Paper 1Unification of Variational and Sensitivity Criteria for Stiffness Extrema: A Coupling Matrix Approach

This paper establishes a formal mathematical bridge between two previously independent frameworks for identifying stiffness extrema in proportionally loaded structures: Mang’s variational criterion (2025) and engineering sensitivity analysis. Using a novel "Coupling Matrix" approach (C = I + dKg/dKe), we prove the analytical equivalence between the stationarity of the potential energy functional and the vanishing of total stiffness sensitivity. This unification demonstrates that stiffness extrema are not merely mathematical curiosities but correspond to physical states where geometric and section stiffness contributions reach a nonlinear equilibrium. The findings provide a rigorous theoretical justification for using sensitivity-based tools to predict structural "turning points" in design practice. 本文在比例加载结构刚度极值的两个独立框架——Mang 变分准则 (2025) 与工程灵敏度分析之间建立了正式的数学桥梁。通过引入创新的“耦合矩阵”方法 (C = I + dKg/dKe),我们证明了势能泛函的平稳性与总刚度灵敏度消失之间的分析等效性。这一统一性证明了刚度极值不仅是数学上的特征点,更对应于几何刚度与截面刚度贡献达到非线性平衡的物理状态。该研究为在工程实践中使用基于灵敏度的工具来预测结构“转折点”提供了严谨的理论依据。