## 5 Nonlinear Kinematic Analyses
The vertical and horizontal components of the roof loads were determined based on the tributary area supported by each wall. The horizontal resistance provided by the tie beams was estimated using Equation (4), where \(t_{wall}\) is the thickness of the wall. Equation (4) was derived through linear regression based on the results obtained from pull-out tests performed on the timber tie beams [5]. A partial safety factor of 2.0 was applied to reduce the capacity of the ties according to [41]. Seismic forces were assumed to act at both the center of mass of the rigid blocks and the application points of the roof load. The load multiplier, \(\alpha_{0}\), was computed using the principle of virtual work by equating the stabilizing moments and the overturning moments. \[F_{\text{fies}}=54.652t_{wall}+2.7346\ (\text{in}\ \text{kN}) \tag{4}\]  
Seismic safety was evaluated by converting the Multi-Degree of Freedom (MDOF) system into the equivalent Single-Degree of Freedom (SDOF) system. The structural capacity was assessed in terms of both force (linear analysis) and displacement (nonlinear analysis) and compared against the corresponding spectral demands following the methodology outlined in [29; 30]. These demands were determined based on the performance requirements of two limit states defined in [29; 30]: Damage Limitation (DL), associated with a return period of 95 years, and Life Safety(LS), also referred to as Significant Damage [42], which is associated with a return period of 475 years. For the return period of 95 years, the reference PGA of the return period of 475 years was scaled by a factor of 0.58, as recommended in [42]. The reference ground acceleration was established according to the Peruvian seismic code for Zone 2 (\(a_{g}=0.25\) g), soil type S\({}_{2}\) (\(S=\)1.2), and an importance factor corresponding to building category B (\(U=1.3\)) [13]. This results in a reference ground acceleration equal to 0.39 g. However, the code permits a reduction in this value based on the characteristics of the structure. In the case of adobe buildings, it refers to the provisions of code E.080 [14], where the reference ground acceleration can be taken as 0.29 g. This reduction aligns with strategies adopted in recommendations for the assessment of heritage structures, where assuming long return periods and high peak ground accelerations, while linked to the importance of the structure, is avoided to prevent excessively invasive interventions [43]. For a more conservative, yet realistic, assessment, the following verifications were carried out, assuming a reference ground acceleration of 0.30 g (\(a_{g}=0.25\) g, \(S=\)1.2). The horizontal load multiplier, \(\alpha_{0}\), is calculated using the Principle of Virtual Work, which equates the external and internal work associated with a virtual displacement configuration. In this approach, the virtual displacement, \(\delta_{k}\), produced by each force in the direction of loading is derived from the virtual rotation, \(\theta_{k}\), applied to the mechanism. The system is assumed to be in equilibrium when the work performed by the external forces equals the internal resisting work, \(L_{fi}\). The equilibrium condition is expressed as  
\[\alpha_{0}\left[\sum_{i=1}^{n}P_{i}\delta_{x,i}+\sum_{j=1}^{n}P_{j}\delta_{x,i }\right]-\sum_{i=1}^{n}P_{i}\delta_{y,i}-\sum_{h=1}^{m}F_{h}\delta_{h}=L_{fi} \tag{5}\]  
Figure 16: Points of load application of the roof and tie beams. where \(P_{i}\) represents the weight of the blocks, \(P_{j}\) is the weight contribution from the floor and roof, \(F_{h}\) includes the internal and external non-mass-related forces, e.g., the horizontal resistance provided by the tie beams, and \(\delta_{x}\), \(\delta_{y}\), and \(\delta_{h}\) are the corresponding virtual displacements in the direction of each force component.