Keywords: multi-layered planar beam, composite structures, inter-layer slip, exact solution, elasticity.

Multi-layered composite systems are widely used in construction as floor and wall elements, and in bridge construction. Composite or multi-layered structures are built up from two or more components made of same or different materials. Each of the constituents fulfills the function for which its material characteristics are most suited. The layers are mutually connected by shear connectors to form an interacting unit. The behaviour of composite multi-layered structures depends on the type of connection between the components. Rigid shear connectors develop a full composite action between the individual components of the member, so that a conventional principles of the solid beam analysis can be employed. Flexible shear connectors, on the other hand, permit the development of only a partial composite action. As a result there exists an inter-layer slip of a sufficient magnitude to have a major effect on the deflections and stresses of the multi-layered system. Thus, the analysis procedures require the consideration of the inter-layer slip between the components of the multi-layered system.

Since the experiments are quite complicated and very expensive to perform, there is a need of development of simple and effective numerical procedures such as finite element methods for calculating composite multi-layered structures. Many researchers [1,2,4] have developed numerical procedures for geometrically and materially non-linear analysis of composite structures with an inter-layer slip. The accuracy of such procedures can easily be assessed by comparing their results to exact solutions presented in this paper.

An exact solution of the mechanical behaviour of a simply supported, multi-layered beam-column is well known in the literature [5,6,7,8]. Unfortunately, there seems to be no report on the exact solution of multi-layered continuous beams over two or more spans. This paper presents the exact analysis of mechanical behaviour of geometrically and materially linear continuous multi-layered composite planar beams over two or more spans. The theory of a layered beam deduced in the paper uses the following assumptions: (1) material is linear elastic, (2) displacements, strains and rotations are small, (3) shear deformations are not taken into account (the "Euler-Bernoulli beam"), (4) strains vary linearly over each layer (the "Bernoulli hypothesis"), (5) the slip modulus of inter-layer connection is constant, (6) the friction between the layers is not taken into account, (7) the curvature is the same for all layers, (8) the number of layers is arbitrary, (9) cross-sections are symmetrical with respect to the plane of deformation and remain unchanged in shape and size during deformation.

We show the validity of the present analytical model and illustrate the efficiency of the derived solution with numerical examples. Kinematic and static quantities, and, in particular vertical deflections of composite beams, are calculated by the present procedure and compared to those obtained by Goodman and Popov [7] and the European code for timber structures Eurocode 5 [3]. Some numerical examples clearly show an excellent agreement between the results calculated by the present analytical solution and those of Eurocode 5 [3].

1

A. Ayoub, "A two-field mixed variational principle for partially connected composite beams", Finite Elements in Analysis and Design, 37, 929-959, 2001. doi:10.1016/S0168-874X(01)00076-2

2

B. Cas, M. Saje, I. Planinc, Nonlinear finite element analysis of composite planar frames with interlayer slip, Computers and Structures, accepted for publication.

3

Eurocode 5, Design of timber structures, Part 1-1: General rules and rules for buildings, ENV 1995-1-1, 1993.

4

C. Faella, E. Martinelli, E. Nigro, Steel and concrete composite beams with flexible shear connection: `exact' analytical expression of the stiffness matrix and applications, Computers and Structures, 80, 1001-1009, 2002. doi:10.1016/S0045-7949(02)00038-X

5

N. Gatessco, "Analytical modeling of nonlinear behavior of composite beams with deformable connection", Journal of Constructional Steel Research, 52, 195-218, 1999. doi:10.1016/S0143-974X(99)00026-7

6

U. A. Girhammar, V. K. A. Gopu, Composite beam-columns with interlayer slip-exact analysis, Journal of Structural Engineering, ASCE, 199(4), 1265-1282, 1993. doi:10.1061/(ASCE)0733-9445(1993)119:4(1265)

7

J. R. Goodman, E. P. Popov, Layered beam systems with interlayer slip, Journal of Structural Division, ASCE, 94(11), 2535-2547, 1968.

8

J. R. Goodman, E. P. Popov, "Layered wood systems with interlayer slip", Wood Science, 1(3), 148-158, 1969.

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