The following are the results of a consolidation test on a soil sample with an initial thickness of 25 mm, a water content of 42%, and a specific gravity (G) of 2.78. Calculate the void ratio corresponding to each pressure increment. The least count of the dial gauge is $10^{-3}$ mm. Determine the coefficient of volume compressibility and the coefficient of compressibility for the pressure increment from 49 kN/m$^2$ to 98.1 kN/m$^2$. (There is no need to plot a graph.)

| Pressure (kN/m$^2$) | 0 | 20 | 49 | 98.1 | 196 | 392.4 |
|-----------------------------|----|----|-----|------|-----|-------|
| Change in dial gauge reading (mm) | 0 | 90 | 250 | 410 | 680 | 760 |

The void ratio corresponding to each pressure increment in the consolidation test can be calculated using the change in dial gauge reading and the initial thickness of the soil sample. The coefficients of volume compressibility and compressibility can be calculated using the change in void ratio and change in pressure.

Explanation:

In the given consolidation test, the void ratio corresponding to each pressure increment can be calculated using the change in dial gauge reading and the initial thickness of the soil sample. The void ratio (e) can be calculated using the formula:

e = (change in dial gauge reading * least count) / (initial thickness)

For each pressure increment, the void ratio can be calculated as follows:

For a pressure increment of 20 kN/m2, the change in dial gauge reading is 90. Using the given least count of the dial gauge (10-3 mm), the void ratio can be calculated as:
e = (90 * 10-3) / 25 = 0.036
Similarly, for pressure increments of 49 kN/m2, 98.1 kN/m2, 196 kN/m2, and 392.4 kN/m2, the void ratios can be calculated as:
e = (250 * 10-3) / 25 = 0.1
e = (410 * 10-3) / 25 = 0.164
e = (680 * 10-3) / 25 = 0.272
e = (760 * 10-3) / 25 = 0.304

To determine the coefficient of volume compressibility (mv) and coefficient of compressibility (mv), we need to calculate the change in void ratio (Δe) and change in pressure (Δσ) for the given pressure increment from 49 kN/m2 to 98.1 kN/m2. The change in void ratio can be calculated as:

Δe = e2 - e1 = (e at 98.1 kN/m2) - (e at 49 kN/m2)

Similarly, the change in pressure can be calculated as:

Δσ = σ2 - σ1 = (pressure at 98.1 kN/m2) - (pressure at 49 kN/m2)

Once we have the values of Δe and Δσ, we can calculate the coefficients of compressibility as follows:

mv = Δe / Δσ

Using the calculated values of Δe and Δσ, we can determine the coefficients of compressibility for the given pressure increment.

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