Question $3$

$($a$)$ Explain why the effective stress path of a drained triaxial compression test has a slope of $1$ on the $s^{\text{'}}-t\text{'}$ plane, and a slope of $3$ on the $q^{\text{'}}-p\text{'}$ plane.

$($b$)$ The initial volume and area of a sandy soil sample $($assume $c^{\text{'}}=0)$ are $200c{m}^3$ and $20c{m}^2$ respectively. A consolidated undrained $($CU$)$ triaxial test is conducted on the sample. The initial void ratio $(e_0)$ of the specimen is $0\mathrm{.}661,$ and volume change after consolidation at effective pressure $({}_3^{\text{'}})$ of $200$kPa is $2\mathrm{.}5ml.$ At failure, the axial shortening is $12\mathrm{.}5mm,$ the axial load is $750N,$ and pore water pressure increased by $100$kPa compared with the beginning of compression stage. Find the volume, area and length after consolidation, and determine the friction angle of the soil.

$(7$ marks$)$

Specimen height $(h^{**})$ and area $(A^{**})$ during consolidation:

$h^{**}$~~$h_0(1-\frac{{}_v}{3})$;$,A^{**}$~~$A_0(1-\frac{2{}_v}{3})$

Specimen area $(A)$ during compression:

$A=A^{**}(\frac{1-{}_v^{**}}{1-{}_a^{**}})$

where $A_0$ is the initial area, ${}_v$ is the volumetric strain during consolidation; ${}_v^{**}$ and ${}_a^{**}$ are volumetric strain and axial strain during compression.

$($c$)$ For the same soil prepared under the same conditions and same initial void ratio as in Q$3($b$),$ the critical state parameters $,$ and $N$ are found to be $0\mathrm{.}10,2\mathrm{.}05$ and $2\mathrm{.}17,$ respectively. Another CU triaxial test is conducted with effective confining pressure $({}_3^{\text{'}})$ of $300$kPa.

i$.$ Estimate the volume change during the consolidation stage.

$(4$ marks$)$

ii$.$ What would be the principal stress difference at failure under undrained compression?

$(4$ marks$)$

$($d$)$ The following parameters are known for a saturated normally consolidated clay: $N=2\mathrm{.}32,=0\mathrm{.}15,=2\mathrm{.}17$ and $M=0\mathrm{.}94.$

Estimate the value of void ratio $(e)$ and principal stress difference $(q)$ at failure if a drained triaxial compression test was conducted on the clay specimen consolidated under cell pressure $({}_3)$ of $400$kPa, with back pressure of $200$kPa. Sketch the effective and total stress paths of the test on the $q-p\text{'}$ plane. $(6$ marks$)$

Equation of ICL:

Equations of CSL:

$v=N-ln{p}^{\text{'}}$

$q^{\text{'}}=M{p}^{\text{'}}$

$v=-ln{p}^{\text{'}}$

$sin{}^{\text{'}}=\frac{3M}{6+M}$

$5$