Q$3.$

The relationship between the relative density, $D_r,$ and the peak angle of friction ${}^{\text{'}}$ of a sand in a drained test can be given as ${}^{\text{'}}=25+0\mathrm{.}18(D_r),D_r$ in $\%.$ A drained triaxial test on the same sand was conducted with an effective chamber confining pressure of $250$ kPa in a sand specimen at a relative density of $50\%.$

a$)$ Calculate the major principal stress at failure.

b$)$ What are the normal and shear stresses on the failure plane?

c$)$ Determine the normal and shear stresses at failure on a plane that makes an angle of $30$ with the major principal plane.