On the Almansi-Michell problem for flexible multibody dynamics

In flexible multibody systems, it is convenient to approximate many structural components as beams. In classical beam theories, such as Timoshenko beam theory, the beams crosssection is assumed to remain plane. While such assuption is capable of capturing the kinetic energy of the system accurately, it cannot represent the strain energy adequately. In the authorss recent paper, an systematic approach was proposed for the modeling of three-dimensional beam problems. The proposed approach is based on the Hamiltonian formalism and leads to an expansion of the solution in terms of extremity and central solutions. This paper extends the previous approach to the "Almansi-Michell problem," i.e., three dimensional beams subjected to distributed loads. Such problems can be represented by non-homogenous Hamiltonian systems, in contrast with Saint- Venants problem, which is represented by homogenous Hamiltonian systems. The solutions of Almansi-Michells problem are not only determined by the Hamiltonian coefficient matrix but also by the applied loading distribution patterns. hence, the contributions of the loading pattern need to be taken into account. A dimensional reduction procedure is proposed and the threedimensional governing equations of Almansi-Michells problem can be reduced to a set of one-dimensional beams equations. Furthermore, the three-dimensional displacements and stress components can be recovered from the one-dimensional beams solution.