Critical hydraulic gradient in soils: If the specific gravity of soil solids is G and the void ratio is e, the critical hydraulic gradient i_c for quick (boiling) condition is given by which expression?

Introduction / Context:
The critical hydraulic gradient i_c marks the onset of a quick (boiling) condition in cohesionless soils during upward seepage, when effective stress at a point reduces to zero. Recognizing i_c helps prevent piping and foundation instability under hydraulic uplift.

Given Data / Assumptions:

• G = specific gravity of soil solids (Gs).
• e = void ratio.
• Upward seepage in saturated soil; unit weight of water = gamma_w.

Concept / Approach:
At boiling, effective unit weight gamma′ becomes zero. For saturated soil, gamma_sat = (G + e) / (1 + e) * gamma_w. Submerged (buoyant) unit weight is gamma′ = gamma_sat − gamma_w = (G − 1)/(1 + e) * gamma_w. The seepage force per unit volume in upward flow equals i * gamma_w. Setting gamma′ − i * gamma_w = 0 at the verge of boiling gives i_c = (G − 1)/(1 + e).

Step-by-Step Solution:

Verification / Alternative check:
For typical sands (G ≈ 2.65, e ≈ 0.65), i_c ≈ (1.65)/(1.65) ≈ 1.0, matching practical rules of thumb for boiling onset.

Why Other Options Are Wrong:

• (G + 1)/(1 + e): arises from confusion with gamma_sat; not the critical gradient.
• (1 + e)/(G − 1) or e/(G − 1): incorrect algebraic inversions that do not satisfy equilibrium at boiling.

Common Pitfalls:
Confusing specific gravity G with unit weights; mixing up e and n (porosity); forgetting that upward seepage reduces effective stress.

Final Answer:
i_c = (G − 1) / (1 + e)