Question

A smooth vertical retaining wall supporting layered soils is shown in the figure. According to Rankine's earth pressure theory, the lateral active earth pressure acting at the base of the wall is \underline{\hspace{2cm} kPa (rounded off to one decimal place).} \includegraphics[width=0.5\linewidth]{image125.png}

Hint:

To calculate the lateral earth pressure, use the Rankine earth pressure formula. For layered soils, calculate the pressure for each layer and then sum them up.

Answer 1

We are given two layers of soil with the following properties: - Layer 1: - Height \( H_1 = 3 \, \text{m} \) - Bulk unit weight \( \gamma_1 = 18 \, \text{kN/m}^3 \) - Angle of internal friction \( \phi_1 = 32^\circ \) - Cohesion \( C_1 = 0 \, \text{kPa} \) - Layer 2: - Height \( H_2 = 4 \, \text{m} \) - Bulk unit weight \( \gamma_2 = 19 \, \text{kN/m}^3 \) - Angle of internal friction \( \phi_2 = 25^\circ \) - Cohesion \( C_2 = 20 \, \text{kPa} \) - Surcharge load \( q = 20 \, \text{kPa} \) The lateral active earth pressure \( P_a \) at the base of the wall is calculated using the Rankine earth pressure theory formula: \[ P_a = \gamma H K_a + q K_a \] Where: - \( \gamma \) is the bulk unit weight, - \( H \) is the height of the soil layer, - \( K_a \) is the Rankine active earth pressure coefficient, - \( q \) is the surcharge load. The active earth pressure coefficient \( K_a \) is given by: \[ K_a = \tan^2 \left( 45^\circ - \frac{\phi}{2} \right) \] Layer 1: For Layer 1, the active earth pressure coefficient is calculated as: \[ K_a = \tan^2 \left( 45^\circ - \frac{32^\circ}{2} \right) = \tan^2 (29^\circ) \approx 0.284 \] Now, calculate the lateral active earth pressure for Layer 1: \[ P_{a1} = \gamma_1 H_1 K_a + q K_a = 18 \times 3 \times 0.284 + 20 \times 0.284 \] \[ P_{a1} = 15.4 + 5.68 = 21.08 \, \text{kPa} \] Layer 2: For Layer 2, the active earth pressure coefficient is calculated as: \[ K_a = \tan^2 \left( 45^\circ - \frac{25^\circ}{2} \right) = \tan^2 (32.5^\circ) \approx 0.436 \] Now, calculate the lateral active earth pressure for Layer 2: \[ P_{a2} = \gamma_2 H_2 K_a + q K_a = 19 \times 4 \times 0.436 + 20 \times 0.436 \] \[ P_{a2} = 33.17 + 8.72 = 41.89 \, \text{kPa} \] Total Lateral Active Earth Pressure: Finally, the total lateral active earth pressure at the base of the wall is: \[ P_a = P_{a1} + P_{a2} = 21.08 + 41.89 = 62.97 \, \text{kPa} \] Thus, the lateral active earth pressure acting at the base of the wall is approximately 63.0 kPa. \[ \boxed{\text{The lateral active earth pressure is } 63.0 \, \text{kPa}.} \]