## 5 Nonlinear Kinematic Analyses
The transformation of the mechanism into an equivalent SDOF system requires the estimation of the equivalent mass, \(M^{*}\), the fraction of participating mass, \(e^{*}\), and the equivalent acceleration, \(a_{0}^{*}\), as defined by the following equations:  
\[M^{*}=\frac{\left(\sum P_{i}\delta_{xi}\right)^{2}}{g\sum P_{i}\delta_{xi}^{2 }} \tag{6}\]  
\[e^{*}=\frac{gM^{*}}{\sum P_{i}} \tag{7}\]  
\[a_{0}^{*}=\frac{g\alpha_{0}}{e^{*}FC} \tag{8}\]  
where \(g\) is the gravity acceleration, and FC is the confidence factor, assumed as 1.0, corresponding to knowledge level LC3, based on the Italian code [29; 30]. According to [29; 30], the verification for the DL limit state is carried out through the linear approach. The verification is satisfied if the acceleration for the activation of the mechanism, \(a_{0}^{*}\), is larger than the demand acceleration evaluated in the DL elastic spectrum at period \(T=0\), i.e., \(a_{g}S\) (for a return period of 95 years, \(a_{g}=0.15g\), \(S=1.2\)):  
\[a_{0}^{*}>a_{g}S \tag{9}\]  
For the LS limit state, the verification in terms of force (linear approach) is satisfied if the \(a_{0}^{*}\) acceleration is larger than the acceleration specified in the LS elastic spectrum (for a return period of 475 years, \(a_{g}=0.25g\), \(S=1.2\)), considering a behavior factor \(q=2.0\)[29; 30]:  
\[t=\frac{\sum P_{i,j}}{2(0.8\sigma_{c}t)} \tag{3}\]0  
The LS limit state verification was also performed through the nonlinear approach, which involved comparing the displacement capacity of the mechanism, \(d_{LS}^{*}\), with the displacement demand, \(\Delta_{d}(T_{LS})\). The verification is satisfied if \(\Delta_{d}(T_{LS})\leq\ d_{LS}^{*}\). The calculation of the center of mass of the system is required to estimate the finite rotation \(\theta\) at which the mechanism loses its stability (angle measured from the vertical). Therefore, the center of mass of the system is given by  
\[t=\frac{\sum P_{i,j}}{2(0.8\sigma_{c}t)} \tag{3}\]1  
The displacement of the control point (\(d_{k0}\)) at which the mechanism loses its stability is estimated as follows:  
\[t=\frac{\sum P_{i,j}}{2(0.8\sigma_{c}t)} \tag{3}\]2  
where \(y_{c}\) is the height of the control point. This displacement is transformed into an equivalent displacement of a SDOF system, \(d_{0}^{*}\):  
\[t=\frac{\sum P_{i,j}}{2(0.8\sigma_{c}t)} \tag{3}\]3  
where \(\delta_{x,k}\) is the virtual displacement of the control point. The displacement capacity for LS limit state is given by [29; 30]:  
\[t=\frac{\sum P_{i,j}}{2(0.8\sigma_{c}t)} \tag{3}\]4with an equivalent acceleration, \(a_{LS}^{*}\):  
\[t=\frac{\sum P_{i,j}}{2(0.8\sigma_{c}t)} \tag{3}\]5  
and equivalent period, \(T_{LS}\):  
\[t=\frac{\sum P_{i,j}}{2(0.8\sigma_{c}t)} \tag{3}\]6  
The displacement demand is calculated using Equation (17), applying the Peruvian seismic code, where \(T_{P}=0.6\) s and \(T_{L}=2.0\) s [13]. \[t=\frac{\sum P_{i,j}}{2(0.8\sigma_{c}t)} \tag{3}\]7  
For the mechanisms of the south and north lateral walls, further considerations were assumed to determine their geometric parameters and hinge location. Given the building location on a slope, the height of the equivalent masonry block was defined using the average height of the wall section. Similarly, the roof load was applied at an average height representative of its actual distribution. Since the buttress is included in the calculation of the hinge location, Equation (3) must be adjusted. The new expression assumes that the buttresses have proper connections to the longitudinal walls, and that the entire section of the buttress falls within the plastic range [44]:  
\[t=\frac{\sum P_{i,j}}{2(0.8\sigma_{c}t)} \tag{3}\]8  
where \(A\) and \(b\) are the area and the depth of the buttress, respectively, and \(t_{e}\) is the effective depth of the area under compression, measured from the outer edge of the buttress to the inner face of the wall. This adjustment enables the calculation of the geometric centroid of the compressed area, which defines the location of the hinge.