The two basic failure mechanisms for failure of a Gravity Dam are sliding and overturning at the downstream toe.

A$)$ First, work out the minimum values of widths $D$ and $W$ in terms of height $H,$ freeboard $h,=\frac{h}{H},$ the angle $,=$ the coefficient of friction between the dam and its foundation, the density of water ${}_w,$ the density of the dam material ${}_c$ and with a factor of safety $F.$

Set up equations of overturning moment and righting moment about the downstream toe of the dam, point $A.$ Remember: water pressure can be represented as a force $P$ acting at a height of $\frac{1}{3^{rd}}$ of the depth of the water above the base. Use the average ground pressure for the sliding calculation.

Your expressions should be in the form:

$\frac{D}{H}>$ "expression $)"-($sliding$)$; $($use the average ground pressure for this calculation.$)$

$\frac{D}{H}>$ "expression $(,,,{}_w,{}_c,F)"-($overturning$)$;

With corresponding expressions for $\frac{W}{H}$

B$)$ For the values below determine the $($range of values$)$ of the angle $$ which would lead to the minimum dam volume.

$=0\mathrm{.}1,=0\mathrm{.}8,{}_w=1000k\frac{g}{m^3},{}_c=2500k\frac{g}{m^3},F=1\mathrm{.}4$