PSI - Issue 78

Alessio Bonelli et al. / Procedia Structural Integrity 78 (2026) 505–512

509

3.2. Cloud Analysis When both the seismic demand, expressed in terms of Engineering Demand Parameters, and the structural limit states are assumed to follow a lognormal distribution, the probability of exceeding a specific damage state can be estimated using the following expression (the complementary probability density function): ( > | ) = 1 − (ln( )−ln( ) √ 2 | − 2 ) (4.1) where (. ) is the standard normal cumulative distribution function, is the median estimate of the structural limit state, is the median estimate of the demand, | is the dispersion of the demand conditioned on the Intensity Measure, and is the dispersion of the structural limit state. The median demand can be estimated using a power-law function, typically expressed as: = ( ) (4.2) where and are regression coefficients derived from the set of maximum displacements (considering for example the sliding case) and the corresponding intensity measures , both obtained from the i-th time history analysis of the equipment subjected to a set of n ground motions. The standard deviation of the residuals (dispersion) can be computed according to Equation (4.4): | = √∑ [ln( ) − ln ( ( ) )] 2 = 1 −2 (4.4) 4. Illustrative example 4.1. Structural description of the equipment As case study an unanchored storage tank, belonging to a petrochemical plant ideally settled Sicily, is chosen for the application of the approach proposed in the previous section. The site is situated in Priolo Gargallo, one of the most active zones of Italy, in terms of earthquake. It is considered the loss of containment event due to the detachment of the flanged joints (context usually associated with the sliding mechanism of the tank). The most common consequences of such circumstances are tensile stresses, plastic deformation and leakage of material from the flanged joints. The equipment used for storing crude oil is composed of a steel shell with an equivalent thickness of 13.0 mm (2 cm for the base thickness) and a yield strength (f y ) of 345 MPa. It has a diameter (D) of 37.96 m, a clear height (H) of 14.0 m, with the maximum filling level which corresponds to 80% of H and a friction coefficient (assumed between the tank base and the soil) of 0.3. The pipe connected to the tank is characterized by an 8 inches diameter and presents discontinuity in the form of a flanged joint, after that it continues its path underground. Due to the difference in terms of dimension and structural behavior, the pipe system is not explicitly modelled. Instead, the focus is placed on recording the relative sliding ln( ) = log( ) + ∙ ( ) (4.3)