A Boundary Integral Formulation for the Dynamic Behavior of a Timoshenko Beam

Mostly, the fourth order differential equation for the deflection is applied in Timoshenko’s beam theory although the second order system combining the deflection and the rotation is much more suitable. Considering this system in the Laplace domain, it is straight forward to determine the respective fundamental solutions by Hörmander’s method and to obtain the corresponding system of integral equations via the weighted residuum method.
Here, avoiding the highly complicated time-dependent fundamental solutions the Convolution Quadrature Method proposed by Lubich is applied resulting in a time stepping formulation for the determination of the time history.
For demonstrating the influence of shear deformation and rotary inertia, examples of a fixed-free and a fixed-simply supported beam are analysed in frequency domain as well as in time domain.