Direct computation of critical equilibrium states for spatial beams and frames

In this paper, explicit formulas for second order derivatives of the residual vector with respect to the state variables for a geometrically exact 3D beam element based on the Reissner's model are presented. These derivatives are required when a direct non‐linear stability eigenvalue problem is solved by the Newton's method. If the external load is parametrized by a single parameter, such an eigenvalue problem consists of solving the critical state variables, the eigenmode, and the critical load parameter from the equation system consisting of the equilibrium equations, the criticality condition, and some auxiliary conditions depending on the type of a critical point. Copyright ©2011 John Wiley & Sons, Ltd.