An Explicit Method for Geometrically Nonlinear Dynamic Analysis of Spatial Beams

Abstract

An explicit method based on the central difference method for nonlinear transient dynamic analysis of spatial beams with finite rotations using corotational total Lagrangian finite element formulation is presented. The kinematics of the beam element is described in the current element coordinate system constructed at the current configuration of the beam element. The beam element has two nodes with six degrees of freedom per node. Three rotation parameters referred to the current element coordinates are defined to determine the orientation of element cross section. A rotation vector is used to represent the finite rotation of a base coordinate system rigidly tied to each node of the discretized structure. Note that the values of nodal rotation vectors are reset to zero at current configuration. The element deformation nodal forces and inertia nodal forces are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory, the d\'Alembert principle and the virtual work principle in the current element coordinates. The standard central difference method is applied to the incremental displacement and rotational vector, and their time derivatives. The orientation of the end cross section of the beam element is updated by the incremental nodal rotation vector. A Numerical example is presented to demonstrate the accuracy and efficiency of the proposed method.