Environmental Engineering Reference
In-Depth Information
Principal Stresses
Stresses acting on any plane passed through a point consist of a normal stress
σ
(com-
pression or tension) and a shearing stress
. (Soil mechanics problems are normally con-
cerned with compressive stresses.) On one particular plane, the normal stress will be the
maximum possible value and the shearing stress will be equal to zero. On one plane per-
pendicular to this plane, the normal stress will be the minimum possible value, with shear
stress also equal to zero. On a second plane perpendicular to this plane, the normal stress
will have an intermediate value and the shearing stress will also be zero. These planes are
termed the principal planes.
The principal stresses are the stresses acting perpendicular to the principal planes
including the maximum (major) principal stress
τ
σ
1
, the minimum (minor) principal stress
σ
3
, and the intermediate principal stress
σ
2
. The relationship between principal stresses
and the normal stress and the shear stress acting on a random plane through a point is
shown in Figure 3.21. The intermediate principal stress is the plane of the paper and, in
soil mechanics problems, is normally considered to be equal to
σ
3
.
The Mohr Diagram
To attain equilibrium, the sum of the forces given in Figure 3.21 should be zero. Therefore,
σ
n
and
τ
can be expressed in terms of the principal stresses and the angle
θ
as
σ
n
[ (
σ
1
σ
2
)/2
(
σ
1
σ
3
)/2 ] cos 2
θ
(3.21)
τ
[(
σ
1
σ
3
)/2] sin 2
θ
(3.22)
If points are plotted to represent coordinates of normal and shearing stresses acting on a
particular plane for all values of
given in equations 3.21 and 3.22, their loci form a circle
which intersects the abscissa at coordinates equal to the major (
θ
σ
3
) principal
stresses. The circle is referred to as the Mohr diagram, or Mohr's circle, given in
Figure 3.22.
σ
1
) and minor (
Applications of Strength Values
Stability Analysis
The values for strength are used in stability analyses; the discussion is beyond the scope of
this topic, except for evaluations of slopes. In general terms, stability is based on plastic equi-
librium or a condition of maximum shear strength with failure by rupture imminent. When
the imposed stresses cause the shear strength to be exceeded, rupture occurs in the mass
along one or more failure surfaces. Analyses are normally based on the
limit equilibrium
σ
n
τ
n
B
σ
3
sin
O
p
A
FIGURE 3.21
Stresses on a random plane through a point (
σ
1
cos
σ
2
is the plane
of the paper).
