$($a$)$ Drained triaxial compression tests are carried out on loose and dense samples of a clean sand. The critical state friction angle $(\backslash (\backslash$phi$\_\{$c$^\{3\}\}\backslash ))$ of $\backslash (27^\{\backslash$circ$\}\backslash )$ is obtained from the tests on the loose sand. The maximum dilatancy ratio determined from the test on dense sand is $0\mathrm{.}7.$ Calculate the friction angle at failure for the dense sand.

$(5$ Marks$)$

$($b$)$ A drained triaxial compression test is carried out on a saturated reconstituted clay sample under an effective net stress of $320$ kPa $.$ The clay tested has the following soil parameters: $\backslash (\backslash$phi$\_\{\backslash$mathrm$\{$cs$\}\}\{\}^\{\backslash$prime$\}=25^\{\backslash$circ$\},\backslash$mathrm$\{$C$\}\_\{\backslash$mathrm$\{$c$\}\}=0\mathrm{.}5,\backslash$mathrm$\{$C$\}\_\{\backslash$mathrm$\{$r$\}\}=0\mathrm{.}05\backslash ),$ and $\backslash (\backslash$Gamma$=3\backslash ).$ The physical meanings of the parameters are given in Pages $6$ and $7.$ At the end of the isotropic consolidation stage $($before the drained shearing$),$ the sample had a void ratio of $0\mathrm{.}87.$ Determine the effective mean stress, void ratio and volumetric strain at failure.

$(6$ Marks$)$

$($c$)$ Assuming the test results in Q$3($b$)$ represent the actual conditions at a site, where the sample is originally taken, estimate the undrained shear strength of the soil at a depth of $10$ m if the groundwater table at the site is located at the ground level and the pre$-$consolidation pressure of the clay is $120$ kPa at the depth of $10$ m $.$ The total unit weight of the soil is $\backslash (18\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash )$ and the unit weight of water is $\backslash (10\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$

$(6$ Marks$)$

$($d$)$ A normally consolidated clay with a $\backslash (\backslash$mathrm$\{$K$\}\_\{0\}\backslash )$ value of $0\mathrm{.}5$ is consolidated in a triaxial cell under a $\backslash (\backslash$mathrm$\{$K$\}\_\{0\}\backslash )$ condition. The given soil parameters for this soil are: $\backslash (\backslash$lambda$=0\mathrm{.}28\backslash ),\backslash (\backslash$mathrm$\{$K$\}=0\mathrm{.}03\backslash )$ and $\backslash (\backslash$mathrm$\{$M$\}=0\mathrm{.}97\backslash ).$ The physical meanings of the parameters are given in Page $7.$ At an effective mean stress of $300$ kPa during the $\backslash (\backslash$mathrm$\{$K$\}\_\{0\}\backslash )$ consolidation stage, the void ratio of the sample is $1\mathrm{.}2.$

$($i$)$ Calculate the critical state void ratio at an effective mean stress of $350$ kPa $.$

$($ii$)$ After the end of $\backslash (\backslash$mathrm$\{$K$\}\_\{0\}\backslash )$ consolidation at the effective mean stress of $300$ kPa $,$ the sample is isotropically consolidated to reach a void ratio of $1.$ Subsequently, the sample is sheard under undrained conditions. Calculate the pore$-$water pressure at failure upon shearing.