This question was previously asked in

GATE CE 2021 Official Paper: Shift 2

CT 1: Ratio and Proportion

2846

10 Questions
16 Marks
30 Mins

**Concept:**

Coefficient of Consolidation:

- It is defined as the parameter used to measure the rate at which the saturated clay or soil undergoes consolidation when subjected to an increase in pressure. It is usually measured in terms of area per unit time.
- It is closely related to the liquid limit of the soil and decreases as the liquid limit of soil increases.
- The coefficient of consolidation is given by, \({{\rm{c}}_{\rm{v}}} = \frac{{\rm{k}}}{{{{\rm{m}}_{\rm{v}}}{{\rm{γ }}_{\rm{w}}}}}\)

where C_{V} is coefficient of consolidation,

k is the coefficient of permeability of the soil,

m_{v} is the coefficient of volume compressibility and γ_{w} is the unit weight of water.

We know, \({{\rm{m}}_{\rm{v}}} = \frac{{{{\rm{a}}_{\rm{v}}}}}{{1 + {{\rm{e}}_{\rm{o}}}}}\)

\({m_v} = \frac{{ - \left( {\frac{{{\rm{\Delta }}e}}{{{\rm{\Delta }}\bar \sigma }}} \right)}}{{1 + {e_o}}}\)

Δe = Change in void ratio of the soil

eo = initial void ratio

\({\rm{\Delta }}\bar \sigma =Change \ in\:effective\;stress\)

The ratio of the coefficient of consolidation of different samples is given by

\(\frac{{{C_{v1}}}}{{{C_{v2}}}} = \frac{{{k_1}}}{{{k_2}}} \times \frac{{{m_{v2}}}}{{{m_{v1}}}}\)

**Calculation**

Given,

There are 2 samples

__For M sample __

Initial Void ratio = 0.575

Final void ratio = 0.510

Change in effective stress = 180- 120 = 60 kN/m^{2}

\({m_v} = \;\frac{{{\rm{\Delta }}e}}{{1 + {e_0}}} \times \frac{1}{{{\rm{\Delta }}\sigma '}}\)

\(= \;\frac{{0.575 - 0.51}}{{1 + 0.575}} \times \frac{1}{{60}}\)

m_{v} = 6.878 × 10^{-4} m^{2}/kN

__For N sample __

Change in effective stress = 60 kN/m^{2}

Initial Void Ratio = 0.600

Final Void Ratio = 0.550

\({m_v} = \;\frac{{{\rm{\Delta }}e}}{{1 + {e_0}}} \times \frac{1}{{{\rm{\Delta }}\sigma '}}\)

\( = \;\frac{{0.6 - 0.55}}{{1 + 0.60}} \times \frac{1}{{60}}\)

m_{v} = 5.208 × 10^{-4} m^{2}/kN

The ratio of Hydraulic Conductivity of M sample to N sample = 0.125

So the coefficient of consolidation ratios is

\(\frac{{{C_{vM}}}}{{{C_{vN}}}} = \frac{{{k_M}}}{{{k_N}}} \times \frac{{{m_{vN}}}}{{{m_{vM}}}}\)

\( = 0.125 \times \frac{{5.20 \times {{10}^{ - 4}}}}{{6.878 \times {{10}^{ - 4}}}}\)

= 0.0946

**∴ The ratio of the coefficient of consolidation is 0.0946**