**Branch :**Civil Engineering

**Subject :**Soil Mechanics

## Solution of Governing Consolidation Equation Using Fourier Series

The solution of any differential equation requires a knowledge of the boundary conditions. By specifi cation of the initial distribution of excess porewater pressures at the boundaries, we can obtain solutions for the spatial variation of excess porewater pressure with time and depth. Various distributions of porewater pressures within a soil layer are possible. Two of these are shown in Figure . One of these is a uniform distribution of initial excess porewater pressure with depth (Figure ). This may occur in a thin layer of fi ne-grained soils. The other (Figure ) is a triangular distribution. This may occur in a thick layer of fi ne-grained soils.

The boundary conditions for a uniform distribution of initial excess porewater pressure in which double drainage occurs are

At the bottom boundary, z 5 2Hdr, Du 5 0, where Hdr is the length of the drainage path.

A solution for the governing consolidation equation, Equation, which satisfi es these boundary conditions, is obtained using the Fourier series,

Tw**o types of excess porewater pressure distribution with
depth: (a) uniform distribution with depth in a thin layer and (b) triangular
distribution with depth in a thick layer.**

where M 5 (p/2)(2m 1 1) and m is a positive integer with values from 0 to ` and

where Tv is known as the time factor; it is a dimensionless term. A plot of Equation gives the variation of excess porewater pressure with depth at different times. Let us examine Equation (9.32) for an arbitrarily selected isochrone at any time t or time factor Tv, as shown in Figure 9.8. At time t 5 0 (Tv 5 0), the initial excess porewater pressure, Duo, is equal to the applied vertical stress throughout the soil layer.

As soon as drainage occurs, the initial excess porewater pressure will immediately fall to zero at the permeable boundaries. The maximum excess porewater pressure occurs at the center of the soil layer because the drainage path there is the longest, as obtained earlier in our experiment in Section 9.3. At time t . 0, the total applied vertical stress increment Dsz at a depth z is equal to the sum of the vertical effective stress increment Ds9z and the excess porewater pressure Duz. After considerable time (t → `), the excess porewater pressure decreases to zero and the vertical effective stress increment becomes equal to the vertical total stress increment. We now defi ne a parameter, Uz, called the degree of consolidation or consolidation ratio, which gives us the amount of consolidation completed at a particular time and depth. This parameter can be expressed mathematically as

The consolidation ratio is equal to zero everywhere at the beginning of the consolidation Duz 5 Duo but increases to unity as the initial excess porewater pressure dissipates.