Stability Analysis of a Simple Strut with Elastic Springs Through Sensitivity Surfaces and Gradients

The global elastic stability of a strut supported by three inclined springs is studied using the exact displacement geometry. The discrete phenomenological model in the theory of stability, described by Thompson and Gaspar, exhibits different postbuckling behaviors, depending on the initial geometry for large imperfections, and this has various applications in engineering. Using the total potential energy method, a system of nonlinear algebraic equations is derived for the nondissipative system. The arc length method is adopted to solve the system of nonlinear equations, considering the stability of equilibrium points at each position, while detecting and traversing the critical points with the possibility of intervention. The characteristic equilibrium paths and their corresponding trajectories are shown and compared with the asymptotic solutions at the first critical point. A parametric analysis is performed, and sensitivity surfaces are constructed for several initial positions of the springs, represented by the initial angle in the horizontal plane. A wide range of independent imperfections with large amplitudes in two directions is considered. The concept of load capacity gradient functions of sensitivity surfaces is introduced and used to qualitatively and quantitatively analyze the stability behavior of the system located far away from the initial position. A few interesting observations and conclusions are obtained from the sensitivity analysis of a simple strut in the postcritical region. The critical combinations of imperfections for each static system are determined through analysis of the gradients. Further, the “stable” and “unstable” regions of the sensitivity surfaces are identified, with some observations made. Finally, the applications of load capacity gradient functions in structural optimization and form findings are reviewed.