Environmental Engineering Reference
In-Depth Information
The dry density of a soil,
ρ
d
, is obtained by eliminating
themassofthewater,
M
w
, from Eq. 2.66:
Changes
in
total
unit
weight
can
be
accommodated
through
incorporation
of
the
SWCC,
which
relates
volumetric water content
θ
to soil suction
ψ
:
G
s
ρ
d
=
e
ρ
w
(2.70)
G
s
+
θ (ψ)(
1
+
e)
1
+
γ
=
γ
w
(2.73)
1
+
e
w
(ψ)
ρ
d
/ρ
w
1
The relationship between total density
ρ
and dry density
ρ
d
for different water contents is presented graphically in
Fig. 2.46. If any two of the volume-mass properties of a
soil (e.g.,
e
,
w
,or
S
) are known, the total density of the
soil,
ρ
, can be computed in accordance with Fig. 2.46 or
Eq. 2.69. The dry density of the soil,
ρ
d
, is computed using
Eq. 2.70 provided the void ratio
e
or the porosity
n
of the
soil are known.
The dry density curve corresponding to a degree of satu-
ration of 100% is called the “zero air voids” curve. The dry
density curves for various degrees of saturation are com-
monly presented in connection with soil compaction data.
G
s
+
γ
=
γ
w
(2.74)
+
e
If a mathematical equation is known for the SWCC (i.e.,
in terms of either volumetric water content or gravimetric
water content), the unit weight of the soil can be written as
a function of the stress state, namely, soil suction. Conse-
quently, variations in the total unit weight due to changes
in water content in the unsaturated soil zone can be taken
into account. Equation 2.73 could also take volume changes
into account during a process (i.e., depending on how the
SWCC is defined), whereas Eq. 2.74 assumes that no volume
change occurs during a process.
2.4.6.2 Unit Weight
The unit weight of a soil mass,
γ
, can be computed by multi-
plying the total density of a soil by gravitational acceleration
g
(i.e.,
ρg
). The unit weight of a soil is often referred to as
the body force involved when gravity acts on the soil. The
above equations for density can be written in the following
unit-weight forms:
2.4.6.4 “Basic” Volume-Mass Relationship
The volume and mass for each phase can be related to one
another using basic relations from the phase diagram. The
mass of water in a soil,
M
w
, is the product of the volume
and density of water:
M
w
=
ρ
w
V
w
(2.75)
G
s
(
1
+
w
)
The volume of water,
V
w
, can also be computed from the
volume relations shown in Fig. 2.45 (i.e., left-hand side):
γ
=
γ
w
(2.71)
1
+
e
G
s
+
Se
V
w
=
SeV
s
(2.76)
γ
=
γ
w
(2.72)
1
+
e
The relationship given in Eq. 2.76 is shown in Fig. 2.45
(i.e., left-hand side). Equation 2.75 can then be rewritten as
When solving soil mechanics problems such as the calcu-
lation of the factor of safety for a slope stability analysis,
the unit weight is used to calculate the weight of a slice
of soil. Likewise, the unit weight of a soil is used when
“switching on” the gravity force (or body force) of a soil.
The unit weight also allows for the calculation of the total
stress state in a soil continuum.
M
w
=
ρ
w
SeV
s
(2.77)
The mass of the water,
M
w
, can also be related to the
mass of the soil solids,
M
s
:
M
w
=
wM
s
(2.78)
2.4.6.3 Unit Weight as Function of Volumetric Water
Content
The unit weight of a soil changes as the amount of water
in a soil changes. The amount of water in a soil can be
defined in terms of gravimetric water content
w
, volumetric
water content
θ
, or degree of saturation
S
. Unsaturated soil
problems involving the calculation of the total stress state
often involve switching on the gravity forces. This means
that the unit weight of the soil is instantaneously applied
and the corresponding total stress states are computed. It is
possible to write the unit weight as a function of the amount
of water for an unsaturated soil rather than designating a
single value for the unit weight of the unsaturated soil. The
water content may fluctuate because of imposed moisture
fluxes at the ground surface or for other reasons.
The mass of the soil solids,
M
s
, is also defined in the
phase diagram in Fig. 2.45:
M
s
=
G
s
ρ
w
V
s
(2.79)
Substituting Eq. 2.79 into Eq. 2.78 yields
M
w
=
wG
s
ρ
w
V
s
(2.80)
Equating Eqs. 2.80 and 2.77 results in a
basic volume-
mass relationship
for soil:
Se
=
wG
s
(2.81)
The basic volume-mass relationship is an equation that is
commonly used to calculate phase relationship properties in
an unsaturated soil.
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