PSI - Issue 78

Pooria Mesbahi et al. / Procedia Structural Integrity 78 (2026) 1839–1846

1841

2.1. Seismic capacity updating (SCU)

Nonlinear static push-over analysis (POA) is usually used for performance assessment Abou-Elfath et al. (2017). Although this procedure presents several shortcomings and provides only a static approximation of the actual dynamic structural behavior, nonlinear POA has been recognized by international codes (ATC 1996, FEMA 1997, CBP 2001, ECS 2005, ATC 2005) as a possible substitute, under certain conditions, for the more accurate nonlinear dynamic analysis of structural systems Barbato et al. (2010). 2.1.1. Bayesian model updating (BMU) using Di ff erential Evolution Markov Chain (DE-MC) Iterative methods are widely used for finite element model updating (FEMU) in structural engineering and are classified as deterministic or stochastic (Ereiz et al., 2022). In this study, a stochastic Bayesian approach is adopted to statistically estimate the probability distributions of unknown parameters. In Bayesian FEMU, uncertain model parameters θ (e.g., material properties, damping ratio) are inferred from measured responses y by combining prior information p ( θ ) with a likelihood function p ( y | θ ), yielding the posterior probability distribution:

p ( y | θ ) p ( θ ) p ( y ) ,

p ( θ | y ) =

(1)

where the evidence p ( y ) reads:

p ( y ) = p ( y | θ ) p ( θ )d θ .

(2)

For nonlinear, high-dimensional FE models, this integral is generally intractable. Therefore, Markov chain Monte Carlo (MCMC) methods are then employed to sample the posterior without explicitly computing p ( y ) (Gelman et al., 2013; Robert & Casella, 2004), enabling posterior summaries and predictive checks.

Likelihood model Considering additive model-data discrepancies, the observation model can be descibed as:

y = g ( θ ) + ε ,

ε ∼N ( 0 , Σ ) ,

(3)

where g ( θ ) is the FE prediction and Σ is the error covariance. Assuming normally distributed errors, the corresponding likelihood is:

1 / 2 exp − 1

1 [ y − g ( θ )] ,

n / 2 |

p ( y | θ ) = (2 π ) −

Σ | −

2 [ y − g ( θ )] ⊤

Σ −

(4)

where n is the dimension of y (Beck & Katafygiotis, 1998; Yin et al., 2021).

Sampler Selection: DE-MC Classical samplers such as Metropolis–Hastings (MH) often su ff er from sampling stagnation in high-dimensional or correlated spaces (Yin et al., 2021). To address this, we employ the Di ff erential Evolution Markov Chain (DE-MC) (Ter Braak, 2006), a population-based MCMC algorithm combining the global search ability of Di ff erential Evolution (DE) with MCMC acceptance rules. DE-MC improves chain mixing, reduces stagnation, and e ffi ciently explores multi-modal posteriors, making it suitable for Bayesian updating of large-scale FE models.