Nonlinear deformation decomposition and mode identification of plane structures <i>via</i> orthogonal theory

Abstract This article presents a novel method for nonlinear deformation decomposition and mode identification of plane structures based on orthogonal theory. The plane deformation of a square element is decomposed into rigid body linear displacements, rigid body rotation, tension-compression, bending, and shearing deformations. A complete orthogonal base vector set is constructed for these basic deformation modes, allowing any deformation to be expressed as a linear combination of these vectors. Compared with the traditional finite element method, the proposed approach demonstrates superior accuracy, convergence, and robustness, especially in identifying bending deformation, which is difficult to isolate using conventional techniques. The method also excels in mode identification, effectively revealing coupled modes such as shear-dominated shapes. It outperforms traditional approaches like animation and mass participation ratio methods by offering more accurate, visual, and comprehensive results. A numerical example involving a plane column validates the feasibility and effectiveness of the method, showcasing its potential in analyzing complex nonlinear deformations and identifying modal behavior in plane structures.