PSI - Issue 78

Tommaso Petrella et al. / Procedia Structural Integrity 78 (2026) 976–983

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2.1. Case of one-story block Let us consider a single-story configuration subjected only to self-weight, as shown in Figure 1a. In this case, within a linear kinematic framework, equating the stabilizing and the overturning moments the critical acceleration multiplier 0 can be calculated using the Eq. (1): 0 = (1) In the non-linear case, both the moments become functions of the rotation angle . Adopting the hypothesis of small displacements, the failure mechanism ’ s evolution can be described through the following relationship: ( − ) = ( + ) (2) Imposing =0 , ultimate rotation reached correspond to the loss of load bearing capacity and equilibrium. In the simplest case of only self-weight load, is equal to 0 as highlighted from Eq. (3): − =0→ = = 0 (3) Focusing on a more general configurations where non-inertial horizontal forces are applied to the block (e.g., thrusts), the equivalence between and 0 no longer holds. In these cases, is typically smaller than 0 , depending on the ratio between the overturning moments produced by the non-inertial and inertial forces. For this purpose, considering a block subjected to: (ai) the self-weight ; (b) a vertical force applied at ( ; ) ; and (c) vertical load together with the horizontal thrust deriving, for instance, from an arch above and applied at ( ; ) . For this configuration, the linear kinematic analysis leads to the following expression for 0 : 0 = + + − + + = (4) where and represent the numerator and denominator respectively of the ratio.The ultimate rotation is given by Eq. (5): Fig. 1. Schematic representation of OOP overturning mechanisms in masonry walls: (a) single rigid block subjected only to self-weight; (b) rigid block with additional vertical load N; (c) block subjected to self-weight and a vertical load S V and horizontal thrust S H , simulating the effect of non-inertial forces such as arch thrusts. a b c