Civil Engineering Reference
In-Depth Information
σ
Combining Eqs. (15.1) and (15.2) and in the limit noting that
q
and
are both
functions of
z
and
t
,
Am
v
∂σ
∂
∂
q
=
(15.3)
∂
z
t
The rate of seepage flow is given by Darcy's law as
q
A
=
V
=
ki
(15.4)
where
V
is the seepage velocity and the hydraulic gradient
i
is
1
γ
w
δ
u
i
=
(15.5)
δ
z
From Eqs. (15.4) and (15.5) and in the limit,
∂
2
u
∂
q
Ak
γ
w
∂
u
Ak
γ
w
∂
=−
=−
(15.6)
∂
z
∂
z
∂
z
∂
z
2
and, from Eqs. (15.3) and (15.6),
2
u
=−
∂σ
∂
m
v
γ
w
∂
k
(15.7)
z
2
∂
t
σ
=
σ
−
The effective stress is given by
(
u
∞
+
u
) and, noting that
u
∞
remains constant,
∂σ
∂
=
∂σ
∂
−
∂
u
(15.8)
t
t
∂
t
The simple and common case is where consolidation takes place after an increment
of undrained loading or unloading so that the total stress remains constant during the
consolidation. Then, from Eqs. (15.7) and (15.8) with
∂σ
/
∂
t
=
0,
2
u
c
v
∂
=
∂
u
(15.9)
z
2
∂
∂
t
where
k
m
v
γ
w
c
v
=
(15.10)
The parameter
c
v
is known as the coefficient of consolidation and has the units of
square metres per year. Values of
c
v
depend on both the permeability
k
and on the
compressibility
m
v
, both of which vary greatly for different soils.
Equation (15.9) is the basic equation for one
-d
imensional consolidation. Solutions
will give the variations of excess pore pressure
u
with d
ep
th
z
and with time
t
. Note
that consolidation theory deals with excess pore pressure
u
and not with absolute pore
pressures.
