118 Ground improvement by deep vibratory methods
cause any shear stresses or horizontal deformations to develop at its outside
boundaries. Consequently, the uniform loading will remain within the unit
cell. Since the stone column is stiffer than the surrounding soil, vertical
stresses are concentrated in the stone column material with an accompanying stress reduction in the less stiff soil surrounding the column. This stress
distribution between column and soil within the unit cell is generally called
stress concentration n and is represented by the ratio of the vertical stresses
σ c in the column and the vertical stress σ s in the soil:
n =
σ
σ
c
s
(4.3)
Equilibrium with the acting vertical stress σ is given by the relationship
σ σ
σ
σ
σ
=
+
−





 = ⋅ + − ⋅
c
c
s
c
c
c
c
s
1
1
A
A
A
A
a
a
(
)
(4.4)
which leads to expressions for σ c and σ s as follows:
σ σ
σ
c
c
c
1
1
= ⋅ + − ⋅
= ⋅
n
n
a
n
( (
) )
(4.5)
with
n
n
n
a
c
c
1
1
= + − ⋅
( (
) )
(4.6)
Q = A • σ
A c = πd
2 /4
A = πd e
2 /4
σ c σ s
d e
d e = 1.05 × b
d e = 1.13 × b
d
d
b
Column grid
Unit cell
b
Figure 4.8 Column grid patterns and unit cell concept.