Geotechnical Engineering

DHARM

256 GEOTECHNICAL ENGINEERING

shown that satisfactory solutions may be obtained for many problems in the field of geotechnical
engineering by two-dimensional analysis, the intermediate principal stress being commonly
ignored.

Let us consider an element of soil whose sides are chosen as the principal planes, the
major and the minor, as shown in Fig. 8.2 (a):

l

A

B

O

tq

s 3

s 1

q

l

q

sq.l

tq.l

sqq 1 lsin cos

sq 1 lcos

sq 1 lcos^2

sq 3 lsin^2

sqq 3 lsin cos

sq 3 lsin

(a) Stress system (b) Force system
Fig. 8.2 Stresses on a plane inclined to the principal planes
Let O be any point in the stressed medium and OA and OB be the major and minor
principal planes, with the corresponding principal stresses σ 1 and σ 3. The plane of the figure is
the intermediate principal plane. Let it be required to determine the stress conditions on a
plane normal to the figure, and inclined at an angle θ to the major principal plane, considered
positive when measured counter-clockwise.

If the stress conditions are uniform, the size of the element is immaterial. If the stresses
are varying, the element must be infinitestinal in size, so that the variation of stress along a
side need to be considered.

Let us consider the element to be of unit thickness perpendicular to the plane of the
figure, AB being l. The forces on the sides of the element are shown dotted and their compo-
nents parallel and perpendicular to AB are shown by full lines. Considering the equilibrium of
the element and resolving all forces in the directions parallel and perpendicular to AB, the
following equations may be obtained:

σθ = σ 1 cos^2 θ + σ 3 sin^2 θ = σ 3 + (σ 1 – σ 3 ) cos^2 θ

= ()()σσ σσ^1313

22

+

+

−. cos 2 θ ...(Eq. (8.3)

τθ = ()σσ^13

2

−. sin 2 θ ...(Eq. 8.4)

Thus it may be noted that the normal and shearing stresses on any plane which is
normal to the intermediate principal plane may be expressed in terms of σ 1 , σ 3 , and θ.
Otto Mohr (1882) represented these results graphical in a circle diagram, which is called
Mohr’s circle. Normal stresses are represented as abscissae and shear stresses as ordinates. If
the coordinates σθ and τθ respresented by Eqs. 8.3 and 8.4 are plotted for all possible values of
θ, the locus is a circle as shown in Fig. 8.3. This circle has its centre on the axis and cuts it at
values σ 3 and σ 1. This circle is known as the Mohr’s circle.