Terzaghi basic differential equation of consolidation can be solved with the following boundary condition:

 `z=0`,`u=0`

`z=2d`,`u=0`

`t=0`,`u=u_0`

 The solution yields,

`int[2u_0/M sin((Mz)/d)]e^(-M^2T_v)`.........(i) from limit `m=0` to` m=â??`

Where,

`m=` an integer

`M= pi/2*(2m+1)`

`u_0=`initial excess pore water pressure.

`T_v=`Time factor which is non dimensional number

`T_v=(C_v*t)/u_0`

A series of isochrones indicating the variation of `u` and `z` can be plotted for different value of `T_v`. The shape of isochrones depend on `u_0` and drainage condition at the boundary of clay layer.

 If both upper and lower boundaries are free draining, the clay layer is known as openlayer. If only one boundary of clay layer is free draining, the layer is called half closed layer.

 The progress of consolidation at any point depends upon the pore water pressure....Show More

Terzaghi basic differential equation of consolidation can be solved with the following boundary condition:

 `z=0`,`u=0`

`z=2d`,`u=0`

`t=0`,`u=u_0`

 The solution yields,

`int[2u_0/M sin((Mz)/d)]e^(-M^2T_v)`.........(i) from limit `m=0` to` m=â??`

Where,

`m=` an integer

`M= pi/2*(2m+1)`

`u_0=`initial excess pore water pressure.

`T_v=`Time factor which is non dimensional number

`T_v=(C_v*t)/u_0`

A series of isochrones indicating the variation of `u` and `z` can be plotted for different value of `T_v`. The shape of isochrones depend on `u_0` and drainage condition at the boundary of clay layer.

 If both upper and lower boundaries are free draining, the clay layer is known as openlayer. If only one boundary of clay layer is free draining, the layer is called half closed layer.

 The progress of consolidation at any point depends upon the pore water pressure `u` at that point. The degree of consolidation at any point `z` is equal to the ratio of the dissipated excess pore water pressure to the initial excess pore water pressure.i.e,

`U_z=(u_0-u_z)/u_0`

`=(1-u_z)/u_0`

Where,`u_z=`excess pore water pressure at time `t`.

The average degree of consolidation for the entire depth of clay layer at any time `t` can be written as:

`U=S_(c(t))/S_c ` 

 $=\int_{0}^{2d} (u_z dz)/u_0$ 

where, `U=`average degree of consolidation

`S_(c(t))=`settlement of layer at time `t`.

`S_c=`Ultimate settlement of the layer.

Substituting of equation (i) yields,

`U=1-sum 2/M^2*e^(-M^2T_v)`

The variation of `U` and `T_v` is shown in figures:

Following plot can be approximated by the expressions given below:

For `U=0` to `60%`,

`T_V=pi/4((U%)/100)^2`

For `U>60%`,

`T_v=1.781-0.933log_(10)(100-U%)`

Or, `T_v=-0.933log_(10)(1-U)-0.085`

Time Factor (`T_v`):

The time factor depends upon the coefficient of consideration (`C_v`) ,time(`t`) and the drainage path `d`. 

`T_v=(C_vt)/d^2`

`=(k/(gamma_w m_v))t/d^2 `

`=(k/( m_v*g*rho_w))t/d^2 `

The drainage path `d` represents the maximum distance that the water has to travel before reaching the free drainage boundary. 

For Open layer (double drainage),

`d=H/2`

For half closed layer( single drainage),

`d=H`

Where, `H=` thickness of the soil layer