Section 4: Soil and Water Conservation Engineering
Soil Mechanics
Subtopic: Active and passive earth pressures
Question Statement:
A smooth vertical retaining wall of height $H = 6.0\text{ m}$ retains a dry cohesionless backfill with horizontal surface. The soil has unit weight $\gamma = 18.0\text{ kN/m}^3$ and internal friction angle $\phi = 30^\circ$. Using Rankine's theory, the total active lateral earth thrust $P_a$ per unit length of wall in $\text{kN/m}$ is ________ (answer in integer).
Answer Key:
108
Step-by-Step Solution & Explanation
1. Rankine's coefficient of active earth pressure ($K_a$):
$K_a = \frac{1 - \sin\phi}{1 + \sin\phi} = \frac{1 - \sin 30^\circ}{1 + \sin 30^\circ} = \frac{1 - 0.5}{1 + 0.5} = \frac{0.5}{1.5} = \frac{1}{3}$
2. Active lateral pressure at depth $z = H = 6.0\text{ m}$:
$p_a = K_a \gamma H = \frac{1}{3} \times 18.0 \times 6.0 = 36.0\text{ kPa}$
3. Total active thrust per meter run of wall (area of triangular pressure diagram):
$P_a = \frac{1}{2} K_a \gamma H^2 = \frac{1}{2} \times \frac{1}{3} \times 18.0 \times (6.0)^2$
$P_a = 3.0 \times 36.0 = 108.0\text{ kN/m}$