Solving the dynamic response and buckling of Timoshenko beams with variable cross-section using the weak form integral equation method

In this paper, free vibration and buckling analysis of functionally graded Timoshenko beams are investigated. Stiffness and mass of Timoshenko beam are assumed to change using an exponential function along the beam length. Fredholm transformation approach is applied to solve the governing equation of motion. The usual method for applying Fredholm transformation is approximation of mode shape function by a power series that this method needs four repetitive integrations to obtain weak form of the governing equation. In this paper, a novelty is introduced into usual approach that is approximation of bending moment acting on the cross section of the beam by a power series. This novelty requires two successive integrations to obtain the weak form. Therefore, the mathematical process required is shorter and simpler. Regarding two integrations, two constants appear in the resulting weak form equation that are determined using appropriate boundary conditions for Timoshenko beam. Approximation of bending moment results in a system of linear algebraic equations that the natural frequencies are determined by calculation of a non-trivial solution for this system of equations. Buckling analysis of Timoshenko beam is also presented and the buckling loads are determined for beams with various end boundary conditions. The efficiency and accuracy of the presented approach are investigated through comparison of the numerical results with those available in the existing literature.