A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams

Abstract

A nonlinear intrinsic formulation for the dynamics of initially curved and twisted beams in a moving frame is presented. The equations are written in a compact matrix form without any approximations to the geometry of the deformed beam reference line or to the orientation of the intrinsic cross-section frame. In accordance with previously published work, when effects of warping on the dynamics and local constraints on the cross-sectional deformation are ignorable, the in- and out-of-plane St Venant warping displacements (which are fully coupled for nonhomogeneous, anisotropic beams) need only be included explicitly in the determination of a suitable elastic law (a two-dimensional problem) and need not be considered explicitly in the one-dimensional equations governing global deformation. In this paper it is presupposed that such an elastic law is given as a one-dimensional strain energy function. Thus, the present equations, which are based on only six generalized strain variables, are valid for beams with closed cross-sections and for which warping is unrestrained. When simplified for various special cases, they agree with similar intrinsic equations in the literature. Although the resulting equations are Newtonian in structure (closely resembling Euler's dynamical equations for a rigid body), the formulation adheres to a variational approach throughout, thus providing a link between Newtonian and energy-based methods. In particular, the present development provides substantial insight into the relationships among variational formulations in which different displacement and rotational variables are used as well as between these formulations and Newtonian ones. For computational purposes, a compact and complete mixed variational formulation is presented that is ideally suited for development of finite element analyses. Finally, a specialized version of the intrinsic equations is developed in which shear deformations is suppressed.