Shear Strength of Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 530

b

τ ff = c ′ + ( σ f - u a ) f tan φ ′ + (u a - u w ) f tan φ

Ordinate intercept

b

φ

b

φ

b

φ

b

φ

c ′ +

( σ f - u a ) f2 tan φ ′

c ′

0 Matric suction, (u a - u w )

Figure 11.12 Horizontal projection of failure envelope onto τ versus u a − u w plane viewed

parallel to σ − u a axis as contour lines of failure envelope on τ versus u a − u w plane.

computed shear strength for the unsaturated soil remains
consistent with the shear strength previously calculated. As
suction goes to zero, the third terms in Eqs. 11.14 and 11.11
disappear, and the pore-water pressure approaches the pore-
air pressure. As a result, both equations revert to the shear
strength equation for a saturated soil. The second term in
both equations should have the same friction angle parame-
ter φ

11.2.9 Mohr-Coulomb and Stress Point Envelopes
The extended Mohr-Coulomb envelope has been defined as
a surface tangent to the Mohr circles at failure. Each Mohr
circle is constructed using the net minor and net major prin-
cipal stresses at failure (i.e., σ 3 f − u af ), as

shown in Fig. 11.14a. The difference between the net minor
and net major principal stresses at failure is called the max-
imum deviator stress.
The top point of a Mohr circle with coordinates
(p f ,q f ,r f ) can be used to represent the stress conditions at
failure. A stress point surface (i.e., stress point envelope) can
be drawn through the stress points at failure (Fig. 11.14b).
The stress point envelope constitutes another representation
of the stress state of the soil at failure conditions.
However, the stress point envelope and the extended Mohr-
Coulomb failure envelope are different surfaces. Neverthe-
less, the stress point envelope is mathematically related to
stress state at failure.
The stress point envelope can be mathematically defined
using the following equation:

u af and σ 1 f −

u w f tan φ ].
Equations 11.14 and 11.11 give the same shear strength
for a soil at a specific stress state;

[i.e., σ f − u a f tan φ and σ f −

therefore, the two

equations can be equated:

tan φ + u a − u w f

tan φ b − u af

tan φ + u a − u w f

tan φ =− u wf (11.15)

Rearranging Eq. 11.15 gives the relationship between the
respective angles of friction:

tan φ = tan φ b tan φ − (11.16)

Equation 11.16 shows that the friction angle φ will gen-
erally be negative since the magnitude of φ b is less than
or equal to φ . Figure 11.13 shows the extended Mohr-
Coulomb failure envelope when failure conditions are plot-
ted with respect to σ

d + tan ψ + tan ψ b q f = p f r f (11.17)

where:

q f = half of the deviator stress at failure [i.e.,

− u w and u a − u w (i.e., Eq. 11.14) and

σ 1 −

σ 3 f / 2],

with respect to σ − u a and u a − u w (i.e., Eq. 11.11).