Geotechnical Engineering

DHARM

260 GEOTECHNICAL ENGINEERING

σθ =

()()σσ σσx+ y x y

+

−

22

. cos 2θ + τxy sin 2θ ...(Eq. 8.14)

τθ =

()σσx− y

2

. sin 2θ – τxy. cos 2θ ...(Eq. 8.15)
Squaring Eqs. 8.14 and 8.15 and adding,

σ

σσ

τ

σσ

θθ− τ

L +

N

M

O

Q

P +=

F −

HG

I

KJ

+

()xy xy

22 xy

2

2

2

(^2) ...(Eq. 8.16)
This represents a circle with centre
()
,
Lσσxy+
N
M
O
Q
2 0 P , and radius
()σσ
xy τxy
F −
HG
I
KJ

• 2
  2
  (^2).
  Once the Mohr’s circle is constructed, the principal stresses σ 1 and σ 3 , and the orienta-
  tion of the principal planes may be obtained from the diagram.
  The shearing stress is to be plotted upward or downward according as it is positive or
  negative. It is common to take a shear stress which tends to rotate the element counter-clock-
  wise, positive.
  It may be noted that the same Mohr’s circle and hence the same principal stresses are
  obtained, irrespective of how the shear stresses are plotted. (The centre of the Mohr’s circle, C,
  is the mid-point of DE, with the co-ordinates
  Fσσxy+
  HG
  I
  (^2) KJ
  and 0; the radius of the circle is CG),
  the co-ordinates of G being σy and τxy.
  The following relationships are also easily obtained:
  σ 1 =
  Fσσxy+
  HG
  I
  2 KJ +
  1
  2
  ()σσxy−+^224 τxy ...(Eq. 8.17)
  σ 3 =
  Fσσxy+
  HG
  I
  (^2) KJ –
  1
  2
  ()σσxy−+^224 τxy ...(Eq. 8.18)
  tan 2θ1, 3 = 2τxy/(σx – σy) ...(Eq. 8.19)
  τmax =
  1
  2
  ()σσxy−+^224 τxy ...(Eq. 8.20)
  Invariably, the vertical stress will be the major principal stress and the horizontal one
  the minor principal stress in geotechincal engineering situations.

8.4 Strength Theories for Soils

A number of theories have been propounded for explaining the shearing strength of soils. Of

all such theories, the Mohr’s strength theory and the Mohr-Coulomb theory, a generalisation

and modification of the Coulomb’s equation, meet the requirements for application to a soil in

an admirable manner.