Dynamic analysis of continuous beams by the boundary element method

Abstract

The main contribution of this work is the development of two Boundary Element Method (BEM) formulations for the dynamic analysis of Euler-Bernoulli continuous beams. The first one employs the static fundamental solution; due to this, it is named D-BEM, with D meaning domain. The other employs the time-dependent fundamental solution, and is called TD-BEM, with TD meaning time-domain. Both of formulations present a domain integral of the load. The D-BEM formulation presents an additional domain integral in the equations, thus requiring the discretisation of the entire domain. The TD-BEM formulation enables one to perform the analysis using only the boundary nodes, in the absence of initial conditions. As the problem is one-dimensional, the boundary is constituted only by the external nodes. At each node, one finds four variables: transverse displacement, cross-section rotation, bending moment and shear force. For this reason, not only one but two BEM equations are necessary to solve the problem. These are the equations related to the displacement and to the cross-section rotation. Both formulations produced results with good agreement between them, as the results of the three examples show. In the last example, a comparison is made between Boundary Element Method and Finite Element Method results.