A co-rotational finite element formulation for buckling and postbuckling analyses of spatial beams Original Research Article

A consistent co-rotational finite element formulation and numerical procedure for the buckling and postbuckling analyses of three-dimensional elastic Euler beam is presented. All coupling among bending, twisting, and stretching deformations for a beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. However, the third-order terms, which are relevant to the twist rate and curvature of the beam axis, are also considered. An incremental–iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A bisection method of the arc length is proposed to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of third-order terms on the buckling load and postbuckling behavior of three-dimensional beams.