Geotechnical Engineering

DHARM

COMPRESSIBILITY AND CONSOLIDATION OF SOILS 229

This solution enables the hydrostatic excess u to be computed for a soil mass under any

initial system of stress ui, at any depth z, and at any time t.

In particular, if ui is considered constant with respect to depth, this equation reduces to

u =

2

1

2

22 4 2

1

u

n

n

nz

H

i enctH

n

v

π

π

π π

(cos)sin− F /

HG

I

KJ

L

N

M

O

Q

P

−

=

∞

∑ ...(Eq. 7.22)

When n is even, (1 – cos nπ) vanishes; when n is odd, this factor becomes 2. Therefore it
is convenient to replace n by (2m + 1), m being an integer. Thus, we have

u =

4

21

21

0 2

u 2122 42

m

m

H

i e

m

mctHv

()

sin

()()/

+

L +

NM

O

= QP

∞

−+

∑ π

π

...(Eq. 7.23)

It is convenient to use the symbol M to represent (π/2) (2m + 1), which occurs frequently:

u =^2

22

0

u

M

M

H

ize MctH

m

Fsin v/

HG

I

KJ

−

=

∞

∑ ...(Eq. 7.24)

Three-dimensionless parameters are introduced for convenience in presenting the re-
sults in a form usable in practice. The first is z/H, relating to the location of the point at which
consolidation is considered, H being the maximum length of the drainage path. The second is
the consolidation ratio, Uz, defined in sec. 7.3, to indicate the extent of dissipation of the hydro-
static excess pressure in relation to the initial value:

Uz = (ui – u)/ui =^1 −

F

HG

I

KJ

u
ui ...(Eq. 7.16)
The subscript z is significant, since the extent of dissipation of excess pore water pres-
sure is different for different locations, except at the beginning and the end of the consolida-
tion process.

The third dimensionless parameter, relating to time, and called ‘Time-factor’, T, is de-

fined as follows:

T =

ct

H

v

2 ...(Eq. 7.25)

where cv is the coefficient of consolidation,

H is the length drainage path,
and t is the elapsed time from the start of consolidation process.
In the context of consolidation process at a particular site, cv and H are constants, and
the time factor is directly proportional to time.

Introducing the time factor into Eq. 7.24, we have

u =

2 2

0

u

M

Mz

H

i e MT

m

Fsin

HG

I

KJ

−

=

∞

∑ ...(Eq. 7.26)

Introducing the consolidation ratio, Uz, we have:

Uz = 112

2

0

−=− F

HG

I

KJ

−

=

∞

∑

u

uM

Mz

H

e

i

MT

m

sin. ...(Eq. 7.27)