a solved problem lateral earth pressure

Lateral earth pressure refers to the pressure exerted by soil against retaining structures, such as walls or foundations, due to the weight of the soil and any additional loads. The two main theories used to calculate lateral earth pressure are Rankine's theory and Coulomb's theory. I will present a solved problem using Rankine's theory, which is simpler and often used for vertical walls under simple conditions. ### Problem Statement Consider a vertical retaining wall that is 6 meters high supporting a backfill of granular soil (cohesionless) with a unit weight of 18 kN/m³. Calculate the total lateral earth pressure exerted on the wall at the base, and determine the resultant force acting on the wall. ### Given Data - Height of the wall (H) = 6 m - Unit weight of the soil (γ) = 18 kN/m³ - Angle of internal friction (φ) = 30° (assuming standard friction angle for granular soils) - Cohesion (c) = 0 (cohesionless soil) ### Step 1: Calculate the Active Earth Pressure using Rankine's Formula Rankine's active earth pressure can be calculated using the formula: \[ P_a = \frac{1}{2} \gamma H^2 K_a \] where \(K_a\) (the coefficient of active earth pressure) can be calculated as: \[ K_a = \tan^2\left(45^\circ - \frac{\phi}{2}\right) \] Substituting the values: \[ K_a = \tan^2\left(45^\circ - \frac{30^\circ}{2}\right) = \tan^2(30^\circ) = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \] Now, we can substitute \(K_a\) into the active pressure formula: \[ P_a = \frac{1}{2} \cdot 18 \, \text{kN/m}^3 \cdot (6 \, \text{m})^2 \cdot \frac{1}{3} \] Calculating: \[ P_a = \frac{1}{2} \cdot 18 \cdot 36 \cdot \frac{1}{3} \] \[ P_a = \frac{1}{2} \cdot 18 \cdot 12 = 108 \, \text{kN/m}^2 \] ### Step 2: Calculate the Total Lateral Force on the Wall The total lateral force can be computed as the area of the triangle, which is given by: \[ F_a = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot P_a \cdot H \] Substituting the known values: \[ F_a = \frac{1}{2} \cdot 108 \, \text{kN/m}^2 \cdot 6 \, \text{m} \] Calculating: \[ F_a = \frac{1}{2} \cdot 108 \cdot 6 = 324 \, \text{kN} \] ### Step 3: Calculate the Location of the Resultant Force The location of the resultant force from the base of the wall can be determined by the centroid of the triangular pressure distribution, which is located at \(\frac{H}{3}\) from the base of the wall: \[ d = \frac{H}{3} = \frac{6 \, \text{m}}{3} = 2 \, \text{m} \] ### Summary of Results - Total Lateral Earth Pressure at the base: \(P_a = 108 \, \text{kN/m}^2\) - Total Lateral Force on the wall: \(F_a = 324 \, \text{kN}\) - Location of the Resultant Force from the base: \(d = 2 \, \text{m}\) This completes the lateral earth pressure problem using Rankine's theory.