A 8.0 m deep ($H$) braced excavation is made in a uniform deposit of sand. The sand has a unit weight ($\gamma$) of $17.5 \text{ kN/m}^3$ and an active earth pressure coefficient ($K_a$) of 0.30. Using Peck's apparent earth pressure envelope for sand, determine the design pressure load on the struts.

A cantilever sheet pile wall is driven into a granular soil to retain a $3.0 \text{ m}$ high excavation ($H = 3.0 \text{ m}$). The soil has $\gamma = 18 \text{ kN/m}^3$ and $\phi' = 30^\circ$. The active earth pressure coefficient is $K_a = 0.333$ and the passive earth pressure coefficient is $K_p = 3.0$. Using the simplified method where the net passive pressure below the dredge line is $p_{net} = \gamma D (K_p - K_a)$, estimate the theoretical depth of penetration ($D$) required for moment equilibrium about the bottom of the pile.

An anchored sheet pile wall retains a $6.0 \text{ m}$ deep excavation in sand ($\gamma = 18 \text{ kN/m}^3, K_a = 0.3, K_p = 3.3$). The anchor tie rod is located $1.5 \text{ m}$ below the ground surface. The pile penetrates $D = 3.0 \text{ m}$ below the dredge line. Assume the passive resistance is fully mobilized at the toe (Free Earth Support assumption). Calculate the required anchor force ($F_A$) per meter run of the wall.

A $5.0 \text{ m}$ deep, wide excavation is made in a soft clay layer having an undrained shear strength ($s_u$) of $25 \text{ kPa}$ and a unit weight ($\gamma$) of $19 \text{ kN/m}^3$. Determine the Factor of Safety against bottom heave using Terzaghi's simplified approach. Assume $N_c = 5.7$.

A deep subway station was being constructed using a braced cut. The excavation was $15 \text{ m}$ deep in a profile of medium-stiff clay. The support system consisted of steel sheet piles braced by horizontal steel struts at $3 \text{ m}$ vertical intervals.

A sheet pile cofferdam was driven into a riverbed consisting of fine sand to allow the dry construction of a bridge pier. The water depth was $4 \text{ m}$, and the excavation depth below the riverbed was $3 \text{ m}$.