Rigid Element Concept for Geometric Nonlinear Analysis of Structures Involving Postbuckling Response

In the incremental-iterative analysis of elastic nonlinear structures, great saving in computation can be achieved if distinction is made between the predictor and corrector phases. The predictor relates to solution of the structural displacements for given load increments, which affects only the number of iterations. For the sake of iteration, the equations used in the predictor need not be exact, but should be accurate to the level not to mislead the direction of iterations. In this study, it will be demonstrated that the use of the linear stiffness matrix [k e ] and a rigid-body qualified geometric stiffness matrix [k g ] in the predictor is sufficient for most purposes. The rigid-body qualified geometric stiffness matrix [k g ] can be derived from the virtual work equation for a rigid displacement field. In contrast, the corrector is concerned with computation of the element force increments and the updating of initial element forces at the end of each incremental step. In a nonlinear analysis, each incremental step can always be made practically small. As such, the element force increments can be computed using only the linear stiffness matrix [k e ], and the initial nodal forces updated by the rigid body rule. For the present purposes, the 2D beam element will be used as the vehicle of illustration of the ideas involved. The procedure will then be extended to treat the 3D beam structures and plates and shells. The effectiveness of the ideas presented herein will be demonstrated in the solution of some nonlinear problems involving the postbuckling response.