A saturated clay layer 3m deep is sandwiched by a pervious material. The soil reaches 90% consolidation in three years. At another site, the same saturated clay layer is bounded by impervious boundary at the bottom and sand at the top. What is the time taken by this layer to reach 90% settlement ?
- A9 years
- B12 yearsCorrect
- C15 years
- D3 years
Solution
Concept
Terzaghi's one-dimensional consolidation theory states that for the same soil and same degree of consolidation, the time factor (Tv) is constant, making time (t) proportional to the square of the drainage path (d²).
Formula
Tv = (Cv × t) / d²; t1 / d1² = t2 / d2²
Step-by-step solution
1. Identify the drainage conditions: Case 1 is double drainage (sand on both sides), so d1 = H/2 = 3/2 = 1.5 m. 2. Case 2 is single drainage (sand on top, impervious at bottom), so d2 = H = 3 m. 3. Use the relation t ∝ d² because Cv and Tv (for 90% consolidation) are constant for the same soil. 4. Set up the ratio: t2 / t1 = (d2 / d1)². 5. Substitute values: t2 / 3 = (3 / 1.5)² = 2² = 4. 6. Calculate t2: t2 = 3 × 4 = 12 years.
Explanation
The time required for a specific degree of consolidation is directly proportional to the square of the drainage path (d²). In the first case (double drainage), d = H/2 = 1.5m, while in the second case (single drainage), d = H = 3m. Since the drainage path doubles, the time required increases by a factor of 2² = 4, resulting in 3 years × 4 = 12 years.
Why the other options are wrong
A: This assumes a 3x increase, which is mathematically incorrect for the d² relationship. C: This assumes a 5x increase, which has no physical basis in consolidation theory. D: This assumes time is independent of drainage conditions, which is incorrect.
Shortcut
If drainage changes from double to single for the same layer thickness, the time taken increases 4 times (2²).
Common mistake
Using the total thickness H for both cases or forgetting to square the drainage path ratio.
Correct answer: B) 12 years