Foundation Settlement and Soil Compression - Elements of Soil Mechanics - page 347

Fig. 11.16 Points of maximum shear stress.

Stress paths can be of many forms and we have already used some: the stress-strain relationships
plotted in τ − σ and p-q space in Chapter 4 to show triaxial test results and the plots in e-log p space
used in Fig. 11.9 to illustrate compression curves, etc. In his analysis of foundation settlement problems,
Lambe ( 1964, 1967) used stress paths of maximum shear.
If a Mohr circle diagram of stress is examined (Fig. 11.16) the point of maximum shear has the
co-ordinates s and t where:

σ + σ σ − σ

1 3 1 3

s = and t =

2 2

σ 1 and σ 3 being the total principal stresses.
In terms of effective stresses, σ 1 and ′ σ 3 , the point of maximum shear has the co-ordinates s ′ and t ′ where: ′

′ + ′

σ σ 1 3

s ′ =

2

′ ′

σ 3 the point of maximum shear stress can be obtained
for each stress circle; the line joining these points, in the order that they occurred, is termed the stress
path or stress vector of maximum shear. Any other point instead of maximum shear can be used to deter-
mine a stress path, e.g. the point of maximum obliquity, but Lambe maintains that the stress paths of
maximum shear are not only simple to use but also more applicable to consolidation work.
Typical effective stress paths obtained from a series of consolidated undrained triaxial tests on samples
of normally consolidated clay together with the effective stress circles at failure are shown in Fig. 11.17.

If a soil is subjected to a range of values of σ 1 and

′ ′

11.8.1 Ratios of σ σ

3

/

1

Consider Fig. 11.17. The line tangential to the stress circles is the strength envelope, inclined at φ ′ to the
normal stress axis. If each Mohr circle is considered it is seen that the ratio σ σ

3

′ / ′ is a constant, to which

1

the symbol Kf f is applied.

The K f line
If the points of maximum shear for each effective stress circle

′

p f and q f are joined together the stress path
of maximum shear stress at failure is obtained. This line is called the Kf f line and is inclined at angle α ′ to
the normal stress axis; obviously tan α ′ = sin φ ′ .