State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 100

Substituting dx and dy ( i.e., Eqs. 3.36 and 3.37) into
Eqs. 3.39 and 3.38, respectively, and multiplying Eq. 3.38
by sin α and Eq. 3.39 by cos α gives

The net normal stress and shear stress at a point can also be
determined using a graphical method. If Eqs. 3.44 and 3.48
are squared and added, the result is the equation of a circle:
σ α −

σ 1 +

u a 2

σ 1 − 2

u a )ds sin 2 α

u a −

σ 3 σ 3

− (σ α − + τ α ds sin α cos α

τ α = − +

2 2

u a )ds sin 2 α + (σ 3 − = 0 (3.40)

(3.49)
The circle is known as the Mohr diagram and represents
the stress state at a point. In saturated soils, the Mohr dia-
gram is often plotted in terms of the principal effective
normal stress on the abscissa and the shear stress on the ordi-
nate. For unsaturated soils, an extended form of the Mohr
diagram can be used as shown in Fig. 3.19.
The extended Mohr diagram uses a third orthogonal axis
to represent matric suction. The circle described in Eq. 3.49
is drawn on a plane with the net normal stress σ

u a )ds cos 2 α − (σ α − − τ α ds sin α cos α

u a )ds cos 2 α + (σ 1 − = 0 (3.41)

Summing Eqs. 3.40 and 3.41 gives

u a )ds( sin 2 α cos 2 α) u a )ds sin 2 α

− (σ α − + + (σ 3 −

u a )ds cos 2 α + (σ 1 − = 0 (3.42)

u a as the
abscissa and the shear stress τ as the ordinate. The center

−

Using trigonometric relations to solve for σ a − u a gives

of the circle has an abscissa defined by σ 1 + σ 3 /2

u a ) 1

− u a and

+ cos 2 α

2

a radius defined by (σ 1 −

σ 3 ) /2.
The matric suction must also be included as part of the
description of the stress state. The matric suction determines
the position of the Mohr diagram along the third axis. Matric
suction goes to zero as the soil becomes saturated, and the
Mohr diagram reverts to a single (σ

σ α − u a = (σ 1 −

u a ) 1

−

cos 2 α
2

+ (σ 3 −

(3.43)

Rearranging Eq. 3.43 gives − u w ) -versus- τ plane.

σ 1 +

u a

σ 1 − cos 2 α

σ 3 σ 3

3.5.1 Construction of Mohr Circles
for Unsaturated Soils
The construction of the Mohr diagram on the (σ

σ α − u a = − + (3.44)

2 2

u a ) -
versus- τ plane is shown in Fig. 3.20. A compressive net
normal stress is plotted as a positive net normal stress in
accordance with the sign convention for the Mohr diagram.
A shear stress that produces a counterclockwise moment
about a point within the element is plotted as a positive
shear stress. This shear stress sign convention is different
from the convention normally used in continuum mechanics
(Desai and Christian, 1977). This convention should only be
used for plotting the Mohr diagram. The major and minor
net principal stresses, σ 1 −

−

The shear stress τ α is obtained by substituting dx and dy
(i.e., Eqs. 3.36 and 3.37) into Eqs. 3.39 and 3.38, respec-
tively, and multiplying Eq. 3.38 by cos α and Eq. 3.39 by
sin α :
− σ α − u a ds sin α cos α + σ 3 − u a ds τ α ds cos 2 α

+

× sin α cos α = 0 (3.45)

− σ α − u a ds sin α cos α + σ 1 − u a ds τ α ds sin 2 α

−

× sin α cos α = 0 (3.46)

u a are plotted on
the abscissa while the center of the Mohr circle is located
at (σ 1 +

u a and σ 3 −

Subtracting Eq. 3.46 from Eq. 3.45 gives
τ a ds sin 2 α

σ 3 ) /2.
The Mohr circle represents the net normal stress and shear
stress on any plane through a point in an unsaturated soil.
The net normal stress and shear stress on any plane can be
determined if the pole point or the origin of planes is known.
Any plane drawn through the pole point will intersect the
Mohr diagram and give the net normal stress and the shear
stress acting on that plane. On the other hand, if the net
normal stress and shear stress on a plane are known and
plotted as a stress point on the Mohr circle, the direction of
the plane under consideration is given by the orientation of
a line joining the stress point and the pole point.
The pole point for the condition shown in Fig. 3.20 is deter-
mined from the known net normal stress and shear stress on
a particular plane. Consider, for example, the case where the
major principal stress acts on a horizontal plane. The stress

σ 3 ) /2 − u a . The radius of the circle is (σ 1 −

cos 2 α + σ 3 − u a ds sin α cos α +

− σ 1 − u a ds sin α cos α

= 0 (3.47)

Using trigonometric relations, it is possible to solve

for τ α :

σ 1 − sin 2 α

σ 3

τ α = (3.48)

2

Equations 3.44 and 3.48 give the net normal stress and
the shear stress on an inclined plane through a point. The
term σ 1 − σ 3 is called the deviator stress and is an indication

of the shear stress in the soil. The largest shear stress for

a given stress state σ 1 − σ 3 / 2 occurs on a plane with an

inclination angle α such that sin 2 α is equal to unity.