Question

A vertical smooth rigid retaining wall is supporting horizontal ground with dry cohesionless backfill having a friction angle of \( 30^\circ \). The inclinations of failure planes with respect to the major principal plane for Rankine’s active and passive earth pressure conditions, respectively, are:

• A. \( 30^\circ \) and \( 30^\circ \)
• B. \( 60^\circ \) and \( 60^\circ \)
• C. \( 30^\circ \) and \( 60^\circ \)
• D. \( 60^\circ \) and \( 30^\circ \)

Hint:

In Rankine’s theory, the inclinations of the failure planes are crucial for correct calculation of earth pressures. Always convert these angles into the respective context of the problem (e.g., horizontal or major principal plane) to ensure correct application.

Answer 1

The Correct Option is B

Step 1: Determine the inclinations of failure planes in Rankine’s theory. Rankine’s earth pressure theory states that the failure plane inclination with respect to the horizontal for active and passive earth pressures are: \[ \theta_a = 45^\circ - \frac{\phi}{2}, \quad \theta_p = 45^\circ + \frac{\phi}{2}, \] where \( \phi \) is the angle of friction. Step 2: Calculate the failure plane inclinations. Given \( \phi = 30^\circ \): \[ \theta_a = 45^\circ - 15^\circ = 30^\circ, \quad \theta_p = 45^\circ + 15^\circ = 60^\circ. \] However, since the wall is smooth and vertical, the inclinations are given with respect to the major principal plane, which is vertical: \[ \text{Active and Passive failure planes} = 90^\circ - \theta_a = 60^\circ, \quad 90^\circ - \theta_p = 60^\circ. \]