PSI - Issue 78

Matteo Tatangelo et al. / Procedia Structural Integrity 78 (2026) 404–411 407 If the time unit is one year, the probability of failure is expressed on an annual basis. Relevant variables, such as resistance and load effect , are referenced to annual extreme values and described by appropriate Probability Distribution Function (PDF). Now, consider a set of loads applied annually on a construction over a given period of time (during its lifetime), where is considered large and for which each load event may be considered an independent event. The load effect Cumulative Distribution Function (CDF) ( ) correspond to the extreme value distribution for one year and consist in the probability that the load effect is lower or equal to a certain value (Freudenthal et al., 1966). If resistance is invariant year-by-year, failure probability for years, i.e. the probability that lower than load applications can be supported and developing in Taylor series, neglecting second-order terms, is expressed with ( ) = [ ,1 ( )] = [1− ̅ ,1 ( )] ≈ 1 − ∙ ̅ ,1 + ∙ ( − 1) ∙ ( ̅ ,1 ) 2 ⁄2−⋯ (1) where ,1 ( ) is the distribution of extreme values in one year. The neglect of second-order terms in is valid only for those values of for which ∙ ̅ ,1 ≪1 . Therefore, the failure probability during the lifetime of a construction is obtained as follows (Melchers and Beck, 2018): ( ) ≈ ∫ 1 − 0 ∞ [1 − ∙ ̅ ,1 ] ∙ ( ) ∙ = ∙ ∫ ̅ ,1 ∙ ( ) ∙ 0 ∞ = ∙ ,1 (2) where ,1 =∫ ̅ ,1 ∙ ( ) ∙ 0 ∞ can be seen as the sum of the failure probabilities over all the cases resistance for which the load exceeds the resistance in the year. The result shows that the lifetime failure probability ( ) can be approximately determined from the annual failure probability ,1 simply by multiplying the latter by the number of years in the designated lifetime (Melchers and Beck, 2018; Wang, 2021). Instead of considering load applications to describe the lifetime of a construction, one might consider making explicit the time in years. Therefore, failure probability may be considered simply a function of the number of statistically independent years to cause failure. Failure probability in a certain period may be obtained as ( ) = ( < ) = 1 − ( ) = {1 − (1 − ,1 ) ≈ ∙ ,1 if ,1 ( 1 )= ,1 ( 2 ) 1−∏ [1− ,1 ( )] =1 if ,1 ( 1 )≠ ,1 ( 2 ) (3) where ( ) is expressed as function of the success probability ( ) over the time, that in turn is expressed, approximately, in terms of failure probability with (2). 3.2. Target reliability indexes over time In order to determine ̅( , ) it is necessary to define preliminarily the initial year . For new constructions, corresponds to the end construction year, from which onwards a reliability verification may be made. Moreover, the target reliability index referred to the nominal life is obtained by assuming that = . It is practical to assume as the origin of the reference time system, i.e. =0 , and therefore, the reliability index ( ) refers to the time interval , being coherent with a time-integrated approach. Similarly, ̅( ) expresses the target reliability index referring to the time interval . Once ̅( ) is known, the target reliability index at year , i.e. ̅( , ) , may be calculated as follows: ̅( , )=− −1 [ ̅ ( , )] (4) where ̅ ( , ) represents the cumulative target failure probability in the time interval conditional on , given by the approximately relationship (Tatangelo et al. 2024): ̅ ( , ) = ∙ ̅ ,1 =( ⁄ ) ∙ [− ̅( )] (5)