From Stability to Safety: The Influence of Dynamic Integrity Metrics on Load Capacity Definition

The elastic instability of imperfection-sensitive structural systems has long been a central concern in engineering. This sensitivity makes it particularly difficult to accurately determine their true load-bearing capacity, posing a persistent challenge for both theoretical modelling and practical design. In this study, we propose a coupled analytical framework that leverages a simplified yet paradigmatic model to emulate the essential behaviour of more complex structures that undergo unstable symmetric bifurcation. The paper framework integrates four traditionally independent axes of analysis: bifurcation theory (Euler load), imperfection sensitivity (Koiter load), dynamic integrity (Thompson load), and reliability assessment (load capacity). While the first two axes are well-established in classical stability theory and require minimal elaboration, the focus of this work is placed on the third axis — dynamic integrity. Here, we evaluate the impulsive/local integrity measure (IIM/LIM) and compare it with the global integrity measure (GIM) of the safe basin of attraction, for several loading conditions. The fourth axis investigates how the choice of different integrity metrics influences the estimation of load capacity, aiming at identifying the most reliable scenario using Bayesian inference. Specifically, we seek to maximize the conditional probability P(S|R), which represents the likelihood of a scenario being reliable given that it is safe. Notably, although the IIM/LIM profile is markedly higher than GIM’s for the system under consideration, both approaches converge to the same load capacity estimate. However, this conclusion should not be generalized, particularly for systems characterized by fractal safe basins, where LIM offers a more accurate and representative integrity measure. The paper concludes by outlining promising directions for future research. Such advancements could significantly improve the safety and reliability of engineering designs.