Civil Engineering Reference
In-Depth Information
so the line from P to the point
δε
z
. (Notice
that the line from the pole to a point on the circle does not give the direction of the
strain but the direction of the plane perpendicular to the normal strain.) In Fig. 2.7(a)
there is an element rotated at an angle
δε
z
gives the plane across which the strain is
θ
and the strains associated with this element
(
2
δγ
1
2
n
,
δε
n
and
δγ
m
,
δε
m
) are given by the points N and M as shown.
2.7 Stress ratio and dilation
We will see later that soils are frictional materials, which means that their strength
(i.e. the maximum shear stress they can sustain) increases with normal stress and so
the stress ratio
is more important than the shear stress alone. Figure 2.8(a) shows
a stressed element and Fig. 2.8(b) is the corresponding Mohr circle of stress with the
pole at P. There are two lines ON which are tangents to the Mohr circle and these
define the points on which the stress ratio is given by
τ
/
σ
τ
σ
=
tan
φ
(2.5)
m
where
φ
m
is the mobilized angle of shearing resistance. From the geometry of Fig. 2.8(b)
1
2
(
1
2
(
t
=
σ
−
σ
h
) and
s
=
σ
+
σ
h
) and
z
z
t
s
=
=
σ
−
σ
h
σ
z
+
σ
h
z
sin
φ
(2.6)
m
or
tan
2
45
◦
+
m
σ
1
+
sin
φ
z
σ
h
=
m
1
2
φ
m
=
φ
(2.7)
1
−
sin
Figure 2.8
Limiting stress ratio and angle of shearing resistance.
