Mechanism of Consolidation: Terzaghi’s Soil-Spring Analogy | Soil Engineering

The mechanism of consolidation was explained by Terzaghi with the help of a spring-dashpot model using soil-spring analogy.

1. Spring-Dashpot Model:

Figure 11.1 shows a dashpot consisting of a cylinder containing a spring supporting a piston. The cylinder is completely filled with water and has an outlet valve to regulate the outflow.

Let the initial stress in the spring be σ’₀ under its own weight and that of the piston, and the spring is deformed due to this stress. Suppose a load is applied on the piston when the valve is in closed condition, so as to cause an additional stress of Δσ on the spring-water system. If the spring has to share this additional load, it has to undergo deformation.

As the water filling the dashpot is incompressible, it will not allow the deformation of the spring. As the spring cannot deform, it cannot take any additional stress and thus the entire additional stress applied is borne by the water in the dashpot.

The stress equation at this initial stage is as follows:

Δσ = Δσ’ + Δu …(11.1)

where Δσ is the additional stress applied on the spring-dashpot system, Δσ’ the additional stress in the spring equal to 0, Δu the additional stress in the water in the dashpot, in excess of hydrostatic pressure.

Now, if the outlet valve is slightly opened and then closed, a little amount of water (which is under stress) will flow out of the dashpot. This will facilitate the deformation of the spring and part of the stress in the water is trans­ferred to the spring.

The stress equation at this intermediate stage is as follows:

Δσ = Δσ’ + Δu …(11.2)

where Δσ is the additional stress applied on the spring-dashpot system, Δσ’ the additional stress in the spring having value of more than zero but less than Δσ, and Δu is the additional stress in the water in the dashpot having value less than Δσ but more than zero.

If the outlet valve is opened for sufficient time, the water from the dashpot flows out until it is completely relieved off the additional stress. This facilitates required deformation of the spring and the additional stress in the water is completely transferred to the spring. The time taken for the complete transfer of stress from the water of the dashpot to the spring depends on the rate of flow of water from the valve.

The stress equation at the final stage is as follows:

Δσ = Δσ’ + Δu …(11.3)

where Δσ is the additional stress applied on the spring-dashpot system, Δσ’ the additional stress in the spring equal to Δσ, and Δu the additional stress in the water in the dashpot equal to 0.

2. Soil-Spring Analogy:

Terzaghi compared the spring-dashpot model to the soil grains – voids filled with water – to explain the mecha­nism of consolidation. Here, the spring is compared to the soil grains and the dashpot is compared to the soil voids filled with water under stress. The valve represents the drainage face (pervious soil).

Let the initial stress in the soil be σ‘₀ under its own weight and the soil is deformed due to this stress. Suppose an additional load is applied on the soil-water system to cause an additional stress of Δσ. If the soil grains have to share this additional load, they have to undergo deformation. As the water filling the soil voids is incompressible, it will not allow the deformation of the soil grains. As the soil solids cannot deform, they cannot take any additional stress and thus the entire additional stress applied is borne by the pore water filling the voids.

The stress equation at this initial stage is as follows:

Δσ = Δσ’ + Δu …(11.4)

where Δσ is the additional stress applied on the soil-water system, Δσ’ the additional stress in the soil grains, known as effective stress increment equal to 0, and Am the additional stress in the water in the voids equal to Δσ.

Now, the additional stress in the pore water, Δu, known as excess hydrostatic pressure or pore water pressure, causes a stress gradient between the pore water within (stress zone) and outside the pressure bulb causing a little amount of water (which is under stress) to flow out of the voids. This facilitates the partial deformation of the soil grains and part of additional stress applied is transferred from the water to the soil solids.

The stress equation at this intermediate stage is as follows:

Δσ = Δσ’ + Δu …(11.5)

where Δσ is the additional stress applied on the soil-water system, Δσ’ the effective stress increment having a value of more than zero but less than Δσ, and Δu the pore water pressure having a value less than Δσ but more than zero.

Now, after lapse of sufficient time, the pore water in the voids of the stressed zone flows out of the pressure bulb until the remaining water in the pressure bulb is completely relieved of the additional stress. This outflow of water facilitates sufficient deformation of the soil, through rearrangement of solid particles and the additional stress in the water is completely transferred to the soil solids.

The time taken for the complete transfer of stress from the pore water to the soil solids depends on the rate of the flow of water, which is permeability of the soil and the avail­ability of suitable drainage face, that is, pervious (sand or gravel) strata. If sand or gravel layer exists at both the top and the bottom of the compressible soil layer (clay), expulsion of pore water will occur faster, both in the upward and downward direction. On the other hand, if the pervious soil layer is only on one side of the compressible soil,

expulsion of pore water takes place only in one direction (either upward or downward) and takes more time. This depends on the geological formation of the soil at the site.

The stress equation at the final stage is as follows:

Δσ = Δσ’ + Δu …(11.6)

where Δσ is the additional stress applied on the soil-water system, Δσ’ the additional stress in the soil solids equal to Δσ, and Δu the pore water pressure equal to 0. The stress system at the initial stage, intermediate stage, and final stage of the consolidation process is shown in Fig. 11.2.

In brief, since water is incompressible, stresses applied to saturated clayey soils are initially taken by the pore water in the form of excess hydrostatic pressure or pore water pressure. The resulting stress gradient in the water within and outside the stress zone causes expulsion of the pore water, further causing transfer of stress from pore water to soil solids, thus increasing the effective stress in soils which undergo compression.

The rate of the expul­sion of pore water, the dissipation of pore water pressure, increase of effective stress, and, hence, the compression of soil depends on the permeability of the clayey soils and the availability of pervious (drainage) layer among other factors. This time-dependent compression of soil, involving expulsion of pore water, is known as consolidation. The end of the consolidation process is signified by the complete dissipation of pore water pressure to zero.