Problem $1$ HKS$11-26.$

In a direct shear test on a specimen of cohesionless sand, the vertical normal stress on the specimen

is $240k\frac{N}{m^2},$ and the horizontal shear stress at failure is $160k\frac{N}{m^2}.$

$($a$)$ Assuming uniform stress distribution within the failure zone and a straight$-$line failure

envelope that goes through the origin $($i$.$e$.,$ cohesion equal to $0).$

$($b$)$ Determine by means of the Mohr diagram the magnitude and direction of the principal

stresses at failure.

$($c$)$ Determine the friction angle.