According to Rankine’s Formula, the minimum depth of foundation when q = 180 kN/m² , γ = 20 kN/m³ and ϕ = 30° is
- A0.5 m
- B1 mCorrect
- C1.5 m
- D2.0 m
Solution
Concept
Minimum foundation depth using Rankine formula
Formula
Dmin = (q / γ) × ((1 - sin(φ)) / (1 + sin(φ)))2
Step-by-step solution
1. Rankine's formula for minimum depth of foundation is Dm = (q / γ) × ((1 - sin ϕ) / (1 + sin ϕ))2. 2. Given bearing capacity q = 180 kN/m², unit weight γ = 20 kN/m³, and angle of internal friction ϕ = 30°. 3. Calculate term: (1 - sin 30°) / (1 + sin 30°) = (1 - 0.5) / (1 + 0.5) = 0.5 / 1.5 = 1/3. 4. Calculate depth: Dm = (180 kN/m² / 20 kN/m³) × (1/3)2 = 9 m × (1/9) = 1 m. Answer: B
Explanation
Rankine's formula for the minimum depth of a foundation is calculated by dividing the allowable soil pressure by the unit weight and multiplying by the square of the passive pressure coefficient term. Substituting the given values yields a minimum depth calculation equal to 1 m. This depth ensures the foundation is set deep enough to prevent lateral movement of the supporting soil.
Why the other options are wrong
A) A depth of 0.5 m is less than the calculated minimum depth of 1 m using Rankine's formula. C) A depth of 1.5 m exceeds the calculated minimum depth requirement of 1 m. D) A depth of 2.0 m is twice the minimum required depth calculated from the formula.
Shortcut
Rankine's Dmin = (q / γ) × [(1 - sinφ) / (1 + sinφ)]²; for φ = 30°, multiply (q/γ) by 1/9.
Common mistake
Forgetting to square the active pressure term in Rankine's minimum foundation depth formula.
Correct answer: B) 1 m