Shear Strength of Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 533

Figure 11.15 Relationships between c, d , φ ,and ψ .

The above equation can be rearranged by substituting

distance between points T and O (i.e., TO ):

Eq. 11.24 for d and substituting u a − u w f for r f in order

to obtain the relationship between, ψ b , φ b , and φ :

d
tan ψ =

c
tan φ

(11.22)

tan ψ b tan φ b cos φ = (11.26)

Substituting Eq. 11.21 into Eq. 11.22 and rearranging
Eq. 11.22 yields

The above relationships can be used to define the
stress point envelope corresponding to an extended Mohr-
Coulomb failure envelope or vice versa. The extended
Mohr-Coulomb failure envelope can be established by test-
ing a soil under saturated and unsaturated conditions. The
Mohr-Coulomb failure envelope for the saturated condition
gives the angle of internal friction φ and the effective
cohesion c .
The cohesion intercept c can be obtained from a single
Mohr circle if a planar failure envelope is assumed at an
angle φ and the matric suction is known. Figure 11.17 illus-
trates the construction of a Mohr circle at failure with its
corresponding p f and q f values. A failure envelope with a
slope angle φ is drawn tangent to the Mohr circle at point
A . The envelope intersects the shear strength axis at point B
and the σ

c cos φ

d = (11.23)

When matric suction at failure is equal to zero (i.e., the
saturated condition), Eq. 11.23 becomes

d = c cos φ

(11.24)

Figure 11.16 shows the intersection lines of the extended
Mohr-Coulomb failure envelope and the stress point
envelope on the shear strength versus matric suction plane.
The intersection lines associated with the extended Mohr-
Coulomb failure envelope and the stress point envelope
are defined by Eqs. 11.12 and 11.18, respectively. The
ratio between the d and c values is always constant and
equal to cos φ (i.e., Eq. 11.23) at various matric suctions.
As a result, the difference between the d and c values
is not constant for different matric suctions. In other
words, the intersection lines are not parallel, or put another
way, φ b

u a axis at point T . The cohesion intercept c is
computed from triangle TBO (Fig. 11.17):

−

p f tan φ

q f
sin φ −

c = (11.27)

ψ b . Substituting Eq. 11.12 and Eq. 11.18 into
Eq. 11.23 gives the following relationship:

=

Rearranging Eq. 11.27 gives

c cos φ + u a − u w f tan φ b cos φ .

(11.25)

d + r f tan ψ b =

q f
cos φ −

p f tan φ c = (11.28)