Problem $5[30$ points$]$

A normally consolidated clay specimen was tested in a consolidated undrained $($CU$)$ axisymmetric triaxial test. At failure, the total stresses were ${}_3=100$kPa and ${}_1=204$kPa. The pore pressure at failure was $u_f=50$kPa. Please do the following:

$($a$)$ The major and minor principal effective stresses at failure $($i$.$e$.,{}_1^{\text{'}},$ and ${}_{3_j}^{\text{'}}).[5$ points$]$

$($b$)$ Noting that the cohesion $(c^{\text{'}})$ is zero for normally consolidated clays, determine the value of the effective friction angle $({}^{\text{'}}).[5$ points$]$

c$)$ The normal effective stress and shear stress $($i$.$e$.,{}_{ff}^{\text{'}}$ and ${}_{ff})$ acting on the plane of failure. $[5$ points$]$

d$)$ The angle $({}_f)$ that the failure plane makes with respect to the ${}_1^{\text{'}},$ direction. $[5$ points$]$

e$)$ A specimen of the same soil is tested in a consolidated drained $($CD$)$ axisymmetric triaxial test. Initially, the major and minor principal stresses are equal to $210$ kPa and $100$ kPa $,$ respectively. When the soil will be loaded, additional compressive stress increments of $$ and $(\frac{}{6})$ will be superimposed on ${}_1$ and ${}_3,$ respectively.

Determine the maximum possible value of $$ such that the shear strength will not be exceeded in the soil element. $[10$ points$]$