Initial insertion of the probe into the ground can be considered as expanding a zero cavity to a diameter the same as that of the probe. Filling of this cavity by stones and inserting the probe further expands the cavity by pushing the stone backfill radially outwards. When the probe fully passes any depth, the cavity at that level would undergo slight contraction. These are schematically shown in Figs.6.4a, b, and c, respectively.
Lifting the probe may cause slight contraction of the cavity, however, it will be prevented by freshly filled gravel. Repeated lifting, filling, and insertion of the probe cause repeated cavity expansions. Consider the soil to be an elastic-perfectly plastic material with an undrained shear strength of Su under an initial horizontal stress of σh0. A, C, and E in Fig.6.4 are material points in a horizontal plane. When the probe is inserted, a cavity of radius Re (= r0, ref. Fig.6.3) is created, and the soil is pushed away radially. Soil between A and E has deformed plastically, while the soil beyond E has undergone only small,
elastic deformations. When the probe passes below the point A, soil unloads elastically until a condition for reverse plasticity is reached at point A, and further unloads until the plastic region just reaches point C. Figs.6.4d and e illustrate the changes in radial and shear stresses within the soil during loading and unloading.
Changes in stress and strain fields during cavity expansion were studied by many researchers, and well-known solutions are available for cavity expansions in clay (Butterfield and Banerjee, 1970; Gibson & Anderson, 1961; Houlsby & Withers, 1988).
Derivations presented in this chapter are based on those reported by Houlsby & Withers (1988). Definitions of radii used in the analysis are schematically presented in Fig.6.4.
A B
B C D
σr
τ E D C
E
σr
τ
σh0
σh0 su
su
su
(d)
(e) (a)
(b)
(c)
A E C
A E C
A E C
Re
Rc
rce
rcc
ree
rec
re0
Cavity Pressure ψc Cavity Pressure ψe
Horizontal Stress σh0
A
Fig.6.4 Definition of Radii Used in the Analysis, and Stress States Around the Probe (Modified after Houlsby & Withers, 1988)
(a) Initial condition; (b) After vibratory probe installation; (c) After the probe passed/removed from the depth of concern, i.e. during cavity contraction; (d) Stress states around the probe corresponding to stage-b; and, (e) Stress states corresponding to stage-c
Small strain elastic analysis:
For a cylindrical cavity,
dr d
r
ε = − ζ
(6.6a)
r εθ = −ζ
(6.6b)
where, ξ is displacement, and εr and εθ are radial and angular strains, respectively.
Assume cylindrical cavity expansion takes place under plane strain conditions. Hence the axial strain εz is zero, and the relationship between stresses is,
z v( r θ)
σ σ σ
∆ = − ∆ + ∆ (6.7)
where, σr, σθ, and σz are radial, angular, and vertical (axial) stresses, respectively, and ν is Poisson’s ratio. The elastic stress-strain relationship is,
1 1 (1 )(1 2 )
r E v v
v v
v v
r
θ θ
σ ε
σ ε
∆ −
⎡ ⎤ ⎡ ⎤⎡
⎢∆ ⎥= + − ⎢⎣ − ⎥⎦⎢
⎣ ⎦ ⎣
⎤⎥
⎦ (6.8)
where, E is Young’s modulus. The radial equilibrium equation is,
( )
= 0 + −
r dr
dσr σr σθ
(6.9)
Substituting Eq.6.8 into Eq.6.9, and eliminating strain terms using definitions 6.6a and b,
2 0
2 + − =
r dr d dr
d ζ ζ ζ
(6.10)
The general solution for the above equation is,
r Ar + B ζ =
(6.11)
For the case of expansion in an infinite material, the displacement at infinity is zero resulting in A = 0. Therefore, the solutions for strains and stresses are:
r2
− B
θ =
ε (6.12a)
εθ
εr = − (6.12b)
θ
θ ε
σ = 2G
∆ (6.12c)
εθ
σr = −2G
∆ (6.12d)
where, G is the shear modulus, and given by, G = E/2(1+ν).
Plastic expansion analysis:
Initial Yield:
The initial stress state in the ground is σr = σθ = σh0 (Fig.6.4). Assuming the axial strain σz is intermediate between σr and σθ at yield, and assuming the Tresca yield condition, initial yield will occur when,
(σr −σθ)= 2Su (6.13)
Therefore, at initial yield (point E in Fig.6.4b),
0 ;
r h Su θ h0 Su
σ =σ + σ =σ − (6.14)
Equilibrium in plastic zone:
Since Eq.6.13 is valid throughout the plastic region for an elastic-perfectly plastic material, the radial equilibrium equation (Eq.6.9) can be rewritten as,
2 u 0
r S
d
dr r
σ + =
(6.15)
Solution for the above equation is:
2 ln( )
r Su r A
σ + = (6.16)
where, A is a constant of integration. Stresses at two different radii can be related as,
2 1 1
2
r r 2 lnu
S r
σ =σ + ⎡ ⎤⎢ ⎥r
⎣ ⎦ (6.17)
Pressure at the cavity wall, i.e. cavity expansion pressure is,
1
1 2 ln
e r u
e
S r
ψ =σ + ⎡ ⎤⎢ ⎥R
⎣ ⎦ (6.18)
Strains and compatibility:
During cavity expansion, incremental radial stress at elastic-plastic boundary point E (Fig.6.4b) is equal to Su. Therefore, by substituting into Eq.6.12d, the hoop strain at this point εθe is given by,
1
2 2
u e
r
S
G I
εθ = − = −
(6.19)
where, Ir (= G/Su) is the rigidity index.
The compatibility condition of no change in volume between points A and E requires that,
0
2 2
e ee
r =r −Re2
(6.20)
By rearranging,
2 2
1 0
e e
ee ee
R r
r
⎡ ⎤ ⎡ ⎤
⎢ ⎥ = −⎢ ⎥
⎣ ⎦ ⎣r ⎦ (6.21)
Using the ‘Hencky’ definition of hoop strain gives εθe = ln[re0/ree] so that,
2 [
1 exp 2
e e
ee
R
r εθ ]
⎡ ⎤
⎢ ⎥ = −
⎣ ⎦ (6.22)
The above equation can be simplified, assuming small strain conditions, to,
2 1
2
e
e
ee r
R
r εθ
⎡ ⎤
= − =
⎢ ⎥
⎣ ⎦ I (6.23)
Noting the radial distance to the point E is ree, and combining Eq.6.12a and Eq.6.23,
2 2
2
1 ; ,
2
e e
e
ee ee 2
R B R
Thus B
r r
εθ = − ⎡⎢ ⎤⎥ = − =
⎣ ⎦ (6.24)
Since the stresses at point E is known, Eq.6.23 can be used with Eq.6.18 to calculate the cavity expansion pressure so that,
( [ ] )
0 1 ln
e h Su r
ψ =σ + + I
)
(6.25)
This result for cavity expansion from zero initial cavity radius applies at all values of expansion radius, so that the expansion phase is expected to take place at constant cavity pressure (Gibson & Anderson, 1961; Houlsby & Withers, 1988).
Excess pore pressure ue due to changes in total stresses can be obtained by,
(
( ) (1 ) ( ) ( ) 3
e r
u r = +v ∆σ r + ∆σθ r (6.26a)
Under undrained conditions ν = 0.5. Combining Eqs.6.17 and 6.25 for the plastic zone, and Eqs.6.12c and d for the elastic zone,
[ ]
( ) 0.5*2 ln 2ln
0
e
e u r
ee
u r S I R for r r
r
for r r
⎛ ⎡ ⎤⎞
= ⎜⎝ + ⎢ ⎥⎣ ⎦⎟⎠
>
≤ ee
(6.26b)
Substituting Eq.6.23 into Eq.6.26b,
( ) 2 ln 0
e u ee
ee
u r S r for r r
r
for r r
= ⎡ ⎤⎢ ⎥⎣ ⎦ ≤
>
ee (6.27)
Cavity contraction:
An interesting addition by Houlsby & Withers (1988) to the cavity expansion problem is that the analysis of changes in states of stresses and strains during cavity contraction following the maximum expansion. During the initial phase of cavity contraction, the whole of the soil, including the plastically loaded section, will unload elastically until the condition for reverse plasticity, i.e., σr - σθ = -2Su, is reached at the cavity wall. Then this plasticity region spreads radially outwards up to point C (Fig.6.4c) when further cavity contraction is not possible. This condition may be reached when either the cavity pressure becomes equal to static pore pressure or there is no space to contract. During stone column installation process, the latter situation may be encountered due to follower tubes while the probe is being inserted into the ground.
Equilibrium conditions within the plastic region would yield relationships similar to Eqs.6.15-17, except for the sign of Su, so that,
1
2 1
2
r r 2 lnu
S r
σ =σ − ⎡ ⎤⎢ ⎥r
⎣ ⎦ (6.28)
The radial stress at point C at the stage of maximum expansion (Fig.6.4b) can be calculated using Eq.6.18. This stress will then reduce by 2Su, when the elastic-plastic boundary just reaches the point C (Fig.6.4c). The radial stress at point C at this stage is,
2 ln ce 2
rc e u u
e
S r S
σ =ψ − ⎡ ⎤⎢ ⎥R −
⎣ ⎦ (6.29)
Hence the current cavity pressure using Eq.6.28 is,
2 ln ce 2 2 ln cc
c e u u u
e c
S r S S r
R R
ψ =ψ − ⎡ ⎤⎢ ⎥− − ⎡ ⎤⎢ ⎥
⎣ ⎦ ⎣ ⎦ (6.30)
The compatibility condition of no change in volume of material between A and C yields,
2 2 2
ce e cc c
r −R =r −R2
(6.31)
The change of hoop strain during unloading of point C from plastic loading to plastic unloading gives,
1 ln ce
r
r I rcc
= ⎢ ⎥⎡ ⎤
⎣ ⎦ (6.32)
The definition of logarithmic cavity strain gives, ln ce
e
cc
r ε ε− = ⎢ ⎥⎡ ⎤r
⎣ ⎦ (6.33)
Using Eqs.6.31-33,
exp 2ln 1
2ln ln
exp 2 1
e c c
cc
r
R R R
r
I
⎡ ⎧⎪ ⎡ ⎤⎫⎪ − ⎤
⎢ ⎨ ⎢ ⎥⎬ ⎥
⎡ ⎤= − ⎢ ⎪⎩ ⎣ ⎦⎪⎭ ⎥
⎢ ⎥ ⎢ ⎧ ⎫ ⎥
⎣ ⎦ ⎢ ⎨ ⎬− ⎥
⎢ ⎩ ⎭ ⎥
⎣ ⎦ (6.34)
Thus,
exp 2 1
exp 2ln 1
c cc
r e c
r R
I R R
= ⎧ ⎫
⎨ ⎬−
⎩ ⎭
⎧ ⎡ ⎤⎫
⎪ ⎪ −
⎨ ⎢ ⎥⎬
⎪ ⎣ ⎦⎪
⎩ ⎭ (6.35)
Again excess pore pressure can be calculated using Eq.6.26a for any distance r.
Under undrained conditions, reduced form of this equation will be,
( 0)
( ) 0.5 ( ) ( ) 2
e r
u r = σ r +σθ r − σh (6.36)
However, since unloading takes place elastically in the region r>rcc, excess pore pressure in this region will remain the same as that just before the contraction phase.
Shear-Induced Excess Pore Pressure
Although Eqs.6.27 and 6.36 pave a way to calculate pore water pressure changes in the ground due to cavity expansion and contraction, in order to account for elasto-plastic behavior of soils, one must also add the shear-induced pore pressure component to the above elastic-perfectly plastic solutions. Thevanayagam et al. [2002c] studied undrained behavior of loose sand and sandy silt mixes and proposed a simple relationship between shear induced pore pressures, relative density, and initial effective confining pressure (σ0’). The shear induced pore pressure normalized by the initial mean effective confining stress was termed as collapse potential (CP). Fig.6.5 shows this relationship. The notation
‘100-OS15’ means test results for the specimen prepared by mixing Ottawa sand (F55, US Silica Co., IL) with 15% by dry weight of silt (sil co sil#40, US Silica Co., IL), and tested under 100 kPa initial effective confining pressure. The shear induced pore pressure component is given by,
0
* '
ush =CP σ
(6.37)
0.0 0.2 0.4 0.6 0.8 1.0
0 20 40 60 80 100
(Drc)eq
CP
100-OS00 100-OS15 100-OS25 400-OS07 400-OS15 100-Fit 400-Fit
?
Fig.6.5 CP vs. Equivalent Relative Density
It should be noted that the above figure only shows the shear induced pore pressure at quasi steady state. Medium dense to dense specimens tend to dilate, at times generating very large negative shear induced pore pressures, following the quasi steady state. At this particular situation, as cavity expansion is also associated with cyclic loadings by the probe, it is very difficult to quantify the dilation. Furthermore, at shallow depths, due to low static pore pressure conditions, the absolute pressure may not go negative.
In this study, shear induced pore pressure is included only for the plastically deformed region. Combining Eqs.6.27 and 6.37, total excess pore pressure during cavity expansion is given by:
( ) 2 ln 0
e u ee sh
ee
u r S r u for r r
r
for r r
= ⎡ ⎤⎢ ⎥⎣ ⎦+
>
≤ ee
(6.38a)
And for cavity contraction,
( )
( )
0 0
( ) 0.5 ( ) ( ) 2 0.5 ( ) ( ) 2
e r h sh
r h
u r r r u for r r
r r for r r
θ θ
σ σ σ
σ σ σ
= + − +
+ − >
cc cc
≤ (6.38b)
The solutions given in the above cavity expansion problem is very sensitive to undrained shear strength Su and rigidity index Ir. For soft, normally consolidated clays, Su
values ranges from 85 – 115 kPa, while Ir is about 135, under effective confining pressures of about 160 – 240 kPa, according to field measurements by Houlsby and Withers (1988). For loose granular soils under initial confining pressure of around 100 kPa, while the Su values at ‘quasi steady state’ are in the vicinity of around 30 – 40 kPa (Yoshimine 1999) they may be several hundreds at ‘steady state’ according to the observations made by Thevanayagam et al. (2003d) on laboratory test specimens.
Therefore, at large strain conditions, the rigidity index values remain around 3.5 – 4. It is still unclear which Su and Ir values are suitable for the simulation of vibro stone column installation due to the associated ground vibrations. However, it is interesting to note that in case Ir = 4, the plasticity region due to cavity expansion would extend up to just 2 times the radius of the cavity, which qualitatively reduces the influence of pore pressures generated by cavity expansion on the post – improvement densification.