Dynamic analyses of plane frames by integral equations for bars and Timoshenko beams

Abstract

Since, especially for higher frequencies, Timoshenko's theory gives more reliable results than Euler–Bernoulli's theory, systems of beams, like frames, under arbitrary dynamic excitations should be analyzed on the basis of this refined theory. In this paper, after deriving the basic fundamental solutions of a lateral unit point force and of a unit single moment, the deflection and the rotation of the beam cross-section are given as integral equation forms of the governing second order differential equation system in the Laplace and the frequency domain. Since in frameworks axial displacements also occur, the well-known integral equations for bars under tension or compression are added to complete the modelling of plane frame structures. From these, the unknown boundary values may be exactly determined via point collocation at the beam ends and solution of the resulting algebraic system. With these boundary values, the exact deflection and rotation can also be found at arbitrary interior points. Finally, the convolution quadrature method is applied to the Laplace domain integral equation formulation in order to analyze the time-dependent wave propagation process. Two examples are presented to demonstrate the method's exactness compared with conventional finite element results: a single beam where the shift of the resonance frequencies when using the Timoshenko theory instead of Euler–Bernoulli's theory is shown, and a two-storey framework where, in addition to the frequence response, the causality of the propagating wave is studied.