Contribution to beam theory based on 3-D energy principle

Abstract

This paper presents a general elastic beam theory, which is consistent with the principle of stationary three-dimensional potential energy. For the sake of simplicity we consider the case of a rectangular cross section. The series expansion of the displacement field up to fourth-order in h (dimension of the cross section) is defined by 45 unknowns. The first variation of the potential energy must be zero but we only impose that each term guarantees an $O(h^5)$ error. By adding supplementary lateral boundary conditions and on two extremities end cross section of the beam, we finally arrive at a well posed system of unidimensional differential equations. A linear algebraic dependence with respect to 16 displacement fields allows us to reduce the unknown to 19 displacement fields. To our knowledge this work is the first contribution to this end when the beam problem is completely three-dimensional.