PSI - Issue 84

Vittorio Palma et al. / Procedia Structural Integrity 84 (2026) 1318–1325 1321 where denotes the permanent load (dead load, mainly self-weight of structural and non-structural elements) and is modelled as a normal random variable, denotes the variable load (imposed live load) and is modelled as a Gumbel random variable, and is the load combination factor, which can take values between 0,1 and 1,0. The probabilistic load effects in flexure are then expressed in general form as: = ∙ 2 ∙ (4) where is a coefficient depending on the static scheme and loading configuration (equal to 1/8 for a simply supported beam under uniformly distributed load). For completeness, the deterministic design load effects are evaluated according to EN 1990 using partial safety factors and , but they are employed only in the definition of the reference configuration and do not enter the probabilistic analysis. The random variables associated with the actions are summarised in Table 1.

Table 1 Random variables and parameters used in the probabilistic model

Parameter (European Commission 2024) (European Commission 2024) (CEN 2023a) (CEN 2023a)

Distribution

Mean

Standard Deviation

Normal Gumbel

1,00 1,00 1,35 1,50

0,10 0,26

- -

- -

3.3. Mechanical model and structural reliability quantification The structural model is formulated within the semi-probabilistic framework of the Eurocode (CEN 2023b, 2023a) to define a deterministic reference configuration. For a given design bending demand (Section 3.2), the flexural capacity of a rectangular cross-section is evaluated using an equivalent rectangular compression block. Since the analysis is restricted to flexural behaviour, the reference configuration is obtained by calibrating the prestressing level through the total tendon area ,tot , while keeping the section geometry ( ,ℎ) and the effective tendon depth fixed. At this stage, the intact state 0 is assumed and a PT-only model is adopted, neglecting the contribution of non prestressed reinforcement. For a given neutral axis depth , the internal forces are ( ) = ; ( ) = ,tot ( ) (5) where the tendon stress ( ) is obtained from strain compatibility assuming a linear elastic law with an upper stress limit. The internal equilibrium condition ( ) = ( ) is solved numerically for . By assuming an internal lever arm = −0.4 , the design flexural resistance is expressed as ( ,tot )= ( ) . The reference configuration is then defined by imposing the design equilibrium condition, ( ,tot )− =0, from which the calibrated total prestressing area ,tot * and the corresponding single-tendon area ,0 * = ,tot * / are obtained through an iterative bisection procedure. In the probabilistic analysis, tendon defectiveness is represented by the discrete system state , where denotes the number of defective tendons. For a given state , the effective total prestressing area is defined as ∗ , ( )= ∗ ,0 [( − ) + ∑ , =1 ] , , ∈ (0,1] (6) where , represents the residual fraction of effective tendon cross-sectional area associated with the -th defective tendon. This factor provides a simplified tendon-level representation of defect severity due to localised corrosion and/or grout-related defects and is consistent with experimental evidence on pitting corrosion in prestressing strands and with residual cross-section indicators proposed in the literature (Franceschini et al. 2022). In the present study, , is modelled as an area-retention factor, whose distribution is selected to reflect corrosion-induced section loss consistent with degradation mechanisms driven by corrosion current density. Let denote the vector of basic random