PSI - Issue 84

Vittorio Palma et al. / Procedia Structural Integrity 84 (2026) 1318–1325

1320

consequences. (Thöns et al. 2025). 3. Probabilistic model

The analysis refers to a representative post-tensioned prestressed concrete beam. The cross-sectional geometry and the stress state at the critical section are defined through a deterministic design procedure enforcing equilibrium between demand and resistance at the ultimate limit state. This configuration is taken as the intact reference state and retained throughout the subsequent analyses. Structural safety is then evaluated probabilistically by propagating uncertainties in actions, material properties, and calculation models, thereby estimating the failure probability and the associated reliability index for the section under consideration. Epistemic uncertainty related to tendon condition is subsequently introduced through the discrete system-state variable , which is updated using inspection evidence. This framework makes it possible to quantify how the inspection sample size and the adopted inspection method affect the system-state probabilities, the estimate of the failure probability , and the associated decision risk. The following sections present the probabilistic models adopted for tendon defectiveness and imperfect inspections, together with those used for actions and structural resistance. 3.1. Probabilistic model of tendon defectiveness and inspection with imperfect detection Consider a finite population of inspectable post-tensioned tendons and a discrete latent random variable , representing the system state associated with defective tendons, with support = { 0 , 1 ,…, } , where = . Prior knowledge on tendon condition is described through a discrete prior distribution on , assumed to follow a beta binomial model with parameters ( 0 , 0 ) (Celati et al. 2025). An inspection campaign selects tendons without replacement and employs an inspection method ∈ℳ , characterized by sensitivity ( ) and specificity ( ) (BDI et al. 2025) . For a given system state , the number of defective tendons in the inspected sample, denoted by , follows a hypergeometric distribution, ∣ ∼ Hypergeom( , , ) . For a given inspection design ( , ) , the observed number of positive inspection outcomes is represented by the discrete random variable , , taking values in the set of admissible information outcomes. The number of positive indications includes both true and false positives and depends on the unobserved number of defective tendons in the inspected sample and on the diagnostic performance of the inspection method. Since is not directly observable, the likelihood of the information outcome is obtained by marginalizing over all possible values of this intermediate quantity, ( , ∣ )=∑ ( =0 , ∣ = ) ( = ∣ , , ) (1) where ( = ∣ , , ) follows a hypergeometric distribution and ( , ∣∣ = ) accounts for imperfect detection through the sensitivity and specificity of the inspection method. The resulting likelihood is used for Bayesian updating of the system-state probabilities, ( ∣ , ) ∝ ( , ∣ ) ( ) (2) The predictive distribution of information outcomes is finally used in the pre-posterior analysis to evaluate the expected informational benefit of the inspection campaign. The explicit modelling of a finite population and sampling without replacement reflects the actual nature of inspection campaigns on post-tensioned tendons (FHWA 2013). 3.2. Loads and load-combination model The load combinations are defined in accordance with EN 1990 (2010), Equations (6.10a) and (6.10b) and Table A1.2(B), consistently with Eurocodes (CEN 2023b). The applied load is expressed through the fundamental vertical load combination adopted in the probabilistic analysis: = (1 − ) ⋅ + ⋅ (3)