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Chapter
5
Shear Strength
5.1 The Mohr-Coulomb Failure Criterion
This
chapter
is
concerned
with
the
resistance
of
a
soil
to
failure
in
shear.
A
knowledge
of
shear
strength
is
required
in
the
solution
of
problems
concerning
the
stability of soil
masses. If at a point on any plane within a soil
mass the shear stress
becomes equal to
the shear strength of the soil, failure will occur
at that point. The
shear strength
(
τ
f
) of a soil at
a point on a particular plane was originally
expressed by
Coulomb as a linear
function of the normal
stress(
σ
f
) on the
plane at the same point:
τ
f
=c+
σ
f
tan
φ
(5.1)
where
c
and
φ
are
the
shear
strength
parameters,
now
described
as
the
cohesion
intercept (or the apparent cohesion)
and the angle of shearing resistance,
respectively.
In accordance with
Terizaghi's fundamental concept that shear stress
in a soil can be
resisted
only
by
the
skeleton
of
solid
particles,
shear
strength
is
expressed
as
a
function of effective normal stress:
τ
f
=c
'+
σ'
f
tan
φ'
(5.2)
where c' and
φ
' are the shear strength
parameters in terms of effective stress. Failure
will thus occur at any point where a
critical combination of shear stress and effective
normal stress develops.
The shear strength of a
soil can also be expressed in terms of the
effective major
and minor principal
stresses
σ
'
1
and
σ
'
3
at failure at the point in question. At failure
the
straight
line
represented
by
Equation
5.2
will
be
tangential
to
the
Mohr
circle
representing the state of stress, as
shown in Fig. 5.1, compressive stress being taken
as
positive. The coordinates of the
tangent point are
τ
f
and
σ
'
f
,
where:
1
τ
f
=
(<
/p>
σ'
1
-
σ'<
/p>
3
)sin2
θ
(5.3)
2
σ'
f<
/p>
=
1
1
(
σ'
1
+
σ'
3
)+
(
σ'
1
-
σ'
3
)cos2
θ
(5.4)
2
2
and
θ
is
the
theoretical
angle
between
the
major
principal
plane
and
the
plane
of
failure.
It is apparent that
?
?
?
?
45
?
?
(5.4)
2
From Fig 5.1 the
relationship between the effective principle
strength
at failure and
the
shear strength parameters can also be obtained.
Now:
1
(
?
?
1
?
?
?
3
)
2
(5.5)
sin
?
?
1
c
?
cos
?
?
?
(
?
?
1
?
?
?
3
)<
/p>
2
Therefore
(
σ'
1
-
σ'
3
)= (
σ'
1
+
σ'
3
)s
in
φ'
+2
c'cos
φ'
(5.6a)
or
?
?
?
?
< br>σ'
1
=
σ'
< br>3
tan
2
(
45
?
?
)+2 c'tan
(
45
?
?
)
(5.6b)
2
2
Equation
5.6
is
referred
to
as
the
Mohr-
Coulomb
failure
criterion.
If
a
number
of
states
of
stress
are
know,
each
producing
shear
failure
in
the
soil,
the
criterion
assumes
that
a
common
tangent,
represented
by
Equation
5.2,
can
be
drawn
to
the
Mohr
circles representing the states of stress: the
common tangent is called the failure
envelope of the soil. A state of stress
plotting above the failure envelope is impossible.
The
criterion
does
not
involve
consideration
of
strans
at,
or
prior
to,
failure
and
implies that the
effective intermediate principal stress
σ
'
2
has no influence on the shear
strength
of the soil. The Mohr- Coulomb failure criterion,
because of its simplicity, is
widely
used in practice although it is by no means the
only possible failure criterion
for
soils. The failure envelope for a particular rail
may not necessarily be a straight
line
but a straight line approximation can be taken
over the stress range of interest and
the shear strength parameters
determined for that range.
By plotting
1
1
(
σ'<
/p>
1
-
σ'
3
) against
(
σ'
1
+
σ'
3
) any state of stress can be represented
2
2
by a stress
point rather than by a Mohr circle, as shown in
Fig. 5.2, and on this plot a
modified
failure envelope is obtained, represented by the
equation:
1
1
(
σ'
1
-
σ'
3
)=a'+
(
σ'<
/p>
1
+
σ'
3
)tan
α'
(5.7)
2
2
where a' and
α
' are the modified shear
strength parameters. The parameters c' and
φ
'
are then given
by:
φ'
=sin
-1
(tan
α'
)
(5.8)
c' = a'
/cos
φ'
(5.9)
Lines drawn from the
stress point at angles of
4
5
°
to the
horizontal, as shown in Fig
5.2,
intersect
the
horizontal
axis
at
points
representing
the
values
of
the
principal
stresses
σ'
1
and
σ'
3
.Fig. 5.2
could also be drawn in terms of total stress, the
vertical and
horizontal
coordinates
being
1
1
(
σ'
1
-
σ'
3
)
and
(
σ
1
+
σ
3
)
respectively.
It
should
be
2
2
noted that:
1
1
(
σ
1
-
σ
3
) =
(
σ'
1
-
σ'
3
)
2
2
1
1
(
σ
1
+
σ
3
) =
(
< br>σ'
1
+
σ'
< br>3
)-u
2
2
In the case of
axial symmetry, a-state of effective stress earl
also be plotted with
respect to
vertical and horizontal coordinates q' and p'
respectively,
q'=(
σ'
< br>1
-
σ'
3
)
(5.10)
1
p'=
(
σ'
1
+2
σ'
3
)
(5.11)
3
The
values
of
these
stresses
(being
functions
of
the
principal
stresses)
are
independent of the
orientation of the coordinate axes, such stresses
being known as
stress invariants. The
corresponding total stresses are:
q=(
σ
1
-
σ
3
)
1
p=
(
σ<
/p>
1
+2
σ
3
)
3
In this case the
relationships between effective and total stresses
are:
q'=q
p'=p-u
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