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.

Rankin 和 Nour-Omid 在 1990 年代率先在共旋转 (CR) 分析中使用非线性投影矩阵（Rankin 和 Nour-Omid，1988 年；Nour-Omid 和 Rankin，1991 年），并且几乎成为了过去三十年的 CR 配方扣除。然而，该矩阵严重依赖于局部位移与全局位移变化的歇斯底里和复杂的推导，导致内力矢量和切线刚度矩阵的复杂表达式，这可能会贬低使用 CR 方法的初衷的简单性和便利性。本文首先讨论现有的与元素无关的 CR 公式，目的是为通用 CR 分析开发一个新的、更简单的框架，避免使用传统的非线性投影矩阵。该方法包括单元内力计算的两个步骤。第一个是获得内力的初步结果。这是通过遵循传统的与单元无关的 CR 公式来完成的，但删除了涉及投影矩阵的项，因此产生了内力和切线刚度矩阵的简单公式。第二个是校正步骤，以获得满足单元自平衡条件的新内力矢量。这一步继承了使用投影矩阵的精神，但直接在全局坐标系中进行，从而避免了局部-全局旋转的复杂纠缠，并且与CR分析中使用的局部CR坐标系的选择无关。这进一步导致了对可以在 CR 框架中协作的不同类型元素的简单统一的表述。修正力的闭合公式以及相关的一致切线刚度矩阵是针对不同的修正方法推导出来的。还表明，现有的用于无穷小旋转分析的线性投影矩阵是当前校正方法的一个特例。多个数值例子涉及各种元素和元素局部 CR 框架的不同选择，以证明所提出框架的性能。结果表明，对于所有示例，结果的准确性与结合传统非线性投影仪矩阵获得的结果的准确性相当。