Environmental Engineering Reference
In-Depth Information
tractions). Body forces act through the centroid of a soil
element and are expressed as a force per unit volume.
Gravitational and interaction forces between phases are
examples of body forces. Surface forces such as external
loads act only on the boundary surfaces of a soil element.
The average value of a surface force per unit area tends
to a limiting value as the surface area approaches zero.
This limiting value is called the stress vector or the surface
traction on a particular face of the element. The component
of the stress vector perpendicular to a face of the element is
defined as a normal stress,
σ
. The stress components parallel
to a face of an element are referred to as shear stresses,
τ
.
There are an infinite number of planes that can be passed
through a
point
in a soil mass. The stress state at a point
can be analyzed by considering all the stresses acting on the
planes that form a cubical element of infinitesimal dimen-
sions. Body forces act through the centroid of the soil ele-
ment and also need to be considered.
A cubical element that is completely enclosed by imagi-
nary, unbiased boundaries can be used as a free-body dia-
gram for a stress equilibrium analysis (Fung, 1977). The
cubical element is referred to as a referential elemental vol-
ume. Figure 3.5 shows a cubical soil element with infinitesi-
mal dimensions of
dx
,
dy
, and
dz
in the Cartesian coordinate
system. The normal and shear stresses on each surface of
the element are shown. The body forces are not shown.
Normal and shear stresses act on every plane in the
x
-,
y
-,
and
z
-directions. The normal stress,
σ
, has one subscript to
denote the plane on which it acts. Soils are most commonly
subjected to compressive normal stresses. In soil mechanics,
a positive normal stress is used to indicate compression in
the soil. All the normal stresses shown on the element are
positive or compressive. Opposite directions would indicate
negative normal stresses or tensions. The shear stress
τ
has two subscripts. The first subscript denotes the plane on
which the shear stress acts, and the second subscript refers
to the direction of the shear stress. As an example, the shear
stress
τ
yz
acts on the
y-
plane and in the
z
-direction. All
of the shear stresses shown have positive signs. Opposite
directions would indicate negative shear stresses. Equating
the summation of moments about the
x
-,
y
-, and
z
-axes to
zero results in the following shear stress relationships:
τ
xy
=
τ
yx
(3.10)
τ
xz
=
τ
zx
(3.11)
τ
yz
=
τ
zy
(3.12)
The stress components can vary from plane to plane across
an element. The spatial variation of a stress component can
be expressed as its derivative with respect to space. The
stress variations in the
x
-,
y
-, and
z
-directions are expressed
as stress fields.
∂σ
y
∂
y
σ
y
+
dy
∂τ
yx
∂
y
τ
yx
+
dy
∂τ
yz
∂
y
σ
z
τ
yz
+
dy
τ
xz
τ
zx
∂τ
xz
∂
x
τ
xz
+
dx
∂τ
zy
∂
z
τ
zy
+
τ
zy
dz
∂σ
x
∂
x
σ
x
σ
x
+
dx
τ
xy
dy
∂τ
zx
∂
z
τ
zx
+
dz
∂τ
xy
∂
x
τ
xy
+
dx
∂σ
z
∂
z
σ
z
+
dz
τ
yz
τ
yx
y
x
σ
y
dx
z
Figure 3.5
Normal and shear surface tractions on cubical soil element of infinitesimal dimen-
sions,
dx, dy
,and
dz
.
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