## 3 Bearing Capacity of Foundation
### Calculation Methodology  
[PERSON] [39] first derived the ultimate bearing capacity calculation formula for strip foundations based on limit equilibrium theory in 1920. [PERSON] [39] assumed a smooth interface between the foundation base and the soil, and that the soil underneath the foundation had no weight, resulting in a sliding surface composed of two sets of planes and logarithmic spiral surfaces transitioning between them. However, due to the unrealistic assumptions made by [PERSON] [39], the results obtained were rough approximations. Subsequently, many scholars made further modifications and developments based on his research. [PERSON] [27] proposed the ultimate bearing capacity formula that considers the weight of the soil underneath the foundation based on [PERSON]'s [39] fundamental principles:  
\[P_{\rm u}=\frac{1}{2}\gamma bN_{\gamma}+qN_{q}+cN_{\rm c} \tag{3}\]  
\[N_{\rm q}=e^{\pi\tan\varphi}\tan^{2}(45^{\circ}+\frac{\varphi}{2}) \tag{4}\]  
\[N_{\rm c}=(N_{\rm q}-1)\cot\varphi \tag{5}\]  
\[N_{\gamma}=\tan(45^{\circ}+\frac{\varphi}{2})[e^{\pi\tan\varphi}\tan^{2}(\frac {\pi}{4}+\frac{\varphi}{2})-1] \tag{6}\]  
Figure 8: Pore pressure at the center line of the embankment.  
where \(P_{\rm u}\) is the ultimate bearing capacity of the foundation, kPa; \(q\) is the overload on both sides of the foundation above the footing level, \(\rm{kN/m^{2}}\); \(\gamma\) is the unit weight of the soil, \(\rm{kN/m^{3}}\); \(b\) is the width of the footing, m; \(N_{\rm c}\), \(N_{\rm q}\), \(N_{\rm\gamma}\) are dimensionless bearing capacity coefficients; \(c\) is the cohesion, kPa; and \(\varphi\) is the friction angle, \({}^{\circ}\).  
[PERSON] et al. [40] noted that when the rate of foundation loading on fully saturated cohesive soil exceeds the dissipation rate of pore water pressure, [PERSON]'s [41] concept of \(\varphi=0\) is commonly used to explain the bearing capacity. In such cases, the net bearing capacity depends solely on the undrained shear strength (\(c_{\rm u}\)) of the soil.  
For fully saturated cohesive soil undergoing undrained loading, where \(\varphi=0\), the bearing capacity factors according to [PERSON] [27] are \(N_{\rm c}=5.71\), \(N_{\rm q}=1\), \(N_{\rm\gamma}=0\), and \(c=c_{\rm u}\). Therefore, the ultimate bearing capacity calculation formula for fully saturated cohesive soil is as follows:  
\[q_{\rm u}=5.71c_{\rm u}+\gamma D \tag{7}\]  
where \(q_{\rm u}\) is the ultimate bearing capacity, kPa; \(D\) is the depth of the foundation, m.  
The net bearing capacity of foundation is then  
\[q_{\rm u}=5.71c_{\rm u} \tag{8}\]  
The formula for the undrained shear strength of soil in the Modified Cam-Clay is as follows:  
\[c_{\rm u}=\frac{M}{2}\exp\left[\frac{(1+e_{1})-(\lambda-\kappa)\ln 2}{\lambda }-\frac{1+e_{0}}{\lambda}\right] \tag{9}\]  
where \(e_{1}\) is the intercept on the isotropic compression consolidation \(e\)-ln\(p\) curve; \(e_{0}\) is the initial void ratio; \(\lambda\) is the compression index; \(\kappa\) is the rebound index; and \(M\) is the critical-state stress ratio.  
In addition, the initial void ratio \(e_{0}\) is related to the intercept \(e_{1}\) as  
\[e_{0}=e_{1}-\lambda\ln p_{\rm c}+\kappa\ln(OCR) \tag{10}\]  
where \(OCR\) is the over-consolidation ratio.