Question

For backfill with sloping surface, the coefficient of active earth pressure is given by ______

(a) \(K_a=\frac{\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)

(b) \(K_a=\frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}} \)

(c) \(K_a=\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)

(d) \(K_a=cosβ\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)

The question was posed to me in final exam.

Asked question is from Active Earth Pressure: Rankine’s Theory topic in portion Failure Envelopes & Earth Pressure of Geotechnical Engineering II

Answer 1

Right answer is (d) \(K_a=cosβ\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)

The explanation is: The vertical pressure in case of backfill with surcharge,

σ= γzcosβ,

Therefore, from the assumption of stresses being conjugate,

\(K_a=cosβ\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}}. \)