## 2 Numerical Modeling
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Sois** & \begin{tabular}{c} **Parameters** \\ \end{tabular} & \begin{tabular}{c} **Value** \\ \end{tabular} \\ \hline \multirow{8}{*}{\begin{tabular}{c} **Sois** \\ \end{tabular} } & Shear wave velocity, \(V_{s}\) (m/s) & 120, 150, 180 \\  & Unit weight, \(\gamma_{\mathrm{s}}\) (kN/m\({}^{3}\)) & 20 \\  & [PERSON]’s ratio, \(\upsilon\) & 0.4 for clays, 0.3 for sands \\  & Calculated Young’s modulus, & \\  & \(E\) (MPa), where \(E=2\)G (\(1+\upsilon\)), & 82, 128, 185 for clays \\  & \(\mathrm{G=}\)\(V_{s}\)\({}^{2}\)\(\mathrm{p_{s}}\), \(\mathrm{p_{s}}\) is the mass density & 76, 119, 171 for sands \\  & of soil = \(\gamma_{\mathrm{s}}\)/g, g = 9.81 m/s\({}^{2}\) & \\  & Calculated strength parameters & \\  & \(V_{s}=23\,s_{u}^{0.475}\) (Refs. [28; 29]) & \\  & \(\phi^{\prime}=3.5\)(N)\({}^{\circ}\) + 22 (Ref. [30]) and & \\  & \(V_{s}=80\)(N)\({}^{0.33}\) (Ref. [31]), & \\  & where N is SPT-N value & \\ \hline \multirow{8}{*}{\
\begin{tabular}{c} **Foundation** \\ \end{tabular} } & Length, \(L\) (m) (\(L=B\)) & 16, 26, 36 \\  & Width, \(B\) (m) (\(B=L\)) & 16, 26, 36 \\  & Thickness, \(d\) (m) & 1 \\  & Unit weight, \(\gamma\) (kN/m\({}^{3}\)) & 24 \\  & [PERSON]’s ratio \(v_{b}\) & 0.13 \\  & [PERSON]’s modulus, \(E_{b}\) (GPa) & 30 \\ \hline \multirow{2}{*}{**Applied Load**} & Uniform load with intensity, \(q\) (kPa) &
\begin{tabular}{c} 264, 100, and 52 for raft with dimensions \\ of 16 m, 26 m, and 36 m \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the numerical model. flexible foundations. Thus, the results can be used to interpret the behaviors of semi-rigid or flexible foundations; the possible influence of foundation thickness was ruled out in this study. 3D Midas-GTS NX analysis [32] was adopted in the study. Such analysis has been used in many engineering design projects. The first author has adopted Midas-GTS analysis to investigate raft foundations and piled raft foundations for many years [33,34]. Agreeable results obtained from the 3D Midas-GTS analysis on elastic and inelastic soils have been successfully compared to those from 3D PLAXIS analysis [35,36]. In the study, the analytic zone of the FE mesh was set up to be 200 m \(\times\) 200 m \(\times\) 60 m (Figure 2). The essential boundary conditions (roller and hinge) were applied to the FE zone. Eight-node isoparametric solid elements were mainly used to discretize the mesh. At some finer regions, 6-node solid elements can also be found. Since linearized shape functions were used for these elements, the FE mesh was made by smaller cubic elements with the dimension of 1m for the foundation and the soils around the ground surface. The convergence and stability of the solutions were ensured. As the maximum differential settlements of the rafts are much less than the distance between the center and the edge of the raft, compatibility condition was assumed at the interface between the slab and the soil. The possible influences of the discrete mesh, the size of the FE zone, and the boundary conditions were clarified to ensure that the solutions are stable and accurate enough in this study. Figures 3 and 4 depict the results of the validations of the FE analytical zone and the solutions compared to PLAXIS analysis on linearly elastic soils and FD analysis with nonlinear soil springs. The effect of the initial state of ground stress was considered in the analysis for more realistic estimation of foundation behavior. The staged construction procedure of the analysis was adopted to model the material nonlinearities. Incremental load was applied at each step in order to reach the total loading pressure. A 10-story building load on the raft measuring 26 m \(\times\) 26 m was presumed. The design structural load pressure is about 100 kPa (10 kPa per story suggested by the design code). Therefore, a total foundation load of 67.6 MN was assumed, and the same amount of load was applied to the rafts with different dimensions.