Problem $2(5$ pts$)$

A saturated specimen of fine sand is isotropically consolidated to a cell pressure of $400$ kPa and then sheared undrained $($CU$).$ The drained $($effective stress$)$ friction angle, ${}^{\text{'}}=37$ and that the undrained $($total stress$)$ friction angle ${}_{cu}=25.$

a$)$ Determine the pore pressure at failure, $u_f$ Does the specimen display expansive or contractive behavior at this consolidation pressure? How would $u_f$ change if the consolidation pressure was increased for another test?

b$)$ Determine the principal stress difference at failure.

c$)$ Calculate the ratio of principal effective stresses at failure $(\frac{{}_{1f}^{\text{'}}}{{}_{3f}^{\text{'}}}).$

d$)$ Calculate $A_f.$

e$)$ Draw effective and total stress Mohr's circles at failure with corresponding failure envelopes.

f$)$ On a $p-q$ diagram, sketch the TSP$,K_f$ line, and a reasonable ESP. Why do we not know the exact ESP?

g$)$ If the test was drained $($CD$),$ what would be the value of ${}_{1f}$ at failure?