**Branch :**Civil Engineering

**Subject :**Soil Mechanics

## Void Ratio and Settlement Changes Under a Constant Load

The initial volume (specifi c volume) of a soil is V 5 1 1 eo (Chapter 3), where eo is the initial void ratio. The change in volume of the soil (DV) is equal to the change in void ratio (De). We can calculate the volumetric strain from the change in void ratio as

Since for one-dimensional consolidation the radial strains and the circumferential strains are zero (εr 5 εu 5 0), then εz 5 εP. We can write a relationship between settlement and the change in void ratio as

where Ho is the initial height of the soil. We can rewrite Equation as

We are going to use rpc to denote primary consolidation settlement rather than Dz, so

The void ratio at any time under load P is

For a saturated soil, eo 5 wGs, where w is the water content. Therefore, we can write Equation as

**Effects of Vertical Stresses on Primary Consolidation:**

We can apply additional loads to the soil and for each load increment we can calculate the fi nal void ratio from Equation and plot the results, as shown by segment AB in Figure. Three types of graph are shown in Figure to illustrate three different arbitrary ways of plotting the data from our test. Figure is an arithmetic plot of the void ratio versus vertical effective stress. Figure is a similar plot except the vertical effective stress is plotted on a logarithmic scale. Figure is an arithmetic plot of the vertical strain (εz) versus vertical effective stress. The segment AB in Figures is not linear because the settlement that occurs for each increment of loading brings the soil to a denser state from its initial state, and the soil’s permeability decreases. Therefore, doubling the load from a previous increment, for example, would not cause a twofold increase in settlement. The segment AB (Figure ) is called the virgin consolidation line, or normal consolidation line (NCL). In a plot of s9z (log scale) versus e, the NCL is approximately a straight line. At some value of vertical effective stress, say, s9zc, let us unload the soil incrementally. Each increment of unloading is carried out only after the soil reaches equilibrium under the previous loading step.

Three plots of settlement data from soil consolidation

When an increment of load is removed, the soil will start to swell by absorbing water from the burette. The void ratio increases, but the increase is much less than the decrease in void ratio for the same magnitude of loading that was previously applied. Let us reload the soil after unloading it to, say, s9zc. The reloading path CD is convex compared with the concave unloading path BC. One reason for this is the evolving soil structure (soil particles arrangement) during loading and unloading. At each loading/unloading stage, the soil particles are reorganized into a different structural framework to resist the load. The average slopes of the unloading path and the reloading path are not equal, but the difference is assumed to be small. We will represent the unloading–reloading path by an average slope BC and refer to it as the recompression line or the unloading–reloading line (URL).

A comparison of the soil’s response with typical material responses to loads as shown in Figures reveals that soils can be considered to be an elastoplastic material. The path BC represents the elastic response, while the path AB represents the elastoplastic response of the soil. Loads that cause the soil to follow path BC will produce elastic settlement (recoverable settlement of small magnitude). Loads that cause the soil to follow path AB will produce settlements that have both elastic and plastic (permanent) components. Once the past maximum vertical effective stress, s9zc, is exceeded, the slope of the path followed by the soil, DE, is approximately the same as that of the initial loading path, AB. Unloading and reloading the soil at any subsequent vertical effective stress would result in a soil’s response similar to paths BCDE.