ce5101 lecture 8 - radial consolidation_ pvd and surcharge (oct 2013) [compatibility mode]
DESCRIPTION
CE5101 Lecture 8 - Radial Consolidation_ PVD and Surcharge (OCT 2013) [Compatibility Mode]TRANSCRIPT
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CE 5101 Lecture 8 – Radial Consolidation and PVD
October 2013
Prof Harry Tan
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Outline
• Radial Consolidation – Barron Theory
• Carillo Theory – Combined vertical and radial Flow
• PVD Design
• Preload Surcharge Design
• FEM Model of PVD and Surcharge
• Some Cases
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�Radial Consolidation - Barron’s Theory (1948)
w
v
w
h cc
where
z
u
r
u
rr
u
t
u
Governing
γγ v
v
v
h
2
2
v2
2
h
m
k,
m
k
c1
c
:coords radialin Equation D-3
==
∂
∂+
∂
∂+
∂
∂=
∂
∂
symmetry) todue s(imperviou 0r
)u(r 3.
0for t 0)u(r 2.
0at t uu 1.
:Conditions
1c
:Only Flow adial
e
w
0
2
2
h
=∂
∂
>=
==
∂
∂+
∂
∂=
∂
∂
Boundary
r
u
rr
u
t
u
R
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( )( ) ( )[ ]
Functions Bessel are ;:
1
41
:Drain) (IdealCondition for
102
2
1
2
0
222
42
1
00
22
UandUd
tcTand
d
dnwhere
αUαnUn)(nα
eαU
u
u;
u
uU
StrainFreeSolution
e
hh
w
e
Tnα
rrr
h
==
−−=−=
−
∑−
only
U
strainFree
Note
r
hT andn of
function a is
fastest settle
drain closest to
soil as settlement
uniform-non
means
:
−
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U Like r
The average degree of radial consolidation coincides with
the local degree of consolidation Ur at ½(D-d) point of soil
cylinder, best place for piezometer to monitor progress of
consolidation
Ur
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2
2
2
2
)(
8
0
)(
8
4
13)ln(
1)(:
;1
:Drain) (IdealCondition qualfor
n
nn
n
nnfwhere
eu
ueU
StrainESolution
nf
T
rnf
T
r
hh
−−
−=
=−=
−
−−
Comparison
show very small
differences
between free-
strain and equal-
strain, esp for
n>10
For n=5,
significant
difference in
first 50% of
consolidation
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What is size of Influence Diameter de or D
sD
D
Square
13.1
4s
:spacing
22
=
=π ( )
sD
D
T
05.1
432/*s/2*s/2*1/2*6
:spacingraingular
2
=
=π
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( )2
2
2
2
2r
4
13)ln(
1:
;8
exp1U
:Drain Vertical )1981(
n
nn
n
nwhere
D
tcT
T
IdealHansbo
hh
h
−−
−=
=−
−=
α
α
( )w
c
c
cs
hh
s
h
q
kzLzm
k
k
m
nwhere
D
tcT
T
E
−+−+=
=−
−=
24
3)ln(
'ln:
;8
exp1U
:ResistanceDrain andSmear offfect
2rz
πα
α
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Effects of Smear and Drain Resistance
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Carillo Theory – Combined vertical and radial Flow
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Combined Flow - Carillo’s Theorem (1942)
( )
( )
problem flow combined thesolutionof a is uu
u osolution t a is ,u
1u osolution t a is ,u
21
2
2
22
2
2
11
then
z
uc
ttzfand
r
u
rr
uc
ttrfIf
v
h
∂∂
=∂∂
=
∂
∂+
∂
∂=
∂
∂=
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )QED
zc
tand
rrrc
t
This
zc
rrrc
tt
zc
rrrc
t
toof
vh
vh
vh
2
2
2
21
2
1
2
1
2
2
2
11
2
1
2
2
2112
2
21
2
21
2
21
2
21
21
uuu1uu
: thatmeans
uu
u1uu
uuuu
uuuu1uuuu
u intouuu Substitute :Pr
∂
∂=
∂
∂
∂
∂+
∂
∂=
∂
∂
∂
∂+
∂
∂+
∂
∂=
∂
∂+
∂
∂
∂
∂+
∂
∂+
∂
∂=
∂
∂
∂∂
=
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( ) ( )( )
theorysHansbo'or sBarron'
theorysTerzaghi'
11U-1
:meanshat
u
u
u
u
u
u
: tolead discussion previous The
0
v
0
h
0
fromU
fromU
UU
T
h
v
vh −−=
=
Combined Flow - Carillo’s Theorem (1942)
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Practical Vertical Drain Design with Plaxis 2D-
FEM
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Outline
• Terzaghi 1D Vertical Flow Consolidation
• Barron 1D Radial Flow Consolidation
• Carillo Combined Flow Consolidation
• Equivalent Plane Strain Consolidation for 2D-FEM
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Terzaghi 1D Vertical Flow Consolidation
5.0..,2.0 ≤≤ vv UeiT
+−−
−≈−≈21.0
442
22
18
1v
vTT
v eeU
ππ
π
πv
v
TU 2≈
For
Then
For
Then
5.0..,2.0 >> vv UeiT
Tv is Time factor
cv is Coeficient of Consolidation
wv
vv
vv
m
kc
H
tcT
γ=
=2
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Barron 1D Radial Flow Consolidation
( )
−+−⋅−
=222
2
4
11
1
4
3ln
1 nnn
n
nµ
µhT
h eU
8
1
−
−=
4
3)ln( −= nµ
w
h
r
hs
q
kzLzs
k
k
s
n)2(
4
3)ln()ln( −+−+= πµ
Th is Time factor
ch is Coeficient of Consolidation
wv
hh
hh
m
kc
D
tcT
γ=
=2
Equal Vertical Strain Condition
For n=D/d > 10
To include smear and drain discharge
Where z = L for single drainage at top,
and z = L/2 for double drainage at top and bottom
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( )2
2
2
2
2r
4
13)ln(
1:
;8
exp1U
:Drain Vertical )1981(
n
nn
n
nwhere
D
tcT
T
IdealHansbo
hh
h
−−
−=
=−
−=
µ
µ
( )w
c
c
cs
hh
s
h
q
kzLzm
k
k
m
nwhere
D
tcT
T
E
−+−+=
=−
−=
24
3)ln(
'ln:
;8
exp1U
:ResistanceDrain andSmear offfect
2rz
πµ
µ
For single drainage at top,
z=L
For double drainage at top and bottom, z=L/2
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Carillo Combined Flow
)1)(1(1 hvvh UUU −−−=
µ
π
h
v
T
h
T
v
eU
eU
8
21.04
1
1
2
−
+−
≈−
≈−
++−
−≈ µπ h
v
TT
vh eU
821.0
4
2
1
For Tv > 0.2
Uv > 50%
For Tv ≤ 0.2
Uv ≤ 50%( ) µπ
hT
vvh eTU
8
/211−
−−≈
From linear superposition
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Equivalent Vertical Permeability for Plane Strain
FEM Model – CUR 191 or Tan 1981
++−
−≈ µπ h
v
TT
vh eU
821.0
4
2
1
+−
−≈21.0
4
'
'
2
1vT
v eU
π
vhv UU ='
Interested only in solution > 50% consolidation
For Axisymmetric Unit Cell
For Equivalent FEM Model
To obtain equivalent vertical consolidation rate
++−
+−
−=−= µππ h
vv
TTT
v eeU
821.0
421.0
4
'
2
'
2
11
hvv
hvv
kD
Hkk
TTT
2
2
2'
2'
32
32
µπ
µπ
+=
+=wv
vv
vv
m
kcand
H
tcT
γ==
2wv
hh
hh
m
kcand
D
tcT
γ==
2
In 2D-FEM only need to replace PVD soil cluster with enhanced vertical kv’ model
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Practical PVD DesignPractical Vertical Drain Design (by Prof Harry Tan SEP 2008)
Terzaghi 1D Vertical Consolidation
H=L single drainage and H=L/2 double drainage
INPUT
Case cv(m2/y) H(m) t(y) Tv Uv
1 2 5 0.25 0.02 0.16
2 2 5 0.25 0.02 0.16
Hansbo/Barron 1D Radial Consolidation
INPUT z=L single drainage and z=L/2 double drainage
Case ch(m2/y) S (m) D(m) t(y) Th d(m) ds(m) kh (m/y) ks (m/y) qw (m3/y) L(m) z(m) n s mu Uh
1 5 1.30 1.365 0.25 0.67 0.050 0.100 0.0050 0.0020 100 10 5 27.3 2 3.61 0.77
2 5 1.50 1.575 0.25 0.50 0.050 0.100 0.0050 0.0020 100 10 5 31.5 2 3.75 0.66
Carillo Combined Flow ConsolidationCase Uv Uh Uvh
1 0.16 0.77 0.81
2 0.16 0.66 0.71
Johnson Surcharge DesignCase Po (kPa) Pf (kPa) Usr=Uvh log[(Po+Pf)/Po] (Po+Pf+Ps/Po) Ps (kPa) Hs (m)
1 100 60 0.81 0.204 1.786 18.6 1.0
2 100 60 0.71 0.204 1.933 33.3 1.9
++
+
==+
0
0
0
0
log
log
P
PPP
P
PP
S
SU
sf
f
sf
f
sr
( )w
h
s
hs
hh
s
h
q
kzLzs
k
k
s
nwhere
D
tcT
T
−+−+=
=−
−=
24
3)ln(ln:
;8
exp1U
:ResistanceDrain andSmear ofEffect Eqn with Hansbo
2h
πµ
µ
)1()1(1 hvvh UUU −−−=
20
Use Excel spreadsheet to determine: Uv, Uh and Uvh for design inputs
If Uvh meets or exceeds requirements, design is adequate
Note: D=1.05s for triangular grid or 1.13s for square grid pattern
and z=L drain at top; or z=L/2 drain top and bottom of PVD
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Preload Surcharge Design – Johnson
ASCE 1970
Assumptions:
a. Primary and secondary compression are separate
b. Instant load applied at end of ½ load period
c. Time rate of settlement determine by Terzaghi theory
21
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Preload Surcharge Design – Johnson
ASCE 1970
'
vσ
22
Objective: To determine amount of surcharge needed to achieve desired
degree of consolidation?
Clay: Ho, Po and Cc
Design Permanent Fill Pf
Surcharge Ps
Pf
Ps
t
S
tsr
Sf
Sf+s
If surcharge is left in place for tsr
(time to removal), then clay will
have compressed by amount
equal to Sf expected under fill
weight alone, ie achieved
U=100% under Pf load alone
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Preload Design
(4) 0.1
log
log
)(
:is surcharge and fillunder ion consolidat of degree required Therefore,
(3) 0.1
log
log
)(
:ision consolidat of degree average tsr,At time
(2) log1
:surcharge and Fill
(1) log1
:only Fill
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
<
++
+
===
=
++
+
==
++
+=
+
+=
++
+
+
P
PPP
P
PP
S
SUU
P
PPPU
P
PP
SU
SU
P
PPPH
e
CS
P
PPH
e
CS
sf
f
sf
f
srsf
sf
sr
f
sfsr
f
f
sfcsf
fcf
23
For Normally Consolidated Clay (NC) of thickness Ho:
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Preload Design Example
large)(very surcharge of m 5.2 94/18 kPa 94160254
54.210100
160
404.0505.0
204.0
100
160log
100
160log
100
160log
log
log
505.0
thenyr, 1after tsr surcharge remove To
ion)consolidat 50%(about 505.02.0
22
0.2 5
1*5
c : theoryTerzaghi
0.408m 100
60100log*10
1.51
0.5 log
1 :only Fill
404.0
0
0
0
0
22
v
0
0
0
0
===−=
==
+
==
+
+
=
++
+
===
=
===
===
=
+
+=
+
+=
+
s
s
s
ssf
f
sf
f
sr
vv
v
fcf
P
P
P
P
P
PPP
P
PP
S
SU
TU
H
tT
P
PPH
e
CS
ππ
24
Clay: Ho, Po and Cc
Design Permanent Fill Pf
Surcharge PsClay 10m thick drained both top and bottom: eo=1.5, Po=100 kPa, Cc=0.5,
cv=5 m2/yr
Fill: Height = 3m with Pf = 60 kPa
Aim: To get 100% consolidation in 1 year, what is Ps needed?
So surcharge alone is not effective
and we need PVD to reduce
surcharge time as well as amount of
surcharge needed
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Preload Design Example
Practical Vertical Drain Design (by Prof Harry Tan SEP 2008)
Terzaghi 1D Vertical Consolidation
H=L single drainage and H=L/2 double drainage
INPUT
Case cv(m2/y) H(m) t(y) Tv Uv
1 2 5 0.25 0.02 0.16
2 2 5 0.25 0.02 0.16
Hansbo/Barron 1D Radial Consolidation
INPUT z=L single drainage and z=L/2 double drainage
Case ch(m2/y) S (m) D(m) t(y) Th d(m) ds(m) kh (m/y) ks (m/y) qw (m3/y) L(m) z(m) n s mu Uh
1 5 1.30 1.365 0.25 0.67 0.050 0.100 0.0050 0.0020 100 10 5 27.3 2 3.61 0.77
2 5 1.50 1.575 0.25 0.50 0.050 0.100 0.0050 0.0020 100 10 5 31.5 2 3.75 0.66
Carillo Combined Flow ConsolidationCase Uv Uh Uvh
1 0.16 0.77 0.81
2 0.16 0.66 0.71
Johnson Surcharge DesignCase Po (kPa) Pf (kPa) Usr=Uvh log[(Po+Pf)/Po] (Po+Pf+Ps/Po) Ps (kPa) Hs (m)
1 100 60 0.81 0.204 1.786 18.6 1.0
2 100 60 0.71 0.204 1.933 33.3 1.9
++
+
==+
0
0
0
0
log
log
P
PPP
P
PP
S
SU
sf
f
sf
f
sr
( )w
h
s
hs
hh
s
h
q
kzLzs
k
k
s
nwhere
D
tcT
T
−+−+=
=−
−=
24
3)ln(ln:
;8
exp1U
:ResistanceDrain andSmear ofEffect Eqn with Hansbo
2h
πµ
µ
)1()1(1 hvvh UUU −−−=
25
Clay: Ho, Po and Cc
Design Permanent Fill Pf
Surcharge PsClay 10m thick drained both top and bottom: eo=1.5, Po=100 kPa, Cc=0.5,
cv=2 m2/yr, ch= 5 m2/yr
PVD parameters: d=0.05m, ds=0.1m, kh=0.005 m/yr, ks=0.002 m/yr, qw=100
m3/yr
Fill: Height = 3m with Pf = 60 kPa
Aim: To get 100% improvement in 3 months, what is Ps needed?
Design requires PVD triangle spacing with 1.3m grid and 1m surcharge or 1.5m grid with 1.9m surcharge
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FEM Modeling of
Embankments on Soft Ground
with PVD
1. Model of single PVD – Axi-symmetric
2. Model of PVD in Plane Strain
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Interface element in PLAXIS used
Impose specified cross-sectional area and
vertical permeability of vertical drain to
simulate well resistance
Effect of smear considered by the
equivalent permeability of surrounding
soils
Method 1 – Using Interface Element for
Vertical Drain
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AXISYMMETRIC
Pore water flow
qw
Soil
rw re
PVD
H
Interface
element
z
r
kh
qw
Soil
rw re
H
z
r
Closed
consolidation
boundary
ti
H
z
r
Soil
qw
rw re
(a) (b) (c)
r
Open Boundary Interface element Drain element
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FEM Axi-Symmetric Model of Single PVD
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FEM Model – Barron Theory
30
Boundary conditions
E_oed=1000 kPa
Cv_soil = 0.01*1000/10 = 1 m2/day
Cv_drain=1*1000/10=100 m2/day
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FEM Model – Barron Theory
31
T=0.1day
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0
10
20
30
40
50
60
70
80
90
100
0.001 0.01 0.1 1 10 100Th
Uh (
%)
Interface Element
Open Consolidation Boundary
Barron's Theory
Radial Consolidation Theory
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ss ss
s m
m
s sss
2ti 2B
∇ Q
P
r
dw
de
QAP
x
∇
2ti
2B or S
CONVERSION FROM AXISYMMETRIC TO
PLANE STRAIN
(c) (d)(b)(a)
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� no drainage (reference)no drainage (reference)no drainage (reference)no drainage (reference)
� drainage with drain element drainage with drain element drainage with drain element drainage with drain element
(sets zero pore pressure conditions)(sets zero pore pressure conditions)(sets zero pore pressure conditions)(sets zero pore pressure conditions)
� drainage with boundary conditiondrainage with boundary conditiondrainage with boundary conditiondrainage with boundary condition
(check on performance of “drain element”)(check on performance of “drain element”)(check on performance of “drain element”)(check on performance of “drain element”)
FEM models investigated:FEM models investigated:FEM models investigated:FEM models investigated:
Axisymmetric modelAxisymmetric modelAxisymmetric modelAxisymmetric model
Plane strain modelPlane strain modelPlane strain modelPlane strain model
� equivalent vertical permeability after CUR 191equivalent vertical permeability after CUR 191equivalent vertical permeability after CUR 191equivalent vertical permeability after CUR 191
� equivalent horizontal permeability after CUR 191equivalent horizontal permeability after CUR 191equivalent horizontal permeability after CUR 191equivalent horizontal permeability after CUR 191
� equivalent horizontal permeability after Indraratna (2000)equivalent horizontal permeability after Indraratna (2000)equivalent horizontal permeability after Indraratna (2000)equivalent horizontal permeability after Indraratna (2000)
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AAA A A AA A
axisymmetric axisymmetric axisymmetric axisymmetric modelmodelmodelmodel
plane strain plane strain plane strain plane strain modelmodelmodelmodel
unit cell for vertical drains placed in pattern of 2x2 m, 5 unit cell for vertical drains placed in pattern of 2x2 m, 5 unit cell for vertical drains placed in pattern of 2x2 m, 5 unit cell for vertical drains placed in pattern of 2x2 m, 5
m highm highm highm high
drain diameter 25 cmdrain diameter 25 cmdrain diameter 25 cmdrain diameter 25 cm
applied loadapplied loadapplied loadapplied load
10 kN/m²10 kN/m²10 kN/m²10 kN/m²
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hvv kD
Hkk ⋅
⋅⋅+=
2
2
2
32´
µπ hh kk =´
( )
⋅−⋅+−⋅
−=
222
2
4
11
1
4
3ln
1 nnn
n
nµ
d
Dn =
CUR 191 equivalent vertical permeabilityCUR 191 equivalent vertical permeabilityCUR 191 equivalent vertical permeabilityCUR 191 equivalent vertical permeability
kv , kh “true“ permeability
kv´ , kh´ equivalent permeability
H drainage length
D equivalent distance of drains
d diameter of drains
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CUR 191 equivalent horizontal permeabilityCUR 191 equivalent horizontal permeabilityCUR 191 equivalent horizontal permeabilityCUR 191 equivalent horizontal permeability
hh kD
Bk ⋅
⋅⋅=
2
2
´µ
α
vv kk =´
( )
⋅−⋅+−⋅
−=
222
2
4
11
1
4
3ln
1 nnn
n
nµ
d
Dn =
U 0,5 0,75 0,9 0,95 0,99
α 2,26 2,75 2,94 3,01 3,09
kv , kh “true“ permeability
kv´ , kh´ equivalent permeability
H ½ the distance of drains in plane strain
D equivalent distance of drains
d diameter of drains
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( )[ ] 2
2
75,0ln
67,0
R
B
nk
k
h
hp ⋅−
=wr
Rn =
Indraratna equivalent Indraratna equivalent Indraratna equivalent Indraratna equivalent horizontal permeabilityhorizontal permeabilityhorizontal permeabilityhorizontal permeability
khp equivalent horizontal permeability for plane strain
kh “true“ horizontal permeability
B ½ distance of drains in plane strain
R equivalent distance of drains
rw diameter of drains
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Excess Pore Pressure after 60% consolidationExcess Pore Pressure after 60% consolidationExcess Pore Pressure after 60% consolidationExcess Pore Pressure after 60% consolidation
Influence of constitutive modelInfluence of constitutive modelInfluence of constitutive modelInfluence of constitutive model
HS HS HS HS ---- ModelModelModelModelLinear Elastic Linear Elastic Linear Elastic Linear Elastic ---- ModelModelModelModel
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time [sec]
1e+3 1e+4 1e+5 1e+6 1e+7 1e+8 1e+9
de
gre
e o
f co
nso
lida
tio
n U
[ -
]
0.0
0.2
0.4
0.6
0.8
1.0
AXI: no drainageAXI: drainage boundary conditionAXI: drainage drain-elementPS: equivalent vertical CUR 191PS: equivalent horizontal CUR 191PS: equivalent horizontal Indraratna
degree of consolidation for different degree of consolidation for different degree of consolidation for different degree of consolidation for different
models (linearmodels (linearmodels (linearmodels (linear----elastic)elastic)elastic)elastic)This image cannot currently be displayed.
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time [sec]
1e+3 1e+4 1e+5 1e+6 1e+7 1e+8 1e+9 1e+10
de
gre
e o
f co
nso
lida
tio
n U
[ -
]
0.0
0.2
0.4
0.6
0.8
1.0
AXI: no drainageAXI: drainage boundary conditionAXI: drainage drain-elementPS: equivalent vertical CUR 191PS: equivalent horizontal CUR 191PS: equivalent horizontal Indraratna
This image cannot currently be displayed.
degree of consolidation for different degree of consolidation for different degree of consolidation for different degree of consolidation for different
models (Hardening Soil model)models (Hardening Soil model)models (Hardening Soil model)models (Hardening Soil model)
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Austrian Case
A
A
B
C
D
E5.0
äußerer Schutzstreifen
WA
SS
ER
KA
NA
L
5.0
C
D
E
B
E1
E2
PW1
A1/1
A1/2A1/3A1/4A1/5A1/6
A2/1
A2/2A2/3A2/4A2/5A2/6A2/7
A2/8
A3/1
A3/2A3/3A3/4A3/5A3/6
A3/7A3/9
A4/1A4/2
A4/3A4/4A4/5A4/6
A4/9 A4/8 A4/7
A5/9A5/1
A6/1
A7/1
A8/1
R/1
A5/2A5/3A5/4A5/5A5/6
A5/8
A6/2A6/3A6/4A6/6A6/7
A7/2A7/3
A8/2
A1/9
A5/7
A6/5
A8/3
A1/8PW3 A1/7
PW4
A2/9
Z3/8
Z4/8
A3/8
RS2/1RS2/2
RS2/3RS2/4
A7/4RS2/5
RS2/6
RS2/7
RS2/8
RS2/9
RS1/3
X
X
Y
Y
Schüttabschnitt 1
Schüttabschnitt 2
Schüttabschnitt 3
LOGISTIK
HALLEUMSCHLAGHALLE
BÜRO
D
D
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soil profile:
1
1
1
peat - undrained
kx = ky = 0,005 m/day ; kx´ = 6,6e-4 m/day
silt, clay - undrained
kx = ky = 0,0001 m/day ; kx´ = 1,3e-5 m/day
silt / silt-clay - undrained
kx = ky = 0,0001 m/day ; kx´ = 1,3e-5 m/day
man made material - drained
γγγγ = 19,5 kN/m3
pre-load - drained
γγγγ = 18 kN/m3
3 m
2,5 m
4,5 m
2 m
14 m
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This image cannot currently be displayed.This image cannot currently be displayed.
FE-MODEL
section D-D
A2/4 A4/4 A6/4
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Results for section D-D
comparison measurement - Plaxis point A2/4
time [days]
0 20 40 60 80 100 120 140
se
ttle
me
nt
[cm
]
-120
-100
-80
-60
-40
-20
0
Plaxismeasurement
calculated final
settlement
139 cm
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Results for section D-D
comparison measurement - Plaxis - point A6/4
time [days]
0 20 40 60 80 100 120 140
se
ttle
me
nts
[cm
]
-50
-40
-30
-20
-10
0
Plaxismeasurements
calculated final
settlement
78 cm
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EXAMPLE EXAMPLE EXAMPLE EXAMPLE ---- EMBANKMENT CONSTRUCTIONEMBANKMENT CONSTRUCTIONEMBANKMENT CONSTRUCTIONEMBANKMENT CONSTRUCTION
influence of consolidation on stability
influence of construction speed is investigated
"fast" construction: 2 days of consolidation per placement of 1 m embankment
"slow" construction: 3 days of consolidation per placement of 1 m layer embankment
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influence of consolidation on stability
"fast": max. excess pore pressure: 100 kPa
"slow": max. excess pore pressure: 86 kPa
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influence of consolidation on stability
"slow": stable
"fast": failure
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influence of consolidation on stability
0 4 8 12 16
-50
-40
-30
-20
-10
0
Time [day]
excess pore pressure [kN/m2]
Chart 1
slow
fast
time [days]
excess pore pressure [kPa]
fast
slow
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influence of consolidation on stability
vertical displacements [m]
fast
slow
0 30 60 90 120
0
0.01
0.02
0.03
0.04
0.05
0.06
Time [day]
Displacement [m]
Chart 1
Point C
Point C
time [days]
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Practical Considerations
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Lateral spreading
Settlement with risk
for downdrag
The Problem — Bridge Foundations
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These photos of bridge
foundations illustrate a common
problem affecting maintenance
($$$!), as well as, on occasions,
one compromising safety
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Photos from in-situ excavation of a pile
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The problem of lateral spreading can be avoided by not installing the piles until the
consolidation is mostly completed, which also would eliminate the risk for excessive
downdrag.
However, the project can rarely wait for the consolidation to develop, and the solution
would be impractical, unless the consolidation can be accelerated by means of vertical
drains. Apart from saving time, accelerating the consolidation also reduces the magnitude
of the lateral spreading and increases soil strength.
In the past, sand drains were used. Since about 25 years, the sand drains have been
replaced with wick drains, which are pre-manufactured bandshaped drains.
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2H
Drainage Layer
Clay Layer
(consolidating)
Drainage Layer
0
1u
u
S
SU t
f
t
AVG −==
v
vc
HTt
2
=
)1(lg1.0 UTv −−−=
where UAVG = average degree of consolidation (U)
St = settlement at Time t
Sf = final settlement at full consolidation
ut = average pore pressure at Time t
u0 = initial average pore pressure (on application of the load at Time t = 0)
where t = time to obtain a certain degree of consolidation
Tv = a dimensionless time coefficient:
cv = coefficient of consolidation
H = length of the longest drainage path
UAVG (%) 25 50 70 80 90 “100”
Tv 0.05 0.20 0.40 0.57 0.85 1.00
Basic Relations for Consolidation
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c/c
d
"Square" spacing: D = √4/π c/c = 1.13 c/c
"Triangular" spacing: D = √(2√3)/π c/c = 1.05 c/c
c/c
h
hc
DTt
2
=
Basic principle of consolidation process in
the presence of vertical drains
h
hUd
DT
−−=
1
1ln]75.0[ln
8
1
hh Ud
D
c
Dt
−−=
1
1ln]75.0[ln
8
2
and
The Kjellman-Barron Formula
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Important Points
Flow in a soil containing pervious lenses, bands, or layers
The consolidation process can be
halted if back-pressure is let to
build-up below the embankment,
falsely implying that the process is
completed
Theoretically, vertical drains operate
by facilitating horizontal drainage.
However, where pervious lenses
and/or horizontal seams or bands
exist, the water will drain vertically
to the pervious soil and then to the
drain. When this is at hand, the drain
spacing can be increased
significantly.
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The Kjellman wick, 1942 The Geodrain, 1972
The Geodrain, 1976
Wick drain types