PSI - Issue 78

Ingrid Boem et al. / Procedia Structural Integrity 78 (2026) 457–464

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3. Procedure The method relies on a mechanical-analytical model based on simplified nonlinear-static (pushover) analysis. The procedure consisted in three main steps: drawing the pushover capacity curves of the buildings, calculating the ground acceleration related to the attainment of specific damage levels, and grouping of the results of the buildings belonging to a sub-typology, to evaluate median values and dispersion of the fragility curve. 3.1. Building’s capacity curve To draw the pushover capacity curves, the “Firstep-M_PRO” tool is used. Firstly, it is necessary to input the building characteristics, namely the in-plan distribution of the shear walls (centroid coordinates, length and thickness), the masonry type(s), the number of storeys and the floor/roof type and orientation. Based on the provided data, the tool, using preset values (which can however still be modified by the user), assigns the mechanical properties to each wall (i.e. Young modulus, compressive and shear strength) and determines the gravitational loads for the seismic combination. Then, the axial stress of each pier (which influences its lateral performance) is computed taking into account the masonry self-weight and the floor loads by influence area. According to the well-known approach (MIT, 2019), the in-plane lateral response of a pier is schematized as elastic– plastic up to ultimate displacement (eventually with residual strength): the secant stiffness is calculated considering both the flexural and shear deformability; the resistance is given by the weaker mechanism between diagonal shear cracking (the Turnšek and Čačović criterion was considered) and bending; the ultimate drift is 0.5% or 1.0%, respectively, for two cases. Actually, where the shear failure occurs, a residual strength of 80% is assumed till a drift of 0.8%. Clearly, the pier behavior strongly depends on its effective height and boundary conditions - both of which are influenced by the coupling action provided by the spandrels. Thus, to get reliable even though approximated predictions by means of the Firstep-M_PRO simplified procedure, some detailed level simulations on several case study school buildings have been carried out. Given to such investigations, the effective pier’s height has been set equal to 0.85 the inter-storey height; besides, a shear-type pattern has been considered in single-storey buildings (strong spandrels), while in multi-storey buildings, an intermediate behavior between a shear-type and a cantilever mechanism has been assumed (partially effective spandrels). The Firstep-M_PRO model works under the hypothesis of global behavior of the structure (i.e. floors sufficiently stiff and effectively connected to the walls); thus, once the control displacement (that of the floor mass center) is known, also the displacement and the corresponding individual force of each pier can be calculated. The initial position of centers of mass and rigidity are estimated, so to consider both translational and torsional effects related to eccentricity. Two distinct analyses are then performed (X, Y): for each, the control displacement at which the former pier attains its elastic limit is determined. Then, the control displacement is increased step by step, updating iteratively the position of the center of rigidity and subsequent floor rotation, based on the current displacements and force levels in the piers. At each step, the sum of the forces carried by the piers in the considered direction gives the floor shear, which is scaled by using a triangular distribution of lateral forces along storeys to get the base shear. This procedure is applied independently to each storey and the base shear related to the weakest storey(s) is identified. To get the global displacement, the displacement of the weaker storey(s) is combined with the elastic displacement of stronger ones. The capacity curve is given by the base shear against the global displacement. 3.2. Ground acceleration Target points referred to the attainment of specific Damage Levels - DL1 slight, DL2 moderate, DL3 heavy, and DL4 very heavy (Grunthal, 1998), are located in the capacity curve of the building (i.e., the multi-degree-of-freedom MDOF system), which is converted into that of an equivalent single-degree-of-freedom (SDOF) system, assuming the structure primarily responds according to its first translational mode. Target points are identified as follows:  DL1 significant reduction in the initial elastic stiffness (by at least 30%);  DL2 beginning of the plastic stage of the curve;  DL3 ¾ of the displacement with sudden resistance decrease or beginning of gradual resistance decrease;  DL4 sudden resistance decrease, or gradual resistance decrease of at least 25%.