Question

For a surcharge angle β, the passive earth pressure is __________

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

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

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

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

The question was posed to me in class test.

The origin of the question is Passive Earth Pressure: Rankine’s Theory topic in section Failure Envelopes & Earth Pressure of Geotechnical Engineering II

Answer 1

Right choice is (d) \(p_p=γzcosβ \frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}}\)

Explanation: If the backfill is having its top surface inclined to the horizontal at an angle β, then the passive earth pressure is given by,

\(p_p=γzcosβ \frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}},\)

where, β= surcharge angle

γ=unit weight of the soil

φ= angle of internal friction.