## 3 Numerical Evaluation
### Material Model and Variables for Shaking Table Test  
The interaction among the expanded units comprising a masonry wall plays an important role in the structural behavior of the wall. Therefore, to model masonry walls, the Drucker-Prager model is applied. This model is advantageous for modeling frictional materials whose yield strengths are higher in compression than in tension [36]. In this study, the input material properties were not assigned separately to the bricks and mortar but were selected as equivalent properties for the expanded unit. These values were referenced in a previous experimental and numerical study [37], and reflect the behavior of typical clay brick masonry under dynamic loading conditions, as shown in Table 1. Material parameters were selected based on values reported in previous published studies involving similar masonry materials tested under dynamic conditions [37].  
To model the mechanical behavior of STPU, the Arruda-Boyce model was adopted as the constitutive formulation using a shell element. Polyurea and similar polymers possess a chain-like molecular configuration and are capable of sustaining large strains exceeding 100%, which results in highly nonlinear and rubber-like hyperelastic responses. Given these characteristics, the Arruda-Boyce model is particularly appropriate, as it captures the material's behavior from a micromechanical perspective, offering high accuracy in representing large deformations [38; 39].  
The model is derived based on statistical mechanics, where a representative volume element is composed of eight polymer chains arranged diagonally within a hexahedral structure, as depicted in Figure 7. The material is treated as incompressible, and its stress  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \begin{tabular}{c} **Density** \\ **(kg/m3)** \\ \end{tabular} & \
\begin{tabular}{c} **Elastic** \\ **Modulus** \\ **(MPa)** \\ \end{tabular} & \(\mathbf{\
u}\) & \(\mathbf{\psi}\) & \begin{tabular}{c} **Flow Stress** \\ **Ratio** \\ \end{tabular} & \
\begin{tabular}{c} **Angle of** \\ **Friction** \\ \end{tabular} &
\begin{tabular}{c} **Yield** \\ **Compressive** \\ **Stress (MPa)** \\ \end{tabular} \\ \hline Expanded unit & 1800 & 2888 & 0.15 & 11.3 & 1 & 36 & 16 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Drucker–Prager material properties of expanded unit for masonry wall [37].  
Figure 6: Finite element modeling methods of masonry wall: (**a**) detailed micro-model; (**b**) simplified micro-model; (**c**) macro-model [34; 35].  
strain relationship is formulated through a strain energy potential function, expressed as Equation (1) [40].  
\[U=\mu\left\{\frac{1}{2}(\overline{h}-3)+\frac{1}{20\lambda_{m}^{2}}\left( \overline{h}_{1}^{2}-9\right)+\frac{11}{1050\lambda_{m}^{2}}\left(\overline{h} _{1}^{3}-27\right)+\frac{19}{7000\lambda_{m}^{6}}\left(\overline{h}_{1}^{4}-81 \right)+\frac{519}{673750\lambda_{m}^{8}}\left(\overline{h}_{1}^{5}-243\right) \right\}+\frac{1}{D}\left(\frac{\overline{h}_{2}^{2}-1}{2}-lnJ_{d}\right) \tag{1}\]  
Here, \(\lambda_{m}\) is the elongation rate in longitudinal direction; \(\overline{h}\) is the deviatoric strain invariant; \(J_{el}\) is the elastic volumetric rate; and \(\mu\) and D are the volume compression controlling material coefficients.  
To reflect the physical properties of STPU in the numerical analysis, the strain-stress data obtained from the tensile strength test were applied to the material model, as shown in Figure 8 [30].  
A shell element was also utilized to model the laminated GFRP. The material model used is the lamina model, which is mainly employed for laminated FRP. The elastic properties of GFRP are summarized in Table 2, and the failure stress used as a sub-option is listed in Table 3 based on a previous study [41].  
Figure 8: Strain–stress graph of STPU [30].  
Figure 7: Eight chain rubber elasticity model: (**a**) undeformed; (**b**) tension; (**c**) compression [40].