Radial Consolidation_ PVD and Surcharge (Oct 2011 Color)

CE5101 Consolidation and SeepageLecture 8 PVD and Surcharge

Prof Harry TanOCT 2010

1

CE 5101 Lecture 8 – Radial Consolidation and PVDConsolidation and PVD

October 2011

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



2

Radial Consolidation - Barron’s Theory (1948)

z

u

r

u

rr

u

t

u

Governing

2

2

v2

2

h c1

c

:coords radialin Equation D-3

wv

wh cc

where

zrrrt

v

v

v

h

m

k,

m

k

1c

:Only Flow adial

2

2

h

r

u

rr

u

t

u

R

symmetry) todue s(imperviou 0r

)u(r 3.

0for t 0)u(r 2.

0at t uu 1.

:Conditions

e

w

0

Boundary

rrrt

Functions Bessel are ;:

1

41

:Drain) (IdealCondition for

102

21

20

222

421

00

22

UandUd

tcTand

d

dnwhere

αUαnUn)(nα

eαU

u

u;

u

uU

StrainFreeSolution

e

hh

w

e

Tnαrr

r

h

strainFree

Note

fastest settle

drain closest to

soil as settlement

uniform-non

means

:

only

Ur

hT andn of

function a is



3

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 pshow very small differences between free-strain and equal-strain, esp for n>10

For n=5, significant difference in first 50% of consolidation



4

What is size of Influence Diameter de or D

sD

D

Square

13.14

s

:spacing 2

2

sD

D

T

05.14

32/*s/2*s/2*1/2*6

:spacingraingular 2

2

2

2

2

2r

4

13)ln(

1:

;8

exp1U

:Drain Vertical )1981(

n

nn

n

nwhere

D

tcT

T

IdealHansbo

hh

h

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



5

Effects of Smear and Drain Resistance

Carillo Theory – Combined vertical and radial Flow



6

Combined Flow - Carillo’s Theorem (1942)

u osolution t a is ,u

1u osolution t a is ,u

2

2

22

2

2

11

z

uc

ttzfand

r

u

rr

uc

ttrfIf

v

h

problem flow combined thesolutionof a is uu 21then

z

crrr

ct

toof

vh

22

112

2112

2

212

21

2

212

21

21

uu1uuuuu

uuuu1uuuu

u intouuu Substitute :Pr

QED

zc

tand

rrrc

t

This

zc

rrrc

tt

vh

vh

2

22

212

12

1

2

21

121

22112

uuu1uu

: thatmeans

uuu

u

:toleaddiscussionprevious The

Combined Flow - Carillo’s Theorem (1942)

11U-1

:meanshat

u

u

u

u

u

u

0

v

0

h

0

UU

T

vh

theorysHansbo'or sBarron'

theorysTerzaghi'

fromU

fromU

h

v



7

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



8

Terzaghi 1D Vertical Flow Consolidation

5.0..,2.0 vv UeiTFor Tv is Time factor

i C fi i t f

v

v

TU 2

Then

For 5.0..,2.0 vv UeiT

cv is Coeficient of Consolidation

vv

vv

m

kc

H

tcT

2

21.0

442

22

18

1v

vTT

v eeU

Then

wvm

Barron 1D Radial Flow Consolidation

hT

U8

1

Th is Time factor

ch is Coeficient of

Equal Vertical Strain Condition

222

2

4

11

1

4

3ln

1 nnn

n

n

h eU 1

4

3)ln( n

ch is Coeficient of Consolidation

hh

hh

m

kc

D

tcT

2

For n=D/d > 10

To include smear and drain di h wvm discharge

w

h

r

hs q

kzLzs

k

k

s

n)2(

4

3)ln()ln(

Where z = L for single drainage at top,

and z = L/2 for double drainage at top and bottom



9

2

2

2

2

2r

4

13)ln(

1:

;8

exp1U

:Drain Vertical )1981(

n

nn

n

nwhere

D

tcT

T

IdealHansbo

hh

h

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

For single drainage at toptop,

z=L

For double drainage at top and bottom, z=L/2

Carillo Combined Flow

)1)(1(1 hvvh UUU 2

From linear superposition

h

v

T

h

T

v

eU

eU8

21.04

1

1

h

vT

T

vh eU

821.0

4

2

1

For Tv > 0.2

Uv > 50%

For Tv ≤ 0.2

Uv ≤ 50%

hT

vvh eTU8

/211



10

Equivalent Vertical Permeability for Plane Strain FEM Model – CUR 191 or Tan 1981

h

vT

T8

21.04

2

Interested only in solution > 50% consolidation

For Axisymmetric Unit Cell

v

vh eU4

1

21.0

4'

'

2

1vT

v eU

vhv UU '

For Axisymmetric Unit Cell

For Equivalent FEM Model

To obtain equivalent vertical consolidation rate

hvv

TTT

821.0

421.0

4

2

'

2

vv

v eeU44

' 11

hvv

hvv

kD

Hkk

TTT

2

2

2'

2'

32

32

wv

vv

vv m

kcand

H

tcT

2wv

hh

hh m

kcand

D

tcT

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 drainageINPUT

Case cv(m2/y) H(m) t(y) Tv Uv1 2 5 0.25 0.02 0.162 2 5 0.25 0.02 0.16 hh

hh

s

h

kL

knh

D

tcT

T

23

)l (l

;8

exp1U

:ResistanceDrain andSmear ofEffect Eqn with Hansbo

2h

Hansbo/Barron 1D Radial Consolidation

INPUT z=L single drainage and z=L/2 double drainageCase 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 0.772 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 0.812 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 1.02 100 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

fsr

w

h

s

hs q

zLzsks

where 24

)ln(ln:

)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



11

Preload Surcharge Design –Johnson ASCE 1970

Assumptions:

a. Primary and secondary compression are separate

b. Instant load applied at end of ½ load period

Ti r t f ttl t d t r i b

21

c. Time rate of settlement determine by Terzaghi theory

Preload Surcharge Design –Johnson ASCE 1970

Objective: To determine amount of surcharge needed to achieve desired degree of consolidation? '

v

Clay: Ho, Po and Cc

Design Permanent Fill Pf

Surcharge Ps

Pf

Ps

ttsr

Sf

Sf+s

22

SIf 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



12

Preload DesignFor Normally Consolidated Clay (NC) of thickness Ho:

(1) log1

:only Fill 00

P

PPH

e

CS fc

f

(3) 0.1

log

log

)(

:ision consolidat of degree average tsr,At time

(2) log1

:surcharge and Fill

1

0

0

0

0

0

00

0

00

P

PPPU

P

PP

SU

SU

P

PPPH

e

CS

Pe

sfsr

f

sfsr

ff

sfcsf

23

(4) 0.1

log

log

)(

:is surcharge and fillunder ion consolidat of degree required Therefore,

0

0

0

0

0

P

PPP

P

PP

S

SUU

P

sf

f

sf

fsrsf

Preload Design Example

Clay: Ho, Po and Cc


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?

160loglog

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

0

22v

0

00

0

f

vv

v

fcf

PP

TU

H

tT

P

PPH

e

CS

24large)(very surcharge of m 5.2 94/18 kPa 94160254

54.210100

160

404.0505.0

204.0

100

160log

100

160log

100log

log

log

505.0

404.0

0

0

0

s

s

s

ssfsf

fsr

P

P

P

P

P

PPP

P

S

SU

So surcharge alone is not effective and we need PVD to reduce surcharge time as well as amount of surcharge needed



13

Preload Design Example

Clay: Ho, Po and Cc


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 kPaFill: Height = 3m with Pf = 60 kPa

Aim: To get 100% improvement in 3 months, what is Ps needed?

Practical Vertical Drain Design (by Prof Harry Tan SEP 2008)

Terzaghi 1D Vertical Consolidation

H=L single drainage and H=L/2 double drainageINPUT

Case cv(m2/y) H(m) t(y) Tv Uv1 2 5 0.25 0.02 0.162 2 5 0.25 0.02 0.16

Hansbo/Barron 1D Radial Consolidation

INPUT z=L single drainage and z=L/2 double drainageCase 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

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

25Design requires PVD triangle spacing with 1.3m grid and 1m surcharge or 1.5m grid with 1.9m surcharge

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 0.772 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 0.812 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 1.02 100 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

fsr

)1()1(1 hvvh UUU

FEM Modeling of Embankments on Soft Ground

with PVD

1. Model of single PVD – Axi-symmetric

2. Model of PVD in Plane Strain



14

Interface element in PLAXIS used

Method 1 – Using Interface Element for Vertical Drain

Impose specified cross-sectional area and vertical permeability of vertical drain to simulate well resistance

Effect of smear considered by the yequivalent permeability of surrounding soils

AXISYMMETRICz

r

z

r

z

r

r

Pore water flow

qw

Soil

PVD

H

Interface element

kh

qw

Soil

H

Closed consolidation

boundary

H

r

Soil

qw

rw re rw re

ti rw re

(a) (b) (c)

Open Boundary Interface element Drain element



15

FEM Axi-Symmetric Model of Single PVD

FEM Model – Barron Theory

Boundary conditions

E_oed=1000 kPa

Cv_soil = 0.01*1000/10 = 1 m2/day

Cv_drain=1*1000/10=100 m2/day

30



16

FEM Model – Barron Theory

T=0.1day

31

0

1020

Interface Element

Open Consolidation Boundary

Radial Consolidation Theory

3040

5060

7080

Uh (

%)

Barron's Theory

90100

0.001 0.01 0.1 1 10 100Th



17

CONVERSION FROM AXISYMMETRIC TO PLANE STRAIN

ss ss

s m

m

s sss

2ti 2B

QP

r

QAP

x

2tdw

de

2ti

2B or S

(c) (d)(b)(a)

no drainage (reference)

FEM models investigated:

Axisymmetric model

drainage with drain element

(sets zero pore pressure conditions)

drainage with boundary condition

(check on performance of “drain element”)

Plane strain modelPlane strain model

equivalent vertical permeability after CUR 191

equivalent horizontal permeability after CUR 191

equivalent horizontal permeability after Indraratna (2000)



18

unit cell for vertical drains placed in pattern of 2x2 m, 5

m high

drain diameter 25 cm

applied load

10 kN/m²

axisymmetric model

plane strain model

hvv kD

Hkk

2

2

2

32´

hh kk ´

CUR 191 equivalent vertical permeability

222

2

4

11

1

4

3ln

1 nnn

n

nd

Dn

kv , kh “true“ permeability

kv´ , kh´ equivalent permeability

H drainage length

D equivalent distance of drains

d diameter of drains



19

CUR 191 equivalent horizontal permeability

hh kD

Bk

2

2

´

vv kk ´

222

2

4

11

1

4

3ln

1 nnn

n

nd

Dn

U 0,5 0,75 0,9 0,95 0,99 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

267,0 Bkhp

Indraratna equivalent horizontal permeability

Rn

275,0ln Rnkh

p

khp equivalent horizontal permeability for plane strain

kh “true“ horizontal permeability

wrn

B ½ distance of drains in plane strain

R equivalent distance of drains

rw diameter of drains



20

Excess Pore Pressure after 60% consolidation

Influence of constitutive model

HS - ModelLinear Elastic - Model

U [

- ] 0.8

1.0

degree of consolidation for different

models (linear-elastic)

degr

ee o

f con

solid

atio

n U

0 2

0.4

0.6

AXI: no drainageAXI: drainage boundary conditionAXI d i d i l t

time [sec]

1e+3 1e+4 1e+5 1e+6 1e+7 1e+8 1e+9

d

0.0

0.2 AXI: drainage drain-elementPS: equivalent vertical CUR 191PS: equivalent horizontal CUR 191PS: equivalent horizontal Indraratna



21

U [

- ] 0.8

1.0

degree of consolidation for different

models (Hardening Soil model)d

egre

e of

con

solid

atio

n U

0 2

0.4

0.6

AXI: no drainageAXI: drainage boundary conditionAXI: drainage drain-element

time [sec]

1e+3 1e+4 1e+5 1e+6 1e+7 1e+8 1e+9 1e+10

d

0.0

0.2g

PS: equivalent vertical CUR 191PS: equivalent horizontal CUR 191PS: equivalent horizontal Indraratna

Austrian Case

WA

SS

ER

KA

NA

L

C

D

E

B

E1

A1/1

A1/2A1/3A1/4A1/5A1/6

A2/1

A2/2A2/3A2/4A2/5A2/6A2/7

A1/9 A1/8PW3 A1/7PW4

Y D

A

A

E2

PW1

A2/3A2/4A2/5A2/8

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

A5/2A5/3A5/4A5/5A5/6A5/8

A6/2A6/3A6/4A6/6A6/7

A5/7

A6/5

A2/9

Z3/8

A3/8

RS2/6

RS2/7

RS2/8

RS2/9

RS1/3

X

XLOGISTIK

HALLEUMSCHLAGHALLE

BÜR

O

B

C

D

E

5.0

äußerer Schutzstreifen

5.0

A7/1

A8/1

R/1

A6/6

A7/2A7/3

A8/2

A6/5

A8/3

Z4/8

RS2/1RS2/2

RS2/3RS2/4

A7/4RS2/5

Y

Schüttabschnitt 1

Schüttabschnitt 2

Schüttabschnitt 3

D



22

soil profile:

pre-load - drained = 18 kN/m3

3 m

1

1

peat - undrained

silt / silt-clay - undrained kx = ky = 0,0001 m/day ; kx´ = 1,3e-5 m/day

man made material - drained = 19,5 kN/m3

2,5 m

4,5 m

2 m

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

14 m

FE-MODEL

section D-Dsection D D

A2/4 A4/4 A6/4



23

Results for section D-D

comparison measurement - Plaxis point A2/4

settl

eme

nt [c

m]

-60

-40

-20

0

calculated finalsettlement139 cm

time [days]

0 20 40 60 80 100 120 140

s

-120

-100

-80

Plaxismeasurement

Results for section D-D

comparison measurement - Plaxis - point A6/4

settl

emen

ts [c

m]

30

-20

-10

0

calculated final

settlement

78 cm

time [days]

0 20 40 60 80 100 120 140

s

-50

-40

-30

Plaxismeasurements



24

EXAMPLE - EMBANKMENT CONSTRUCTION

influence of consolidation on stability

influence of construction speed is investigatedinfluence 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


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

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



25


"slow": stable

"fast": failure


-50

excess pore pressure [kN/m2]

Chart 1

slow

fast

excess pore pressure [kPa]

fast

-40

-30

-20

slow

0 4 8 12 16

-10

0

Time [day]

time [days]



26


vertical displacements [m]

0.06

Displacement [m]

Chart 1

Point C

fast

slow0.03

0.04

0.05

Point C

0 30 60 90 1200

0.01

0.02

Time [day]

time [days]

Practical Considerations



27

The Problem — Bridge Foundations

Lateral spreading

Settlement with risk for downdrag

These photos of bridge foundations illustrate a common

bl ff ti i tproblem affecting maintenance ($$$!), as well as, on occasions, one compromising safety



28

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.



29

2H

Drainage Layer

Clay Layer (consolidating)

0

1u

u

S

SU t

f

tAVG

where UAVG = average degree of consolidation (U)

S l Ti

Basic Relations for Consolidation

Drainage Layer

vv c

HTt

2

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 consolidationwhere 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

)1(lg1.0 UTv

c/c

d

"Square" spacing: D = 4/π c/c = 1.13 c/c

"Triangular" spacing: D = π c/c = 1.05 c/c

c/c

Basic principle of consolidation process in the presence of vertical drains

D 11D2

hh Ud

DT

1

1ln]75.0[ln

8

1

hh Ud

D

c

Dt

1

1ln]75.0[ln

8

2

and

hh c

DTt

The Kjellman-Barron Formula



30

Important Points

Build-up of Back Pressure

The consolidation process can be halted if back-pressure is let to build-up below the embankment

Flow in a soil containing pervious lenses, bands, or layers

build-up below the embankment, falsely implying that the process is completed

Theoretically, vertical drains operate by facilitating horizontal drainage. H h i lHowever, 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

Radial Consolidation_ PVD and Surcharge (Oct 2011 Color)

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