PSI - Issue 84

Alessandro Lipari et al. / Procedia Structural Integrity 84 (2026) 615–622

617

= 1− 1 1/ ≈− ( )

(2)

For instance, if a certain level of an action z has F ( z ) = 0.98 in any one year ( n = 1), then its probability of exceedance is 0.02 and such level is expected to be attained on average every 50 years (=1/0.02). Conversely, T ( z ) = 1000 years corresponds to F ( z ) = 0.999 in any one year ( n = 1), 0.951 in 50 years and 0.905 in 100 years. Note that if extreme values are not recorded yearly, T must be expressed in the relevant time unit and F ( z ) will take different values. For instance, if extreme values are recorded daily, T must be expressed in days and F refers to the probability of non-exceedance in any one day. It must be emphasised that T is fundamentally different from the reference period t ref . The abovementioned normal (or log-normal) distribution generally fits well material properties and permanent actions. However, variable actions, such as traffic loading, climatic actions and earthquakes, are best approached with the extreme value theory . To this end, the Gumbel distribution has been traditionally used, which is actually a special case of the Generalised Extreme Value (GEV) distribution, whose CDF is: ( ) = {− [1 + ( − )] − 1 } (3) in which μ , σ and ξ are the location, scale and shape parameters, respectively. The cases ξ > 0 and ξ < 0 are named Fréchet and Weibull distributions, or Type II and III, respectively. When ξ = 0, the GEV distribution reduces to the Gumbel distribution (or Type I): ( ) = {− [− ( − )]} (4) The location and scale parameters of the Gumbel distribution can be found from the mean and standard deviation of the sample, μ’ and σ’ , as = ′ − 0.5772 ′ and = ′√6/ . For Type II and III distributions, the parameters may be estimated through Maximum Likelihood Estimation (Lipari et al., 2012). The GEV distribution can be conveniently depicted on Gumbel probability paper plots, in which the ordinate is the Standard Extremal Variate ( ) = − [− ( ( ))] . In such rescaled plots, the Gumbel distribution appears as a straight line. 3. Eurocode 1 The Load Model 1 (LM1) in EC1-2 consists of a tandem system (TS) and uniformly distributed loads (UDL) (Fig. 1) (European Committee for Standardization, 2023a). It is generally used for global verifications of bridges with loaded length from about 7 m to 200 m. Notably, the load for Lane 1 is substantially heavier than that for other lanes. The adjustment factors α should be selected based on the expected traffic. They are given in the National Annexes and values equal to 1 correspond to heavy industrial international traffic. In all cases, for bridges without road signs restricting vehicle weights, α Qi ≥ 0.8 and for i ≥ 2, α q1 ≥ 1. The Italian structural design code Norme Tecniche per le Costruzioni (NTC hereinafter) endorses the LM1, but without the factors α (Ministero delle Infrastrutture e dei Trasporti, 2018). The LM1 was calibrated with about one week of WiM data collected on the A6 Motorway near Auxerre (France) (Prat, 2001) and its characteristic values were computed for a 1000-year return period (approximately probability of exceedance of 5% in 50 years) (European Committee for Standardization, 2023a). For assessment purposes, there are no indications on the characteristic values or return periods to be used. The informative Annex C “Reliability analysis and code calibration” in European Committee for Standardization (2023b) states that a characteristic value “may be taken as a specified p -fractile value from the statistical distribution chosen to represent the basic variable”, that is , the probability of non-exceedance F . For time-variant actions, a typical value of p = F = 0.98 is suggested when referring to the distribution of yearly extreme values, that is T = 50 years (Eq. 2). Indeed, the characteristic snow and wind loads in Eurocode 1, parts 1-3 (EC1-1-3) and 1-4 (EC1-1-4) respectively, are based on T = 50 years, i.e., “annual probability of exceedance of 0.02” (European Committee for Standardization,