Multiple bifurcation paths visualized by a computational asymptotic stability theory

Abstract

Multiple bifurcation (MB) is a compound stability problem of nonlinear structures, in which the singular system stiffness matrix at the stability point coincidentally undergoes two or more zero eigenvalues. The corresponding critical eigenvectors are generally coupled in the actual buckling modes, as frequently observed in symmetric structures.This paper presents a practical diagnosis to visualize all secondary paths branching from the compound stability point when the stiffness matrix is deficient in rank by two. To find post-critical equilibria around an MB point (MBP), asymptotic expansions are assumed to consist of two homogeneous and one particular solution of the singular stiffness equations. Furthermore, hyper-dual numbers are introduced to numerically evaluate the derivatives of the system stiffness with respect to the nodal degrees-of-freedom. The resulting bifurcation equations are a set of three simultaneous cubic polynomial equations with unknown perturbation parameters that can be solved by a popular graphical software. The number and location of existing equilibria above and beneath the MBP at the load level can exactly indicate each type of branching path, such as asymmetric, unstable, or stable symmetric bifurcation paths.Two numerical examples demonstrate that the proposed asymptotic theory can reliably diagnose MB. The first one is the Augusti model, i.e., a simple rigid column supported by elastic springs that shows that the numerical prediction by the proposed method is consistent with Augusti’s analytical results. The second example was computed using the plate and shell finite element (FE) program to verify that the asymptotically expanded and visually solved bifurcation equations work well and can be implemented in existing FE codes for stability analysis, including imperfection-sensitivity and optimization.