Intro Doc
$\underline{Introduction}$ $\\$ Terzaghi's theory requires the following input: $B$, $D$, $c'$, $\phi'$, $\gamma$, $\gamma_{sat}$, $D_w$ and foundation type. The bearing capacity will be calculated using the following equations based on the foundation type. $\\$ $\bullet$Continuous foundation: \begin{equation}\label{eq1} q_u=c' \cdot N_c+q \cdot N_q+\frac{1}{2}\cdot \gamma \cdot B \cdot N_{\gamma} \end{equation} $\bullet$Square foundation: \begin{equation}\label{eq2} q_u=1.3 \cdot c' \cdot N_c+q \cdot N_q+ 0.4 \cdot \gamma \cdot B \cdot N_{\gamma} \end{equation} $\bullet$Circular foundation: \begin{equation}\label{eq3} q_u=1.3 \cdot c' \cdot N_c+q \cdot N_q+0.3 \cdot \gamma \cdot B \cdot N_{\gamma} \end{equation} $\\$ The bearing capacity factors $N_c$, $N_q$, and $N_{\gamma}$ can be calculated using the following equation. \begin{equation}\label{e4a} N_c=\cot \phi' \left[ \frac{e^{2(3\mathrm{\pi}/4-\phi'/2) \tan \phi'}}{2 \cos^2 (\mathrm{\pi}/4+\phi'/2)} -1 \right] \end{equation} \begin{equation}\label{e4b} N_q= \frac{e^{2(3\mathrm{\pi}/4-\phi'/2) \tan \phi'}}{2 \cos^2 (\mathrm{\pi}/4+\phi'/2)} \end{equation} \begin{equation}\label{e2c} N_{\gamma} \approx \frac{2 (N_q+1) \tan \phi'} {1 + 0.4 \sin (4 \phi')} \end{equation} The calculations of $q$ and $\gamma$ needs to consider the groundwater conditions. Assuming the depth of water table is $D_w$, $q$ and $\gamma$ can be calculated using the following equations. \begin{equation}\label{eq5} q= \begin{cases} \gamma D , \space \space\space\space\space\space\space\space\space \text{if } D_w \ge D\\ \gamma D_w +(\gamma_\text{sat}-\gamma_\text{w})(D-D_w), \text{if } D_w \lt D \end{cases} \end{equation} \begin{equation}\label{eq6} \gamma= \begin{cases} \gamma^\prime \space\space\space\space\space \text{if } D_w \lt D+B \space \text{and } D_w \lt D\\ \overline{\gamma} \space\space\space\space\space\space \text{if } D_w \lt D+B \text{ and } D_w \ge D\\ \gamma \space\space\space\space\space \text{ if } D_w \ge D+B \end{cases} \end{equation} where, $\gamma^\prime = \gamma_\text{sat} - \gamma_\text{w} \text { and} \\ \overline{\gamma} = \gamma^\prime + \frac{D_\text{w}-D}{B} (\gamma-\gamma^\prime). \\$ $\\$ For the Allowable Stress Design (ASD), the bearing capacity meets the requirements if the following equation holds \begin{equation}\label{e7} SF=\frac{q}{p} \ge SF_\text{design} \end{equation} where SF is the safety factor, SF is the safety factor adopted for design, and $p$ is the loading (distributed load or pressure).