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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.
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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