Intro Doc
$\underline {Introduction}$ $\\$ The web-based tool will be developed to predict the sliding resistance of the shallow foundation. The end users will enter their input for their design via text boxes and a dropdown menu provided on a home page. This input will be passed to the back end (server) and used for the calculation based on the theory. The calculation results will be displayed on a web page, which will be different from the homepage (dynamic web pages). $\\$ $\\$ $\underline {Rankine's \space Active\space and\space Passive\space Pressure}$ $\\$ As shown in the above figure, active and passive pressures will develop if the horizontal stress pointing to the right is applied to the foundation, respectively. Rankine's earth pressure theory can be used to calculate the active and passtive pressures. \begin{equation}\label{eq1} p_a=K_\mathrm{a} \sigma^\prime _1 - 2 \sqrt{K_\mathrm{a}} \cdot c \end{equation} \begin{equation}\label{eq2} p_p=K_\mathrm{p} \sigma^\prime _1 + 2 \sqrt{K_\mathrm{p}} \cdot c \end{equation} where $K_\mathrm{a}$ and $K_\mathrm{p}$ are the active and passive pressure coefficients. In Rankine's theory, these two coefficients can be calcualted using the effective internal friction angle as \begin{equation}\label{eq3} K_\mathrm{a}=tan^2(45-\frac{\phi}{2}) \end{equation} \begin{equation}\label{eq4} K_\mathrm{p}=tan^2(45+\frac{\phi}{2}) \end{equation} The total forces resulting from the active and passtive pressures are as follows when the sliding occurs along the width (B) direction, \begin{equation}\label{eq5} F_\mathrm{a}=(\frac{1}{2} K _\mathrm{a} \gamma H^2 -2 \sqrt{K_\mathrm{a}}c H) \cdot B \end{equation} \begin{equation}\label{eq6} F_\mathrm{p}=(\frac{1}{2} K _\mathrm{p} \gamma H^2 +2 \sqrt{K_\mathrm{p}}c H) \cdot B \end{equation} If sliding occurs along the length direction, then replace $B$ in the above equations with $L$. $\\$ $\\$ $\underline {Sliding\space Resistance}$ $\\$ $\underline {Cohesionless\space soils}$ $\\$ Here, we have used the force version of the Allowable Stress Design. \begin{equation}\label{eq7} \frac{F_\mathrm{u}}{F}=\frac{0.5 F_\mathrm{p}+ F \cdot \mu}{S+F_\mathrm{a}} \ge SF_\mathrm{design} \end{equation} where 0.5 is added in front of $F_\mathrm{p}$ to take into consideration of the fact that not all the passive pressure can be mobilized due the large deformation required for the full mobilization of $p_\mathrm{p}$. $\\$ $\underline {Cohesive\space soils}$ $\\$ The undrained shear strength instead of friction and Rankine pressures control the sliding resistance, accordingly, \begin{equation}\label{eq8} \frac{F_\mathrm{u}}{F}=\frac{0.5 C_\mathrm{u} D B+ C_\mathrm{u} \cdot B L}{S} \ge SF_\mathrm{design} \end{equation}