Large Spatial Deformations of Rods Using Generalized Variational Principles

Summary

General equations and corresponding incremental formulations for the analysis of rods undergoing large spatial deformations are presented. The derivation is based on a generalized three-dimensional variational principle with the increments of the Lagrangian stresses, the deformed position vector and the rotation as independent variables. “Engineering” strains and conjugate “Jaumann”-stresses are related by a semilinear material law. The principle allows assumptions for the stresses and for the kinematical variables simultaneously. This is used in the reduction to the one-dimensional description for general rods. Warping effects and arbitrary cross sections are considered in the fully spatial and geometrically nonlinear description. The resulting generalized principle in terms of seven stress resultants and conjugate displacements and rotations is given in its general and incremental forms. It could be used for a mixed or hybrid finite element approach. In the paper “exact” stiffness matrices are obtained via the integration of the local equations of the principle in the form of a system of first-order ordinary differential equations. Numerical results of sample problems are given.