Geotechnical Engineering

DHARM

SHEARING STRENGTH OF SOILS 259

σ 1 /σ 3 =

1

1

+

−

F

HG

I

KJ

sin

sin

β

β

m

m

...(Eq. 8.12)

σ for the plane of maximum obliquity,
σcr = σ 3 (1 + sin βm) ...(Eq. 8.13)
In case the normal and shearing stresses on two mutually perpendicular planes are
known, the principal planes and principal stresses may be determined with the aid of the
Mohr’s circle diagram, as shown in Fig. 8.4. The shearing stresses on two mutually perpen-
dicular planes are equal in magnitude by the principle of complementary shear.

txy

sx

txy

sx

sy

Minor

principal

plane

ssxy>

(assumed)

Major

principal

plane

q

q 1

sy

(a) General two-dimensional stress system

t sq H

tq

2 qJ D B

2 q 1

G

tmax

EA

txy

C

O (Origin

of planes)

p

F

M

s 3

sy

ssxy+

sx

s 1

ssxy–

2

(b) Mohr’s circle for general two-dimensional stress system

s

Fig. 8.4 Determination of principal planes and principal stresses from Mohr’s circle
Figure 8.4 (a) shows an element subjected to a general two-dimensional stress system,
normal stresses σx and σy on mutually perpendicular planes and shear stresses τxy on these
planes, as indicated. Fig. 8.4 (b) shows the corresponding Mohr’s circle, the construction of
which is obvious.
From a consideration of the equilibrium of a portion of the element, the normal and
shearing stress components, σθ and τθ, respectively, on a plane inclined at an angle θ, meas-
ured counter-clockwise with respect to the plane on which σx acts, may be obtained as follows: