Shear Strength of Soils - Elements of Soil Mechanics - page 125

3

(2) q = ( σ − σ )

1

Similar expressions apply for effective stress:

1
3

′ = ′ + ′

p ( σ 2 σ ) (3)

1 3

′ = ′ − ′

q ( σ σ ) (4)

1 3

The specific volume, v was defined in Chapter 1 and is the total volume of soil that contains a unit volume
of solids:

v = + ( 1 e ) (5)

The advantage of the p and q parameters is their association with the strains that they cause. Changes
in p ′ are associated with volumetric strains and changes in q with shear strains.
For the general three-dimensional state, Equations (1) to (4) have the form:

1
3

p = ( σ + + σ σ )

1 2 3

1
2

q = [( σ − σ )

2

+ ( σ − σ )

2

+ ( σ − σ ) ]

2

1 2 2 3 3 1

4.14.1 Stress paths in three-dimensional stress space

We have considered two-dimensional stress paths and we must now examine the form of these paths if
they were plotted in three-dimensional space defined by p ′ , q and v.

Undrained tests

If we consider the plane q-p ′ then we can plot the effective stress paths for undrained shear in a manner
similar to the previous two-dimensional stress paths. Remember that q = σ 1 − σ 3 and that

σ + 2

3

σ 1 3

p ′ =

The resulting diagram is shown in Fig. 4.35a . The points A 1 , A 2 and A 3 lie on the isotropic normal con-
solidation line and their respective stress paths reach the failure boundary at points B 1 , B 2 and B 3 . As the
tests are undrained, the values of void ratio at points B 1 , B 2 , B 3 are the same as they were when the soil
was at the stress states A 1 , A 2 and A 3 respectively. Knowing the e values we can determine the values of
specific volume and prepare the corresponding plot on the v-p ′ plane (Fig. 4.35b ).
It is seen that the failure points B 1 , B 2 and B 3 lie on a straight line in the q-p ′ plane and on a curve,
similar to the normal consolidation curve, in the v-p ′ plane.

Drained tests

The effective stress paths for drained shear are shown in Fig. 4.36 . For the q-p ′ plane the plot consists
of straight lines which are inclined to the horizontal at tan − 1 (3). The reason why is illustrated in Fig. 4.36.
The points C 1 , C 2 and C 3 represent the failure points after drained shear, so the void ratio values at
these points are less than those at the corresponding A points.
The stress paths in the v-p ′ plane are illustrated in Fig. 4.36b . As with the undrained case, the failure
points C 1 , C 2 and C 3 lie on a curved line similar to the normal consolidation line.