Geotechnical Engineering

DHARM

SHEARING STRENGTH OF SOILS 261

8.4.1 Mohr’s Strength Theory
We have seen that the shearing stress may be expressed as τ = σ tan β on any plane, where β
is the angle of obliquity. If the obliquity angle is the maximum or has limiting value φ, the
shearing stress is also at its limiting value and it is called the shearing strength, s. For a
cohesionless soil the shearing strength may be expressed as:

s = σ tan φ ...(Eq. 8.21)
If the angle of internal friction φ is assumed to be a constant, the shearing strength may
be represented by a pair of straight lines at inclinations of + φ and – φ with the σ-axis and
passing through the origin of the Mohr’s circle diagram. A line of this type is called a Mohr
envelope. The Mohr envelopes for a cohesionless soil, as shown in Fig. 8.5, are the straight
lines OA and OA′.

O

+f

• f

D

F

+t

• t

EB s

II

I

III

A

A¢

C

Fig. 8.5 Mohr’s strength theory—Mohr envelopes for cohesionless soil

If the stress conditions at a point are represented by Mohr’s circle I, the shear stress on

any plane through the point is less than the shearing strength, as indicated by the line BCD;

BC represents the shear stress on a plane on which the normal stress is given by OD.BD,

representing the shearing strength for this normal stress, is greater than BC.

The stress conditions represented by the Mohr’s Circle II, which is tangential to the

Mohr’s envelope at F, are such that the shearing stress, EF, on the plane of maximum obliq-

uity is equal to the shearing strength. Failure is incipient on this plane and will occur unless

the normal stress on the critical plane increases.

It may be noted that it would be impossible to apply the stress conditions represented by

Mohr’s circle III (dashed) to this soil sample, since failure would have occurred even by the

time the shear stress on the critical plane equals the shearing strength available on that plane,

thus eliminating the possibility of the shear stress exceeding the shearing strength.

The Mohr’s strength theory, or theory of failure or rupture, may thus be stated as follows:

The stress condition given by any Mohr’s circle falling within the Mohr’s envelope represents

a condition of stability, while the condition given by any Mohr’s circle tangent to the Mohr’s

envelope indicates incipient failure on the plane relating to the point of tangency. The Mohr’s

envelope may be treated to be a property of the material and independent of the imposed