This document discusses methods for updating reliability models of statically loaded structures based on measured response data. It extends the first-order reliability method (FORM) and inverse FORM to incorporate sensor measurements using Bayesian techniques like Markov chain Monte Carlo simulations. Numerical examples on linear and nonlinear frames are used to demonstrate the proposed procedures for combining finite element models with reliability analysis modules.

1. Updating reliability models of statically
loaded instrumented structures

2. 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.
2

3.  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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4. Problem statement
We consider a finite element model for an existing,
statically loaded structure and write the equilibrium
equation as
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6. 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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8. In further work, for the sake of simplicity, we write
8
simply as
We introduce the notation
to denote the 1 x Ns row vector of measurements.

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10. Normalized sensitivities
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11. 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

12. forms a sequence of independent random vectors for
k = 1,2,..., n . Consequently, the likelihood function
is given by
Following this, the posterior probability of failure
is obtained as
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14. 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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16. An exactly solvable case :
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17. Estimation of posterior failure probability
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