## 3 Outline of the Simulation Method  
### Rigid Plastic FEM Constitutive Equation  
The Rigid Plastic Finite Element Method (RPFEM) was developed by [PERSON] et al. [19; 20] and upgraded by [PERSON] et al. [21] and [PERSON] et al. [22]. The RPFEM calculates the limit load without any assumption of a potential failure mode. The method has been successfully used to examine geotechnical problems [23; 24]. The RPFEM using the rigid plastic constitutive equation with the Drucker-Prager yield function necessitates only a few soil parameters such as the unit weight, the cohesion, the shear resistance angle, and the dilatancy angle, which is its advantage. The Drucker-Prager yield function is used in plane strain conditions with an associated flow rule. The Drucker-Prager yield function is expressed as follows:  
\[f(\sigma)=\alpha I_{1}+\sqrt{J_{2}}-\kappa=0 \tag{1}\]  
where, \(I_{1}=tr(\sigma)\) is the first invariant of the stress tensor \(\sigma\) and \(J_{2}=\frac{1}{2}S:S\) is the second invariant of deviator stress tensor \(S\). \(\alpha\) and \(\kappa\) are soil parameters corresponding to the shear resistance angle and cohesion, respectively, and are expressed as  
\[\alpha=\frac{\tan\phi}{\sqrt{9+12\text{tan}^{2}\phi}},\ \ \kappa=\frac{3c}{\sqrt{9+12\text{tan}^{2}\phi}} \tag{2}\]  
where \(c\) is the cohesion and \(\phi\) is the shear resistance angle. The volumetric strain rate \(\dot{\varepsilon}_{v}\) for the rigid plastic constitutive equation is expressed as follows:  
\[\dot{\varepsilon}_{v}=tr\big{(}\dot{\varepsilon}\big{)}=tr\bigg{(}\lambda \frac{\partial f(\sigma)}{\partial\sigma}\bigg{)}=tr\bigg{(}\lambda\bigg{(} \alpha\mathbf{I}+\frac{S}{2\sqrt{J_{2}}}\bigg{)}\bigg{)}=\frac{3\alpha}{\sqrt{3 \alpha^{2}+\frac{1}{2}}}\dot{\varepsilon} \tag{3}\]  
where \(\lambda\) is the plastic multiplier and \(\dot{\varepsilon}\) is the norm of the strain rate \(\dot{\varepsilon}\). I express the unit stress tensors. The strain rate \(\varepsilon\), which is purely a plastic component, should satisfy the volumetric constraint condition on the dilation property of the soil as  
\[h\big{(}\dot{\varepsilon}\big{)}=\dot{\varepsilon}_{v}-\frac{3\alpha}{\sqrt{3 \alpha^{2}+\frac{1}{2}}}\dot{\varepsilon}=\dot{\varepsilon}_{v}-\eta\dot{ \varepsilon}=0 \tag{4}\]  
Any strain rate that is compatible with the Drucker-Prager yield criterion must satisfy the kinematical constraints of equations Equation (4). \(\eta\) is one coefficient determined by Equation (4), which is one of the dilation characteristics. The dilation constitutive equation is expressed by the Lagrange method after Tamura [25] as follows:  
\[\sigma=\frac{\kappa}{\sqrt{3\alpha^{2}+\frac{1}{2}}}\frac{\dot{\varepsilon}} {\dot{\varepsilon}}+\beta\bigg{(}\mathbf{I}-\eta\frac{\dot{\varepsilon}}{\dot{ \varepsilon}}\bigg{)}=0 \tag{5}\]  
The first term expresses the stress component, uniquely determined for the yield function. The second term expresses the indeterminate stress component along with the yield function. The indeterminate parameter \(\beta\) remains unknown until the boundary value \begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties** & **Cohesive** & **Intermediate** \\ \hline Cohesion \(c\) (kPa) & \(10,50,75\) & \(10,50\) \\ Shear resistance angle \(\phi\) (deg) & \(0\) & \(10,20,30,40\) \\ Unit weight \(\gamma\) (kN/m\({}^{3}\)) & \(16\) & \(16\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Soil parameters. problem is solved. Equation (6) introduces the constraint condition on the strain rate into the constitutive equation by using the penalty method. \[\sigma=\frac{\kappa}{\sqrt{3\alpha^{2}+\frac{1}{2}}}\frac{\hat{\varepsilon}}{e}+P (\dot{\varepsilon}_{v}-\eta\dot{e})\bigg{(}\mathbf{I}-\eta\frac{\hat{\varepsilon}}{e }\bigg{)}=0 \tag{6}\]  
where \(P\) is the penalty constant. This technique makes the computation faster and more stable. It is worth mentioning that the RPFEM is developed based on the upper bound theory of the limit analysis and is effective in the calculation of the bearing capacity of strip footings on all types of soil.