Keywords: composite beam, interlayer slip, Timoshenko beam theory, locking, finite element method.

Multi-layered structures have been playing an increasingly important role in different areas of engineering practice, perhaps most notably in civil, automotive, aerospace and aeronautic technology, to name a few. Within the domain of civil engineering, classic cases of such structures are steel-concrete composite beams in buildings and bridges, wood-concrete floor systems, coupled shear walls, concrete beams externally reinforced with laminates, sandwich beams, and many more. It is well known, that the behaviour of these structures largely depends not only on different materials of individual components, but also by the type of their connections. Many ways are available to connect the components. Usually, mechanical shear connectors are employed to provide a desired composite action. With the use of rigid shear connectors, a full shear connection and full composite action between the individual components can be achieved. Consequently, conventional principles of the solid beam analysis can be employed. Unfortunately, the full shear connection can hardly be realized in practice and thus only an incomplete or partial interaction between the layers can be accomplished. Consecutively, an interlayer slip often develops. This is of such magnitude as to significantly affect the mechanical behaviour of composite systems.

Hence, the inclusion of the interlayer-slip effect into multi-layered beam theory is essential for optimal design or accurate and realistic representation of the actual mechanical behaviour of multi-layered structures with partial interaction between the components. Much effort and a large number of research studies have been devoted to obtaining and to understanding the solution of the aforementioned problem. Early studies on beams with partial interaction between the layers were based on the assumptions of linear elastic material models and the Euler-Bernoulii hypothesis of plane sections. Perhaps the first but certainly the most quoted partial action theory was developed by Newmark et al. [1]. Up to now, a number of theories have been developed and presented in professional literature, e.g. [2]. Such complex problems are usually solved using numerical methods such as finite element methods. It is well known, that finite element models often experience so-called slip and shear locking, especially for high values of stiffness of the shear connection [3]. Besides, one of the basic assumptions of all aforementioned models with partial interaction between the layers was the most commonly used Euler-Bernoulli beam theory for each individual layer, respectively. It is well known that the main shortcoming of the classical Euler-Bernoulli beam theory is that transverse shear deformation is not allowed. Thus the application of the Timoshenko beam theory is indispensable for accurate prediction of the mechanical behaviour of the aforementioned structures. In the literature no reports exist on the finite element formulation of Timoshenko composite beams with an interlayer slip. In the present paper, we aim to fill this gap.

The objective of this paper is two-fold. Firstly, we present a new locking-free deformation-based finite element formulation for the linear static analysis of two-layer planar beams with a significant interlayer slip. Secondly, the incorporation of the transverse shear deformation into the two-layer composite beam theory with an interlayer slip. Consequently a new locking-free deformation-based finite element formulation for the numerical treatment of linear static analysis of two-layer planar Timoshenko composite beams with a significant interlayer slip has been proposed. In this formulation, the modified principle of virtual work has been employed as a basis for the finite element discretization. Only a deformation field vector remains the only unknown function to be interpolated in the finite element implementation of the principle. In contrast with many of the displacement-based, force-based and mixed finite element formulations of the composite beams with an interlayer slip, the present formulation is completely locking-free. Any kind of locking (shear, slip, curvature), poor convergence or stress oscillations are absent in these finite elements. Besides, the re-formulation of the composite beam theory with the consideration of the Timoshenko beam theory for the individual component of a composite beam represents one of the major improvement in the field of analysis of non-slender composite beams with an interlayer slip. As only a few finite elements are required to describe a composite beam of a frame with great precision, the new finite element formulation is perfectly suited for practical calculations.

1

N. M. Newmark, C. P. Siest, C. P. Viest, Test and analysis of composite beams with incomplete interaction, Proceedings of the Society for Experimental Stress Analysis, 1, 75-92, 1951.

2

S. Schnabl, I. Planinc, M. Saje, B. Cas, G. Turk, "An analytical model of layered continuous beams with partial interaction", Structural Engineering and Mechanics, 22(3), 263-278, 2006.

3

 A. Dall'Asta, A. Zona "Slip locking in finite elements for composite beams with deformable shear connection", Finite Elements in Analysis and Design, 40, 1907-1930, 2004. doi:10.1016/j.finel.2004.01.007

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