Updating reliability models of statically loaded instrumented structures

Updating reliability models of statically loaded instrumented structures

The study extends the first order reliability method (FORM) and inverse FORM to update reliability models for existing, statically loaded structures based on measured responses. Solutions based on Bayes’ theorem, Markov chain Monte Carlo simulations, and inverse reliability analysis are developed. The case of linear systems with Gaussian uncertainties and linear performance functions is shown to be exactly solvable.

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FORM and inverse reliability based methods are subsequently developed to deal with more general problems. The proposed procedures are implemented by combining Matlab based reliability modules with finite element models residing on the Abaqus software. Numerical illustrations on linear and nonlinear frames are presented.

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Problem statement

We consider a finite element model for an existing, statically loaded structure and write the equilibrium equation as

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Here is the probability measure. Furthermore, we assume that the structure is instrumented with a set of s sensors and a set of measurements from these sensors is available for N episodes of loading conditions. These measurements could be on structural strains, displacements, or reactions transferred to the supports and the model for measurements is expressed as

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In further work, for the sake of simplicity, we write

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simply as

We introduce the notation

to denote the 1 x Ns row vector of measurements.

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Normalized sensitivities

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Analysis :

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The posterior failure probability is given by

be the posterior pdf of X after the measurements have

been assimilated. By applying Bayes’ theorem, we get

Where is the posterior pdf, C is the normalization

constant is the likelihood function, and is the prior pdf. As has been noted, the noise term appearing in

forms a sequence of independent random vectors for k = 1,2,..., n . Consequently, the likelihood functionis given by

Following this, the posterior probability of failure is obtained as

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It is important note that the MCMC based methods do not

require the explicit determination of the normalization

constant C. Also, we note that the determination of the

quantity can be construed as the system

identification step in which the probabilistic model for the

load and system parameters are updated based on the

measurements made. In the present study we focus on

extending concepts based on FORM to characterize

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An exactly solvable case :

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Estimation of posterior failure probability

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