A footing 2·5 m × 2·5 m carries a pressure of 400 kN/m² at a depth of 1 m in sand. The saturated unit weight of the sand is 20 kN/m³ and the unit weight above the water table is 17 kN/m³. The design shear strength parameters are c′ = 0 and φ = 40° (Nq = 64; N_γ = 95). What is the factor of safety with respect to shear failure, if the water table is 5 m below ground level ?
- A1·5
- B6
- C7Correct
- D3
Solution
Concept
Terzaghi bearing capacity of a square footing in sand
Formula
qu = q Nq + 0.4 γ B N_γ; FS = (qu − q)/(qapplied − q)
Step-by-step solution
1. q = γ D = 17 × 1 = 17 kN/m². The water table is 4 m below the footing base (> B = 2.5 m), so it does not affect the γ-term. 2. qu = 17 × 64 + 0.4 × 17 × 2.5 × 95 = 1088 + 1615 = 2703 kN/m². 3. Net qu = 2703 − 17 = 2686; net applied = 400 − 17 = 383. 4. FS = 2686 / 383 ≈ 7.0.
Explanation
For a square footing the width term is 0.4γBNγ. Since the water table is deeper than B below the base, the full unit weight 17 kN/m³ is used. The net factor of safety is about 7.
Why the other options are wrong
1.5 and 3 are far below the computed value. The net factor of safety is 7.0 and the gross value is 6.8, so 6 does not match either.
Shortcut
Square footing: 1.3cNc + qNq + 0.4γBNγ. Check where the water table is relative to the base.
Common mistake
Reducing γ for the water table although it is deeper than B below the base.
Correct answer: C) 7