, creep and shrinkage is not considered. The linear viscoelasticity model can be described using either a differential formulation with internal variables or an integral formulation, the latter being the most commonly used by engineers. In addition, with concrete being an aging material, constitutive parameters are timedependant as well. This makes the analysis more complicated since Laplace transform techniques cannot be used. For a regular beam,

∗

Corresponding author. E-mail address: [email protected] (M. Hjiaj).

0141-0296/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2010.05.009

the continuous initial boundary value problem is governed by a coupled system of integral–differential equations which cannot be solved analytically for realistic concrete constitutive models. In composite steel–concrete beams with flexible connectors, a slip at the steel–concrete interface occurs, which cannot be neglected in the analysis and the design of composite structures. In contrast with the full interaction case or regular beams, time-dependent analysis of composite beams accounting for partial interaction requires the structural system to be considered even for a simply supported beam. Any solution procedure requires the use of a numerical integration procedure in order to transform the hereditary integral-type relationship into a time-discretized constitutive relationship which, despite the presence of a hereditary term, is more easy to handle (analytically) in solution algorithms. A key difference between the algorithms proposed in the literature lies in the method used to evaluate the history integrals. The most straightforward method is to numerically integrate the constitutive relation in which the history integral is approximated by a sum according to the trapezoidal or the midpoint rule. This method, called the general method, has been proposed by Bažant [1]. With this method, the entire stress-history at each point needs to be stored. In contrast, the so-called algebraic methods, consider a single step, which means that stress-history

Q.-H. Nguyen et al. / Engineering Structures 32 (2010) 2902–2911

is ignored [2–4]. Obviously, the algebraic methods give different levels of accuracy depending on the degree of refinement of the quadrature formulas adopted [5]. Another single step method, called the equivalent time method, has been proposed by Acker [6]. This method has been later improved by Boeraeve [7]. In this method, the hereditary integral-type relation is replaced by a single term that is supposed to characterize the whole stress history. Compared to the general method, this method underestimates the creep in concrete under increasing loading. In order to avoid the storage of the whole stress history, Bažant proposed the rate-type method (incremental approach), in which the integral-type stress–strain relationship is approximated by a rate-type stress–strain relation based on a Kelvin chain springdashpot model of aging viscoelasticity [8]. The main drawback of rate-type formulations is the identification of the Kelvin chain parameters which appears to be quite complicated. In the very beginning, time-dependent analysis of composite beams was carried out assuming full interaction between the concrete slab and the steel beam that is no slip occurs at the steel/concrete interface. This leads to analytical procedures for time-dependent analysis of composite steel–concrete cross sections in full interaction that have been proposed by Gilbert [9]. Early papers on the time-analysis of composite beams taking into account partial shear interaction were published by Bradford and Gilbert [10] and Tarantino and Dezi [11,12]. Bradford and Gilbert [10] proposed a method for the time-dependent response of simply supported steel–concrete composite beams where the age-adjusted effective modulus method (AAEM) has been used. Tarantino and Dezi [11] adopted the step-by-step approach to discretize the constitutive concrete model and combine it with the finite-difference method for the space discretization. Closed-form solutions based on an algebraic method have been proposed by Faella et al. [13] and Ranzi and Bradford [14]. In 1990, Boerave implemented the fictitious loading age method (FLAM) in the general Finite Element program FINELG where a 10 DOF corotational displacement-based beam element is used to model [7]. This method has been later improved by Somja and de Ville de Goyet [15]. Virtuoso and Vieira [16] developed an forcebased finite element formulation using the rate-type method for the modeling of time-dependent behavior of the concrete. With the same method, Jurkiewiez et al. [17] proposed an analytical solution for simply supported composite beams. It is worth mentioning that all cited research papers are based on linear viscoelasticity ignoring the interaction between time effects and concrete cracking. In this article, a space-exact, time-discretized formulation for the time-analysis of composite beams is proposed. By adopting a step-by-step procedure for the discretization in time of the concrete constitutive relationship, the time-discretized governing equations of the problem are analytically solved. The solution is therefore approximate in time and exact in space. Based on the analytical expressions of the displacements and the stress resultants, the space-exact expression of the stiffness matrix and the vector of equivalent nodal forces are deduced for a generic composite beam element. The proposed space-exact finite element model is used in a displacement-based procedure for the timeanalysis of continuous beams with partial interaction. The concrete cracking effects in the slab are taken into account by neglecting the concrete’s contribution along 15% of the beam length on each side of the internal support as suggested by Eurocode 4 [18]. The organization of the paper is as follows. In Section 2, the governing equations for a composite beam with partial interaction are presented. Section 3 is devoted to the constitutive models of each component. At the end of this section, the timediscretized constitutive relationship for concrete is derived. The exact analytical solution of the time-discretized initial boundary value problem and the resulting space-exact stiffness are derived

2903

Fig. 1. Free body diagram of an infinitesimal composite beam segment.

in Section 4. Two numerical applications dealing with a two-span continuous composite beam are presented in Section 5 in order to assess the performance of the proposed model and to study the effects of creep and shrinkage on the structural behavior and support the conclusions in Section 6. 2. Field equations The behavior of a deformable body at the instant t must satisfy three basic conditions. These conditions are equilibrium, compatibility, and material constitutive relationships. In this section, we recall the field equations for a composite beam with partial shear interaction in a small displacement setting. We assume that plane sections remain plane and normal to the centroid after deformation (Euler–Bernoulli’s assumption). The constitutive relations are presented in the next section. All variables subscripted with c belong to the concrete slab section and those with s belong to the steel beam. Quantities with subscript sc are associated with the shear connectors and those with subscript sr are related to the reinforcing bars. 2.1. Equilibrium The equilibrium equations are derived by considering a differential element dx located at an arbitrary position x (see Fig. 1). Since the behavior of the concrete slab is time-dependent, the equilibrium equations for the composite steel–concrete element must be satisfied at every instant t:

∂x Nc (x, t ) + Dsc (x, t ) = 0 ∂x Ns (x, t ) − Dsc (x, t ) = 0 T (x, t ) = ∂x (M (x, t ) + HNs (x, t ))

(1)

∂x2 M (x, t ) + H ∂x Dsc (x, t ) + p0 = 0

(4)

(2) (3)

where – ∂xi • = di • /dxi – H = Hc + Hs the distance between the reference axis of the concrete element and the steel element – T = Tc + Ts and M = Mc + Ms – Ni , Ti , Mi (i = s, c ) are the axial forces, the shear forces and bending moments at the centroid of cross-section ‘‘i’’ – Dsc is the bond force per unit length (see Fig. 1) – p0 is the external uniformly distributed load. 2.2. Compatibility The curvature and the axial deformation at any section are related to the beam displacements through kinematic relations. Under small displacements and neglecting the relative transverse displacement between the concrete slab and the steel beam, these

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where J (t , τ ) is the creep function which is defined as the strain at time t produced by a unit sustained stress applied at age τ . For a realistic expression of the creep function, the above integral cannot be solved analytically and a time stepping algorithm should be applied in order to get a time-discretized constitutive model. The most straightforward method for its numerical solution is to replace the integral by a finite sum using a numerical scheme. The time interval of interest is partitioned into n − 1 sub-intervals of size 1t, not necessarily equal, according to t1 < t2 < · · · < ti < ti+1 < · · · < tn . Thus Eq. (10) can be approximated using a trapezoidal rule as n−1

1X (n) ε (n) ∼ Jn,i+1 + Jn,i σ (i+1) − σ (i) = εsh + σ (1) Jn,1 +

Fig. 2. Kinematics of a composite beam.

relations are as follows:

εc (x, t ) = ∂x uc (x, t ) εs (x, t ) = ∂x us (x, t )

(5)

κ(x, t ) = −∂ v(x, t )

(7)

dsc (x, t ) = us (x, t ) − uc (x, t ) + H ∂x v(x, t )

(8)

(6)

2 x

with u the longitudinal displacement, v the transversal displacement, εc the strain at the concrete section centroid, εs the strain at the steel section centroid, κ the curvature and dsc the relative slip between the concrete slab and the steel beam (see Fig. 2).

in which ε (i) = ε(ti ), σ (i) = σ (ti ) and Ji,j = J (ti , tj ). Finally, the time-dependent behavior of concrete can be expressed as follows n −1   X (n) Ψn,i σ (i) σ (n) ∼ = E (n) ε(n) − εsh +

(12)

i =1

where E (n) =

2

(13)

Jn,n + Jn,n−1

and

3. Constitutive relationships The cross-section considered in this paper is formed by a concrete slab, reinforcing bars, a steel beam and shear connectors. The steel reinforcement and the steel beam are assumed to be linear elastic while the concrete slab experiences creep and shrinkage. The constitutive relations are obtained simply by integrating over each cross-section the appropriate uniaxial constitutive model (only the effect of normal stresses is considered). The constitutive model does not take into account the concrete cracking and the tension stiffening effects. More advanced models have been proposed in the literature where these effects are considered (see, for example [19]). 3.1. The concrete slab

Ψn ,i

 J −J n,2 n ,1   Jn,n + Jn,n−1 = J − Jn,i−1   n,i+1 Jn,n + Jn,n−1

ε(t ) = εel (t ) + εcp (t ) + εsh (t ) (9) in which εel (t ) is the instantaneous strain, εcp (t ) the creep strain and εsh (t ) the shrinkage strain. Time effects are described using linear viscoelastic models. These models can be formulated either using a hereditary integral or internal variables. In this paper, the former is used. As a consequence of creep and shrinkage, the stresses in redundant structures vary with time even if the load is constant. It is well known that under service conditions, the maximum stress in a concrete structure seldom exceeds 45% of the compressive strength and the creep strain may be assumed to be proportional to the stress [20]. The evolution of the viscous strain due to a stress history is derived using the principle of superposition. The total strain history ε(t ) is expressed by the Volterra integral equation as follows [21]

Z

t + t1

J (t , τ ) dσ (τ )

(10)

if i = 1 (14)

if i > 1.

The relation (12) gives the stress at tn in terms of the current strain and the stresses at previous time instants. Using the time-discretized uniaxial constitutive relation (12), the axial force and the bending moment in the reinforced concrete slab at time instant tn are given by Nc(n) (x) =

Z

σc(n) (x, zc ) dA +

X

Ac

Mc(n) (x) =

The strain in the concrete comprises an instantaneous and a time-dependent part. The instantaneous strain, also called elastic strain, develops instantaneously when the structure is loaded. The time-dependent part grows gradually with time. The timedependent strain is due to creep and shrinkage of the concrete. Considering a concrete specimen subjected to a uniaxial load at a constant ambient temperature, the total strain in the specimen at instant t may be decomposed into three strain components:

ε(t ) = εsh (t ) + σ (t1 ) J (t , t1 ) +

(11)

2 i=1

Z

σsr(n) (x, zsr )Asr

(15)

nr

zc σc(n) (x, zc ) dA +

X

Ac

zsr σsr(n) (x, zsr )Asr

(16)

nsr

where σsr is the stress in the reinforcing bar and nsr the number (n) of reinforcing bars. Substituting σc by its expression (12) in relations (15) and (16), one obtains (n)

Nc(n) = (EA)(cn) εc(n) − (EA)(con) εsh + (EB)c(n) κ (n) +

n −1 X

(i) Ψn,i Nco

(17)

i =1

(n)

Mc(n) = (EB)c(n) εc(n) − (EB)(con) εsh + (EI )c(n) κ (n) +

n −1 X

(i) Ψn,i Mco (18)

i =1

where

(EA)co(n) = Ec(n) Ac ; (EA)c(n)

(EB) (n) = Ec(n) Bc ; X co = (EA)co(n) + Esr Asr ; nsr

(n)

(EB)c

(n)

= (EB)co +

X

zsr Esr Asr ;

nsr

(EI )c(n) = Ec(n) Ic +

X

zsr2 Esr Asr .

nsr

(i)

Bc is the first moment of area of the concrete section; Nco and (i) Mco are the axial force and the bending moment resisted by the concrete only (without reinforcement), respectively.

Q.-H. Nguyen et al. / Engineering Structures 32 (2010) 2902–2911

(i)

3.2. The steel beam and the shear connectors

2905

(i)

consider that Nco and Mco have the following expressions:

For the steel beam, the linear elastic constitutive model is adopted as well as for the shear connectors. If we consider the effects of the normal stress σ only, we can identify the axial force as

(i)

(i)

(i)

(i) Nco = α1 x2 + α2 x + α3 i

 X  (i,j) + β1 sinh(µj x) + β2(i,j) cosh(µj x)

(26)

j =1

Ns(n) (x) =

Z

σs(n) (x, zs ) dA = (EA)s εs(n) (x)

(19)

As

(i)

(i)

(i)

(i) Mco = α4 x2 + α5 x + α6 i

X

+

and the bending moment as

 β3(i,j) sinh(µj x) + β4(i,j) cosh(µj x) .

(27)

j=1

Ms(n) (x) =

Z

zs σs(n) (x, zs ) dA = (EI )s κ (n) (x)

(20)

As

where (EA)s = Es As , (EI )s = Es Is in which Es is the steel’s Young’s modulus and As , Is are the area and the second moment of area of the steel section, respectively. The relationship between the relative slip and the shear force is given by D(scn) (x) = ksc d(scn) (x)

(21)

The above expressions are exact for i = 1 since they correspond to the analytical expressions of the axial force and the bending moment as given by the elastic solution of Newmark et al. [22]. In what follows, we will prove that these expressions are exact for any instant ti . To do so, we first assume that the relations (26) and (27) (i) (i) represent the exact expressions for Nco and Mco at instant tn−1 . If (i) we prove that at instant tn , we obtain the same expressions for Nco (i) and Mco , we can then conclude that these expressions are valid for (i) any ti . It should be noted that in the above expressions, α1..6 and

β1(i..,j4) are constants, updated at the end of each step ti .

where ksc is the shear stiffness.

By using the following formula

4. Derivation of the space-exact stiffness matrix

n −1

X The analytical solution for the instantaneous response of composite beam with partial interaction has been given by Newmark et al. [22]. If a one-step scheme, such as algebraic or ratetype methods, is used to approximated the viscoelastic behavior of concrete (affine constitutive model), the Newmark’s solution can be easily extended to cover the time-analysis of composite beams [13,14,17]. Although these solutions provide some insight into the time-dependent behavior of composite beams, they remain approximate as the stress-history is not properly taken into account. The general step-by-step method provides a more accurate solution but requires the storage of the whole stresshistory. In this paper, an analytical solution of the time-discretized governing equations is derived. Based on this solution, the spaceexact stiffness matrix is established. Making use of the compatibility Eqs. (5)–(8) and the timediscretized constitutive relationships (17)–(21), the equilibrium equations (1), (2) and (4) are expressed in terms of displacements: 5 (n) x us

∂

2 3 (n) n x us

−µ ∂

(n)

(n)

= ζ1 + ζ2

n −1 X

n −1 X

∂

X (n)

∂x3 v (n) = ζ4(n) ∂x4 u(sn) + ζ5(n) ∂x2 u(sn) + ζ6

=

i =1

! Ψn ,j β

(j,i)

sinh(µi x)

j=i

(28) one obtains n −1 X

(i) Ψn,i Nco =

i =1

3 X

λ(kn) x3−k +

n −1  X

η1(n,i) sinh(µj x)

i =1

k =1

 + η2(n,i) cosh(µj x) n −1 X

(i) Ψn,i Mco =

i =1

3 X

λ(kn+)3 x3−k +

k=1

n −1  X

(29)

η3(n,i) sinh(µj x)

i =1

+ η4(n,i) cosh(µj x)



(30) (i,j)

(i)

where the expressions of λ1..6 and η1..4 are given in Appendix. Substituting Eqs. (29) and (30) into Eq. (22), one obtains

(22)

n−1

(i) Ψn,i ∂x Nco

(EA)s 2 (n) u(cn) = u(sn) − ∂x us + H ∂x v (n) ksc

X

µ2i (ζ2(n) η4(i,j) + ζ3(n) η2(i,j) ) cosh(µi x).

(31)

i=1

(23)

(n)

Eq. (31) can be analytically solved for us following closed-form expression:

i=1

(n)

sinh(µj x)

j=1

+ n−1

β

n −1 n −1 X X

i=1

(i) Ψn,i ∂x2 Nco

i =1

where ζ1 given by

i =1

! (i,j)

∂x(5) u(sn) − µ2n ∂x(3) u(sn) = (ζ1(n) + 2ζ2(n) λ(4n) + 2ζ3(n) λ(1n) ) n−1 X + µ2i (ζ2(n) η3(i,j) + ζ3(n) η1(i,j) ) sinh(µi x)

(i) Ψn,i x2 Mco

i =1

+ ζ3n

Ψn ,i

i X

(24)

(n)

(n)

(n)

so we obtain the (n)

(n)

u(sn) = C1 sinh(µn x) + C2 cosh(µn x) + C3 x2 + C4 x + C5

· · · ζ6(n) are constants, defined in Appendix, and µn is

+ ζ7(n) x3 +

n −1 X

a(sn,i) sinh(µi x) + b(sn,i) cosh(µi x)




(32)

i =1

v   u u u  H 2 (EA)(cn) + (EI )(n) − 2H (EB)(cn) 1  µn = u . h i2 + tksc  ( EA )s ( n ) ( n ) (EI )(n) (EA)c − (EB)c

where (25)

a(sn,i) =

ζ2(n) η3(n,i) + ζ3(n) η2(n,i) µi (µ2i − µ2n )

The solution of the system of Eqs. (22)–(24) requires the (i) (i) expressions of Nco and Mco at all steps ti prior to step tn . Let us

b(sn,i) =

ζ2(n) η4(n,i) + ζ3(n) η1(n,i) . µi (µ2i − µ2n )

and

(33)

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Q.-H. Nguyen et al. / Engineering Structures 32 (2010) 2902–2911

Remark. If J (tn , ti ) approaches J (tn , ti+1 ) then µi approaches µi+1 . Thus the denominator of the above equations approaches zero which will cause numerical problems. For this reason, the choice of two successive discrete times ti and ti+1 that leads to close values of the creep function J (tn , ti ) and J (tn , ti+1 ) must be avoided. By substituting Eq. (32) into Eq. (23) and solving for v (n) , one obtains

v

(n)

=

(n)

ζ4

ζ (n) + 5 µn

!

n −1 X

 Fig. 3. Nodal forces and displacements of a composite beam element.

!

+ ζ9(n) x3 + C6(n) x2 + C7(n) x + C8(n) + ζ8(n) x4

3

+

(n)

C1 cosh(µn x) + C2 sinh(µn x)

ζ5(n) C3(n)

+

(n)



a(vn,i) sinh(µi x) + b(vn,i) cosh(µi x)




(34)

i =1

(n)

in which C1..8 are constants of integration related to the nodal (n)

(n)

(n)

(n,i)

(n,i)

(n)

After some algebraic manipulations, one can see that Nco has the same expression as (26) and this achieves the proof. The same (n) (n) (n) can be said for Mco . By substituting the expressions of us uc (n) and v into the compatibility relations (5)–(7) and making use the constitutive relations (17)–(20), one obtain the closed-form (n) (n) expressions for the axial forces Nc and Ns , the bending moment M (n) and the shear force T (n) :

displacements, and ζ7 , ζ8 , ζ9 , as , bs , a(vn,i) and b(vn,i) are coefficients defined in the Appendix. (n) By setting the equations in Box I us and v (n) can be expressed in a vector form as

(n) (n) Nc(n) = YNc C(n) + RNc (x)

u(sn) = X(sn) C(n) + Zs(n) (x)

(n) (n) Ns(n) = YNs C(n) + RNs (x)

+

n−1 X

a(sn,i) sinh(µi x) + b(sn,i) cosh(µi x)




+

n−1  X

+

(35)

n−1  X

(n)

(36)

(n) Xθ C(n)

+

n−1  X

= −∂x X(cn)

aM

cosh(µi x)



(42)

(n,i)

(n,i)

sinh(µi x) + bθ

(n)

Zθ

cosh(µi x)



(37)

= −∂x Zc(n) (n,i)

u(cn) = X(cn) C(n) + Zc(n) (x)

n −1

X




(38)

i =1

(n,i)

sinh(µi x) + bT



cosh(µi x) .

(43)

In the above expressions, several coefficients have been introduced. The exact meaning of these coefficients is given in the Appendix. From the closed-form expressions for the displacement field and the force field, one can easily deduce the expression of the exact stiffness matrix. Let us consider a composite beam element of length L (see Fig. 3). Applying the kinematic boundary conditions at x = 0 and x = L, one obtain the relationship between the vector of constants of integration C(n) and the vector of nodal displacements q(n) : (44)

where q(n) = u(c1n)

v1(n) θ1(n) u(c2n) u(s2n) v2(n) h h iT iT T (n) X(n) = X(cn) (x = 0) ··· Xθ (x = L)  T Z(n) = Zc(n) (x = 0) · · · Zθ(n) (x = L) . 

where

(EA)s

∂x(2) X(sn) + H ∂x X(vn) ksc (EA)s (2) (n) Zc(n) = Zs(n) − ∂x Zs + H ∂x Zv(n) ksc (n,i)

and the coefficients ac and bc are given in the Appendix. At this stage, we have derived the closed-form expressions for the displacements uc , us and v n at instant tn and we are able de derive the analytical expressions for the stress resultants. For instance, (n) the axial force resisted by the concrete only Nco can be computed using the following relation: (n) Nco = (EA)co(n) ∂x u(cn) .

(n,i)

aT

q(n) = X(n) C(n) + Z(n)

ac(n,i) sinh(µi x) + b(cn,i) cosh(µi x)

(n,i)

(n)

(n)

i =1

(n,i)

X(cn) = X(sn) −

(n,i)

sinh(µi x) + bM

T (n) = YT C(n) + RT (x)

+

and the coefficients aθ and bθ are given in Appendix. Substituting Eqs. (35) and (36) into Eq. (24), one obtains the (n) solution for uc as follows

+

(n,i)

X

+

(n) Zθ (x)

aθ

n −1 X

(41)

i=1

i=1

(n)



(n)

n−1

where Xθ

(n,i)

M (n) = YM C(n) + RM (x)

The derivative of the transverse displacement (36) yields the crosssection rotation θ (n)

+

(n,i)

(40)

aNs sinh(µi x) + bNs cosh(µi x)

i =1

=



i=1

v (n) = X(vn) C(n) + Zv(n) (x) n −1 X
+ av(n,i) sinh(µi x) + b(vn,i) cosh(µi x) .

θ

(n,i)

i=1

i=1

(n)

(n,i)

aNc sinh(µi x) + bNc cosh(µi x)

(39)

(n)

us1

θ2(n)

T

The nodal displacements being independent variables, the inverse of X(n) always exists: C(n) = X(n)



−1

q(n) − Z(n) .




(45)

Next, we apply the static boundary conditions at the beam ends and obtain the relationship between the vector of constants of integration C(n) and the vector of nodal forces Q(n) (Fig. 3): Q(n) = Y(n) C(n) + R(n)

(46)

Q.-H. Nguyen et al. / Engineering Structures 32 (2010) 2902–2911

C(n) = C1(n)

(n)



(n)

C2

X(sn) = sinh(µn x)

"

(n)

ζ4

=

C4

cosh(µn x)



X(vn)

(n)

C3

ζ (n) + 5 µn

(n)

(n)

C5 x2

C6 x

!

1 (n)

cosh(µn x)

ζ4

2907

 (n) T

(n)

C7

C8

0

0

ζ (n) + 5 µn

!



0

sinh(µn x)

ζ5(n) 3

# 0

0

x

2

x

1

(n)

Zs(n) = ζ7 x3 (n)

(n)

Zv(n) = ζ8 x4 + ζ9 x3 Box I.

Fig. 4. Geometry of the composite beams tested by Gilbert and Bradford [23].

where (n) Q(n) = Nc1

Y(n) =

(n)

(n)



Ns1

hh

iT −Y(Nnc) (x = 0)

h

(n)

T1

(n)

R(n) = −RNc (x = 0)

(n)

M1

h

···

···

(n)

Nc2

(n)

Ns2

(n)

T2

 (n) T

M2

iT iT

(n)

YT (x = L)

iT

RT (x = L)

.

Substituting C(n) of the expression (45) into the Eq. (46), one obtains the relationship between the nodal displacement vector q(n) and the nodal force vector Q(n) : (n)

K(n) q(n) = Q(n) + Q0 in which K

(n)

=Y

(n) (n)

(47)



X

 (n) −1

represents the exact stiffness matrix

(n) (n)

at instant tn and Q0 = K Z − R(n) represents the nodal forces vector due to internal distributed load and stress history. The exact stiffness matrix K(n) will be utilized in a displacement-based finite element program for the time-dependent analysis of continuous composite beams with arbitrary support and load conditions. 5. Applications In this section, the proposed space-exact time-discretized finite element model is used to study the time-dependent behavior of two-span continuous composite beams. We begin with a comparison between the long-term deflections predicted by the present computational model and those obtained by earlier experimental tests. For that, the two-span composite beams tested by Gilbert and Bradford [23] is considered. Experimental results related to the long-term behavior of continuous beams are fairly rare. The experimental data provided by Gilbert and Bradford [23] seem to be the only ones available in the literature. Next, we consider two-span continuous beam studied by Dezi and Tarantino [24] and we compare the predictions of the proposed model against those obtained by the numerical model of Dezi and Tarantino [12]. Finally, we investigate the effects of creep and shrinkage on the behavior of the same continuous composite beam. 5.1. Application I: comparison with experimental data The purpose of this application is to assess the capability of the proposed model to predict satisfactorily the long-term structural

behavior of composite beams with partial interaction. We investigate the long-term deflection of two identical continuous beams (B1 and B2) with two equal spans. The beam B1 was subjected only to its self-weight, i.e. p0 = 1.92 N/mm, while the beam B2 carried an additional superposed sustained load of 4.75 N/mm, i.e. p0 = 6.67 N/mm. These continuous beams were tested by Gilbert and Bradford [23] over a period of 340 days. The dimensions of the tested beams, the loading and the geometric characteristics of the cross-section are shown in Fig. 4. For each beam, two analyses were carried out. The first one included an ‘‘uncracked analysis’’ where the concrete cracking in the slab is ignored. The second one included a ‘‘cracked analysis’’ as suggested by Eurocode 4 [18]. In this analysis, concrete cracking is taken into account by neglecting the concrete’s contribution along 15% of the span length on each side of the internal support. The material parameters of steel used in the computer analysis are: Es = 2 × 105 MPa; Esr = 2 × 105 MPa. The shear stiffness (ksc ) is not given in the Ref. [23] and we have assumed a value of 104 MPa for ksc . The concrete’s compressive strength obtained from the experiments [23] is equal to 27 MPa. Accordingly, we can consider for the computer analysis that the concrete is of grade C30. Shrinkage is assumed to start at 7 days. Three wellknown models are selected to predict the creep function and shrinkage strain of concrete: (1) CEB-MC90 model [25]; (2) ACI209 model [26]; and (3) B3 model [27]. Table 1 presents the input values for each concrete model. It should be noted that these values have been taken in order that the predicted shrinkage strain at 340 days be nearly equal to the measured one, i.e. εsh (340, 7) ≈ 0.00052 [23]. The following discrete times are considered for the computer analysis:

[7

30

50

80

120

170

230

300

365] days.

The mid-span deflections obtained by the proposed model, using 2 elements for the ‘‘uncracked analysis’’ and 4 elements for the ‘‘cracked analysis’’, are compared against the experimental results in Fig. 5 for both beams. These deflections are computed using creep and shrinkage functions given in CEB-MC90 model [25]. As can be seen from this figure, compared to the ‘‘uncracked analysis’’, the numerical time-deflection curves given by ‘‘cracked analysis’’ are closer to the experimental curves. With ‘‘uncracked analysis’’, the model underestimates the mid-span deflections. These results indicates that the concrete cracking effects must be taken into account for the continuous composite beams even under serviceability loads. Furthermore, Fig. 5 shows that the final deflections obtained with ‘‘cracked analysis’’ are in reasonable agreement with experimental results, however, some disagreements can be observed for time interval up to 200 days. These disagreements can be explained by the fact that the values of creep function and shrinkage strain, predicted by CEB-MC90 model [25], at this time interval may not agree with measured values of the test specimens. By using different above-mentioned models for creep and shrinkage predictions, the mid-span deflection versus time curves obtained with ‘‘cracked analysis’’ are plotted and compared with experimental curves in

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Q.-H. Nguyen et al. / Engineering Structures 32 (2010) 2902–2911

Table 1 Input values for different creep and shrinkage prediction models. CEB-MC90 model input values fck (MPa) 30

RH (%) 40

h (mm) 70

Ciment type 2

ACI-209 model input values

(fc0 )28 (MPa) 38

H (%) 50

ω (kg/m3 ) 2400

c (kg/m3 ) 500

s (mm) 25.4

Ψ (%)

α (%)

50

6

v/s (mm) 70

780

(εsh )u (10−6 )

gct 0.043

Curing Moist

c (kg/m3 ) 500

a/c 2.96

m 0.5

n 0.1

v/s (mm) 70

E (28) (MPa) 29 182

Ciment type 2

Curing Moist

B3 model input values fc0 (MPa) 38 ∗

h (%) 50

w (kg/m3 ) 192

Note: All symbols are defined in the corresponding cited reference.

Fig. 7. Geometry of the composite beam analyzed by Dezi and Tarantino [24].

Fig. 5. Numerical–experimental comparison of mid-span deflection versus time curves.

Fig. 6. Mid-span deflection versus time curves for different creep and shrinkage models.

Fig. 6. It is observed that, compared to the two other models, the B3 model [27] seems to perform better the time-deflection curves. This is due to the fact that the B3 model [27] probably provides the most accurate predictions for the creep functions and shrinkage strain. The accuracy of the B3 model [27] may be justified by its relative complexity and was clarified by Fanourakis and Ballim [28] in their comparative study of the different creep and shrinkage models. 5.2. Application II: comparison with existing model The purpose of this application is to compare the predictions of the proposed model against the numerical results obtained by

Dezi and Tarantino [12]. The long-term response of the two-span composite beam, previously studied by [24], is considered. Next, the effects of creep and shrinkage on the long-term behavior of this continuous composite beam are analyzed. The dimensions of the beam as well as the geometry of the cross-section are described in Fig. 7. The beam is subjected to a uniformly distributed load p0 = 64.56 N/mm at t1 = 30 days from the casting time of the concrete slab. Shrinkage is assumed to start at tsh = 30 days and the final time of the viscoelastic analysis is 25 550 days, which is equivalent to 70 years. The elastic modulus adopted for steel is 2.1 × 105 MPa and a stiffness of 400 MPa is considered for the connectors. The concrete grade is assumed to be C30. Creep and shrinkage functions recommended by the CEB-MC90 [25] and adopted in the original study; are considered with the following assumptions: the cement is normal hardening; the humidity (RH) is about 50%. The time-analysis of the continuous beam is carried out using 2 elements (one element per span) and the following discrete times) are considered (see Box II) The time-evolution of the internal support reaction is plotted in Fig. 8. Fig. 9 shows the internal support reaction versus the stiffness of the shear connectors. The numerical comparison shows a good agreement between the results obtained with the proposed model and those obtained by Dezi and Tarantino by means of the finite difference method using 40 segments [24]. The evolution internal support reaction (R) observed is the result of the bending moment redistribution caused by the creep and the shrinkage of concrete (see Fig. 12). Indeed, the bending moment at the internal support increases with time. To maintain the beam in equilibrium, the reaction at the end support must decrease and therefore will cause an increase of the reaction at the internal support. The effect of creep and shrinkage on the beam’s deflection is now examined. The deflection distributions along the beam axis at 30 days and 25 550 days are depicted in Fig. 10 considering first the effects of creep only and next the combined action of creep and shrinkage. Since creep tends to decrease the secant modulus of the concrete, we can expect that the deflection will increase with time as observed in Fig. 10. The maximum deflection increases by about 27% compared to these at 30 days. The increase of the maximum deflection is more significant (66%) when creep and shrinkage act simultaneously. Indeed, shrinkage tends to reduce the length of the concrete slab which causes a positive bending moment that one can translate by a remarkable redistribution of axial force in the concrete slab in Fig. 12.

Q.-H. Nguyen et al. / Engineering Structures 32 (2010) 2902–2911

[30

32

35

45

65

100

200

500

800

2100

4000

10 000

2909

25 550] days

Box II.

Fig. 8. Time evolution of intermediate support reaction. Fig. 11. Distributions of axial force in the concrete slab.

Fig. 9. Mid-support reaction versus stiffness of the shear connectors.

Fig. 12. Effect of creep and shrinkage on the bending moment.

limited variations of the internal support reaction when only creep is considered (see Fig. 8). On the other hand, the timedependent redistribution of the bending moment due to shrinkage is significant. As can be seen from Fig. 11, shrinkage causes an increase of the bending moment at mid-support of 70% after 70 years. It should be noted that in this case the shrinkage strain at the final time is equal to 6 × 10−4 (negative) which is rather important and explains such an increase of the deflection as well as the bending moment. 6. Conclusions

Fig. 10. Effects of creep and shrinkage on the deflection.

The bending moment distributions along the beam length at the initial time t1 = 30 days and final time tf = 25 550 days taking into account creep and both creep and shrinkage are presented in Fig. 12. As can be seen, there is almost no redistribution of bending moment due to creep. It can be explained by the very

In this article, a space-exact, time-discretized solution for the time-dependent analysis of composite beams with partial interaction has been presented. This solution has been obtained by deriving a closed-form solution of the time-discrete equilibrium equations expressed in terms of the displacements. Based on the analytical expressions of the displacements and the internal forces, the space-exact expression of the stiffness matrix and the vector of equivalent nodal forces have been deduced for a generic composite beam element. The proposed space-exact stiffness matrix has been utilized in a displacement-based procedure for the time-analysis of continuous beams with partial interaction. The model is capable

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Q.-H. Nguyen et al. / Engineering Structures 32 (2010) 2902–2911

to perform both cracked and uncracked analysis as defined in Eurocode 4 [18]. The proposed model has been successfully applied to twospan continuous composite beams under a static distributed load. Fairly good agreement between the predicted deflections and the measured deflections has been found. The numerical results have been indicated that the effects of concrete cracking must be taken into account for continuous composite beams even under serviceability loads. Furthermore, it has been observed that the choice of the creep and shrinkage models has a certain influence on the time-deflection responses. The comparison between the predicted deflections obtained using three different creep and shrinkage models (i.e. CEB-MC90 [25], ACI-209 [26] and B3 [27] models) and the measured ones showed that the B3 model [27] appears to provide the most accurate predictions up to 340 days. The predictions of the model were also compared against the numerical results obtained by Dezi and Tarantino [12]. The results show that good agreement with the results obtained by Dezi and Tarantino [24] (by means of the finite difference method using 40 segments) has been achieved with only 2 space-exact finite elements. The effects of creep and shrinkage on the behavior of a two-span continuous composite beam were studied. The numerical results show that shrinkage of the concrete slab plays an important role in the behavior of composite beams. It leads to a large increase in deflection and a significant redistribution of internal forces.

(n)

 − (EB)(cn) p0 ksc =  ;  2  (n) (n) ( n ) (EA)c − (EB)c (EA)s (EI )   (EI )(n) − H (EB)(cn) p0 ksc =  ;  2  (n) (n) ( n ) (EA)c − (EB)c (EA)s (EI )

ζ1(n)

ζ3(n)

H (EA)c

ζ4(n) = ζ5(n) =

(EA)(cn) (EA)s ksc H (EA)c

λ(1n..)6 =

(n)

H (EA)c

(n)

− (EB)c

;

ζ6(n) λ(2n) 3 n −1 X

p0

ζ6(n) =

−1 (n)

H (EA)c

ζ8(n) =

− (EB)(cn) (n) (n)

3ζ5 ζ7

; (n)

Ψn,i α1..6 ;

η1n..,i4 =

n−1 X

(i,j)

Ψn,j β1..4 ;

j =i

(ζ4(n) µ2i + ζ5(n) )µi b(sn,i) + ζ6(n) η1(n,i) ; µ2i

(ζ4(n) µ2i + ζ5(n) )µi a(sn,i) + ζ6(n) η2(n,i) ; µ2i   (EA)s 2 (n,i) = 1− µi av + H µi b(vn,i) ; ksc   (EA)s 2 (n,i) = 1− µi bv + H µi a(vn,i) ;

b(vn,i) = a(cn,i) b(cn,i)

ksc

= (EA)(cn) µi b(cn,i) − (EB)(cn) µ2i a(cn,i) + η1(n,i) ;

(n,i)

= (EA)s µi bs(n,i) ;

aN s

= −µi b(vn,i) ;

(n,i)

= −µi a(vn,i) ;

bθ

(n,i)

aN c

(n,i)

aθ

(n,i)

= (EB)(cn) µi b(cn,i) − (EI )(n) µ2i a(cn,i) + η3(n,i) ;

(n,i)

= µi (b(Mn,i) + Hb(Nns,i) );

(n,i)

= (EB)(cn) µi a(cn,i) − (EI )(n) µ2i b(cn,i) + η4(n,i) ;

(n,i)

= µi (a(Mn,i) + Ha(Nns,i) );

bM bT

(n)

YNc = (EA)(cn) ∂x X(cn) − (EB)(cn) ∂x2 X(cn) ;

(n)

YNs = EAs ∂x X(sn) ;

(n)

YM = (EB)(cn) ∂x X(cn) − (EI )(n) ∂x2 X(vn) ; (n)

YT

= ∂x Y(Mn) + H ∂x Y(Nns) ;

(n)

RNs = EAs ∂x Zs(n) ;

(n)

RT

= ∂x R(Mn) + H ∂x R(Nns) ;

(n)

RNc = (EA)(cn) ∂x Zc(n) − (EB)(cn) ∂x2 Zv(n)

(n) + λ(1n) x2 + λ(2n) x + λ(3n) − Ec(n) Ac εsh ;

(n)

RM

= (EB)(cn) ∂x Zc(n) − (EI )(n) ∂x2 Zv(n) (n) + λ(4n) x2 + λ(5n) x + λ(6n) − Ec(n) Bc εsh .

References

;

12

= (EA)s µi a(sn,i) ;

aT

;

+ ζ6(n) λ(1n)

(n,i)

aM

;

ζ1(n) + 2λ(4n) (ζ2(n) + ζ3(n) ) ; 6µ22

i =1

a(vn,i) =

− (EB)(cn)

(EA)(cn) + (EA)s

ζ7(n) = − ζ9(n) =

(n)



ζ1(n)

ζ2(n) =

= (EA)(cn) µi a(cn,i) − (EB)(cn) µ2i b(cn,i) + η2(n,i) ;

bN s

Appendix. Detailed expressions



(n,i)

bN c

;

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