PSI - Issue 84

132 Simone Celati et al. / Procedia Structural Integrity 84 (2026) 127–134 state function is defined: ( ) = Θ ( )−Θ .The structural reliability is evaluated through the following equation with consideration of the ship impact probability ( ) and the failure probability due to the ship impact ( , ℎ ( ) ). The computed total prior reliability is always higher than 3.7, which is the limit reliability for high consequences and high cost of interventions. ( )=−Φ −1 { ∙ , ℎ ( )} ; , ℎ ( ) = Φ[− ℎ ( )] (14) Table 2. Probabilistic models for the initiation phase Variable Distribution min max [K] (fib 2015) N 281 5 - - [K] (fib 2015) D 293 - - - (fib 2015) N 4800 700 - - (fib 2015) B 0,3 0,12 0 1 ,0 [mm 2 /year] (fib 2015) N 280,67 56,13 - - ℎ (Straub, Malioka et al. 2009) LN 1 0,05 [mm] * D 80 - - - [wt.-%/c] (fib 2015) U 3 0,58 2 4 , [wt.-%/c] (fib 2015) LN 0,28 - - [wt.-%/c] ** (Lindvall 2003) B 2 0.5 0.2 3 [mm] D 0 - - - N: Normal; D: Deterministic; B: Beta; LN: Log-normal; U: Uniform. *From design requirements; ** Distribution, standard deviation and minimum value taken to match reference (fib 2015). Based on the data-driven chloride diffusion model provided in Section 3, the probabilistic chloride diffusion model is updated. Measurement uncertainties are disregarded as the chloride penetration profile measurement was performed in a laboratory setting. The results from the studied caisson at a depth of -2 meters (submerged) are presented here and used to calibrate the model for chloride ion ingress. These data were acquired in 2010 by the Danish Technological Institute (DTI), after 10 years of service of the structure. The 21 measurements refer to the chloride concentration at different depths in year 10. The regression parameters, namely the apparent chloride concentration and diffusion coefficients, evaluated with the relative standard deviation and the correlation coefficient, are reported in Table 3. Using the likelihood model, the predicted chloride concentration at 10, 20, 50, and 100 years at the cover depth (80 mm) are shown as cumulative distribution functions (CDFs) in Fig. 1 (right), while the corresponding prior models are shown in Fig. 1 (left). The figure shows that the data-driven model (Eq. (8)) can predict the chloride concentration over time, preserving the time dependency observed in the prior model. Indeed, in both cases, the expected value of the chloride concentration evolves towards the right, and as the forecasting horizon increases, the uncertainty also grows, however, with a significantly lower increase than the prior model. The calibrated chloride diffusion coefficient (Eq. (8)) serves as the likelihood function even though the variable is not directly measured but inferred, accounting for data and model uncertainties. The posterior distribution of the chloride diffusion coefficient and the apparent surface concentration of chlorides are then calculated using Eq. (9) and Eq. (10), respectively. Then, Eq. (11) is utilised to calculate the predictive distribution of ℎ incorporating the measurements and the updated parameter distributions. The predictive distribution ( ℎ ′′ ( | ̅, ̅) ), along with the prior and the likelihood (defined using ,Δ ( ) and ,0 ( 0 ) ) to estimate the data-driven model through Eq. (1)), are illustrated in Fig. 2. The figure shows that the high chloride concentration measured results in both a lower mean value and reduced standard deviation, i.e., the uncertainty, associated with the posterior distribution of the time to initiation. Table 3: Evaluated parameters' expected value, squared error, and correlation coefficient Parameter Expected value Squared error Correlation coefficient 4.4. Updating of chloride diffusion model

,Δ , [ . −%/ ] , (10) [mm 2 /year]

3.6923 52.6980

0.1481 10.40

−0.6994