Derivation of the exact stiffness matrix for a two-layer Timoshenko beam element with partial interaction
Résumé
This paper presents the full closed-form solution of the governing equations describing the behaviour of a shear-deformable two-layer beam with partial interaction. Timoshenko's kinematic assumptions are considered for both layers, and the shear connection is modelled through a continuous relationship between the interface shear flow and the corresponding slip. The limiting cases of perfect bond and no bond are also considered. The effect of possible transversal separation of the two members has been neglected. With the above assumptions, the present work can be considered as a significant development beyond that available from Newmark et al.'s paper [4]. The differential equations derived considering the above key assumptions have been solved in closed form, and the corresponding "exact" stiffness matrix has been derived using the standard procedure basically inspired by the well-known direct stiffness method. This "exact" stiffness matrix has been implemented in a general displacement-based finite element code, and has been used to investigate the behaviour of shear-deformable composite beams. Both a simply supported and a continuous beam are considered in order to validate the proposed model, at least within the linear range. A parametric analysis has been carried out to study the influence of both shear flexibility and partial interaction on the global behaviour of composite beams. It has been found that the effect of shear flexibility on the deflection is generally more important for composite beams characterized by substantial shear interaction.