## 2 Formulation of Analytical Model
### Modeling Strategy of SIM Panel
#### 2.3.1 Type 1 SIM Panel with an Open Gap  
In Type 1 construction, the SIM panel is positioned tightly against the columns, featuring a gap between the top of the panel and the frame, as illustrated in Figure 8. This configuration implies that the frame interacts with the SIM panel solely through the columns. The gap remains open during a seismic event, and no compressive force on the panel originates from the top of the frame. As the frame experiences lateral loads, frictional forces emerge on the bed joints of the panel due to its self-weight. The distribution of the friction force can be conceptualized as a triangular force, as depicted in Figure 8.  
The maximum theoretical frictional force at the bottom of the panel (Fmax) is determined by Equation (5) [23], with the notations for the equation already outlined in Table 1.  
\[F_{max}=H_{p}L_{p}t_{p}\rho_{SIM}\beta H_{SIM} \tag{5}\]  
It is crucial to compare the theoretical Fmax value with the experimental results. For comparison, panels with an open gap (Panel TO and Panel MO) and panels with foam  
Figure 7: Modeling of traditional masonry panel: (**a**) Geometry of bare frame (not to scale); (**b**) Model of traditional masonry panel in Strand7.  
(Panel TF and Panel MF) are categorized as Type 1 SIM panels with a gap; experimental details can be found [9; 13]. It is noteworthy that the F\({}_{\text{max}}\) remains constant, independent of the displacement or storey drift of the panel. For the tested panels, F\({}_{\text{max}}\) has a constant value of 7.28 kN, resulting in a total frictional force (which represents the area of the triangle in Figure 8) of 7.28 kN for the 2 m \(\times\) 2 m experimentally tested panel. The response of the Type 1 SIM panel (SIM panel with a gap) to lateral loads in this study is depicted in Figure 9.  
It is noted that the theoretical value obtained from Equation (5) aligns well with the experimental results of up to 40 mm displacement. However, beyond 40 mm of in-plane displacement (2% of the storey drift), the behavior of open-gap panels becomes similar to that of the Type 2 SIM panel (SIM panel without a gap). This transition occurs because, after 40 mm of in-plane displacement (2% of the storey drift), panels with a gap and panels with foam experience an additional lateral load due to the development of extra frictional forces on each SIM bed joint, attributed to the 'Vertical Clamping' of the panel, as detailed  
Figure 8: Type 1 SIM panels with gap, adopted from [PERSON] [23].  
Figure 9: Comparison of experimental and analytical results for Type 1 SIM panel.  
in Section 2.3.2. This study proposes the following equation to model the behavior of SIM panels with a gap (Type 1 SIM panel).  
\[F_{max}=H_{p}L_{p}t_{p}\rho_{SIM}g\mu_{SIM}+4\times 10^{-5}\big{\langle}\Delta_{ t}-0.002H_{p}\big{\rangle}^{-1.017}\times\frac{E_{SIM}\Delta_{t}{}^{2}L_{p}t_{p} \mu_{SIM}}{4H_{p}{}^{2}} \tag{6}\]  
where \(\Delta_{t}\) and E\({}_{SIM}\) are the in-plane displacement of the panel and modulus of elasticity of SIM panel, respectively.  
It is essential to highlight that the proposed Equation (6) for the Type 1 SIM panel with a gap cannot account for the shear force resulting from the cracking of the units, as evidenced in Figure 9 after a displacement of \(-80\) mm. Nevertheless, the proposed equation (Equation (6)) demonstrates a good correlation with the experimental results. Notably, F\({}_{\text{max}}\) is calculated at 21.85 kN, and the total frictional force is 32.78 kN for the 4 m \(\times\) 3 m panel. The idealized response of the Type 1 SIM panel, up to 40 mm in-plane displacement, to the lateral load in this study is depicted in Figure 10.