A square foundation (B x B) has to be constructed 1.2 m below the ground surface. Assume that 
following properties:
𝛾 = 16.50 kN/m3
𝛾𝑠𝑎𝑡 = 18.45 kN/m3
∅
′ = 30 Degree
C’ = 10 kN/m2
The water table is 0.50 m below the bottom of the footing. The gross allowable load with factor of 
safety of 3.00 is 665 kN. Determine the size of the foundation.

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.

A square foundation (B x B) has to be constructed 1.2 m below the ground surface. Assume that following properties: γ = 16.50 kN/m3, γsat = 18.45 kN/m3, ∅' = 30 Degree, C' = 10 kN/m2. The water table is 0.50 m below the bottom of the footing. The gross allowable load with factor of safety of 3.00 is 665 kN. Determine the size of the foundation.

Determine the size of a square foundation with a footing depth of 1.2 m and water table 0.5 m below the footing. The problem involves calculating the ultimate bearing capacity using Terzaghi's equation, with properties like soil unit weight, cohesion, friction angle, and a factor of safety. The gross allowable load is 665 kN. Suitable for undergraduate civil engineering students.

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