A Unified Probabilistic Framework for the Reliability and Robustness Assessment of Series Structural Systems: Axiomatic Foundations and Computational Approximations

This paper develops a unified probabilistic framework for the simultaneous assessment of reliability and robustness of series structural systems. Part I formulates an axiomatic theory of structural robustness: the robustness factor Frob is defined as the negative decadic logarithm of the ratio between the damaged-state system failure probability and a normalised target failure probability, and five formal properties are proven—normalisation, monotonicity in damage severity, exact combination bounds under independent damage scenarios, invariance under reliability-preserving transformations, and boundedness. Part II describes the computational core, the evaluation of series system failure probability: the principal approximation methods (simple bounds, the complement-product reformulation, Ditlevsen’s narrow bounds) are formally derived, and a geometric mean-bound approximation over a dominant-element subset is introduced, with four propositions establishing consistency with bounds, monotonicity, convergence, and a complete characterisation of the error direction under positive equicorrelation, including a provably conservative regime above a threshold correlation. Part III synthesises both parts into a computational framework that evaluates Frob from first-order reliability method (FORM) outputs alone, avoiding repeated multi-dimensional numerical integration in the inner loop of design optimisation, with an explicit, computable error bound guaranteeing invariance of the robustness classification; a two-sided Slepian envelope further extends this guarantee to systems with non-equicorrelated, non-negative inter-element correlation. The framework is illustrated through a numerical example and validated against a published case study of a glued–laminated timber truss.