DHARMSHEARING STRENGTH OF SOILS 257Ms 3
ss 13 +
2ss 13 –
2
sq
s 1
C¢ C 1GG
–2q(^2) q^2 qm
F
E 180° D
b
bm
Op
- q
q45°JH
CtqsFig. 8.3 Mohr’s circle for the stress conditions illustrated in Fig. 8.2
The Mohr’s circle diagram provides excellent means of visualisation of the orientation
of different planes. Let a line be drawn parallel to the major principal plane through D, the
coordinate of which is the major principal stress. The intersection of this line with the Mohr’s
circle, Qp is called the ‘Origin of planes’. If a line parallel to the minor principal plane is drawn
through E, the co-ordinate of which is the minor principal stress, it will also be observed to
pass through Op; the angle between these two lines is a right angle from the properties of the
circle. Likewise it can be shown that any line through Op, parallel to any arbitrarily chosen
plane, intersects the Mohr’s circle at a point the co-ordinates of which represent the normal
and shear stresses on that plane. Thus the stresses on the plane represented by AB in Fig. 8.2
(a), may be obtained by drawing Op C parallel to AB, that is, at an angle θ with respect to OpD,
the major principal plane, and measuring off the co-ordinates of C, namely σθ and τθ.
Since angle COpD = θ, angle CFD = 2θ, from the properties of the circle. From the geom-
etry of the figure, the co-ordiantes of the point C, are established as follows:
σθ = MG = MF + FG
=()()σσ σσ 13 13
22+
+−. cos 2θ
τθ = CG =()σσ 13
2−. sin 2θ
These are the same as in Eqs. 8.3 and 8.4, which prove our statement.
In the special case where the major and minor principal planes are vertical and horizon-
tal respectively, or vice-versa, the origin of planes will be D or E, as the case may be. In other
words, it will lie on the σ-axis.
A few important basic facts and relationships may be directly obtained from the Mohr’s
circle:
- The only planes free from shear are the given sides of the element which are the
principal planes. The stresses on these are the greatest and smallest normal stresses.