## 5 Nonlinear Kinematic Analyses  
The seismic safety evaluation of the church was assessed in terms of displacement demand, following the kinematic approach outlined in the Italian code [29, 30]. This is a simplified procedure that takes into account the expected failure mechanisms, which, in this case, considered the collapse mechanisms identified in the nonlinear pushover analyses. The structural elements are modeled as rigid bodies, and their equilibrium capacity under horizontal seismic loads is evaluated through kinematic limit analysis. The models include a portion of the south lateral wall with a hinge at the top of the foundation layer (Figure 14a); the whole north lateral wall, considering a horizontal hinge at ground level (Figure 14b); the main facade with its buttresses (Figure 15a); and the whole rear facade (Figure 15b). The verification procedure was carried out using both linear and nonlinear approaches. In the linear method, the activation of each collapse mechanism was assessed by computing the horizontal load multiplier, \(\alpha_{0}\), i.e., the minimum seismic coefficient required to trigger body motion. The nonlinear approach, on the other hand, involves constructing a load-displacement capacity curve (\(\alpha-d_{c}\)), which describes the progressive capacity of the rigid block until equilibrium loss (collapse). Typically, these curves exhibit a linear reduction of \(\alpha\) as the displacement increases, eventually reaching zero. The methodology outlined in [29; 30] is based on the assumptions that (1) masonry exhibits no tensile strength, (2) sliding between blocks does not occur, and (3) macro-blocks behave as monolithic rigid bodies. These assumptions result in a linear capacity curve with infinite initial stiffness up to the activation of the collapse mechanism, which is defined by the load multiplier \(\alpha_{0}\). While effective for simplified analysis, this rigid-body approach tends to overestimate the horizontal load multiplier \(\alpha_{0}\) when compared with deformable body models [37; 38; 39]. However, such an overestimation is more significant in the case of free-standing cantilever walls than in walls that develop flexural out-of-plane resistance before the activation of an overturning mechanism [29; 30], as was observed in the case of the church (Figure 12). Once the mechanism is activated, the rigid-body curve approximates well the deformable-body curve [40]. Figure 14: Geometric configuration of the block and distribution of forces for the out-of-plane collapse mechanism of the (**a**) south lateral wall and (**b**) north lateral wall (dimension in meters). Although the kinematic limit analysis treats the blocks as rigid, the limited compressive strength of adobe masonry can be considered by adjusting the location of the horizontal hinge to account for compression stresses at the wall toe. This modification shortens the lever arm associated with gravity loads and, consequently, reduces the stabilizing moment, allowing for a more realistic representation of the material's mechanical limitations within the rigid-body framework. Considering the previous hypotheses and the assumption of a uniform distribution of stress, the location of the rotation hinge, t, can be calculated as  
\[t=\frac{\sum P_{i,j}}{2(0.8\sigma_{c}t)} \tag{3}\]  
where \(\sum P_{i,j}\) is the total vertical load acting on the mechanism, \(\sigma_{c}\) is the compressive strength of the material, and \(l\) is the length of the hinge. The self-weight of the blocks was applied at their respective centers of mass. The roof weight was calculated by considering the timber elements, the canes, the earthen layer, and the roof tiles. These loads were applied at two locations, i.e., at the top (center of  
Figure 15: Geometric configuration of the block and distribution of forces for the out-of-plane collapse mechanism of the (**a**) main facade and (**b**) rear facade (dimension in meters). the wall) and bottom (inner face) chords of the timber trusses (Figure 16).