## 2 Formulation of Analytical Model
### Modeling Strategy of SIM Panel
#### 2.3.2 Type 2 SIM Panel without a Gap (Close Gap)  
The Type 2 SIM panel is constructed firmly against all sides of the frame without any gap between the top of the panel and the frame, as illustrated in Figure 11. This design enables the SIM panel to withstand additional lateral loads generated by the development of extra frictional forces on each SIM bed joint, attributed to the'vertical clamping' of the panel. The response to this additional frictional force is similar to the idealization of an equivalent compressive strut for traditional masonry (refer to Section 2.2). However, it is crucial to note that the interaction between the frame and panel differs for traditional masonry and SIM, and the equivalent compressive strut model is not applicable to the SIM panel [23]. The additional frictional force at each SIM bed joint of the panel, denoted as F\({}^{C}(\Delta t)\), due to the vertical clamping effect of the panel, is determined by Equation (7), as recommended by [PERSON] [23]. \[F^{C}(\Delta_{t})=\frac{E_{SIM}\Delta_{t}{}^{2}L_{p}t_{p}\mu_{SIM}}{4H_{p}{}^{2}} \tag{7}\]  
Figure 10: Idealized lateral load response for Type 1 SIM panel up to 2% story drift. The total frictional forces for the Type 2 SIM panel without a gap can be computed using the following equation, as outlined in [PERSON] [23]. \[F_{max}=H_{p}L_{p}t_{p}\rho_{SIM}g\mu_{SIM}+\frac{E_{SIM}\Delta_{t}{}^{2}L_{p}t_ {p}\mu_{SIM}}{4{H_{p}}^{2}} \tag{8}\]  
The comparison between the theoretical frictional force and the experimental results is presented in Figure 12. It is essential to note that the vertical scaling of the graphs in Figure 12 is not the same. The observed discrepancy between the theoretical (based on a previously developed equation by [PERSON] [23]) and experimental values is substantial, with the theoretical values often exceeding the experimental ones by more than 100 times. However, a closer alignment is observed for smaller displacements, typically around 1 mm. This disparity is attributed to assumptions made during the derivation of Equation (8) and the impact of additional frictional forces due to vertical clamping, which significantly relies on the second-order displacement component. To address this discrepancy, a reduction factor is introduced to better align the theoretical predictions with the experimental results. The proposed equation (Equation (9)) in this study aims to predict the shear force response for the Type 2 SIM panel without a gap. \[F_{max}=H_{p}L_{p}t_{p}\rho_{SIM}g\mu_{SIM}+R_{Type2}\frac{E_{SIM}\Delta_{t}{} ^{2}L_{p}t_{p}\mu_{SIM}}{4{H_{p}}^{2}} \tag{9}\]  
where RType2 is the reduction factor to match the experimental results. RType2 was determined based on the collective variation observed in Panel TO, Panel TF, Panel MO, and Panel MF for both push and pull directions of shear forces. The correlation between the reduction factor and the collective variation of in-plane displacement is illustrated in Figure 13.  
The determined value of RType2 is substituted into Equation (9), and the resulting equation for calculating the total frictional force for the Type 2 SIM panel is presented in Equation (10). Figure 14 depicts the variation of the proposed theoretical model with the experimental results for the Type 2 SIM panel without a gap. \[F_{max}=H_{p}L_{p}t_{p}\rho_{SIM}g\mu_{SIM}+4\times 10^{-5}\Delta_{t}{}^{-1.017} \times\frac{E_{SIM}\Delta_{t}{}^{2}L_{p}t_{p}\mu_{SIM}}{4{H_{p}}^{2}} \tag{10}\]  
Figure 14 illustrates that the proposed theoretical model cannot predict the drop in shear force due to unit cracking. Furthermore, the additional frictional force due to the clamping of the panel is also a function of the displacement/story drift of the structure. To achieve the correct response to the additional frictional force due to the clamping of the  
Figure 11: Type 2 SIM panels without a gap, adopted from [PERSON] [23]. panel, an iterative process was adopted in static analysis.