Solution strategy and rigid element for nonlinear analysis of elastically structures based on updated Lagrangian formulation

Three phases are essential to the incremental–iterative analysis of elastically nonlinear structures: the predictor, corrector and error-checking phases. The predictor relates to solution of the structural displacements for given load increments, which affects only the number of iterations. The corrector is concerned with recovery of the element forces for given element displacements, which governs the accuracy of solution. By choosing a robust incremental–iterative scheme, the use of only the linear stiffness matrix [ k e ], via the predictor and corrector, is good enough for solving a wide range of moderately nonlinear problems, and this is sufficient for most practical purposes. For highly nonlinear problems, i.e., for those with winding loops in the postbuckling responses, a rigid-body qualified geometric stiffness matrix [ k g ] should be added in the predictor to ensure proper directions of iteration. The geometric stiffness matrix [ k g ] that is rigid-body qualified is derived from the virtual work equation by assuming the displacement field to be of the rigid type. The above ideas are demonstrated in the solution of several nonlinear problems.