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In the theory of compression, soil compression is assumed to occur by the decrease in the volume of voids, the volume of solids remaining constant throughout the process. Also, higher the effective stress in soils, higher is the resulting compression. Hence, the compressibility of soils is expressed in the terms of a plot between void ratio on the y-axis and effective stress on the x-axis. Void ratio is used to represent compression because it is a ratio of the volume of voids to the volume of solids, the former being reflective of compression and the latter being constant in soil compression.

The construction of any structure takes place in stages causing stresses in soils also in increments. To represent this field condition, the laboratory consolidation test is conducted by applying the stress in increments on an undisturbed soil specimen, allowing sufficient time for the complete consolidation to take place under each stress increment.

A typical laboratory consolidation test consists of loading an undisturbed soil specimen with a given stress increment and recording the vertical compression of the specimen with a dial gauge at regular time intervals for sufficient time. The next stress increment is applied only after the compression of the soil is completed under the previous stress increment and the process is repeated.

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**There are mainly the following two types of results that are obtained from a laboratory consolidation test: **

i. Compression of soil as a function of effective stress increment expressed as a relation between the final void ratio (also known as equilibrium void ratio) on the y-axis versus effective stress on the x-axis as shown in Fig. 11.3, which is more commonly presented as e-logσ ‘curve, as shown in Fig. 11.5.

ii. Compression of soil as a function of time at the given stress increment expressed as compression (ΔH) or dial gauge reading (R) on the y-axis versus time on the x-axis.

**1. Void Ratio-Pressure Relations****: **

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Figure 11.3 shows a typical void ratio-pressure relation obtained from a laboratory consolidation test, when both void ratio and pressure are plotted on an arithmetic scale. The following two important terms are obtained from this curve.

**i. Coefficient of Compressibility****: **

The coefficient of compressibility, a_{v}, is the slope of the void ratio-pressure curve when both are plotted on an arithmetic scale –

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a_{v} = – Δe/Δσ’ …(11.7)

Thus, the coefficient of compressibility is defined as the ratio of the change in the void ratio to the change in effective stress. The negative sign is used to indicate that the void ratio decreases with the increase in effective stress.

**ii. Coefficient of Volume Compressibility****: **

The coefficient of volume compressibility, m_{v}, is defined as the ratio of change in the volume of soil to the change in the effective stress per unit original volume –

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m_{v} = ΔV/V_{1 }× 1/Δσ’ …(11.8)

Figure 11.4 shows the three-phase diagram of the soil. Here V_{s} is the volume of the soil solids and V_{v} the volume of the voids. The volume of air V_{a} = 0, because the theory of consolidation is applicable for fully saturated soils only.

Void ratio –

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e = V_{v}/V_{s}

Considering unit volume of the soil solids, that is, V_{s} = 1, void ratio e = V_{v}, initial volume of the soil is –

V_{1 }= V_{a} + V_{v} = 1 + e_{1}

Now –

ΔV/V_{1}= Δe/(1 + e_{1})

Assuming constant loaded area, A, we have –

ΔV/V_{1 }= (V_{1}/V_{2})/V_{1}

**where V _{f} is the final volume of the soil after a given stress increment **

**Δσ**

**’:**

Combining the above two equations, we have –

**2. Compression Index****: **

Figure 11.5 shows the more commonly used void ratio-pressure relationship, where the effective stress is plotted on a log scale as the e-logσ’ relationship is observed to be linear for typical normally consolidated clays.

Compression index is the slope of the e-logσ’ plot and is defined as the ratio of change in the void ratio to the corresponding change in the effective stress. It is given by the mathematical relation –

where e_{1} is the void ratio at effective stress σ_{1}‘and e_{2} is the void ratio at effective stress σ_{2}‘. The negative sign is used to indicate that the void ratio decreases with the increase in effective stress, and to make the compression index positive –** **

where Δe is the decrease in the void ratio when the effective stress increases from σ_{1 }‘to σ_{2}‘.

The compression index is usually determined from the e-logσ’ curve obtained from a laboratory consolidation test. In the absence of the consolidation test data, compression index may be computed from Skempton’s equation as follows –** **

For undisturbed clay –

C_{c} = 0.009(a) (w_{L} 10) …(11.12)

For remolded clay –

C_{c} = 0.007(w_{L}– 10) …(11.13)

where ω_{ L} is the liquid limit of the soil specimen. The properties of a_{v}, m_{v}, and C_{c} are useful to determine the ultimate consolidation settlement of the soil under any given effective stress increment.

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