PSI - Issue 75

Thomas Constant et al. / Procedia Structural Integrity 75 (2025) 660–676 Author name / Structural Integrity Procedia 00 (2025) 000–000

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˜ g B , ˜ d B Kriging surrogates of the X B -component of the functional decomposition of g and d , respectively c A , c B Computational cost of model m A and m B , respectively X A c Random variable associated to the c -th load configuration n dof Number of degrees of freedom of a system M Mass matrix C Damping matrix K Sti ff ness matrix u Displacement vector f External force vector

2. Mathematical statement of the durability problem and of its associated reliability problem

Without loss of generality, let us consider that m A corresponds to a group of FE models, that produces a set of local stress tensors at any node of a zone of interest supposedly critical for the system. The critical zone is assumed to be discretized by a set of nodes, denoted hereafter by N c . Themodel m B is a fatigue model that maps this stress fields σ k ∈N c to the damage d . In the following, x A ∈ R n A denotes for the input parameters of m A and corresponds to a realization of the random vector X A solely related to external excitations, while x B ∈ R n B is a realization of the random vector X B gathering the structural and material characteristics of the critical location. Thus, the input vector that enters the numerical chain m B ◦ m A takes the form x = ( x A , x B ) ∈ R n A + n B . The discrete time-dependent output of the model m A given the inputs x A , gathers a set of temporal stress tensors, i.e. for all nodes k ∈N c of the zone of interest and reads:

m A ( x A ) = { σ k ( x A , t ) } t

(1)

∈ 0 , T , k ∈N c

Then, the model m B computes the local damage in the critical region of interest based on this set N c of temporal stress tensors provided by m A .

d ( x A , x B ) = m B ◦ m A ( x A , x B ) = m B ( { σ k ( x A , t ) } t

x B )

(2)

∈ 0 , T , k ∈N c ,

A more detailed description of the models m A and m B is proposed in the following subsection, providing the necessary theoretical framework for the subsequent formulation of the structural reliability problem.

2.1. Mathematical description of model m A

In the most general case, the model m A represents a chain of linear or non-linear dynamic finite element (FE) models. However, for the sake of brevity, we here restrict m A to a single modal superposition dynamic FE analysis. A comprehensive review of FE analysis of system can be found in e.g. (Imbert (2000)) but is not the subject of this paper. Under the assumption of linear elasticity, the output of m A corresponds to the solution of the following motion equation:

M ¨ u ( x A , t ) + C ˙ u ( x A , t ) + Ku ( x A , t ) = f ( x A , t )

(3)