A Comprehensive Study and Insight in Co-Rotational Approach Based Geometrically Nonlinear Analysis With Planar Solid Elements

The co-rotational (CR) approach is a widely adopted and efficient strategy for geometrically nonlinear analysis. The Classic Formalism, which derives full variations from local element to global coordinates, has been refined over the years and is generally regarded as theoretically sound. This study presents an insight into CR planar solid elements. To isolate these issues from element-specific effects, we focus on the simplest planar solid elements: four-node quadrilaterals and three-node triangles. A comprehensive investigation is conducted to provide a unified overview and systematic comparison of commonly used local element frame construction methods. The findings indicate that the Classic Formalism may exacerbate unrealistic asymmetric responses in symmetric problems, particularly when the local element frame lacks precision. To address this, we propose a direct force correction formalism that introduces a correction term to directly enforce spin equilibrium. On the one hand, the derivation process eliminates the need for a second variation in computing the tangent stiffness matrix; on the other hand, the corrected equal weights act on each node of elements and are directly related to the rotation matrix, rather than a variant of the rotation matrix. As a result, this formalism may reduce the asymmetry in symmetrical cases better than Classic Formalism. Importantly, the proposed formalism is broadly applicable across various element types and frame construction methods.