Question $2(7$ points$)$

Consider the trapdoor problem described in $[1],$ usually encountered in mining. Using OptumG$2($adaptive Lowe Bound and Upper Bound solutions$),$

I. find the stability number $(N)$ for values of $\frac{H}{B}=1\mathrm{.}0,2\mathrm{.}0,4\mathrm{.}0,8\mathrm{.}0,12\mathrm{.}0,16\mathrm{.}0,20,$ assuming a smooth interface $(4$ point$),$

compare your results by plotting them versus the stability numbers given on Figure $20$ in $[1](3$ point$),$

Question $3(9$ points$)$

Consider the anchored sheet pile problem shown in Figure $1,$ which supports an excavation site for an underground metro tunnel in Melbourne, and answer following questions using Optum G$2($Lower Bound and Upper Bound analyses$).$

I. Assuming an anchor of infinite strength, determine the smallest possible wall yield moment $(2$ points$).$

II$.$ Assuming a wall of infinite strength, determine the smallest possible anchor force $(2$ points$).$

III. Determine a sufficient number of solutions in between those found in Parts I and II to determine the range of allowable combinations of wall yield moment and anchor strength $(5$ points$).$ You need to generate a plot for yield moment versus anchor strength.

Question $4(10$ points$)$

For the soil profile shown in Figure $2.$a$,$ find the maximum unsupported depth of excavation in long term $(FS=1\mathrm{.}50)$ for

I. a vertical cut as in Figure $2.$b $(4$ point$).$

II$.$ an inclined cut as in Figure $2.$c $(6$ point$).$

Use Optum G$2($Limit Analysis only$)$ and clearly explain your strategy.

$[1]$ Sloan, S$.$W$.,$ Assadi. A$.,$ and Purushorthaman, N$.(1990).$ Undrained stability of a trapdoor, Geotechnique,