If the specific gravity of the soil is represented by G and the void ratio is e, the hydraulic gradient i is expressed as:

Q34: If the specific gravity of the soil is represented by G and the void ratio is e, the hydraulic gradient i is expressed as:

A. (G-1)/(1+e)

B. (G+1)/(1-e)

C. (1-G)/(1+e)

D. (1+G)/(1+e)

Correct Answer: A. (G-1)/(1+e)

📝 Detailed Explanation

This question asks for the exact formula connecting a soil’s specific gravity and void ratio to its critical hydraulic gradient — the threshold value introduced conceptually in Q33.

✅ Why “(G-1)/(1+e)” Is Correct

The critical hydraulic gradient is given by i_c = (G − 1)/(1 + e), where G is the specific gravity of the soil solids and e is the void ratio. This formula emerges directly from balancing the submerged unit weight of the soil against the seepage force per unit volume at the point of critical failure.

❌ Why the Other Options Are Wrong

(G+1)/(1-e): has both the wrong sign structure — G+1 instead of G−1 — and would behave incorrectly as e approaches 1, blowing up rather than smoothly decreasing as intended.
(1-G)/(1+e): would give a NEGATIVE critical gradient for any realistic specific gravity (G is always greater than 1 for real soil solids, typically 2.6-2.75), which is physically meaningless.
(1+G)/(1+e): uses the wrong sign in the numerator (1+G instead of G−1), overstating the critical gradient relative to the correct formula.

🔑 Key Point

i_c = (G−1)/(1+e) is structurally identical to the submerged (buoyant) unit weight formula, γ_sub = (G−1)γ_w/(1+e), divided through by γ_w — recognizing this connection makes the formula much easier to recall and verify.

💡 Key Concepts for Students

This formula directly explains Q33’s qualitative trend: since e appears only in the denominator, increasing e (all else equal) necessarily decreases i_c.
For a typical sand (G ≈ 2.65, e ≈ 0.6-0.7), i_c works out to roughly 1 — a useful benchmark figure worth recognizing.
This same (G−1)/(1+e) structure appears in the submerged unit weight formula covered in the Index Properties topic — recognizing the shared structure between the two formulas helps with both.
See Q33’s Going Deeper section for the full physical mechanism (quicksand, loss of effective stress) that this critical gradient formula is designed to predict.

📚 Going Deeper: Deriving the Critical Hydraulic Gradient Formula

The critical hydraulic gradient formula follows directly from setting the upward seepage force per unit volume equal to the submerged (buoyant) unit weight of the soil — the exact balance point where effective stress drops to zero.

Submerged unit weight of the soil: γ_sub = (G−1)γ_w/(1+e) (see Index Properties topic). The upward seepage force per unit volume, at the critical condition, equals i_c·γ_w. Setting these equal: i_c·γ_w = (G−1)γ_w/(1+e), and dividing both sides by γ_w gives i_c = (G−1)/(1+e) directly.

Variable	Meaning	Typical Value
G	Specific gravity of soil solids	≈ 2.6 to 2.75 for common inorganic soils
e	Void ratio	Varies widely by soil and density state
ic	Critical hydraulic gradient	Typically around 0.9 to 1.1 for common sands

This derivation is a good example of how many soil mechanics formulas aren’t independent facts to memorize in isolation, but follow directly from combining two more basic relationships (here, submerged unit weight and the seepage-force balance) — recognizing these connections makes the whole formula set far more manageable than memorizing each one separately.