Civil Engineering Reference
In-Depth Information
Figure 2.9
Angle of dilation and zero extension lines.
The planes, shown by double lines, on which this stress ratio occurs are at angles
α
and
β
as shown and, from the geometry of the figure,
45
◦
+
1
2
α
=
β
=
φ
(2.8)
m
For frictional materials these correspond to the planes on which the most critical
conditions occur and they should be the planes on which failure will occur.
When the major and minor principal strains have opposite signs the origin of the
axes is inside the Mohr circle, as shown in Fig. 2.9(b). There are two planes, shown by
broken lines in Fig. 2.9(b), across which the normal strains are zero, and so there are
two directions, shown by double lines, at angles
along which the strains are
zero as shown in Fig. 2.9(a). These planes are defined by an angle of dilation
α
and
β
ψ
. From
1
2
(
1
2
(
=
δε
+
δε
h
) and
g
=
δε
−
δε
h
), and if the volumetric
Fig. 2.9(b), the lengths
v
z
z
strain is
δε
v
=
δε
z
+
δε
h
then the angle of dilation is given by
ψ
=−
δε
v
δγ
tan
(2.9)
or
ψ
=
δε
+
δε
h
δε
z
−
δε
h
z
sin
(2.10)
where
is the increment of shear strain across the plane. (The negative signs are
required in Eqs. (2.9) and (2.10) so positive angles
δγ
are associated with dilation or
negative volumetric strains.) From the geometry of the figure,
ψ
45
◦
+
1
2
α
=
β
=
ψ
(2.11)
