Geotechnical Engineering

DHARM

258 GEOTECHNICAL ENGINEERING

2. The maximum or principal shearing stress is equal to the radius of the Mohr’s circle,
   and it occurs on planes inclined at 45° to the principal planes.
   τmax = (σ 1 – σ 3 )/2 ...(Eq. 8.5)


3. The normal stresses on planes of maximum shear are equal to each other and is equal
   to half the sum of the principal stresses.
   σc = (σ 1 + σ 3 )/2 ...(Eq. 8.6)


4. Shearing stresses on planes at right angles to each other are numerically equal and
   are of an opposite sign. These are called conjugate shearing stresses.


5. The sum of the normal stresses on mutually perpendicular planes is a constant (MG′

• MG = 2MF = σ 1 + σ 3 ). If we designate the normal stress on a plane perpendicular to the plane
  on which it is σθ as σθ′ :

σθ + σθ′ = σ 1 + σ 3 ...(Eq. 8.7)

Of the two stresses σθ and σθ′, the one which makes the smaller angle with σ 1 is the

greater of the two.

6. The resultant stress, σr, on any plane is στθθ^22 + and has an obliquity, β, which is
   equal to tan–1 (τθ/σθ).
   σr = στθθ^22 + ...(Eq. 8.8)
   β = tan–1 (τθ/σθ) ...(Eq. 8.9)


7. Stresses on conjugate planes, that is, planes which are equally inclined in different
   directions with respect to a principal plane are equal. (This is indicated by the co-ordinates of
   C and C 1 in Fig. 8.3).


8. When the principal stresses are equal to each other, the radius of the Mohr’s circle
   becomes zero, which means that shear stresses vanish on all planes. Such a point is called an
   isotropic point.


9. The maximum angle of obliquity, βm, occurs on a plane inclined at

θcr F=°+

HG

I

KJ

45

2

βm with respect to the major principal plane.

θcr = 45° + βm

2

...(Eq. 8.10)
This may be obtained by drawing a line which passes through the origin and is tangen-
tial to the Mohr’s circle. The co-ordinates of the point of tangency are the stresses on the plane
of maximum obliquity; the shear stress on this plane is obviously less than the principal or
maximum shear stress.
On the plane of principal shear the obliquity is slightly smaller than βm. It is the plane
of maximum obliquity which is most liable to failure and not the plane of maximum shear,
since the criterion of slip is limiting obliquity. When βm approaches and equals the angle of
internal friction, φ, of the soil, failure will become incipient.
Mohr’s circle affords an easy means of obtaining all important relationships. The follow-
ing are a few such relationships :

sin βm = σσ

σσ

13

13

−

+

F

HG

I

KJ

...(Eq. 8.11)