A square shallow foundation is $1.5 \text{ m} \times 1.5 \text{ m}$ in plan. It is placed at a depth ($D_f$) of $1.0 \text{ m}$ in a dense sandy soil. The soil parameters are: $\gamma = 18.0 \text{ kN/m}^3$, $c' = 0 \text{ kPa}$, and $\phi' = 30^\circ$. The groundwater table is very deep. Using Terzaghi's equation, determine the ultimate bearing capacity ($q_u$) and the allowable bearing capacity ($q_{allow}$) with a factor of safety of 3.0.

(Given Terzaghi bearing capacity factors for $\phi' = 30^\circ$: $N_c = 37.16, N_q = 22.46, N_\gamma = 19.13$)

A rectangular footing ($B = 2.0 \text{ m}$, $L = 3.0 \text{ m}$) is constructed at a depth of $D_f = 1.5 \text{ m}$ in a uniform sand deposit. The water table is very deep. The soil properties are $\phi' = 30^\circ$, $c' = 0$, and $\gamma = 18.5 \text{ kN/m}^3$. Using Meyerhof's general bearing capacity equation, calculate the ultimate bearing capacity ($q_u$). Assume vertical loading ($i_q = i_\gamma = 1.0$) and level ground/base ($d_q, d_\gamma$ are ignored for simplicity here, assume $= 1.0$). The bearing capacity factors are $N_q = 18.4$, $N_\gamma = 22.4$. The shape factors are $s_q = 1 + \frac{B}{L}\tan\phi'$ and $s_\gamma = 1 - 0.4\frac{B}{L}$.

A continuous wall footing ($B = 2.0 \text{ m}$) carries a vertical load of $P = 400 \text{ kN/m}$ and a moment of $M = 100 \text{ kN}\\cdot\text{m/m}$. The footing is embedded $1.0 \text{ m}$ in clay ($s_u = c = 120 \text{ kPa}$, $\phi = 0$, $\gamma = 18 \text{ kN/m}^3$). Determine the effective width ($B'$) and calculate the ultimate bearing capacity ($q_u$) using Terzaghi's equation ($N_c = 5.7, N_q = 1.0, N_\gamma = 0$).

A square footing ($B = 2.5 \text{ m}$) is founded at $D_f = 1.0 \text{ m}$ in sand ($c'=0$, $\phi'=35^\circ$, $\gamma_{dry} = 17 \text{ kN/m}^3$, $\gamma_{sat} = 20 \text{ kN/m}^3$). The groundwater table rises from a deep location to exactly the ground surface. Evaluate the percentage reduction in the ultimate bearing capacity ($q_u$). (Assume $N_q=33.3$, $N_\gamma=48.0$).

A temporary grandstand was erected on large, square timber footings resting on a deposit of very loose silty sand. The designers calculated the allowable bearing capacity using general shear failure parameters, assuming a rigid, dense soil response.

A tall, heavy cylindrical steel grain silo was constructed on a thick, very soft, highly compressible clay layer underlain by bedrock. A thick concrete mat foundation was used to spread the load.