The direct force correction based framework for general co-rotational analysis

Abstract

The use of nonlinear projector matrix in co-rotational (CR) analysis was pioneered by Rankin and Nour-Omid in 1990s (Rankin and Nour-Omid, 1988; Nour-Omid and Rankin, 1991), and has almost became a standard manner for CR formulations deduction over the past thirty years. This matrix however relies heavily on a hysterical and sophisticated derivation of the variation of the local displacements to the global ones, leading to complicated expressions for the internal force vector and the tangent stiffness matrix, which may devalue the simplicity and convenience for the original intention of using CR approach. This paper begins by making a discussion on existing element independent CR formulation and the objective is to develop a new and simpler framework for general CR analysis that avoids using conventional nonlinear projector matrix. The methodology consists of two steps in the element internal force calculation. The first one is to obtain a preliminary result of the internal force. This is done by following the conventional element-independent CR formulation but dropping the terms involving projector matrix and therefore yields simple formulations of the internal force and the tangent stiffness matrix. The second one is a correction step to obtain a new internal force vector that satisfies the element self-equilibrium condition. This step inherits the spirit of using projector matrix but is conducted directly in the global frame, thus avoiding complicated entanglement of local–global rotation and is independent of the choice of the local CR frame used in the CR analysis. This further leads to a simple and unified formulation for different kinds of elements that can be cooperated in CR framework. Closed formulation of the correction force as well as the related consistent tangent stiffness matrix is derived for different correction approaches. It is also shown that the existing linear projector matrix used for infinitesimal rotation analysis is a special case of the current correction approaches. Multiple numerical examples involving various kinds of elements and different choices of element local CR frame are presented to demonstrate the performance of the proposed framework. The outcomes show that for all the examples the accuracy of the results is comparable with those obtained in conjunction with conventional nonlinear projector matrix.