Environmental Engineering Reference
In-Depth Information
dσ
3
=
infinitesimal increase in isotropic pressure,
σ
−
u
a
and
u
a
−
u
w
change as the soil is compressed. The
volume of the soil changes in accordance with the con-
stitutive relation for the soil structure. Volume change is
primarily the result of compression of the pore fluid since
the soil solids are essentially incompressible. The volume
of the voids,
V
ν
,
referenced to the initial total volume of
the soil,
V
0
(i.e.,
V
ν
/V
0
), is a function of the stress state
variables. The volume change constitutive relationship can
be linearized for finite changes in stress state. The linear
compressibility form of equation for volume change was
proposed by Fredlund and Morgenstern (1976):
B
w
=
tangent pore pressure parameter for the water pres-
sure during isotropic, undrained compression, and
du
w
=
increase in pore-water pressure due to an infinites-
imal increase in isotropic pressure,
dσ
3
.
The concept of a
B
tangent pore pressure parameter
was previously used (i.e., Eq. 15.13), where there was an
infinitesimal increase in total stress. The
B
secant pore
pressure parameters are particularly useful in computing the
final pore-air and pore-water pressures after a large change
in total stress. The
B
tangent pore pressure parameters
can also be used to estimate the final pore pressures under
large changes in total stress by using a marching-forward
technique with finite increments of total stress. The total
stress is incrementally increased, commencing from an
initial condition and proceeding to the final condition under
investigation. As saturation is approached, the pore-water
pressure approaches the pore-air pressure (i.e.,
u
w
→
u
a
or
B
w
→
B
a
) and the air bubbles dissolve in the water.
At saturation, a change in total stress causes an almost
equal change in pore-water pressure. The small difference
between the change in total stress and the change in pore-
water pressure at saturation can be ignored due to the low
compressibility of water relative to the compressibility of
the soil structure (Skempton, 1954). In other words, the
B
tangent parameter becomes equal to 1.0 (i.e.,
B
a
=
B
w
=
1
.
0). The convention commonly adopted in soil laborato-
ries assumes the soil is saturated when the
B
pore pressure
parameter reaches a value of 1.0 (Skempton, 1954). It is
possible for the
B
secant pore pressure parameter to be less
than 1.0 even though the soil has reached saturation. Also,
values of
B
a
and
B
w
are not equal at saturation since the ini-
tial pore-air pressure was greater than the initial pore-water
pressure (Fig. 15.11).
The derivations for tangent pore pressure parameters for
various loading conditions are presented in the following
sections. Hilf's analysis is an exception where a secant-type
pore pressure parameter is derived. A prime sign is assigned
to the secant parameters (e.g.,
B
) to differentiate between
the secant and the tangent pore pressure parameters.
dV
ν
V
0
=
m
1
d(σ
−
u
a
)
+
m
2
d(u
a
−
u
w
)
(15.23)
where:
dV
ν
/V
0
=
volume change referenced to the initial
total volume of the soil,
V
ν
=
volume of soil voids,
V
0
=
initial total volume of the soil,
m
1
=
coefficient of soil volume change with
respect to a change in net normal stress,
d(σ
−
u
a
)
=
change in net normal stress,
m
2
=
coefficient of soil volume change with
respect to a change in matric suction, and
d(u
a
−
u
w
)
=
change in matric suction.
The continuity of a referential, unsaturated soil element
requires that overall volume change must be equal to the sum
of the changes in volume associated with the air and water
phases which fill the pore voids. The air phase constitutive
equation can be expressed as
dV
a
V
0
=
m
1
d(σ
−
u
a
)
+
m
2
d(u
a
−
u
w
)
(15.24)
where:
dV
a
/V
0
=
change in the volume of air referenced to the
initial total volume of the soil,
V
a
=
volume of air,
m
1
=
coefficient of air volume change with respect
to a change in net normal stress, and
15.4.2 Necessary Volume Change Constitutive
Relations
The derivation of pore pressure parameters requires vol-
ume change constitutive relations for an unsaturated soil.
These constitutive relations describe the volume changes
that take place under
drained
loading. The volume changes
are expressed in terms of the stress state variable changes.
These constitutive relations were formulated and explained
in Chapter 13 on volume-mass constitutive relations. This
section briefly summarizes the constitutive relations required
for deriving the pore pressure parameter equations.
Consider an unsaturated soil specimen that undergoes one-
dimensional, drained compression. The stress state variables
m
2
=
coefficient of air volume change with respect
to a change in matric suction.
The constitutive equation for the water phase can be writ-
ten as
dV
w
V
0
=
m
1
d(σ
−
u
a
)
+
m
2
d(u
a
−
u
w
)
(15.25)
where:
dV
w
/V
0
=
change in the volume of water referenced to
the initial volume of the soil,
Search WWH ::
Custom Search