This problem involves comparing the net ultimate bearing capacities of two types of shallow foundations, a circular footing and a square footing, both resting on pure clay soil. We are given that the diameter of the circular footing is equal to the side length of the square footing.

Understanding Bearing Capacity in Pure Clay

Pure clay soil is characterized by having an angle of internal friction ($\phi$) equal to zero. In such conditions, the bearing capacity is primarily governed by the soil's undrained cohesion ($c_u$). The general bearing capacity equation, considering factors like shape and depth, is often expressed as:

$$q_u = c N_c s_c d_c + q N_q s_q d_q + 0.5 \gamma B N_\gamma s_\gamma d_\gamma$$

Where:

• $q_u$ = Ultimate bearing capacity
• $c$ = Cohesion (here, $c_u$)
• $q$ = Effective surcharge pressure at foundation level ($q = \gamma D_f$)
• $\gamma$ = Unit weight of the soil
• $B$ = Width of the foundation
• $D_f$ = Depth of foundation
• $N_c, N_q, N_\gamma$ = Bearing capacity factors
• $s_c, s_q, s_\gamma$ = Shape factors
• $d_c, d_q, d_\gamma$ = Depth factors

For pure clay ($\phi=0$), the bearing capacity factors are typically taken as $N_c = 5.14$, $N_q = 1$, and $N_\gamma = 0$. The equation simplifies because the term involving $N_\gamma$ becomes zero.

The net ultimate bearing capacity ($q_{net}$) is calculated as $q_{net} = q_u - q$. Substituting the values for $\phi=0$ and the $N_q$ term:

$$q_{net} = c N_c s_c d_c + q N_q s_q d_q - q$$

$$q_{net} = c_u (5.14) s_c d_c + q (1) s_q d_q - q$$

Analyzing Shape and Depth Factors

Shape factors ($s_c$) depend on the ratio of the foundation's width ($B$) to its length ($L$). For $\phi=0$ conditions, a common form for the shape factor $s_c$ is:

$$s_c = 1 + 0.2 \frac{B}{L}$$

Depth factors ($d_c$) often depend on the ratio of foundation depth ($D_f$) to width ($B$):

$$d_c = 1 + 0.2 \frac{D_f}{B}$$

Let's apply these to both footing types, given that the diameter ($D$) of the circular footing equals the side ($B$) of the square footing ($D=B$).

Circular Footing

• For a circular footing, the width and length are effectively the same, so $B_{circ} = D$ and $L_{circ} = D$. The ratio $B/L = D/D = 1$.
• Shape factor $s_{c, circ} = 1 + 0.2 (1) = 1.2$.
• The depth factor depends on $D_f/B_{circ} = D_f/D$. So, $d_{c, circ} = 1 + 0.2 (D_f/D)$.
• The term $s_q d_q$ is generally 1 for $\phi=0$ in many formulations.
• The net ultimate bearing capacity is:

  $$q_{net, circ} = c_u (5.14) (1.2) (1 + 0.2 \frac{D_f}{D}) + q (1) (1) - q$$

  $$q_{net, circ} = 6.168 c_u (1 + 0.2 \frac{D_f}{D}) + q (1 + 0.2 \frac{D_f}{D}) - q$$

  $$q_{net, circ} = 6.168 c_u (1 + 0.2 \frac{D_f}{D}) + q (0.2 \frac{D_f}{D})$$

Square Footing

• For a square footing, $B_{sq} = B$ and $L_{sq} = B$. The ratio $B/L = B/B = 1$.
• Shape factor $s_{c, sq} = 1 + 0.2 (1) = 1.2$.
• The depth factor depends on $D_f/B_{sq} = D_f/B$. So, $d_{c, sq} = 1 + 0.2 (D_f/B)$.
• The net ultimate bearing capacity is:

  $$q_{net, sq} = c_u (5.14) (1.2) (1 + 0.2 \frac{D_f}{B}) + q (1) (1) - q$$

  $$q_{net, sq} = 6.168 c_u (1 + 0.2 \frac{D_f}{B}) + q (1 + 0.2 \frac{D_f}{B}) - q$$

  $$q_{net, sq} = 6.168 c_u (1 + 0.2 \frac{D_f}{B}) + q (0.2 \frac{D_f}{B})$$

Calculating the Ratio

We are given that the diameter of the circular footing ($D$) is the same as the side of the square footing ($B$), meaning $D=B$. Comparing the expressions for the net ultimate bearing capacities:

$$q_{net, circ} = 6.168 c_u (1 + 0.2 \frac{D_f}{D}) + q (0.2 \frac{D_f}{D})$$

$$q_{net, sq} = 6.168 c_u (1 + 0.2 \frac{D_f}{B}) + q (0.2 \frac{D_f}{B})$$

Since $D=B$, the terms $D_f/D$ and $D_f/B$ are identical. Therefore, the entire expressions for $q_{net, circ}$ and $q_{net, sq}$ are equal:

$$q_{net, circ} = q_{net, sq}$$

The ratio of their net ultimate bearing capacities is:

$$\frac{q_{net, circ}}{q_{net, sq}} = \frac{q_{net, sq}}{q_{net, sq}} = 1$$

Conclusion on Bearing Capacity Ratio

The ratio of the net ultimate bearing capacities of the circular footing and the square footing is 1, meaning they are equal under the given conditions (pure clay and equal characteristic dimensions).