UNDERGROUND ENGINEERING - Introduction to Soft Soil Geotechnique: Content, Context and Application - page 254

At the interface r = r p , beyond which the elastic zone starts, the displacement is
relatively small and also here

r = c u and w p = c u r p / 2 G . This gives with (15.7)

q = c u (1 + 2ln( r p /r 1 )) (15.8)

( r p 2 - ( r p - w p ) 2 ) , which
gives approximately r 1 2 = 2w p r p , omitting terms r 0 2 and w p 2 . With w p = r p c u / 2 G on
finds r 1 2 = ( c u /G ) r p 2 or

( r 1 2 - r 0 2 ) = 2 Using constant volume (Vesi): dV = 2

( r p /r 1 ) 2 = G/c u = I r

(15.9)

The ratio I r is called the rigidity index. Expressions (15.8) and (15.9) provide the
corresponding cavity maximum pressure, beyond which the system collapses, i.e.

q crit = c u (1+ ln I r ) (15.11)

If, for example, G = 20 MPa and c u = 50 kPa, then for q = 50(1+ ln(20000/50))
= 350 kPa, a cohesive soil behaves fully plastic in a region with a radius r p = r 1 I r ½
= 20 r 1 . This represents more or less a case, where a concrete foundation pile is
driven through this soil. Here, r 1 can be associated with the pile radius. Its weight
causes a pressure at least ten times more than 350 kPa and the pile will sink easily
down by its own weight, see Fig 12.6, which shows a blow count zero from soil
depth of 3 to 12 m.

z

at r > r p

at r = r p

w p

elastic zone

w 0

at r = r 0

plastic zone

r

r 1

r p

r 0

r

q

c

(a) cylindrical cavity (b) Mohr circle
Figure 15.6 A contracting cylindrical cavity in cohesive soil

Contracting cavity
There may be a zone where soil behaviour is plastic. In this plastic zone, r 0 < r <
r p , the larger tangential stresses will squeeze the soil into the cavity by a radial

plastic flow until the cavity is closed (Fig 15.6). Also here, for the plastic zone r 0 <
r < r p distinction is made between two cases: (1) a large cavity or relatively small
deformations, i.e. |w 0 | << r 0 and (2) a small cavity or relatively large deformations,
i.e. | w 0 | ~ r 0 .
In the case of a large contracting cavity or relatively small deformations, i.e. | w 0 |
<< r 0 , elaboration of (15.1) with yield condition (15.3) and boundary condition

r

= q at r = r 0 gives