Problem $2$ HKS$11-27.$

A sand specimen is tested in direct simple shear. The stress conditions are shown. Initial conditions: ${}_v=312$kPa,$K_0=0\mathrm{.}5{}_h=156$kPa

At failure: ${}_v=312$kPa,${}_{hv}=180$kPa

$($a$)$ Draw the Mohr circles for both initial and final stress conditions.

$($b$)$ Clearly show the locations of the poles of these circles.

$($c$)$ Determine the magnitude and orientation of the principal stresses at failure.

$($d$)$ Determine the friction angle, assuming cohesionless soils $).$