Geoscience Reference
In-Depth Information
projection of the radius into the
x
-axis (from the center to
n
) has the value [(
surfaces inclined at angle
with respect to the normal to
1
. The cube on the left (Fig. 3.71a) shows an anticlock-
wise angle, whereas the cube on the right (Fig. 3.71b)
shows a clockwise angle. We will proceed up the circle if
the angle is anticlockwise (from
, which added to
the mean normal stress gives the equation for the normal
stress component of a given traction (Equation 27;
Fig. 3.69e). Following the same line of thought, it is easy
to see that the distance
1
3
)/2] cos 2
1
to the normal of the
surface), as in case (a) which corresponds to a left-handed
shear stress, and down the circle, into the negative area, if
the angle is clockwise corresponding in this case to a right-
handed shear stress. From the points intercepting the cir-
cle at the end of the radius we can read the values at the
coordinate axis.
A numerical example to illustrate the representation of
stresses and how to use Mohr circles is illustrated in
in the diagram has the value
[(
, which is the equation for the shear
stress component (Equation 29 in Fig. 3.69e).
To represent and calculate the shear and normal compo-
nents of any traction acting over a surface inclined at angle
1
3
)/2] sin 2
starting at
the zero value located at the point occupied by
, we have to draw a radius forming an angle 2
1
. In
Fig. 3.71, two cubes projected in a plane (2D view) show
(a)
The Mohr circle
(b)
t
s
1
−
s
3
2
t
= sin 2
u
t
+
t
s
1
−
s
3
2
t
t
2
u
s
n
s
n
s
1
s
3
180°
2
u
= 0°
2
u
_
s
n
+
s
n
s
n
s
3
s
1
s
1
+
s
3
2
,
t
=0
s
1
+
s
3
2
Mean
normal
stress
_
s
1
−
s
3
2
t
cos 2
u
Differential stress
(
s
1
-
s
3
)
s
1
+
s
3
2
s
1
−
s
3
2
s
n
=
+ cos 2
u
Fig. 3.70
(a) The Mohr circle is a tool to visualize and calculate stress components. It is represented as a circle positioned in a reference
coordinate axis (
n
,
) and centered at a point whose values are the average normal stress (
1
3
)/2 and
0. (b) Stress at a point and rela-
tionship to the fundmental stress equations.
(a)
(b)
s
1
s
1
t
u
u
A
s
n
t
t
s
3
s
3
s
n
s
3
s
3
s
n
2
2
u
s
3
s
1
B
σ
1
s
1
Left-handed or positive
shear stress
Right-handed or negative
shear stress
Fig. 3.71
Representation of clockwise and anticlockwise
angles in the Mohr circle for positive and negative shear stresses.
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