Static analysis of cracked Timoshenko beam resting on elastic foundation using Galerkin’s method

In this paper, the static behavior of a cracked Timoshenko beam resting on an elastic foundation is investigated through the approximate method of Galerkin. The crack is simulated using a rotational spring whose stiffness is obtained in terms of geometric and material characteristics of the cracked structures. The governing differential equations are derived by considering a discontinuity in the axial direction of the beam. In the solution of the problem by use of the Galerkin’s method, the beam is divided into two parts, as the first part is defined from the support of the left-hand side to the crack position and the second is assumed from the crack position to the right-hand end of the beam. For each part, a deflection function is considered with five unknown parameters. Ten unknown parameters for two parts are determined using four boundary conditions, four continuity conditions in the crack point and two equations of the Galerkin integral. The deflection equation throughout the beam is obtained from the known parameters. The results are reported in two boundary conditions including simply supported-simply supported (SS-SS) and clamped-free (C-F) and the validity of the presented solution is accepted with some references. The results show that the elastic foundation decreases the side effect of the crack in the structure.