CE 5101 Lecture 8 – Radial Consolidation and PVD

October 2013

Prof Harry Tan

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

�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

( )( ) ( )[ ]

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

:

−

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

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

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

( )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

Effects of Smear and Drain Resistance

Carillo Theory – Combined vertical and radial Flow

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

∂

∂=

∂

∂

∂

∂+

∂

∂=

∂

∂

∂

∂+

∂

∂+

∂

∂=

∂

∂+

∂

∂

∂

∂+

∂

∂+

∂

∂=

∂

∂

∂∂

=

( ) ( )( )

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)

Practical Vertical Drain Design with Plaxis 2D-

FEM

Outline

• Terzaghi 1D Vertical Flow Consolidation

• Barron 1D Radial Flow Consolidation

• Carillo Combined Flow Consolidation

• Equivalent Plane Strain Consolidation for 2D-FEM

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

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

( )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

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

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

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

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

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

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:

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

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

FEM Modeling of

Embankments on Soft Ground

with PVD

1. Model of single PVD – Axi-symmetric

2. Model of PVD in Plane Strain

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

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

FEM Axi-Symmetric Model of Single PVD

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

FEM Model – Barron Theory

31

T=0.1day

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

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)

� 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)

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

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

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

( )[ ] 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

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

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.

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)

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

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

This image cannot currently be displayed.This image cannot currently be displayed.

FE-MODEL

section D-D

A2/4 A4/4 A6/4

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

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

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

influence of consolidation on stability

"fast": max. excess pore pressure: 100 kPa

"slow": max. excess pore pressure: 86 kPa

influence of consolidation on stability

"slow": stable

"fast": failure

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

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]

Practical Considerations

Lateral spreading

Settlement with risk

for downdrag

The Problem — Bridge Foundations

These photos of bridge

foundations illustrate a common

problem affecting maintenance

($$$!), as well as, on occasions,

one compromising safety

Photos from in-situ excavation of a pile

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.

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

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

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.

The Kjellman wick, 1942 The Geodrain, 1972

The Geodrain, 1976

Wick drain types