Question Bank • QB_SWCE_ADV_110 NAT • 2 Marks

Section 4: Soil and Water Conservation Engineering

Soil Mechanics

Subtopic: Terzaghi's one dimensional soil consolidation theory

Question Statement:

A saturated clay stratum of thickness $H = 4.0\text{ m}$ is situated between two permeable gravel layers (two-way drainage). The coefficient of consolidation is $c_v = 2.40 \times 10^{-7}\text{ m}^2/\text{s}$. For $50\%$ average consolidation, the theoretical time factor is $T_v = 0.197$. The time required to attain $50\%$ consolidation in days is ________ (round off to one decimal place).

Answer Key:

38.0

Step-by-Step Solution & Explanation

**Method 1: Direct Terzaghi Drainage Formulation** 1. Drainage path length $d$ for two-way drainage: $d = \frac{H}{2} = \frac{4.0\text{ m}}{2} = 2.0\text{ m}$ 2. Time factor relationship: $T_v = \frac{c_v \cdot t}{d^2} \implies t = \frac{T_v \cdot d^2}{c_v}$ 3. Substituting values: $t = \frac{0.197 \times (2.0)^2}{2.40 \times 10^{-7}} = \frac{0.197 \times 4.0}{2.40 \times 10^{-7}} = \frac{0.788}{2.40 \times 10^{-7}} = 3.2833 \times 10^6\text{ seconds}$ 4. Converting seconds to days: $t = \frac{3.2833 \times 10^6}{86400\text{ s/day}} = 37.999\text{ days} \approx 38.0\text{ days}$ **Method 2: Parabolic Approximation for $U \le 60\%$** $T_v = \frac{\pi}{4} U^2 = \frac{3.14159}{4} \times (0.50)^2 = 0.19635$ $t = \frac{0.19635 \times 4.0}{2.40 \times 10^{-7} \times 86400} = 37.88\text{ days} \approx 38.0\text{ days}$
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