nonlinear analysis of composite bridges by the finite element method

Computm & Sttnchue~ Vol. 40, No. 5, pp. 1151-l 167, 19!31 00453949pi s3.00 + 0.00 Printed inGm.ttBritain. WnP-pk

NONLINEAR ANALYSIS OF COMPOSITE BRIDGES BY THE FINITE ELEMENT METHOD

J. J. Lr~,t M. FAFARD,~ D. BRAULI~W~ and B. M~ss~co~~tzj

tCivi1 Engineering Department, Laval University. Quebec, Canada GlK 7P4 @colt Polymchnique, Montreal, Canada

Ahatraet-The study of the nonlinear behaviour of composite bridges is quite complex. The difficulties can be attributed to the use of various materials and structural components and to their behaviour under different loading conditions. A sophisticated numerical tool is therefore required to improve our understanding of the structural behaviour of bridges. The finite element method is a powerful one which can be adopted to fulfill this task. In this paper, a plate/shell element (called DLTP), a shear connector element and a contact element for the nonlinear analysis of composite bridges are presented. The finite element procedure is based on small elasto-plastic strains and updated Lagrangian formulation to account for the large displacements and rotations of the structures. Various material models are developed to simulate the behaviour of steel, concrete and interface media (shear connectors, contact and friction) as well as phenomena encountered in the analysis of composite bridges. Numerical examples are presented for the validation of the proposed analytical procedure.

NOTATION

area coefficients of Yam-Chapman’s empirical formulation strain displacement matrix nodal displacement vectors at the interface between concrete and steel virtual deformation gradient tensor Young’s modulus secant modulus of concrete Green-Lagrange deformation tensor virtual Green-Lagrange deformation tensor at configuration C* deformation gradient between configurations C’ and C2 global nodal force vector yield function interface element nodal force ultimate compressive strength tensile strength of concrete yield strength of steel shear modulus of concrete elastic matrix of plate element elasto-plastic matrix of plate element assumed iterations for convergence stiffness matrix interface element axial and tangential stiffness ratio of Young’s modulus of steel to concrete normal and tangential unit vectors, respectively transformation matrix residual vector Piola-Kirchhoff stress tensor in configuration C’ global displacement vector arc length virtual work

Greek symbols OL ratio of principle stresses

INTRODUCTION

;r h 8 K

relative displacement vector at the interface virtual displacement at the interface

Thousands of short- and medium-span bridges, con-

strain strutted in and after the 1950s in North America, are

rotation or referred to rotation as a subscript concrete slab-on-steel girder bridges. As traffic vol- hardening parameter ume and truck loads increase, some of the existing

1151

cu a(K) d

Superscripts I(

e

P T

Subscripts S

C

cr

n

t

eq

ep

ext int

FT U

Y

stiffness matrix related to relative interface displacement and nodal force sets loading factor Coulomb’s friction coefficient Poisson’s ratio of concrete initial Poisson’s ratio of concrete principal stresses cracking stress of concrete ultimate concrete compressive strength in biaxial state represents components of Cauchy stress tensor (uz = 0 along the thickness n) ultimate concrete strain corresponding to u, uniaxial yield stress deviatoric stress tensor

refers to virtual refers to element ith iteration pth load step indicates transformation

refers to steel or stud refers to concrete refers to cracking refers to normal direction tangential or tangential direction refers to equivalent value refers to elasto-plastic external internal refers to softening effect refers to stiffening effect refers to ultimate value refers to yielding

1152 J. J. LIN et al.

bridges may be overloaded and their structural responses may be propelled beyond the elastic level into the nonlinear state. The current methods provided by numerous design codes cannot adequately predict these bridge responses due to the high structural indeterminacy and the nonlinearity introduced by overloading. Therefore developing nonlinear analytical models to reevaluate the complete behaviour of existing bridges up to failure under modern traffic loadings has become one of the high priorities in current bridge research [ 11.

Slab-on-girder bridges can be taken as structures essentially composed of thin plate/shell concrete and steel elements with composite action provided for by means of mechanical connections such as studs. Finite element analysis of these bridges, with the approximation of taking the deck as an eccentrically strengthened plate, have shown a poor coincidence with test results, especially in the nonlinear range [2,3]. The discrepancy is mostly due to the insuffi- ciency of the proposed approach in modelling the material plastification through the bridge deck and its inability to simulate geometric nonlinearities. Hence, a reliable and detailed analysis of such structures by the finite element method requires an appropriate family of elements which are able to model the complex behaviour of concrete plates and steel girders as well as of the interface medium between them. Also, the study of the complete response of this type of bridges at all stages of loading up to the ultimate load requires the proper consideration of both material performance and geometrical nonlinearity due to large deformations.

In this study, a family of elements, which includes concrete and steel plate/shell elements, a mechanical interface connection (stud) element, as well as a contact element, is adopted. The plate/shell element used for both concrete and steel is the DLTP element [4] (six node triangular) composed of DKTP (discrete Kirchhoff triangle plus) for bending and the classical LST element (linear strain triangle-six nodes) for membrane. The stud and contact elements are represented by a bar element with either six or twelve degrees of freedom (DOF). The formulation is based on the small elasto-plastic strain theory and the updated Langrangian (U.L.) configuration for the representation of the deformation and stress states. Both geometrical and material nonlinearities are considered in the formulation of constitutive relations. The adopted elements have been successfully used in the study of various types of structures with geometrical and material nonlinearities [4-81.

Different constitutive relationships and yield cri- teria are defined to account for various material nonlinearities encountered in analysis. The von Mises yield criterion incorporating normality flow and hardening rules is used in the representation of the plasticity of steel. The concrete characteristics are specified by an incremental hypoelastic plane stress material model. An uniaxial stress-strain curve is

assumed for the representation of the concrete behaviour. The proposed curve also depicts strain softening of concrete at cracks and tension stiffening effect between cracks during the cracking process to allow for material weakening as stresses progress. Concrete failure in plane stress state is monitored by an implemented biaxial failure envelope. Moreover, a stress-rotating crack model is employed for modelling the crack propagation under loading and unloading. Yam-Chapman’s empirical relationship between the interface shear force and slip is used to describe constitutive behaviour of the shear connector. The behaviour of the contact element, subjected to Coulomb’s friction law, is defined by a bilinear relationship between the tangential rigidity of the element and the interface slip increment.

UPDATED VARIATIONAL MODEL

For an oriented structure, the variational representation of equilibrium relations in the current configuration C(t) in a local orthogonal coordinate system (see Fig. 1) [6] is written as follows, in tensor notation, for all admissible test functions (virtual displacements (u”)

W = tr([D: ][a,]) dV - W,,, = 0, (1)

where [or] represents components of Cauchy stress tensor (0, = 0 along the thickness n), [OF] is the virtual deformation gradient tensor, (u*) = (u*v *w*) is the virtual displacement field and IV,,, represents the virtual work due to applied forces.

From an algorithmic point of view, we indicate the known equilibrium configuration at the beginning of a step by C’ and the estimation of desired configuration C(t) by C’. The variational problem expressed in an incremental form becomes

W = W(c2 + A WI,., = 0, (2)

where WI,., defines the contribution of residual forces in the estimated configuration C2 and AWlr2 is the improvement to obtain an equilibrium for the step C(t) - c2.

Y

t

X

Fig. 1. Definition of the Cauchy stress approximation.

Nonlinear analysis of composite bridges 1153

The expression for W in the C2 configuration is

w;= s

tr(PiVd) dV* - Whx,, (3) y2

where W: corresponds to W for the configuration C2 (superscript) using the C2 geometrical space descrip tion (subscript). In the updated formulation, eqn (2) is written in the C2 description. We thus have

W= W;+(AW),=O, (4)

where W: is given by eqn (3) and

AW,= s

WW~lb~l + PiW~l) do2 - Whx, (5) c?

in which [E;] is the virtual Green-Lagrange deformation tensor in the configuration C2. Note that from a practical point of view, the integration is carried on V’ instead of V2. The small strain deformation assumption justifies the changes of integration volume (V’ x V’).

To update the stresses, [of] can be obtained from the stress tensor [S]] given by the following equations [5-71

[S:] = det[F][q-‘[a2][Fl-r (6)

with the deformation gradient [F] between C2 and C’ defined by

au au au --- ax, ah 82, I 1 rl=v1+ g 5 E = VI + [A I,

I

aw aw aw zgay,aZ, 1

(7)

where II = u:, v = vi, w =w:. One may show from eqn (6) that for small strains

(Fig. 1)

Pm = m local in C’ local in C2

(8)

since

det[l;l z 1.

By polar decomposition

where [R] is the rotation matrix between the local

coordinates of configurations C’ and C2. The constitutive relations are written in the incremental form using

[ST] = [a’] + [S] - ‘1. (10)

For elastic behaviour, in vectoral notation with the elasticity matrix [H,], we have

S;-‘= [H,]E;-' (11)

with

E- ‘I = fwTv?l - m.

For plastic material, the increment [$-‘I is cor- rected according to proper constitutive laws for different materials to respect the corresponding yield criterion.

FINITE ELEMENT MODELS

The general variational relations are discretixed by a finite element approximation to give

w=A, W’. (12)

The general presentation of eqn (2), in discretixed form, is

W = 2, u*'(f - [P'J Asf) = 0,

where the superscript e refers to element, incremental form, is

(13)

or, in

[Kr] AU = R, (14)

where [Kr] is the tangent stiffness matrix and R, the residual vector.

Plate /shell element

We use the same type of element (DLTP) to simulate the steel plate and the concrete slab of composite bridges. The DLTP element is obtained by the superposition of a discrete Kirchhoff model (DKTP) for bending and a linear strain triangular element (LST) for membrane (see Fig. 2). The DLTP element has six nodes with (u, v, w) at mid-side nodes and (u, u, w, 0x, I+, 02) at corner nodes [4,6]. The approximations for the displacement field are given

by


(Plate bending, DKTP) (Membrene, LST)

(Flat shell, DLTP)

(a) Definition of plate element IXTP

Steel bare in concrete plate Smeared layer of reinforcement of equivalent thickness

(b) Representation of reinforcement

Fig. 2. Adopted plate/shell elements.

with

The functions (N, ’ . . N6) represent classical quadratic approximations [9, IOJ and Hi, H.F are given elsewhere [4,6]. It has been shown that such an approximation for w,,~ and w,). has provided an efficient nonlinear element [6].

The Green-Lagrange tensor is defined with quadratic variation for u and v and linear variation for w in the calculation of nonlinear components. The geometrical stiffness matrix in the tangent stiffness matrix is obtained with a linear interpolation of (u, 0, w).

The elementary stiffness matrix is integrated at six Gauss points in the plane of the element with a proper number of gauss or radau points through the thickness of the element. In order to avoid a singular- ity in the assembled global matrix for tlat elements, the local dement matrices are increased by a fictitious local variable 6, at each corner [9]. Derivation of the element stiffness matrix and the residuals can be found elsewhere [4,6].

The representation of reinforcement is one of the particular aspects of concrete plate elements. In the present study, each set of reinforcement is idealized as a smeared two-dimensional membrane layer with equivalent thickness, ~sitioned at a constant coordinate with respect to the normal axis in the local system, as shown in Fig. 2b. The reinforcement is treated as an integral part of the element. Its stiffness is added to that of the concrete plate to obtain the stiffness of the element as

[K’] = [K::] + [KC]

and the elementary residual as

RY=R:+R:

with R: defined as

(1.3

(16)

where Sz represents the elementary uniaxial stresses of the reinforcement and [B] is the strain-displacement


matrix. The subscripts c and s denote concrete and positive value of d” and nonxero value of d,, for a steel bars, respectively. If an angle exists between the given joint, imply separation and occmmm~ of slip,

steel bars and the global coordinates, the stiffness respectively, of the two contact nodes of the joint

contribution of the reinforcement can be evaluated as concerned.

where t is the equivalent thickness of the reinforcement layer and [R] is a rotational transformation matrix from the local to the global system. The matrix [JY,] is the elementary elasto-plastic matrix of reinforcement between the membrane and bending deformations and their relevant internal forces in the local coordinates.

Interface element model

The composite action between the concrete deck and the steel girders in composite bridges is described by two interactions at the interface: traction and shear due to the presence of mechanical connections, such as studs. If no shear connectors are provided, it is also possible to consider the beneficial contribution of adherence and friction to the development of a certain degree of composite action. The composite action provided by interface friction deteriorates as the interface slip or separation of the two surfaces, take place.

To simulate the aforementioned phenomena at the interface, we first consider a set of generic two-dimensional joints, with paired nodes, which separate the two solid bodies in contact, as shown in Fig. 3a. The relative tangent and normal displacement sets at interfaces, d, and 4, respectively, are given by

d,=(d,-d,).t (18)

dn = (d, - d,) * n, (19)

where d, and d, are the sets of updated displacements of the contact joints associated with concrete and steel, respectively. Vectors t and n are unit vectors in the positive tangential and normal directions. A

(a) Ncunendature for intedaa de&ptbn

Fig. 3. Finite element modeling of the interface.

The following assumptions are made: (a) no media penetration (d,, 3 0) and no traction at interface if no mechanical connections are provided, (b) all interface deformations of a contact joint depend only on the relative deformations of its paired contact nodes, and the displacements in the normal and tangential directions are independent; (c) the components of interface forces, which follow some proper constitutive laws, are proportional to the corresponding relative displacement increments of the two contact nodes.

Let [Kr] be the global stiffness matrix and F the global nodal force vector of a system with two solid bodies S and C having no interaction at their interface. In this case, the system can be partitioned into two independent components with [K,] and F, representing the concrete body and [K,] and F, representing the steel body. We therefore have two independent groups of equations. If interface interaction does exist, we need to modify [Kr] and F to include the contribution of the interface media. This can be done by introducing equilibrium conditions at the common boundary of concrete and steel. It is assumed that the following equilibrium condition is satisfied at interface r

Uy)+C,=O, (20)

where L is a simple linear operator according to assumption (c) and Co is the vector of external forces acting on interface r. The vector y represents the relative interface displacement set at the interface, which can be written as y =(&,d,)T=dc-d,, if a discrete form is adopted. We can introduce eqn (20) by writing another variational formulation for the entire system

W= IV,+ W,+ &Ir(L,(y)+C,,)dr, s

(21) r

where WC and W, are the virtual work related to concrete and steel, respectively. Their discretixed form is given separately by eqn (13) if no linkage exists at the interface. The vector 61 is a virtual displacement set and has to be boundary admissible. Here, we make 1 identical to the relative interface displacement set y

s Grl*(L(y)+C,)dr =6y: L(y)dr

r s r

+6yf s C,dr. (22) r

Assumptions (b) and (c) indicate that L(y) = [oh, = [~](d< - 4) if a linear interpolation between the two


nodes of a joint at the interface is adopted. The matrix [FC] is the constitutive matrix of the interface. Therefore, we have the virtual work of the whole system in discretized form

W= W,+ W,+(fid,-c5d,)7‘tK](d,-dJ)

+ (ad, - SdJTCo. (23)

Let U, and U, he the separate displa~ment fields of two solid bodies (concrete and steel) on interface r. Since d,aU, and d,oU,, the third and fourth terms in eqn (23) will only modify the elements in the global stiffness matrix and the global nodal force vector, which are associated with d, and d,, without any increase or decrease of the number of equations of the global system [9].

We can partition displa~ment sets U, and U, as U, = (d, , d,>’ and U, = (4, dzjT and corresponding nodal force vectors F, and F, as F, = (f, , fcjT and F# = jfs, f,]‘. Invoking equilibrium for the whole system in the discretized form of eqn (23), by sett- ing virtual work W = 0, and recalling that SU, = (ad,, sd,}’ and SU, = (6d,,6d,}T are arbi- trary virtual displacement sets, we can obtain

i- f, -l

-_ f, -‘C,

~ i fs-Co ’

(24)

f*

where f, and f, are modified as f, - CO and f, + Co, respectively. In eqn (24), the cells modified by K, (=[rc]) represent the interactions at the interface between bodies S and C. If (K] is a zero matrix, the stiffness matrix will return to [KT] with two sets of independent equations for the solid bodies since no interaction exists between them.

In this study, the contact mechanisms and mechanical connections are simulated by the use of a bar element with two nodes i and j, as shown in Fig. 3b. The formulation of this element is based on the small relative displa~ment theory. According to assumptions (bf and (c) made earlier, the bar element can be seen as two independent linear springs having a stiffness k, perpendicular to the longitudinal axis of the bar and k, parallel to the longitudinal axis of the bar. The element stiffness can be easily obtained by applying the principle of virtual work to three springs defined in the three-dimensional local coordinate system, one in the normal direction and two in the transverse directions of the bar element. The elemen-

tary stiffness matrix has the following form

k, 0 0 -k, 0 0 0 k, 0 0 -k, 0 1

1 0 0 -k, 0 -k, 0 0 0 k, 0 k, 0 1

(25)

The corresponding elementary nodal displacement vector is (d:, d:)‘: For the element connected to the summits of the plate element, the elementary stiffness matrix has to be expanded to a 12 x 12 matrix to accommodate the rotational displacements at each node. In this ease, a unique fictitious elementary rotational stiffness $ is assigned to the corresponding rotational ~spla~ments. The elementary matrix, eqn (25), can be transformed from the local to the global system by using a transformation matrix [RJ

[GIG = Wl&I~~l. (26)

For each interface element, the stiffness matrix and nodal forces are updated on the basis of current displa~ment volume. Depending on the values cho- sen for k, and k, , numerous iterations may be needed until equilibrium at the interface and assumption (a), made earlier, are satisfied for a given degree of tolerance.

MATERIAL MODELLING

In this section, the constitutive performance of concrete, steel, interface friction and mechanical connections are described in a form suitable for numerical computation. Essentially, material modelling includes the definition of the constitutive relationships, of the onset of material yielding, and of material post-yielding behaviour.

Steel model

The von Mises yield criterion with normality flow rule is used to represent the material plasticity of steel

F&r’], [6], K) = J($ tr[a’]*) - I = 0, (27)

where [a’] is the deviatoric stress tensor, ci(~) the uniaxial yield stress and K the hardening parameter.

For subsequent yielding, the isotropic hardening rule is used. The plasticity matrix [H,], with first order plastic consistence, is used for calculating fKT]

[H,,] = [He] - ; vvT (28)

with

2CT

VT = % [He]


(a) Uniaxial stress - atrain law for concreta

(b) Biaxial failure envelope

Fig. 4. Concrete constitutive law and failure envelope.

and

aF' a’7;+-$yv,

where [II&] is the elastic matrix and i; is the slope of the uniaxial stress-strain curve.

A layered approach is used to consider the development of plastification through the thickness for both concrete and steel plate elements. The element thickness at an integration point is interpreted by a number of radau or gauss sampling points; a layer being defined by the reference coefficient of the radau or gauss point and its corresponding weight. The stresses at each sampling point are calculated according to the plasticity criterion defined for steel by eqn (U), and the proper constitutive relationships described later for concrete in the plane stress state.

Concrete model

The reinforced concrete model for plane stress conditions used in this analysis was developed to predict the global behaviour of concrete structures using the finite element method [ll]. The various assumptions used in the derivation of the model can be summar+& in the following five points:

1. It is an hypoelastic ineremen tal model for plane stress situations.

2. Concrete is assumed isotropic up to failure, either by crushing in compression or cracking in tension.

3. After cracking concrete is treated as an orthotropic material, using a smeared crack approach.

4. Two cracks may form at one point in orthogonal directions and their orientations follow either a principal stress or a principal strain orientation, called stress rotating or strain rotating model, respectively.

5. After cracking or/and crushing, stresses undergo strain softening and the tangent modulus is set to zero in the direction associated with failure.

Various features and relationships defining the concrete model are presented in the following sections.

Prefailure relutiotkships. The uniaxial stress-strain curve for compression, proposed by Saenx [12] is adopted. If a, is the ultimate strength and c, the associated strain (both negative), the stress-strain relationship is expressed as

0= 0

(29) 1+

with

and in which E, is the initial tangent modulus. Equation (29) describes the stress-strain relationship of concrete up to peak stress (Fig. 4a). The post-peak relationship, or the strain softening branch beyond Lo, will be described later. The tangent modulus E is given by

The Saenz equation adequately depicts the usual concrete stress-strain relationship. However, it is valid only for ratios of E, to E, larger than or equal to 2.0. Since it is common to have steeper stress- strain curves for high-strength concrete, it is proposed to define a pseudo-ultimate strain, e *, obtained by isolating L in eqn (29) for a ratio of E, to E,, equal to 2.0, so that the Saenx equation could be used [l I] with 6, replaced by c* in eqns (29) and (30).

In a plane stress state, the biaxial stress failure envelope is defined in three regions in the principal stress coordinate system (Fig. 4b): biaxial tension, biaxial compression and tension-compression. In the biaxial tension region, concrete is assumed to cracking at f; for any principal stress ratio. In the biaxial compression state of stress, it is commonly accepted that the failure envelope is a function of the stress

1158 J. J. L.XN et al.

ratio and the Kupfer and Gerstle description is used. The mla~onship proposed by Kupfer and Gerstle can be expressed as [13]

1 + 3.65cr cc,= - (1 +e)Z f;

with principal stresses ur and c2, CT, being algebraically larger or equal to c2

GQ.0. *2

In eqn (31), o, is negative and f: is the uniaxial compressive strength (positive by convention). In the case of ~nsion~omp~~ion, a bilinear failure envelope, similar to the one proposed by Balak~shnan and Murray [14], is adopted. A distinction by a value of a = cx, is made to identify a failure either in tension or in compression (Fig. 4b). The stress ratio defining this limit is written as

23f; a,,= -34fI. (32)

For a value of a algebraically larger than LX,,, the failure will be the crushing of concrete at a strength determined by the limit value of ~7~ on the failure envelope between points a and b. For values of c1 smaller than a,,, tensile failure will occur at a stress level determined by the failure envelope between points b and c.

fi t...h

Fig. 5. Model of concrete in tension.

t Y. fty

x.‘tx

~

I 4 4 Z.fn -_-_--_._-_

t + t

(a) Three dimensional Coulomb’s law

(b) Nonnai load.dis~ment curve

Fig. 6. Constituti~e ~lationships for contact element.

Poisson’s ratio in tension and at low compressive stresses remains unchanged. However, its value increases for compressive stresses about 75% of the ultimate. Based on experimental observations [13] and on practical considerations, the relationship used for Poisson’s ratio is given by

0 J

v=v0+(0.5-vo)+ 2 (1 -a) (33) d EU

in which v, is the initial Poisson’s ratio, @z is the minimum principal stress and u, is determined by eqn (29). The factor (I - CI) is equal to 1 .O in an uniaxial state of stress, and reduces progressively to zero for equal 6, and tr2 stresses by taking into account the confinement effect of 6, on c2.

Post-failure moaklling in tension. The post-failure model for concrete, either plain or reinforced, was developed to predict the behaviour of concrete after cracking in the direction of the principal tensile stresses. The essence of the model is the consideration of tension softening effect of concrete at cracks (called crack process zone) and tension stiffening effects of concrete between cracks of a reinforced concrete member [ 151. The shape of the strain softening curve for plain concrete (Fig. Sa) was based on several experimental studies in which the energy


dissipation during cracking was reported. For the tension stiffening effect, the derivation of some of the parameters were based on assumptions of the CEB model [16]. A detailed description of this model can be found elsewhere [I 1, 151. The final stress-strain curve, describing the average contribution of concrete in a reinforced concrete specimen is illustrated in Fig. 5b. This curve is not unique but varies as a function of the reinforcement ratio and also the orientation of the crack to reinforcement. The average tensile stress carried by concrete after cracking is called uST since it includes tension stiffening effects.

In a biaxial stress situation, the shear modulus is also reduced progressively as a function of the stress reduction at the crack after tensile failure in one direction. The post-cracking shear modulus is given as

>G,,=O.,~ 2(1 + va)

(34)

in which o,, is the cracking stress and E2 is the tangent modulus in the direction parallel to the crack. If two cracks exist, G,, is equal to G,,,, which has a shear retention factor of 0.1. Similarly, when the average stress normal to the crack reaches zero at the end of the tension softening curve (Fig. 5a), the crack is assumed completely open and G, is reduced to G,,,,.

Post-failure modelling in compression. Unlike tensile failure where only the properties perpendicular to the crack change, when compressive failure occurs at one point, concrete in all direction at this point is assumed crushed. The tangent moduli in the two directions and the Poisson’s ratio are set equal to zero while the shear modulus is set equal to G,, , given by eqn (34). If both directions are in compression before crushing, both undergo strain softening. When compression failure occurs in a tension-compression state (along line a-b in Fig. 4b), only that portion of concrete stressed in compression before the failure occurs undergoes strain softening; the stress in the other direction being set equal to zero.

The softening branch in compression is assumed linear. It starts at the peak point of a (E,,, a,) and ends at point b (ecmaX, Q,,,) (see Fig. 4a), with L,,,

and =,, defined by

Unloading from the softening branch is allowed along a path defined by a point d on the softening branch, where unloading starts, and a focal point with coordinates (Q, 6,) given by

u,= 4.lU~” (37)

cr,, 4= &” - 1.1 z. (38)

E

The model presented herein has been successfully used to predict the behaviour of various structural components tested in the laboratory. The types of members to which the model response has been compared include reinforced concrete specimens in direct tension, prestressed concrete panels in biaxial tension, plates subjected to various plane stress conditions, and panels loaded axially and laterally. In these analyses a degenerated plate-shell element has been used [l 1,15,17]. The agreement was very satis- factory for strength, displacements, crack patterns and failure modes. From these comparisons, the model appears to be greatly reliable in predicting the behaviour of laboratory specimens.

Contact-no connectors

If no mechanical connection is provided at the interface of concrete plate and steel girders, composite action is developed, in a limited manner, by interface friction. In a constitutive approach to the contact m~hanisms at the interface, the foIlo~ng set of incremental formulations are applied on the basis of updated configuration with Ad, evaluated iteratively at load step p $1

d; + ’ = d{ + Ad, 2 0 * separation, so

dp+ 1 = df: + Ad,, < 0 + contact, n

then/;+’ =f;+k,Ad,<O

andf$‘+‘=fr+k,Ad,#O,

where fn and f, are normal and tangential contact forces at interface, k,, and k, are updated normal and tangential rigidity; dn and d, are relative displacements at a contact point, respectively. The superscripts p and p + 1 denote loading steps p and p + 1.

Coulomb’s law, 9 =f; - ,L& = 0, is used as yield function to define the onset of interface slip. Then

If~+‘l6~]ff:+‘l~no interface slip

IfC+‘] > rlfi+‘l *interface slip occurs

and k, = 0,

where p is the coefficient of friction. Its value depends upon the materials of two contact bodies and surface conditions. For a three-dimensional contact problem, Coulomb’s law defines a yield surface as shown in Fig. 6a and can be expressed as in eqn (39). It is assumed that fi lies in the x-y plane with two components, I, and & and f. is in the z-direction

Ii60 J. J. LIN et al.

(see Fig. 3b)

~~=[(f~~‘)2+cf~~+:f)2]-~~~+‘( =o. (39)

The rigidity k, has to be much greater than in the local structural stiffness so as to assure no ~netration at the interface. For k,, a reasonably large value has to be assumed to make the interface act as a linear spring before slip. A proper choice of values for these two parameters depends upon the type of structure analysed and indi~dual numerical experience.

Special considerations have to be given to the interface during separation and slipping, which indicate zero values for k, and k, and may introduce rigid body motions, which cause the singularities in solving a problem. To avoid such potential problem, k, and k, can be defined by some relatively small values, rather than zero, but without causing numerical inaccuracy. This can be done by multiplying k, and k, by a small value, called multiplier C. The adopted load-deflection curves for k,, and k, are given in Figs 6b and c, respectively.

Shear connectors

When shear connectors are provided at the interface, two different constitutive laws are used separately to describe the normal and tangential behaviour. In the normal direction, the stiffness of a stud is evaluated as an usual steel bar element with k, = lT?A, Jh,, where Es is elastic modulus, A, the area of cross-section and h, is the height of the stud. The material plasticity is represented by the von Mises yield criterion with normality flow rule, given by eqn (27).

On the tangent surface, the constitutive behaviour is defined by the typical shear-slip function proposed by Yam and Chapman [18] and shown in Fig. 7

Q = a(1 - e-h), (40)

where y is the interface slip, equivalent to d,, defined before and a and b are constants. By choosing two points on the excremental curve so that the relationship, y2 = 27,) can be satisfied, the constants a and b can be defined by

Q: a=ze, (41)

Fig. 7. Typical load-&p curve for flexible shear connector.

(42)

The stiffness in the tangential direction can be expressed as (Fig. 7)

k de=.&e-h’ ‘=dy

If eqn (43) is written in incremental form, we have

AQ = k, by. (44)

The following empirical load-slip equation reported by Ollgaard et al. [19] has been used in this study to calculate Q, and Q,

g = (1 _ e-0.7a9y/S

”

with Q, evaluated by the following equation adopted by the CSA Sl6-88 Code [20]

The equation is valid for studs with h, 2 4d, where h, and d are the length and diameter of the shear connector, respectively. In eqn (46), f, is the ultimate strength of steel in MPa and A,, is the cross-section area of the shear connector in mm2.

NONLINEAR SOLUTION STRATEGY

The solution algorithms in the nonlinear analysis of composite bridges need to be carefully planned since high nonlinear responses can be expected. Such nonlinearities mostly come from the load history dependent characteristics of materials used in composite bridges and may occur at low load levels due to interface slip, separation or concrete cracking, In order to stay as close as possible to the real response of the structure, measures are taken as follows in the solution of a problem

R(U, I ) = d F - R,,, PJ)t (47)

where U represents a n-degree displacement field and /1 is a load factor. Writing them in an iterating form, we have

Ui+lsU~+AU (48)

II’+’ = A’ f AL. (49)

The Newton-Raphson (N-R) method is used to evaluate the correction AU in eqn (14). This strategy is also implemented by the arc length method, in which a constant As is imposed for two sequential loading steps in the analysis. This approach is further enhanced by making the arc length at current load-


ing step p + 1 adjustable. The adjustment is done according to the number of previous iterations (I,,) and the average number of iterations (Id) expeoted for convergence. The arc length at step p + 1 can be expressed as

Asp+ ’ = A~P~(r~/Z~). (50)

The solution scheme is therefore accelerated when the structural response is more linear at lower loads. When higher nonlinearity is encountered, loading increment is adjusted by eqn (50) so that better and faster convergence can be expected.

NUMERICAL EXAMPLES

Various experimental examples are considered to illustrate the applicability of the proposed finite element model and material models described in this paper. It is often that only f: is reported in the published results of experiments. In the following analyses, EC and E= values are evaluated according to the equations proposed by AC1 Committee 363 [ZI] (identified by superior plus sign) whilef ; is evaluated by the relationship proposed by Collins and Mitchell [22] (identified by super asterisk), based on the given value off :.

Composite beams of Imperial College

A number of simply supported and continuous composite beams were tested at Imperial College [IS, 231, Two of these specimens are analysed to validate the models.

Simply supported single-span beam [ 181. The simply supported beam selected from the test series is beam El 1, loaded at midspan. The beam consists of an 152 mm thick concrete slab and an I-section steel girder, 304 x 152mm x O.l96kN/m (12 x 6in x 44 lb/ft BSB), connected by 100 uniformly distributed head studs, 19 x 100 mm (314 x 4 in). The geometric come-on of the heam is given in Fig. 8a. The beam failed at an ultimate load of 5 17 kN (51.9 tons).

The material properties are

steel 4 = 265 MPa (17.2 ton/S) Es = 2 x IO5 MPa (13,300 ton/ir$) K = 0.022 (hardening factor) a = 48 kN [eqn (4111 b = 2.9 (l/mm) [eqn (4211

concrete f : = 50 MI% (7250 psi) f; = 2.3 hiPa* E,=3.0 x lO’MPa+ cm = O&028+ v. = 0.2 (assumed).

Only one quarter of the beam is considered in the analysis by taking advantage of the double symmetry of the specimen. The finite element mesh used is shown in Fig. 8b. The load-midspan deflection and interface slip at 448 kN (45 tons) are shown in Figs 9a and b, respectively. Very good coincidence with the ex~~rnen~l results are observed.

J_ it- 334 _ (12in)

(a) Elevation and cmss section

(b) Finite element ~~

Fig. 8. Imperial. College simple span composite beam Ei 1.


G=-

- Numerical - Expedmental

Deflection (inches)

(a) Midspan deflection

0.0 910 18.0

(b) interface slip dist~b~t~ along the span

Fig. 9. Imperial College simple span composite beam El I (1 ton = 9.95 kN, 1 in = 25.4 mm, I ft = 304 mm).

Two-span conrin~ous beam [23]. The continuous beam, named CBI and shown in Fig. IOa, had two equal spans with two point loads placed at midspan of each span. The girder is a rolled I-steel section 152 x 76 mm x 0.0053 kN/m (6 x 3 in x 12 lb/ft). Two rows of shear studs of dimension 9 x 50 mm (3/S x 2 in), are uniformly spaced at 146 mm (5.74 in) along the girder. The ultimate load measured was 150 kN (15.1 tons) at which shear connector failure occurred followed by the spalling of concrete on one span.

The material properties used are

Steel -6, = 300 MPa (19.5 ton/in*)

= 320 MPa (20.8 ton/in’) (for reinforcement)

Es = 2 x 10’ MPa (13,300 ton/in*) K = 0.005 (hardening factor) a = 17.8 kN [eqn (4111 b = 2.90 (1 /mm) [eqn(42)]

Concrete f : = 47.6 MPa (6900 psi) f ; = 2.3 MPa (assumed)* EC = 3.0 x lo” MPa+ ecu = 0.0027+ v0 = 0.2 (assumed)

The finite element discretization of one half of the beam is shown in Fig. lob. It should be noted that only one quarter of the beam has been considered in the analysis due to symmetry with respect to the

* - ’ (lln) -i- (Et, -’ (3 in)

(b) Finite element idealization

Fig. 10. Imperial College two span continuous beam CBI (1 ton = 9.95 kN, 1 in = 25.4 mm, 1 ft = 304 mm).

middle support. Figures I la-c compare both the analytical and experimental deformed configuration of the beam, the interface slip and the steel strain of the outmost fibre of the bottom flange at a load level equal to 121 kN (12.2 tons), respectively. The pre- dicted ultimate capacity is 144.3 kN (14.5 tons), which is different from the experimental value of 150.2 kN (15.1 tons) by only 3%.

In the above two examples, the excellent agreement with the experimental results demonstrates the validity of concrete, steel and shear connector elements presented previously for the analysis of composite beams. The validity of the models is further illustrated by the following applications.

Composite bridge model of Car&on University [24]

A third-scale single-span bridge model with three steel beams of W250 x 39 sections was tested at Carleton University. The model was 6 m long and 2.08 m wide. Each beam had 80 studs (15 mm in diameter) in two rows distributed uniformly along the span. The loadings and geometric layout are shown in Fig. 12a. The three point loads applied are aimed at simulating a real truck load and are placed on central girder B, as shown in Fig. 12b. The measured ultimate loading capacity of the specimen is 761 kN.

Material properties:

steel f, = 300 MPa (I-section)

= 400 MPa (reinforcement) .E,=2 x 105MPa

rc = 0.005 (hardening factor) a = 45.0 kN [eqn (4111 b = 2.90 (l/mm) [eqn (42)1

Concrete f I=42.1 MPa f ; = 3.5 MPa E, = 3.0 x 104 MPa+ L, = 0.0026’ v, = 0.2

Nonlinear analysis of composite bridge-s 1163

tie half of the structure is analysed. The finite element mesh for the cross-section is illustrated in Fig. 12~. Twenty groups of elements are used for the concrete plate and the steel beams along the span. The numerical failure load for the model is equal to 725 kN, which is about 5% different from the experimental value. Comparisons between numerical and experimental results for load vs maximum deflection curves, strain variation in the bottom flange of steel beams and strain in the top surface of the concrete plate are given in Figs 13,14 and 15 for girders A and B, respectively. Very good agreement can be observed from these figures. The discrepancies observed at certain load levels can be explained by the fact that unloading and reloading were necessary during the test at load levels 533 kN because the loading actua- tor slid to one side and at load level 711 kN because the top flange of girder B buckled. This certainly has affected the loading performance of the model.

Co~~~~~o~ bridge modei of Win~or ~n~ers~t~ [3]

A two-span continuous bridge model, tested at Windsor University, is simulated to illustrate the adaptability of the proposed model to different types of bridges. The bridge model was constructed at a scale of one tenth. The bridge has two equal spans of 1980 mm in length and 950 mm in width. The steel frame consists of four longitudinal steel beams (S5 x 10) and three welded diaphragms (S4 x 7.7) in each span and one at each support. The concrete deck is 51 mm thick and has a reinforcement ratio of 0.2%. Shear studs, 12 mm in diameter and 38 mm in length ($ x 1 i in) are spaced at 100 mm) (3 studs/ft). No stud is provided near the middle support over one fifth of the span length. The bridge model is loaded monotonically to failure using a simulated truck loading located at centre of each span but eccentrically with respect to the Iongitudinal axis of the model, as shown in Fig. 16a.

The material properties used

See1 fy = 325.0 MPa (steel beam)

= 400 MPa (steel bar, assume) E, = 2 x 10’ MPa (13,300 ton/W) K = 0.005 (hardening factor) a = 28.7 kN [eqn (4111 b = 2.90 (l/mm) [eqn (42)]

Concrete f: = 36.0 MPa f : = 2.0 MPa* EC = 2.7 x 10’ MPa+ cW = 0.00024 + v. = 0.2 (assumed)

One span is analysed because of the symmetry about the middle support. For simplicity, the transverse diaphragms are aligned with the bottom of the

p-122ton p-12.2LIMrr

i 2%%% I 0.0 55 11.0 10.5 22.0

0.0

-0.2

-0.4

-0.8

-0.8

-1.0 t - Numelkal l

+ Expsma-

400.0

-2000.0

0.0

2000.0

4000.0 0.0 5.5 $1.0 le.5 22.0

11. Continuous beam CBI at P = 12.2 tons (1 ton = 9.95 kN, 1 in = 25.4 mm, 1 ft = 304 mm).

longitudinal steel beams. The finite element idealization is shown in Fig. 16b. Figure 16c shows the analytical and experimental load vs midspan curves of girders A, B and C. It can be seen that the numerical and experimental curves for each girder are very close. The analytical failure load is 625 kN (140.5 kips) while the experimental ultimate load is 623 kN (140.0 kips).

Numerical bridge model with contact elements

The contact element developed previously is aimed at the simulation of the contact mechanism at the interface of concrete slab and steel girder when no m~hani~al shear connectors are provided. Due to the lack of test results for this type of bridge, the Carleton University bridge model was used again but with all stud elements replaced by contact elements. The main purpose of this exercise is to see how the behaviour of bridges can be affected by the interface media if no studs are provided. Moreover by using the same bridge model, the adequacy of the analysis can be


1 l%P

+---I-l ~~~~ 16x2 plate elements /group In 8 cross section 20 groups abng the span

(c) Finite element idealization

Fig. 12. Carleton University bridge model.

checked by comparing the numerical results to the known performance of the tested modd.

The finite element mesh is the same as that shown in Fig. 12~. The contact properties before the occur- rence of interface deformation are k, = lO’* and

r 800.0

z 60Q.0

3 400.0

ii I-

200.0

- Numerical

.Ok__ik-ci-i 0.0 30.0 60.0 SC.0 120.0

bairn delbcibn (mm)

(a) Girder A

% 600.0

Ii t

400.0

- NumerIcal

E 200.0

0.0

Maximum $efbcIlM (mm)

(b) Glrdar B

Fig. 13. Load vs maximum deflation of Carleton Unive~ity bridge model.

Distance from left support (m)

(a) Girder A

6000.0

4WC.0

2wo.o

0.0 J 0.0 1.0 2.0 3.0 4.0 5.0 6.0

Distance from left suppod (m)

(b) Girder B

Fig. 14. Variation of steel strain along the bottom flanges of girders.

k, = IO5 N/mm. Numerical experience indicates that the initial magnitude of these two parameters in a reasonable range f 107-10’0 fork, and 104 - 10’ for k,) does not affect structural behaviour. However, a low

1 8

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Distaw fmm tat support (m)

(a) Concrete strain along Girder A

1WO.O

1000.0

500.0

0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0

Distance from bit suppod (m)

(b) Concrete atrain along Girder B

Fig. 15. Variation of ion~tu~nai strain on the top surface of the concrete slab.

Nonlinear analysis of composite bridges

Table 1. Comparison of numerical results for different values of C and fi

1165

Maximum Load had Total Coulomb’s deflection carried by carried by Numerical

friction at failure girder A at girder B at axial load Multiplier coefficient of girder failure failure at failure

C P B (nun) (kN)t orN)t (kN)t

96.5 254.5 351.0 10-a 0.0 103.8 Vi’7

(12.8)

‘336:.“1

(47.5)

y;b’

10-E 0.5 66.7 108.4 251.6

o&30)

IO-6 0.0 98.4 (14.2) (33.1) (47.3) 95.6 356.3 451.9

10-6 0.5 69.4 (12.7) (46.6) (59.3)

7 Tbe values in parentheses are the percentages of the numerical values to the experimental ultimate capacity of the specimen with studs (761 kN).

value results in a slower convergence rate. At the onset of interface separation or/and sliding, the new rigidity of a contact element was evaluated on the basis of Ck, and Ck, (see Figs 6b and c). Obviously the choice of C depends on the initial values of k, and k, used and has to be relatively small enough to the initial values of k,, and k,. Two values of C, equal to 10m6 and 10-s, are considered to see the influence of C value on the structural behaviour. If a contact element connects to the paired summit nodes of concrete and steel plate elements, its elementary rotational rigidity k. is set equal to 10m6. Two values

(a) Cross section and loading position

(b) Finite element idealization

f

Desecllon (hilee)

(12) Load vs midapan curves

Fig. 16. Windsor University two span continuous bridge model (1 kip = 4.45 kN, 1 in = 25.4 mm).

of Coulomb’s friction coefficient, p = 0.0 (no friction) and p = 0.5 are considered to evaluate the effect of interface friction on the hehaviour of the bridge. The choice of the second value is based on the lower value of the experimental results obtained by Rabbat and Russel[25].

The numerical rota1 load vs maximum deflection curves are presented in Figs 17a and b for girders A and B, respectively. The experimental load vs maximum deflection curves with full composite action plotted in the figures in order to be compared to the curves obtained for non-composite and partial composite action provided by interface friction. The minimal influence of multiplier C on the structural behaviour can also he seen from the respective curves in Fig. 17 and in Table 1.

The Coulomb’s friction coefficient p has a notable influence on both the ultimate capacity and the deformation of the structure., as shown in Fig. 17. The total analytical ultimate load is about 46% of the tested ultimated capacity of the model with studs for p = 0.0 and 60% for p = 0.5. By examining Fig. 17b, it can be seen that the curves of the model for p = 0.0 deviate immediately from the experimental one with studs after loading since no composite action is developed. For the numerical model with 1 = 0.5, the deflection curves are almost the same at the initial loading stage due to the composite action provided by interface friction. The analytical curves then gradually deviate from the experimental one due to the deterioration of the interface friction as load increases. When the load carried by each girder is examined, it can be found that the load carried by girder A does not change very much in all cases considered in Table 1, while the load carried by girder B changes with the value of p. This indicates that the partial composite action is mainly developed at the interface of girder B. This is reasonable since loads are all applied to girder B only. As far as deformations are concerned, larger deflections are observed when p = 0.0 than when p = 0.5 at all load levels. Also girder A with p = 0.0 snapped back at certain load steps, as shown in Fig. 17a, due to lateral slip at the interface.


800.0

Maximum defltiMn (mm)

(a) Load va maximum deftection of Girder A

.

Numerical (with cotli~ elemerlis)

-o- v = 0.0

- p = 0.5 >

+ p = 0.0

-D- p=o.s >

EXpfHiIlWN~l (with shrds)

c= 10-6

&lo-@

Numerical (with contact elements)

Maximum deflection (mm)

(b) Load vs m~imum deflection of Girder B

Fig. 17. Load vs maximum deflection curves with varying p and C parameters

2.0 4.0

Distance from left suppon (m)

(a) Interface slip

2.0 4.0

Distance fmm left suppxi (m)

fb) Interface separation

Fig. 18. Comparison of relative displacement at interfaces.

Figure 18a shows that the lateral slip at the interface of girder A and longitudinal slip at the interface of girder B for p =O.O and p =0.5, with C = 1O-s. Much larger Iaterai and longitudinal slips can be observed when p = 0.0 which therefore Leads to a larger deflection. Figure 18b indicates that the model with p = 0.0 atso shows larger interface separation than the model with ~1 = 0.5.

It is important to point out that in all cases considered here failure started with yielding of girder B. For the case p = 0.5, the load drops near failure, The concrete crushing near the loading positions and loss of the interface friction are observed.

CONCLUSIONS

A family of finite element models for the analysis of reinforced concrete slab-on-steel girder bridges has been presented in this paper. The analytical procedure is based on the updated Lagrangian formulation in order to trace large deformations and rotations of the composite bridges. These models have been verified by comparison with the results of experimental tests from different sources and very good agreement has been obtained for all cases

studied. The material models have been proven to be effective in the ~muIation of concrete, steel, shear connector and interface contact behaviour at differ-


ent load levels. The analyses with assumed properties for describing interface contact mechanisms have given acceptable results. However further experimental verifications are suggested. The proposed analytical procedure can be readily used for the study of the behaviour of composite bridges, for the evaluation of existing bridges and is also a perfect tool for carrying parametric studies. By extension, it can also be adapted to the analysis of composite floors in build- ings.

Acknowledgements-The authors wish to thank the Quebec Ministry of Transportation for funding this project and Lava1 University for providing the computer system which was used to carry out the numerical analysis.

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REFERENCES

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nonlinear analysis of composite bridges by the finite element method

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