PSI - Issue 78

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

1321

̈ ( ) > (1 − ̈ ( ) ) ( )

(1)

where g is the gravity acceleration and  is equal to: = ( )

(2)

Once the wall begins to rock, its response to seismic excitation can be analysed using the well-known differential equation that governs the motion of a rigid rocking body: ̈( ) = − 2 (1− ̈ ( ) ) [ ( ( )) − ( )] + 2 ̈ ( ) [ ( ( )) − ( )] (3) where ( ) is the rotation around the hinge O when assuming positive values, or about hinge O′ when assuming negative values. For a rectangular block, the parameter p is given by: =√ 34 (4) During rocking, the wall impacts the base, causing the motion to reverse direction around the alternate hinge (i.e., switching from O to O′, or vice versa). This impact results in energy dissipation, manifested as a reduction in angular velocity. The amount of this reduction depends on the geometry and material properties and typically ranges between 0.8 and 1.0 for unreinforced masonry. Assuming no local material failure, this reduction is quantified by the well known coefficient of restitution introduced by Housner, defined as: √ = ̇ ̇ =1− 3 2 ( ( )) 2 (5) where ̇ and ̇ are the angular velocities immediately before and after impact, respectively.

Fig. 1. Geometrical configuration of the free-standing masonry wall (left) and schematic representation of its rocking response under seismic excitation (right). 3. Selection and Characteristics of the Input Ground Motion In this study, ten ground motion records from the Central Italy earthquake are considered, with their main parameters summarized in Table 2. Each record includes three acceleration components: East – West, North – South, and vertical (Up – Down), as illustrated in Fig. 2. No seismic input amplification is applied in this study. Since the analysis focuses exclusively on the out-of-plane response of the masonry wall, a single horizontal seismic input is considered, assumed to act at an angle β with respect to the East direction (Fig. 3). Consequently, the effective out -of plane horizontal accelerogram is computed as: