Yqe re due to Pressure Resultant ight due to ws pressure

(ii) New embankment with super load

(i) Clay layer at depth

Pressure due to soil weight re due to Pressure Resultant ight due to ws pressure

(ii) New embankment with super load

(c) Trapezoidal Fig. 10.2 Forms of initial excess pore pressure.

10.3 Terzaghi's theory of consolidation

Terzaghi's first presented this theory in 1925 and most practical work on the prediction of settlement rates is now based upon the differential equation he evolved. The main assumptions in the theory are as follows.

(i) Soil is saturated and homogeneous.

(ii) The coefficient of permeability is constant.

(iii) Darcy's law of saturated flow applies.

(iv) The resulting compression is one dimensional.

(v) Water flows in one direction.

(vi) Volume changes are due solely to changes in void ratio, which are caused by corresponding changes in effective stress.

The expression for flow in a saturated soil has been established in Chapter 2. The rate of volume change in a cube of volume dx.dy.dz is:

For one-dimensional flow (assumption v) there is no component of hydraulic gradient in the x and y directions, and putting kz = k the expression becomes:

a2 h

The volume changes during consolidation are assumed to be caused by changes in void ratio.

Porosity

Another expression for the rate of change of volume is therefore: J- (dx.dy.dz dt V 1 +e,

Equating these two expressions: ^h _ 1 Oe dz2 ~~ 1 + e dt

The head, h, causing flow is the excess hydrostatic head caused by the excess pore water pressure, u.

With one-dimensional consolidation there are no lateral strain effects and the increment of applied pressure is therefore numerically equal (but of opposite sign) to the increment of induced pore pressure. Hence an increment of applied pressure, dp, will cause an excess pore water pressure of du (= —dp). Now: de hence a =

or de = a du Substituting for de:

k N ^u du d2u du where cv = the coefficient of consolidation and equals k k

7wa 7wmv

In the foregoing theory z is measured from the top of the clay and complete drainage is assumed at both the upper and lower surfaces, the thickness of the layer being taken as 2H. The initial excess pore pressure, Uj, = —dp. The boundary conditions can be expressed mathematically:

when z = 0, u = 0 when z = 2H, u = 0 when t = 0, u = Uj

A solution for d2u_du Cvd^~dt that satisfies these conditions can be obtained and gives the value of the excess pore pressure at depth z at time t, uz:

0 0