PSI - Issue 78

Simona Coccia et al. / Procedia Structural Integrity 78 (2026) 1318–1325

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4. Response of archetype masonry walls under selected excitations A parametric analysis is carried out on three masonry walls with different heights ( H = 3 m, 5 m, and 10 m), while maintaining a constant slenderness ratio α (= 0.1). The motion described by Equation (3) is numerically solved using MATLAB’s built -in ode45 solver. The vertical ground motion component is kept constant, whereas the two horizontal components are combined according to Equation (6), considering incidence angles β of 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135° and 157.5°, in order to account for all possible directional variations. Collapse is assumed to occur when the rotation at the end of the seismic motion surpasses the slenderness angle α . Figure 4 presents the time histories of the dimensionless rotation ratio θ(t)/α for the 3 and 10 m-high walls subjected to ground motions 2, across some considered β values. As shown, in only a few cases does the vertical component of seismic excitation influence the collapse condition — either preventing it (e.g., β = 22.5° H =3 m) or triggering it (e.g., β =0°, H =3 m) — depending on the direction of the vertical impulse relative to the block’s rotation. Moreover, the presence of the vertical component can affect the vibration frequency of the rocking block (see, for example, Fig. 4, β = 22.5° or β = 90°, H = 10m). It is worth noting that the relevance of these effects also depends on the geometry of the considered element, as highlighted in Fig. 4, which shows a significantly lower dimensionless rotation ratio θ(t)/α for the 10 m -high wall. In general, taller walls exhibit negligible influence of the vertical component on vibration frequency, peak rotation amplitude, and collapse occurrence.

Fig. 4. Rocking behavior of the 3 and 10m-high walls under input motion 2, with (red line) and without (blue line) and without consideration of the vertical component of the seismic action.

To quantify the influence of the vertical component on the dynamic response, the number of collapses resulting from the numerical integration of Eq. (3) is reported in Table 3. As confirmed by the results — and in agreement with findings reported in the literature (Makris and Kampas, 2016; Wang et al., 2023; Chen et al., 2025) — the number of collapses decreases with increasing wall height. Nevertheless, when the vertical component is included, several configurations that would otherwise remain stable under horizontal excitation alone are observed to collapse. In cases where collapse does not occur, Fig. 5 presents the ratio between the maximum rotation obtained from the integration of the motion including the vertical component (  max1 ) and that obtained without it (  max2 ), for all three wall geometries, as a function of the horizontal PGA of the seismic input. As shown in the figure, the deviation of this ratio from