Question

As per the Rankine's earth pressure theory, which of the following statements is/are FALSE?

• A. For the active earth pressure, the inclination of failure plane is \((45^{\circ} + \phi/2)\) with respect to the major principal plane.
• B. For the active earth pressure, the inclination of failure plane is \((45^{\circ} - \phi/2)\) with respect to the major principal plane.
• C. For the passive earth pressure, the inclination of failure plane is \((45^{\circ} + \phi/2)\) with respect to the major principal plane.
• D. For the passive earth pressure, the inclination of failure plane is \((45^{\circ} - \phi/2)\) with respect to the major principal plane.

Hint:

The failure plane always makes \(45^{\circ}+\phi/2\) with the major principal plane in Rankine's theory, for both the active and passive cases.

Answer 1

The Correct Option is B, D

Rankine's theory finds the orientation of the failure plane using the Mohr-Coulomb criterion, which gives the same angular relationship, $\left(45^{\circ} + \phi/2\right)$ measured from the plane carrying the major principal stress, whether we are in the active state or the passive state. Only the identity of which stress is major switches between the two cases. Let's check each statement against this.

1. (A) Active case, angle $= 45^{\circ}+\phi/2$ from the major principal plane: In the active state the vertical stress is major and acts on the horizontal plane. The Mohr-Coulomb result gives exactly $45^{\circ}+\phi/2$ from that plane. This statement is TRUE, not false.
2. (B) Active case, angle $= 45^{\circ}-\phi/2$ from the major principal plane: This does not match the Mohr-Coulomb result found above; the correct angle is $45^{\circ}+\phi/2$, not $45^{\circ}-\phi/2$. This statement is FALSE.
3. (C) Passive case, angle $= 45^{\circ}+\phi/2$ from the major principal plane: In the passive state the horizontal stress becomes major, acting on the vertical plane, but the Mohr-Coulomb geometry still gives $45^{\circ}+\phi/2$ from that vertical major principal plane. This statement is TRUE, not false.
4. (D) Passive case, angle $= 45^{\circ}-\phi/2$ from the major principal plane: Same issue as (B); the correct angle from the major principal plane is $45^{\circ}+\phi/2$, so this statement is FALSE.

The trap in this question is remembering that active failure planes look steep and passive ones look flat when both are measured from the horizontal ground surface. That difference is real, but it happens because the major principal plane itself rotates 90 degrees between the active and passive cases, not because the $45^{\circ}+\phi/2$ rule changes.

Let's summarize:

• Failure plane angle from the major principal plane is always $45^{\circ}+\phi/2$ in Rankine's/Mohr-Coulomb theory, in both active and passive states.
• (A) and (C) apply this rule correctly, so they are true statements.
• (B) and (D) both wrongly use $45^{\circ}-\phi/2$, so they are the false statements.

The false statements are (B) and (D).