Shear Strength of Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 524

524 11 SHEAR STRENGTH OF UNSATURATED SOILS
net minor normal stresses at failure [i.e., σ 1 − u a f and

σ 3 −

u a f , respectively].

The principal stress ratio is defined as σ 1 − σ 3 f / σ 3

u w f (Bishop et al. 1960) and can also be used as a
failure criterion when pore-water pressures are measured
during the test. A plot of the principal stress ratio versus
the axial strain for an undrained triaxial test on compacted
shale is shown along with the corresponding stress versus
strain curve in Fig. 11.6a. The maximum deviator stress,

−

σ 1 −

σ 3 max , and the maximum principal stress ratio,

σ 1 −

u w f , may not occur at the same axial
strain in the undrained test, as illustrated in Fig. 11.6a.
The maximum principal stress ratio is a function of the
pore-water pressure measured during an undrained test
(Fig. 11.6b). On the other hand, the maximum deviator stress
is not a direct function of the pore pressures. For the results
presented in Fig. 11.6a, the authors selected the maximum
principal stress ratio as the failure criterion since it occurred
prior to the maximum deviator stress. It is not presently
clear whether pore-air pressure or pore-water pressure
should be used in calculating the principal stress ratio. It is
also possible that other definitions of principal stress ratios
could be used to define the state of failure under undrained
conditions.
The above-mentioned failure criteria describe some maxi-
mum stress combinations that the soil can resist. Researchers
have also suggested that the amount of strain in the soil
specimen should play a role in assessing the state of fail-
ure. Consideration of amount of strain at failure is some-
what related to whether peak or critical state stresses should
be used when selecting failure conditions for engineering
design purposes.
A plot of stress versus strain does not always exhibit a dis-
tinct maximum value even at quite large strains, as shown in
Fig. 11.7. In this case, an arbitrary strain (e.g., 12%) may be
selected to represent the failure criterion. The shape of triax-
ial test specimens can become quite distorted at high strain
values, and this can jeopardize the quality of the test data.
A limiting displacement definition for failure is sometimes
used when interpreting direct shear test results.
The above-mentioned failure criteria have been proposed
for the analysis of shear strength data for unsaturated soils,
often with limited corroborating evidence. In general, the
different failure criteria produce quite similar shear strength
parameters. Probably the largest difference in shear strength
parameters occurs as a result of the difference between peak
conditions and critical state conditions.

σ 3 max / σ 3 −

(a)

(b)

(c)

Figure 11.6 Undrained triaxial tests on compacted shale:
(a) stress versus strain curves; (b) pore pressure versus strain
curves; (c) soil volume change versus strain curves (after Bishop
et al., 1960).

Figure 11.7 Defining failure criterion in terms of limiting strain
value.

11.2.4 Shear Strength Equations for Unsaturated Soils
The shear strength constitutive relationship provides a math-
ematical equation relating the normal and shear components
of the stress tensors. Any one of several shear strength fail-
ure criteria could be used to extend the saturated soil shear
strength representation to unsaturated soil conditions. The

Mohr-Coulomb failure criterion was extended to embrace
unsaturated soils by Fredlund et al., (1978). The extended
shear strength equation for an unsaturated soil can be written
as follows:

τ = c + u a ) tan φ + (σ n − (u a − u w )f 1 (11.3)