$1\mathrm{.}1$ A "standard" triaxial compression test involves increasing the vertical stress $($i$.$e$.{}_v$ is $+),$

while maintaining a constant horizontal stress $({}_h=0),$ giving a stress path with a $3$:$1$

gradient "Standard" undrained triaxial compression tests are carried out on $2$ samples of a

kaolin clay, prepared by isotropic consolidation from a slurry state.

Sample A: Normally consolidated to $p^{\text{'}}=100$kPa

Sample B: Normally consolidated to $p^{\text{'}}=500$kPa, and then unloaded isotropically

to $p^{\text{'}}=100$kPa.

For each case, determine:

$($a$)$ the $($ultimate or critical state$)$ undrained shear strength

$99\mathrm{.}4$kPa

$($b$)$ the excess pore water pressure $(u)$ at this stage $($assuming the initial

backpressure $u_o$ is zero$).(\{$:$u_{fA}=63\mathrm{.}41$kPa;$u_{fB}=-32\mathrm{.}6$kPa$)$

$($c$)$ Draw a sketch or sketches of the stress paths in $q-p^{\text{'}}$ and $v-p^{\text{'}}$ space for each case.

$1\mathrm{.}2$ If drained compression tests $($with the same "standard" stress path$)$ had been carried out

instead on the same two samples, determine :

$($d$)$ the ultimate drained shear strength, $s_d$; and; $(s_{dA}=S_{dB}=75$kPa$)$

$($e$)$ the volumetric strain due to shearing to this state. Indicate if volumetric strain is

positive $($compression$)$ or negative $($dilation$).($A: $+8\mathrm{.}6\%$ contraction; B: $-2\mathrm{.}7\%$

dilation$)$