PSI - Issue 84

130 Simone Celati et al. / Procedia Structural Integrity 84 (2026) 127–134 where ̂ ( ̅, ̅) is the estimated response vector, is the sample size, and 2 is the number of parameters. The covariance matrix of the parameters ( Σ p ) can is calculated as ∑ = 2 [ ] −1 , where is the Jacobian matrix of the model function evaluated at the evaluated parameter values. To account for the time-dependent variability of the diffusion process, the evaluated apparent diffusion coefficient is used to calculate a new distribution for ,0 using the formulation provided in (fib 2015), where ( ̅) is the ageing function evaluated at time ̅ : ,0, = , ( ̅) ( ̅) (7) This calibrated coefficient enables the calculation of the apparent diffusion coefficient at any time accounting for concrete ageing and observed data. Accordingly, Eq. (4) is modified as: , ( , ) = ,Δ , [1−erf( −Δ 2√ ,0, ( )∙ )] (8) A probability density function (PDF) can be associated to the calibrated parameters ,Δ , and , ( ̅) . These PDFs are used as likelihood function of ,Δ , ( ,Δ , → ,Δ ( )~ ( , ; ) ) and , ( ̅) ( , ( ̅) ( )~ ( , ; ) ) as they incorporate the data and their uncertainties. and represent the expected value of the calibrated ,Δ , and , ( ̅) , respectively, while and are estimated using Σ p . Eq. (7) can be used to define the likelihood function for ,0, , denoted as ,0 ( 0 ) . This function does not follow any parametric probability distribution and must be evaluated numerically or through simulations. 3.2. Updating of the corrosion model The likelihood functions of ,Δ , and ,0, , defined in Section 3.1, can be used to perform Bayesian updating for the apparent surface chloride concentration and the chloride diffusion coefficient parameters. The prior distribution of the parameters, ,0 ′ ( 0 ) and ,Δ ′ ( ) , can be modelled based on relevant literature sources (for example, fib 2015). The updated (posterior) distributions ,0 ′′ ( 0 ) and ,Δ ′′ ( ) can be calculated using the following equations: ,0 ′′ ( 0 )= 1 ,0 ( 0 ) ∙ ,0 ′ ( 0 ) (9) ,Δ ′′ ( )= 1 ,Δ ( ) ∙ ,Δ ′ ( ) (10) where and are normalization constants. The posterior parameter distributions ,0 ′′ ( 0 ) and ,Δ ′′ ( ) can be used to calculate the predictive distribution of the time to initiation of corrosion ( ℎ ′′ ( | ̅, ̅) ). In particular, the total probability theorem is leveraged to integrate out the uncertainties relative to the parameters ,Δ and ,0 , now described by their updated distributions (see Eqs. 9 and 10), through the following equation. ℎ ′′ ( | ̅, ̅) = ∫ ℎ ′ ( | , 0 ) ,Δ ′′ ( | ̅, ̅) ,0 ′′ ( 0 | ̅, ̅)dC l dD 0 [C l ,D 0 ] (11 ) 4. Case study: Pier caisson of the Øresund Bridge A submerged pier section of the Øresund bridge is selected for demonstration as here chloride diffusion is relevant for the time to corrosion initiation (Lindvall 2003, Melchers and Chaves 2018, Björnsson, Thöns et al. 2024). The reliability of the pier's external wall over time is evaluated for potential failure due to ship impact. The utilisation ratio for the horizontal reinforcements and their details ( 25/150 ) are derived from the relevant design documentation. A time-dependent reliability analysis is conducted for the caisson of one of the east approach bridge at -2 meters section, which, according to the measurements, has shown the highest chloride concentration after 10 years.