PSI - Issue 78

Matteo Tatangelo et al. / Procedia Structural Integrity 78 (2026) 404–411

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a)

b)

c)

Fig. 2. Schematization of time intervals for the reliability assessment: a) is known and < − ; b) is unknown and = − ; c) is unknown and = Once ̅ 0 ( ) and ̅ ( ) are known, and is defined (Fig. 2.a), the corresponding target reliability indexes ̅ 0 ( , ) and ̅ ( , ) may be calculated according to the Eq. 6. However, if is known and ≪ , such as for very old existing constructions having − ≥ , or else is unknown, then it may be conventionally derived as depicted in Fig. 2.b and Fig. 2.c. In detail, Fig. 2.b reports the case when = − is assumed. Note that the estimated reliability index is known starting from and its trend may be known in the following years by means of construction monitoring. In the years before , if no monitoring is available, the capacity may be assumed constant. While, Fig. 2.c refers to the case when = is assumed and, therefore, = . By converting the target reliability indexes ̅( ) into ̅( , ) , the procedure proposed permits of performing a reliability analysis according to either Level 2 or Level 1 method (Tatangelo et al. 2024). 3.3. Alarm thresholds Reliability checks are made comparing the reliability target indexes ̅ 0 ( , ) and ̅ ( , ) , calculated by means of (4), with the estimated one ( ) of a construction (or element), by knowing for a given . In order to estimate a reliability index of a construction for the i-th year, one may consider the PDFs of its structural resistance ( ) and load effect ( ) . The resistance ( ) is well-known through degradation laws, like those related to steel corrosion, which predict its change over time. For the structural load effect ( ) , its PDF represents the maximum values within a given time interval, and it can be derived from a PDF established for a more fundamental time unit. The resulting reliability index, denoted as ( ) , provides a measure of the cumulative probability of failure over the interval − , and it is estimated using the relationship provided. ( ) = − −1 {1 − ∏ 1 − [− 1 ( )] =1 } (6) where Φ[− 1 ( )] = 1 ( ) is the annual failure probability. As an example, Fig. 3 presents several reliability checks performed on an existing building. It visually compares the estimated reliability index of the structure with pre-defined target reliability levels. These evaluations were conducted assuming that is a known value and > has been determined, specifically for three distinct CCs and a collapsed span length of =100m. The estimated reliability index ( ) may exceed the target ones ̅ 0 ( , ) and ̅ ( , ) in two different years. During this period, i.e. when ̅ 0 ( , ) < ( ) < ̅ ( , ) , the estimated reliability will be higher than the minimum ̅ 0 ( , ) but still fall short of the reliability target ̅ ( , ) required for existing constructions. The intersection of ( ) with ̅ ( , ) identify the attention year , corresponding to the year when ( )= ̅ ( , ) . When ( ) = ̅ 0 ( , ) , the limit years is identified, after which the construction is no longer reliable. The period between and is called Interval of Attention ( ), where ̅ 0 ( , ) < ( ) < ̅ ( , ) . While, the interval from to may be defined as Residual Life ( ), that is the period for which an existing construction may be used satisfying the minimum reliability criteria. It may be calculated as follows: = − (7) The reliability analysis in Fig. 3.a show that is obtained. However, in Figs. 3b and 3c, cannot be calculated because is not reached, meaning the building is reliable across the entire range in these latter two cases.