Intro Doc

• rankine and coulomb.tex

$\underline{Rankine\space Theory\space for\space Granular\space Backfill}$ $\\$ The coefficients for the active and passive pressures are calculated as follows $$\begin{equation}\label{eq1} K_\mathrm{a}=\cos \alpha \frac{\cos \alpha - \sqrt{\cos^2 \alpha -\cos^2 \phi^\prime}}{\cos \alpha + \sqrt{\cos^2 \alpha -\cos^2 \phi^\prime}} \end{equation}$$ $$\begin{equation}\label{eq2} K_\mathrm{p}=\cos \alpha \frac{\cos \alpha + \sqrt{\cos^2 \alpha -\cos^2 \phi^\prime}}{\cos \alpha - \sqrt{\cos^2 \alpha -\cos^2 \phi^\prime}} \end{equation}$$ $\\$ $\\$ $\underline{Coulomb\space Theory\space for\space Granular\space Backfill}$ $\\$ $$\begin{equation}\label{eq3} K_\mathrm{a}=\frac{\sin^2(\beta+\phi^\prime)}{\sin^2 \beta \cdot \sin(\beta-\delta^\prime) [1+\sqrt{\cfrac{\sin(\phi^\prime+\delta^\prime) \sin(\phi^\prime - \alpha)}{\sin(\beta-\delta^\prime) \sin(\alpha+\beta)}}]^2} \end{equation}$$ $$\begin{equation}\label{eq4} K_\mathrm{p}=\frac{\sin^2(\beta-\phi^\prime)}{\sin^2 \beta \cdot \sin(\beta+\delta^\prime) [1-\sqrt{\cfrac{\sin(\phi^\prime+\delta^\prime) \sin(\phi^\prime + \alpha)}{\sin(\beta+\delta^\prime) \sin(\alpha+\beta)}}]^2} \end{equation}$$ $\\$ $\\$ $\underline{Forces\space caused\space by\space the\space earth\space pressure}$ $\\$ The active and passive forces are calculated using the following equation by overlooking the tensile stresses: $$\begin{equation}\label{eq5} F_\mathrm{a}=\frac{1}{2} \gamma H^2 \cdot K_\mathrm{a} \end{equation}$$ $$\begin{equation}\label{eq6} F_\mathrm{p}=\frac{1}{2} \gamma H^2 \cdot K_\mathrm{p} \end{equation}$$

App

• Rankine and Coulomb Earth Pressure.py