Shear Strength of Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 526

where: an extension of the shear strength equation for a saturated

soil. Two stress state variables are used to describe the shear
strength of an unsaturated soil while only one stress state

S = degree of saturation of the soil at failure.

variable [i.e., effective normal stress σ f − u w f ] is required

Wheeler and Sivakumar (1995) suggested a critical shear
strength equation where each of the shear strength properties
is a function of matric suction:

for a saturated soil.
The proposed shear strength equation for an unsaturated
soil exhibits a smooth transition to the shear strength
equation for a saturated soil. The pore-water pressure u w
approaches the pore-air pressure u a as the soil approaches
saturation and the matric suction u a −

q = M s (p − u a ) + μ s (11.10)

u w goes to zero. As
the matric suction component vanishes, Eq. 11.11 reverts
to the equation for a saturated soil.

where:

M s = material characteristic that is a function of suction,

and

11.2.7 Extended Mohr-Coulomb Failure Envelope
The failure envelope for a saturated soil is obtained by
plotting a series of Mohr circles corresponding to failure
conditions on a two-dimensional plot as shown in Fig. 11.2.
The line tangent to the Mohr circles is called the failure
envelope and is defined by Eq. 11.1. In the case of an
unsaturated soil, Mohr circles corresponding to failure con-
ditions can be plotted in a three-dimensional manner, as
illustrated in Fig. 11.9. The three-dimensional plot has the
shear stress τ as the ordinate and the two stress state vari-
ables σ

μ s = material characteristic that is a function of suction.

A summary of critical state shear strength models can be
found in Leong et al. (2003a).

11.2.6 Approximate Linear Shear Strength Equation
A linear form of the shear strength for an unsaturated soil
can be formulated in terms of independent stress state vari-
ables (Fredlund et al., 1978). Any two of three possible
stress state variables can be used to write an appropriate
shear strength equation. The stress state variables σ

u w as abscissas. The frontal plane
represents saturated soil conditions where matric suction is
zero. The σ

− u a and u a −

− u a

and u a − u w have been found to be the most practical combi-

nation of stress state variables for solving practical engineer-
ing problems. The linear form of the shear strength equation
can be written as follows when using σ

u w axis on the
frontal plane since the pore-air pressure becomes equal to
the pore-water pressure at saturation.
The Mohr circles for an unsaturated soil are plotted with
respect to the net normal stress axis σ

− u a axis reverts to the σ −

− u a and u a − u w

as the stress state variables:

u a in the same
manner as the Mohr circles are plotted for saturated soils
with respect to effective stress axis σ

−

c + σ f − u a f tan φ + u a − u w f tan φ b τ ff = (11.11)

u w . The location
of the Mohr circle plot in the third dimension is a function
of the matric suction (Fig. 11.9). The surface tangent to
the Mohr circles at failure is referred to as the extended
Mohr-Coulomb failure envelope for unsaturated soils. The
extended Mohr-Coulomb failure envelope defines the shear
strength of an unsaturated soil. The intersection line between
the extended Mohr-Coulomb failure envelope and the frontal
plane is the failure envelope for saturated conditions.
The inclination of the failure plane is defined by joining
the tangent point on the Mohr circle to the pole point. The
tangent point on the Mohr circle at failure represents the
stress state on the failure plane at failure.
The extended Mohr-Coulomb failure envelope may be a
planar surface or it may be somewhat curved. The theory is
first presented with the assumption that the failure envelope
is planar and can be described by Eq. 11.11. A curved failure
envelope can also be described by Eq. 11.11 for limited
changes of the stress state variables. Techniques for handling
the nonlinearity of the failure envelope are discussed later.
Figure 11.9 shows a planar failure envelope that inter-
sects the shear stress axis, giving a cohesion intercept c .
The envelope has slope angles of φ and φ b with respect
to σ

−

where:

c =

intercept of the “extended” Mohr-Coulomb
failure envelope on the shear stress axis
where the net normal stress and the matric
suction at failure are equal to zero, also
referred to as “effective cohesion,

σ f −

u a f

= net normal stress state on the failure plane

at failure,

u af = pore-air pressure on the failure plane at fail-

ure,

φ = angle of internal friction associated with

the net normal

stress state variable,

σ f −

u a f ,

u a −

u w f

= matric suction on the failure plane at fail-

ure, and

φ b

= angle indicating the rate of increase in shear

strength with respect to a change in matric

suction, u a − u w f .

A comparison of Eqs. 11.1 and 11.11 reveals that the shear
strength equation for an unsaturated soil takes the form of

− u a and u a − u w axes, respectively. Both angles are