Qualified Geometric Stiffness for Linear Buckling and Second-Order Nonlinear Analysis of Framed Structures

There exist various potential energy formulations dealing with the linear buckling and second-order nonlinear analysis of framed structures with different degrees of refinement in the kinematic model. However, the geometric stiffnesses derived often give rise to different structural behaviors, which indeed represents a confusion regarding their qualified usage in bifurcation and post-buckling analysis. This study aims to carry out a comprehensive evaluation of the validity of the geometric stiffness for use at the predictor and corrector phrases of an incremental analysis based on the rigid-body motion test. To remove the unbalanced element forces caused by the nonqualified geometric stiffness, a supplementary correction matrix is developed according to simple kinematic and static analysis in the context of rigid rotations. An updated Lagrangian approach-based force recovery procedure (FRP) is presented for updating the element forces with improved reliability and efficiency, when using relatively large step size. Some benchmark problems which exhibit compound three-dimensional nonlinear behavior of framed structures are solved to clarify the capabilities of rigid-body qualified and nonqualified geometric stiffnesses along with the existing FRPs for different mesh sizes, step sizes and load patterns. It is shown that the proposed procedure can be adopted to predict correct buckling loads and post-buckling equilibrium paths without adding extra computational costs.