To determine the size of the foundation for the square footing, we need to calculate the ultimate bearing capacity of the soil and use it to find the required footing size, $B$.

Step-by-Step Approach:

1. Given Data:

   • Foundation depth: $D_f = 1.2 \, \text{m}$
   • Soil unit weight (above the water table): $\gamma = 16.50 \, \text{kN/m}^3$
   • Saturated soil unit weight (below the water table): $\gamma_{\text{sat}} = 18.45 \, \text{kN/m}^3$
   • Friction angle: $\phi' = 30^\circ$
   • Cohesion: $C' = 10 \, \text{kN/m}^2$
   • Factor of safety: $F_s = 3.00$
   • Gross allowable load: $Q_{\text{allow}} = 665 \, \text{kN}$
   • Water table depth below footing: $0.5 \, \text{m}$

2. Determine the Ultimate Bearing Capacity (Using Terzaghi's Equation for Square Footing):

   The ultimate bearing capacity $q_u$ for a square footing is given by the formula:

   $q_u = C'N_c + \gamma D_f N_q + 0.5 \gamma' B N_{\gamma}$

   Where:

   • $N_c$, $N_q$, $N_{\gamma}$ are bearing capacity factors dependent on the friction angle $\phi'$.
   • $\gamma'$ is the effective unit weight of the soil.

   Let's calculate each term step by step.

3. Bearing Capacity Factors:

   Using typical values for $\phi' = 30^\circ$:

   • $N_c = 30.14$
   • $N_q = 18.4$
   • $N_{\gamma} = 15.1$

4. Effective Unit Weight $\gamma'$:

   Since the water table is 0.5 m below the footing, the soil below the water table will be partially submerged. The effective unit weight $\gamma'$ is calculated as:

   $\gamma' = \gamma_{\text{sat}} - \gamma_w$ Where $\gamma_w$ is the unit weight of water, typically $9.81 \, \text{kN/m}^3$.

   $\gamma' = 18.45 - 9.81 = 8.64 \, \text{kN/m}^3$

5. Ultimate Bearing Capacity $q_u$:

   The formula for ultimate bearing capacity is applied for the unknown size $B$ of the footing. Now, let's assume a size $B$, and solve the equation.

I'll calculate the value of $B$ next.The size of the square foundation, $B$, is approximately 20.38 meters.

Would you like more details or have any other questions?

Here are some additional questions that might help expand on this topic:

1. How would changes in the water table level affect the foundation size?
2. What are the implications of using a different soil friction angle $\phi'$?
3. How does the factor of safety influence the design of the foundation?
4. What happens to the foundation design if the cohesion $C'$ increases?
5. How can you adjust the footing design if the load were to increase beyond 665 kN?

Tip: Always consider the position of the water table when designing foundations, as it significantly impacts soil behavior and bearing capacity.