PSI - Issue 84

Simone Celati et al. / Procedia Structural Integrity 84 (2026) 127–134 129 effect on the apparent chloride diffusion coefficient and accounting for the temperature effect as described in fib 2015. To address the uncertainties associated with the model, a model uncertainty variable ℎ is introduced. The variable ℎ is fully defined by the model described in Eq. (1), denoted as Μ T ( ) . This means that if all parameters within are deterministically specified, then ℎ is also deterministic. The vector consists of the parameters: ℎ , , , , ,0 , ,Δ , with Δ is assumed to be zero, each associated with a prior distribution. Therefore, the prior distribution of ℎ can be defined as the stochastic outcome of the model Μ T ( ) provided the prior distributions of the parameters. Because only the parameters ,0 and ,Δ are going to be updated, the prior distribution of ℎ can be conveniently expressed conditionally as: ℎ ′ ( )~M T ( ℎ , , , | ,0 , ,Δ ) .As the chloride concentration at the reinforcements level exceeds a critical threshold, the passive layer on the reinforcements breakdown and corrosion begins, as reported in Lindvall 2003. Thus, following the definitions provided in Björnsson, Thöns et al. 2024, and Celati, Natali et al. 2025, corrosion initiation event ( ) can be defined for submerged elements as: ∶ ( ) = ≥ ℎ , whereas ̅ is its complementary event, i.e., corrosion not initiated. As the event is realized corrosion propagation begins. Corrosion is considered to progressively reduce the steel cross-section and thus the capacity over time. The time-dependent residual reinforcement steel area ( ) , in the propagation phase of corrosion, is calculated with following equation (Celati, Natali et al. 2025): ( )={ 2 4 ; ̅ ( − ( − ℎ )) 2 4 ; (2) In Eq. (2), ( ) is the reinforcements’ residual cross section, is the initial diameter of the reinforcement, the pitting factor and the corrosion rate. The general limit state function ( ) can then be modified to specifically account for the corrosion process: ( ) = { ( = 0, ) − ≤ 0 ̅ ( , ( )) − ≤ 0 (3) 3. Updating of the chloride diffusion 3.1. Likelihood modelling of chloride diffusion To consider the measured chloride concentration within concrete, a non-linear regression analysis is performed to fit the model in Eq. (4) to the collected data. The parameters calibrated during the regression are ,Δ , and , , ensuring numerical stability. The model equation for the non-linear regression analysis is as follows: ( , ̅) = ,Δ ∙ [1−erf( −Δ 2√ , ( ̅)∙ ̅ )] (4) The term , ( ̅) represents the apparent chloride diffusion coefficient evaluated at the year the measurements were collected, ̅ . A non-linear least square method is employed to fit the model to the data, i.e.: ( , ̅) ≈ , ( , ̅) = ([ ̅, ̅], ) + (5) In Eq. (5), ([ ̅, ̅], ) is the model function (Eq. 4), and [ ̅, ̅] is a ×2 predictor matrix comprising the depth into the concrete where the i-th measure ( ̅ , ̅) was taken ( ̅ ), and the inspection year (fixed at ̅ ). The parameter vector includes ,Δ and , ( ̅) , and parameter evaluation is carried out using the iterative Levenberg– Marquardt algorithm (Seber and Wild 2003). The error term is modelled as normally distributed with zero mean and standard deviation estimated through Eq. (6): 2 = ( ( ̅, ̅)− ̂ ( ̅, ̅)) ( ( ̅, ̅)− ̂ ( ̅, ̅)) −2 (6)