Direct Evaluation of the Sensitivity of the Nonlinear Critical Point to Geometrical Imperfections

Abstract

A generalized path-following method is proposed for the direct evaluation of the critical point sensitivity to geometrical imperfections expressed as combination of given shapes. After a finite element discretization, the critical point is defined by a system of nonlinear algebraic equations imposing equilibrium and critical condition according to the null vector method. An arc-length equation is added to follow a path of critical points in the imperfection space. The solution of the extended system by a Newton scheme requires a Jacobian involving also the derivatives of the critical condition with respect to the FE DOFs and the derivatives of all the problem equations with respect to the imperfection parameters. An analytical and efficient computation of the Jacobian is achieved by a solid-shell model and a strain-based model of the imperfection. In addition, a mixed stress-displacement iterative scheme is devised for a highly efficient and robust solution. Finally, the derivatives of the critical point with respect to the imperfection parameters are also obtained analytically and can be used to generate a gradient-based critical path for a quick search of the worst-case imperfection. A compressed coldformed lipped channel beam is analyzed as a case study.