Improvement of fine-grained and cohesive soils 159
With Equations 4.2 and 4.5, the above equation can be rewritten as
m
n a
n
a
=
⋅
+ − ⋅
c
c
( (
) )
1
1
(4.67)
Introducing Equation 4.12, we get
m
n a
a
=
⋅ =
− +
c
c
β
β
β
1
(4.68)
Equation 4.68 denotes the maximum load ratio attributable to the stone
columns. Conservatively, m can be reduced to m′ by simply neglecting
the area ratio a c . The load ratio ′
m 1 may then be calculated from the
settlement improvement factor β 1 according to Priebe (Equation 4.52),
which includes the correction for column compressibility by the following equation
′ =
−
m 1
1
1
1
β
β
(4.69)
On the other hand, it might be unsafe to calculate the average shear strength
based only on the load ratio of the stone columns, because doing so would
apply the stress concentration also to the unit weight of the in-situ soil,
which may not be conservative, especially with deep slip circles. In Kirsch
and Sondermann (2003), it is therefore recommended to calculate the shear
strength of the improved soil as the mean of the strength values on the basis
of either the area ratio or the load ratio.
Priebe (2003) recommends modeling each individual stone column in a
two- dimensional slip circle analysis with vertical slices, and factorizing the
load on top of the columns by the stress concentration factor and simultaneously reducing the load on the soil in between. This can, for example, be
achieved by adjusting the unit weight of the embankment. With large systems, this approach can be quite laborious. Therefore Priebe alternatively
proposes calculating average shear strength values on the basis of the load
ratio but reducing them with depth by applying a reduction factor being the
quotient of the load on top of the improved ground Q top and the total load
Q total at the respective depth. The maximum reduction of the load ratio is
reached when the columns are unloaded. In this case, the strength of the
improved ground is the average of the strength of the soil and the columns,
weighted by the area ratio according to Priebe adopted for the compressibility of the column material:
′′ =
+ ′ −
⋅
m A A m A A
Q
Q
1
1
c
c
top
total
/
/
(
)
(4.70)
With Equations 4.2 and 4.5, the above equation can be rewritten as
m
n a
n
a
=
⋅
+ − ⋅
c
c
( (
) )
1
1
(4.67)
Introducing Equation 4.12, we get
m
n a
a
=
⋅ =
− +
c
c
β
β
β
1
(4.68)
Equation 4.68 denotes the maximum load ratio attributable to the stone
columns. Conservatively, m can be reduced to m′ by simply neglecting
the area ratio a c . The load ratio ′
m 1 may then be calculated from the
settlement improvement factor β 1 according to Priebe (Equation 4.52),
which includes the correction for column compressibility by the following equation
′ =
−
m 1
1
1
1
β
β
(4.69)
On the other hand, it might be unsafe to calculate the average shear strength
based only on the load ratio of the stone columns, because doing so would
apply the stress concentration also to the unit weight of the in-situ soil,
which may not be conservative, especially with deep slip circles. In Kirsch
and Sondermann (2003), it is therefore recommended to calculate the shear
strength of the improved soil as the mean of the strength values on the basis
of either the area ratio or the load ratio.
Priebe (2003) recommends modeling each individual stone column in a
two- dimensional slip circle analysis with vertical slices, and factorizing the
load on top of the columns by the stress concentration factor and simultaneously reducing the load on the soil in between. This can, for example, be
achieved by adjusting the unit weight of the embankment. With large systems, this approach can be quite laborious. Therefore Priebe alternatively
proposes calculating average shear strength values on the basis of the load
ratio but reducing them with depth by applying a reduction factor being the
quotient of the load on top of the improved ground Q top and the total load
Q total at the respective depth. The maximum reduction of the load ratio is
reached when the columns are unloaded. In this case, the strength of the
improved ground is the average of the strength of the soil and the columns,
weighted by the area ratio according to Priebe adopted for the compressibility of the column material:
′′ =
+ ′ −
⋅
m A A m A A
Q
Q
1
1
c
c
top
total
/
/
(
)
(4.70)