Geotechnical Engineering

DHARM

262 GEOTECHNICAL ENGINEERING

stresses. Also, the Mohr’s circle of stress depends only upon the imposed stresses and has
nothing to do with the nature and properties of the material.
To emphasise that the stresses in Eq. 8.21 are those on the plane on which failure is
incipient, we add the subscript f to σ:
s = σf tan φ ...(Eq. 8.22)
It is possible to express the strength in terms of normal stress on any plane, with the aid
of the Mohr’s circle of stress. Some common relationships are :

σf = σ 3 (1 + sin φ) = σ 1 (1 – sin φ)

=

σσ φ

φ

13

2

2

F −

HG

I

KJ

.

cos

sin

...(Eq. 8.23)

s = σf tan φ = σ 3 tan φ (1 + sin φ)

= σ 1 tan φ (1 – sin φ) =

σσ 13

2

F −

HG

I
KJ. cos φ ...(Eq. 8.24)
The primary assumptions in the Mohr’s strength theory are that the intermediate prin-
cipal stress has no influence on the strength and that the strength is dependent only upon the
normal stress on the plane of maximum obliquity. However, the shearing strength, in fact,
does depend to a small extent upon the intermediate principal stress, density speed of applica-
tion of shear, and so on. But the Mohr theory explains satisfactorily the strength concept in
soils and hence is in vogue.

It may also be noted that the Mohr envelope will not be a straight line but is actually
slightly curved since the angle of internal friction is known to decrease slightly with increase
in stress.

8.4.2 Mohr-Coulomb Theory

The Mohr-Coulomb theory of shearing strength of a soil, first propounded by Coulomb (1976)

and later generalised by Mohr, is the most commonly used concept. The functional relation-

ship between the normal stress on any plane and the shearing strength available on that plane

was assumed to be linear by Coulomb; thus the following is usually known as Coulomb’s law:

s = c + σ tan φ ...(Eq. 8.25)

where c and φ are empirical parameters, known as the ‘apparent cohesion’ and ‘angle of shear-
ing resistance’ (or angle of internal friction), respectively. These are better visualised as ‘pa-
rameters’ and not as absolute properties of a soil since they are known to vary with water
content, conditions of testing such as speed of shear and drainage conditions, and a number of
other factors besides the type of soil.

Coulomb’s law is merely a mathematical equation of the failure envelope shown in Fig. 8.6
(a); Mohr’s generalisation of the failure envelope as a curve which becomes flatter with in-
creasing normal stress is shown in Fig. 8.6 (b).

The envelopes are called ‘strength envelopes’ or ‘failure envelopes’. The meaning of an
envelope has already been given in the previous section; if the normal and shear stress compo-
nents on a plane plot on to the failure envelope, failure is supposed to be incipient and if the
stresses plot below the envelope, the condition represents stability. And, it is impossible that
these plot above the envelope, since failure should have occurred previously.