PSI - Issue 78

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

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transformers. In the last twenty years the Housner’s formulation was applied for different typologies of free -standing elements, among them minarets, columns, art objects and historical structures under earthquakes (Sorrentino et al., 2008; Berto et al., 2012; Makris and Vassiliou, 2012; Lagomarsino and Cattari, 2015, Coccia and Como, 2025). Most studies on the seismic rocking of rigid blocks primarily consider the horizontal component of ground motion, often overlooking the potential impact of vertical excitation. However, vertical acceleration can significantly influence the rocking response, as shown by Yim et al. (1980), and the probability of overturning increases notably when the vertical component exceeds the horizontal one (Dimentberg et al., 1993). These findings highlight the need to incorporate vertical excitation in the seismic analysis of rocking systems (Taniguchi, 2002). The significance of this influence depends strongly on the block geometry, the vertical-to-horizontal peak ground acceleration (V/H) ratio, and the frequency content of the vertical component (Makris and Kampas, 2016; Linde et al., 2020). At the level of individual ground motions, the inclusion of the vertical component can lead to highly variable effects on rocking behavior — from negligible to substantial — depending on the specific characteristics of the record (Lachanas et al., 2022). This variability arises from the inherently nonlinear and sensitive nature of rocking dynamics, where small variations in input parameters can cause disproportionate changes in response. The effect of vertical excitation also varies with the size and slenderness of the block. Smaller blocks tend to exhibit a more pronounced rocking response and are more susceptible to overturning under vertical excitations (Wang et al., 2023). While slight rocking is influenced by both vertical and horizontal inputs, moderate rocking and overturning are predominantly governed by the horizontal component (Chen et al., 2025). On a broader statistical basis, neglecting the vertical component does not introduce significant bias in the estimation of exceedance probabilities for moderate to large rocking amplitudes or overturning. However, vertical excitation can influence the initiation of uplift, particularly in stocky blocks with large slenderness angles approaching the onset of rocking. In such cases, the vertical component contributes to the equations of motion with a weighting factor of similar magnitude to that of the horizontal component. Furthermore, increases in peak vertical ground acceleration proportionally reduce the horizontal acceleration required to initiate uplift. This paper investigates the effects of vertical ground acceleration on three archetypal masonry walls subjected to two-sided rocking motion. The results are compared with those obtained when only horizontal ground acceleration is considered. Ten natural records from the 2016 Central Italy Earthquake are applied to walls of varying heights (3, 5, and 10 meters), all having the same slenderness ratio. Additionally, the combined effects of both horizontal components of ground motion are considered. Nomenclature ̈ ( ) horizontal seismic excitation ̈ ( ) vertical seismic excitation direction of the horizontal seismic excitation with respect to the wall alignment ( ) wall rotation around the hinge H height of the wall B Wall thickness  slenderness of the wall R distance from the centre of gravity to the rotation hinges p structural pulsation of the wall r coefficient of restitution of the wall 2.Rocking motion modelling of a free-standing masonry wall under combined horizontal and vertical excitations The reference element is a masonry wall with height H and thickness B , subjected to vertical and horizontal seismic inputs. Assuming no material ruptures, the wall can be modelled as a rigid body. Neglecting the occurrence of sliding failure, the block undergoes rocking motion about the corner hinge O or O′ , as illustrated in Fig. 1. The rocking motion initiates when the following condition is satisfied: