Numerical determination of consolidation rates

When a consolidating layer of clay is subjected to an irregular distribution of initial excess pore water pressure, the theoretical solutions are not usually applicable unless the distribution can be approximated to one of the cases considered. In such circumstances the use of a numerical method is fairly common.

A brief revision of the relevant mathematics is set out below.

Maclaurin's series

Assuming that f(x) can be expanded as a power series: y = f(x) = ao + aix + a2x2 + a3x3 + ■ • ■ anxn

^ = f'(x) = ai + 2a2x + 3asx2 + 4a4x3 -|----nanxn~'

^ = f'(x) = 2a2 + 2.3a3x + 3.4a4x2 + • ■ • n(n - l)anxn"2 dxz d3v

-4 = f"'(x) = 2.3a3 + 2.3.4a4x + ■ • • n(n - l)(n - 2)anxn~3 dxJ

etc.

Generally fn(0)

Substituting these values:

This is the Maclaurin's series for the expansion of f(x).

Taylor's series

If a curve y = f(x) cuts the y-axis above the origin O at a point A (Fig. 10.11) we can interpret Maclaurin's expression as follows:

Let P be a point on the curve with abscissa x. Let the values of f(x), f'(x), f"(x), etc., at A be:

Let the value of f(x) at P be yp. Then x2vii x3vi." f(x)atP = yp = y0+xy;+^ + ^+...

This is a Taylor's series and gives us the value of the co-ordinate of P in terms of the ordinate gradient, etc., at A and the distance x between A and P.

Gibson and Lumb (1953) illustrated how the numerical solution of consolidation problems can be obtained by using the explicit finite difference equation. The differential equation for one-dimensional consolidation has been established:

Consider part of a grid drawn on to a consolidating layer (Fig. 10.12a). The variation of the excess pore pressure, u, with the depth, z, at a certain time, k, is shown in Fig. 10.12b, and the variation of u at the point O during a time increment from k to k + 1 is illustrated by Fig. 10.12c. In Fig. 10.12b: from Taylor's theorem:

. , Az2 „ Az3 ,„ u2,k = Uo,k - Azu0 k + — UQ k - — u0k + • ■ •

. , Az2 „ Az3 ,„ U4,k = Uo,k + Azu0>k + — Uo_k + ^Tu0,k + • • •

Fig. 10.11 Taylor's series.

(a) Consolidation grid (b) Variation of u with z

(c) Variation of u with t

(a) Consolidation grid (b) Variation of u with z

(d) Schematic form of equation Fig. 10.12 Explicit recurrence formula (general).

Adding and ignoring terms greater than second order:

By Taylor's theorem:

Ignoring second derivatives and above:

.". Uo,k+i = r(u2k + u4>k - 2u0,k) + u0,k where cvAt cvAt

The schematic form of this expression is shown in Fig. 10.12D. Hence if a series of points in a consolidating layer are established, Az apart, it is possible by numerical iteration to work out the values of u at any time interval after consolidation has commenced if the initial excess values u,, are known.

Impermeable boundary conditions

Figure 10.13a illustrates this case in which conditions at the boundary are represented by

Hence between the points 2k and 4k:

The equation therefore becomes:

uo,k+i = 2r(u2>k - uo,k) + u0,k and is shown in schematic form in Fig. 10.13b.

The boundary equation can also be used at the centre of a double drained layer with a symmetrical initial pore pressure distribution, values for only half the layer needing to be evaluated.

Errors associated with the explicit equation

A full discussion of this subject was given by Crandall (1956), but briefly errors fall into two main groups: truncation errors (due to ignoring the higher dz 2Az i.e.

(b) Schematic form of equation

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