## 5 Numerical Validation of the Model
### Dynamic Analysis of a Masonry House  
#### 5.2.1 The Studied House  
The next phase in validating the DFM for transient dynamic analysis involved the numerical simulation of a dynamic test performed on a structure. The DFM enables 3D transient dynamic analysis of various types of masonry structures. For this validation, the model's performance was evaluated through the simulation of a unidirectional shake table test performed on a half-scale masonry house, as performed by [PERSON] et al. [31]. It is worth noting that only a few 2D or 3D shake tables exist, and the experimental data from such tests remain limited. Despite the unidirectional nature of the shake table test, 3D effects may arise within the structure. The DFM has demonstrated its ability to capture these effects, such as vertical modes, as confirmed by the modal analysis.  
To design the half-scale house model, [PERSON]'s similitude law was applied, ensuring an accurate scale-down of the prototype's properties to those of the model. This relationship is outlined in Table 3, which illustrates the proportional transformation of the properties of the model (M) relative to the prototype (P). For this shake table experiment, the selected similitude parameters were \(\lambda=0.5\), \(e=1\), and \(\rho=1\).  
The model configuration is shown in Figure 16, with specific designations for cardinal orientation: the south wall was positioned on the left side of the figure, while the east and west walls were identified as IP-loaded walls. \"Door wall 2\" refers to the central partition, and the north wall is at the opposite end. The model had a symmetrical design.  
The house was constructed using mud mortar and extruded adobe bricks (175 mm \(\times\) 125 mm \(\times\) 50 mm). Its dimensions are shown in Figure 16. The total mass of the masonry structure was about 3.8 tons.  
The results presented in this section pertain to a slightly reinforced masonry house. Figure 17a shows the house during construction, where the only reinforcement consists of timber frames around the doors and windows. In Figure 17b, a speckle pattern can be seen painted on the walls for digital image correlation (DIC) techniques.  
Two additional tests were conducted on houses with different seismic strengthening techniques [31]; however, these were not analyzed in the present paper. The focus here was on the structure that exhibited the most significant damage to validate the nonlinear DFM model for masonry walls.  
The roof truss was built with timber and covered with galvanized iron (GI) sheeting. It was secured with 10 metal cables--8 along the west and east walls, and 2 along the north and south walls--which passed through the lintel level and were pre-stressed, as seen in Figure 17b. The roof's total mass was approximately 185 kg.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Quantity** & **Symbol** & **Similitude** \\ \hline Length & L & L\({}_{p}=\lambda\) L\({}_{M}\) \\ Modulus of elasticity & e & E\({}_{p}=\) e E\({}_{M}\) \\ Density & \(\rho\) & \(\rho p=\rho\)\(\rho\)\(\rho\)\(M\) \\ Area & A & A\({}_{p}=\lambda^{2}\)A\({}_{M}\) \\ Volume & V & V\({}_{p}=\lambda^{3}\)V\({}_{M}\) \\ Mass & m & m\({}_{p}=\rho\)\(\lambda^{3}\) m\({}_{M}\) \\ Velocity & v & v\({}_{p}=\) e\({}^{1/2}\)\(\rho^{-1/2}\)v\({}_{M}\) \\ Acceleration & a & a\({}_{p}=\) e \(\lambda^{-1}\)\(\rho^{-1}\) a\({}_{M}\) \\ Force & F & F\({}_{p}=\) e \(\lambda^{2}\)F\({}_{M}\) \\ Stress & \(\sigma\) & \(\sigma_{p}=\) e \(\sigma_{M}\) \\ Time & t & t\({}_{p}=\) e\({}^{-1/2}\)\(\lambda\)\(\rho^{1/2}\) t\({}_{M}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Cauchy similitude law.