PROBLEM $1(35$ points$)$

For the sheet pile wall below, find :

$-$ The required depth of the pile, D$,$ and the point of rotation, z$.(15$ points$)$

$-$ The location and magnitude of the maximum moment $($the plane of zero shear$)(10$ points$)$

$-$ If $\backslash (\backslash$mathrm$\{$S$\}=\backslash$mathrm$\{$M$\}\_\{\backslash$max $\}/\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$b$\}\},\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$b$\}\}=25,000\backslash$mathrm$\{$psi$\}\backslash )$ and $\backslash (\backslash$mathrm$\{$S$\}=11\mathrm{.}1\backslash$mathrm$\{$in$\}^\{3\}/\backslash$mathrm$\{$lf$\}\backslash ),$ would the pile be adequate? $(10$ points $\backslash ()\backslash )$

Assumptions:

$-$ The active and passive pressure coefficients do not change below the ground water table

$-$ The water pressure below the excavation is in equilibrium

$-$ The pile is allowed to move the requisite distances to reach the passive and active pressure

$-$ The soil is cohesionless and there is no adhesion between the pile and the soil

Expectations for Problem $1$

$-$ Provide a diagram of forces.

$-$ Briefly explain how you solved the system of simultaneous equations. If you used Excel, describe how you set up the equations and how you interpreted the results.