Structural reliability analysis of a stable equilibrium point based on the dynamic integrity of its safe basin

In continuation to as reported by Mazzilli et al. (in: Lacarbonara, Balachandran, Leamy, Ma, Terneiro Machado, Stepan (eds) Part I. Advances in nonlinear dynamics, Springer, Cham 2023), this paper gives a further step towards an improved reliability analysis of a dynamical system stable equilibrium point, based on the dynamic integrity of its safe basin. Although the ideas are applicable to a broader class of problems within dynamical systems theory, they will be illustrated in the discussion of the load carrying capacity of a structure liable to buckle. The analysis uses the joint probability density function that can be constructed from the statistical properties of both the buckling strength and the applied load, plus the probability density function of the dynamic integrity measure. Specification of the probability for the stable equilibrium point to achieve at least the reference buckling strength, associated with the reference integrity measure, will lead to the threshold of the action’s expected value (Thompson’s load) considering the basin’s integrity profile. Alternatively, once defined the thresholds for both the action and the dynamic integrity measure, coherently with the integrity profile, it can be evaluated the non-failure probability. Assuming the structural system is imperfection sensitive, the analysis proposes to depart from the probability density function of the system’s imperfection parameter itself, as the primary input variable, to obtain the probability density function of its buckling strength, as the output variable. Additionally, from the integrity profile, the dynamic-integrity measure’s probability density function can be obtained as the new output variable from that of the buckling strength, now taken as the input variable. Hence, a reliability measure can be made available to help engineers deciding whether a particular design is reliable, given it is safe, contributing to the improvement of current design practices.