A Family of Orthogonal Iteration Methods for Tracing the Nonlinear Equilibrium Path of Structures

Nonlinear structural analysis serves as a fundamental tool for accurately predicting structural bearing capacity and ultimate strength. The incremental-iterative solution scheme represents the prevailing methodology for tracing nonlinear load–displacement responses and is implemented in most commercial finite element software. To enhance the robustness and computational efficiency of existing schemes, this paper first revisits the incremental-iterative framework, providing a detailed analysis that clarifies the distinct roles of the load increment factor in the predictor and corrector phases. Subsequently, a novel framework of updated orthogonal iterative schemes (UOIS) is established. Within this framework, the current generalized stiffness parameter (CGSP) and a cumulative indicator Si are introduced in the predictor phase to adaptively control the magnitude and sign of the load increment, respectively. In the corrector phase, four enhanced orthogonal iteration strategies are formulated. Furthermore, to improve computational efficiency, a novel acceleration strategy is proposed, which embeds a secant prediction operator in the predictor phase, thereby circumventing the costly assembly and inversion of the tangent stiffness matrix. The results demonstrate that: (1) compared to the conventional generalized stiffness parameter (GSP), the proposed CGSP exhibits superior stability in tracking stiffness variations, offering a more reliable indicator for adaptive step-size control; (2) the cumulative indicator Si reliably identifies load limit points and accurately distinguishes between loading and unloading regimes; (3) the UOIS framework demonstrates strong convergence in tracing complex equilibrium paths with multiple critical points and exhibits significantly superior robustness under large increment sizes compared to the generalized displacement control method (GDCM); and (4) the secant-prediction acceleration strategy achieves substantial improvements in computational efficiency without compromising solution accuracy.