Raft Foundation Elastic Settlement Calculation

This problem requires us to calculate the elastic settlement of a rigid circular raft foundation placed on sandy soil. The elastic settlement, also known as immediate settlement, occurs due to the elastic deformation of the soil immediately after the application of the load. Since the raft is considered rigid, the settlement will be uniform across its entire area.

Given Parameters for Raft Foundation Settlement

We are provided with the following parameters:

• Raft Diameter (\(D\)): 20 m
• Raft Thickness: 1.6 m (This parameter is not used as the raft is specified as rigid)
• Bearing Pressure (\(q\)): 110 kPa
• Soil Young's Modulus (\(E_s\)): 30 MPa
• Soil Poisson's Ratio (\(\nu_s\)): 0.3
• Raft Material (Concrete) Properties: Young's Modulus (\(E_c\)) = 30 GPa, Poisson's Ratio (\(\nu_c\)) = 0.15 (These parameters are not used as the raft is specified as rigid)

Before proceeding with the calculation, it's important to ensure all units are consistent. We will convert all values to SI base units (Pascals for pressure and modulus).

• \(q = 110 \text{ kPa} = 110 \times 10^3 \text{ Pa}\)
• \(E_s = 30 \text{ MPa} = 30 \times 10^6 \text{ Pa}\)

Elastic Settlement Formula for Rigid Circular Rafts

For a rigid circular foundation resting on an elastic half-space, the immediate (elastic) settlement \(S_e\) can be calculated using the following formula:

\[ S_e = \frac{q D I_f (1 - \nu_s^2)}{E_s} \]

Where:

• \(S_e\) is the elastic settlement.
• \(q\) is the bearing pressure applied by the foundation.
• \(D\) is the diameter of the circular foundation.
• \(I_f\) is the influence factor for settlement. For a rigid circular footing, a commonly used value for \(I_f\) is approximately 0.80.
• \(\nu_s\) is Poisson's ratio of the soil.
• \(E_s\) is Young's modulus of the soil.

Step-by-Step Settlement Calculation

Now, let's substitute the given values into the formula:

The influence factor \(I_f\) for a rigid circular footing is typically taken as 0.80.

\[ S_e = \frac{(110 \times 10^3 \text{ Pa}) \times (20 \text{ m}) \times 0.80 \times (1 - (0.3)^2)}{30 \times 10^6 \text{ Pa}} \]

First, calculate the term \((1 - \nu_s^2)\):

\[ 1 - (0.3)^2 = 1 - 0.09 = 0.91 \]

Now, substitute this back into the settlement formula:

\[ S_e = \frac{(110 \times 10^3) \times 20 \times 0.80 \times 0.91}{30 \times 10^6} \]

Calculate the numerator:

\[ \text{Numerator} = 110000 \times 20 \times 0.80 \times 0.91 \] \[ \text{Numerator} = 2200000 \times 0.80 \times 0.91 \] \[ \text{Numerator} = 1760000 \times 0.91 \] \[ \text{Numerator} = 1601600 \]

Now, perform the division:

\[ S_e = \frac{1601600}{30 \times 10^6} \] \[ S_e = \frac{1601600}{30000000} \] \[ S_e = 0.0533866... \text{ m} \]

Final Elastic Settlement Result

The calculated settlement is in meters. To convert it to millimeters, multiply by 1000:

\[ S_e = 0.0533866... \times 1000 \text{ mm} \] \[ S_e \approx 53.39 \text{ mm} \]

Rounding to two decimal places, the elastic settlement is approximately 53.39 mm. Comparing this to the given options, the closest value is 53.36 mm. The slight difference is likely due to rounding in the influence factor or the option value itself.

Therefore, the elastic settlement of the rigid circular raft foundation is approximately 53.36 mm.