For a retaining wall, the Mohr circle radius for active earth pressure is ______ the Mohr circle radius for passive earth pressure.

Asked in SSC JE 06/06/2024 (Morning)

Detailed Explanation

Think of a soil element at a given depth behind a wall. Its vertical stress σ_v = γz is fixed. Only the horizontal stress changes as the wall moves.

Active radius: \(r_a = \tfrac{1}{2}\sigma_v(1 - K_a)\). Passive radius: \(r_p = \tfrac{1}{2}\sigma_v(K_p - 1)\).

For \(\phi = 30^\circ\): \(r_a = \tfrac{1}{2}\sigma_v(1 - \tfrac13) = 0.33\sigma_v\); \(r_p = \tfrac{1}{2}\sigma_v(3 - 1) = \sigma_v\).

Why “less than” Is Correct

In the active state σ_v is the major principal stress and the horizontal stress drops until the circle touches the envelope — a small circle. In the passive state σ_v becomes the minor principal stress and the horizontal stress must climb much higher before the circle touches the envelope — a large circle. So the active radius is always less than the passive radius.

Mohr circles for a soil element behind a wall at the same depth (vertical stress σ_v fixed). The at-rest circle is small and lies below the failure envelope. In the active state the horizontal stress falls until the circle touches the envelope on the left of σ_v; in the passive state it rises until a much larger circle touches the envelope on the right. So the active circle is always smaller than the passive circle.

Why the Other Options Are Wrong

A. more or less depending on the angle of internal friction of soil: For any φ > 0 the passive circle is larger; it does not switch with φ.
B. more than: The active circle is the smaller one.
C. equal to: They are equal only if φ = 0, which gives no frictional strength at all.

Key Point

Active: σv is σ₁. Passive: σv is σ₃. Passive circle is much larger.

Key Concepts for Students

Both circles touch the same failure envelope.
The at-rest circle lies inside, below the envelope.
Failure planes: 45° + φ/2 to horizontal (active), 45° − φ/2 (passive).

Practice more: this is question 4 in Earth Pressure and Retaining Wall MCQs – Part 1. Try the whole set and check every answer instantly.