Environmental Engineering Reference
In-Depth Information
Figure 11.9 Extended Mohr-Coulomb failure envelope for unsaturated soils.
matric suction u a
u w f . At zero net normal
stress, the intercept can be referred to as the “total
cohesion intercept.”
assumed to be constants. The cohesion intercept c and the
slope angles φ and φ b are the strength parameters used to
relate shear strength to the stress state variables. The shear
strength parameters represent many factors which have influ-
enced the results of the shear strength test. Some of these
factors are density, void ratio, degree of saturation, mineral
composition, stress history, and strain rate. In other words,
these factors have been combined and are expressed math-
ematically in terms of the shear strength parameters.
The mechanical behavior of an unsaturated soil is affected
differently by changes in net normal stress than by changes
in matric suction (Jennings and Burland, 1962). The increase
in shear strength due to an increase in net normal stress
is characterized by the friction angle φ . The increase in
shear strength caused by an increase in matric suction is
characterized by the angle φ b when assuming linear fail-
ure conditions. The value of φ b appears to be consistently
equal to or less than φ , as indicated in Table 11.1 The soils
represented in Table 11.1 are from a variety of geographic
locations.
The failure envelope intersects the shear stress versus
matric suction plane along a line of intercepts, as illus-
trated in Fig. 11.10. The line of intercepts represents the
increase in strength as matric suction increases. The shear
strength increase with respect to an increase in matric suc-
tion is defined by the angle φ b . A linear form for the line
of intercepts can be written as follows:
The extended Mohr-Coulomb failure envelope can be pre-
sented as a horizontal projection onto the shear strength
τ versus σ
u a plane. The horizontal projection can be
shown for various matric suction values u a
u w f .The
horizontal projection of the failure envelope onto the τ ver-
sus σ
u a plane results in the series of contours shown in
Fig. 11.11. The lines have cohesion intercepts that depend
upon the magnitude of the corresponding matric suctions.
The cohesion intercept reverts to the effective cohesion c
when matric suction goes to zero. All lines of equal suction
have the same slope angle φ as long as the failure plane
is planar. The equation for each of the contour lines can be
written as
+ σ f
u a f
tan φ
τ ff =
c
(11.13)
where:
c
=
total cohesion intercept.
Substituting Eq. 11.12 into Eq. 11.13 yields the equation
for the extended Mohr-Coulomb failure envelope. Equation
11.11 is the same as Eq. 11.13 and Fig. 11.11 is a two-
dimensional representation of the extended Mohr-Coulomb
failure envelope. The failure envelope projections illustrate
the increase in shear strength as matric suction is increased at
a specific net normal stress. Equation 11.13 is a convenient
form of the shear strength equation to use when performing
simple analytical studies involving unsaturated soils.
c + u a
u w f
tan φ b
c
=
(11.12)
where:
c
=
intercept of the extended Mohr-Coulomb failure
envelope with the shear stress axis at a specific
 
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