Analytical solution for a heterogeneous Timoshenko beam subjected to an arbitrary dynamic transverse load

Abstract

This paper presents the analytical solution for dynamic response of a simply supported heterogeneous Timoshenko beam. Elastic constants and material density are assumed to vary in the thickness direction according to an arbitrary even function. First, the governing equations are established and the dependence of Timoshenko shear coefficient on material functions is derived for rectangular cross-section. Once the equations of motion are derived, they are solved using the classical integral transform method and the Laplace transforms of the beam deflection and of the slope of the beam are presented for an arbitrary type of dynamic load. Then a particular impact problem of a three-layered beam is solved and the analytical results obtained by means of the numerical inverse Laplace transform are confronted with the results of numerical simulations. The correctness and the validity of derived analytical solution are then discussed based on this comparison. Analysis made show that using the presented solution the response of symmetric laminate and functionally graded beams and of symmetric layered beams with functionally graded layer(s) to an arbitrary dynamic load, impact included, can be investigated. The presented approach is suitable for solving the optimisation problems due to its efficiency. • The analytical solution for dynamic response of a heterogeneous beam is derived. • The problem was solved for arbitrary type of transverse load, impact included. • The shear correction factor significantly depends on the beam composition. • Results can be used for symmetric laminate, functionally graded and layered FG beams. • Presented approach is very effective and suitable for solving optimisation problems.