1D Consolidation - Terzhagi Theory (SEP 2011)

CE5101 Consolidation and SeepageLecture 6

Prof Harry TanOCT 2011

1

CE 5101 Lecture 6 – 1D ConsolidationConsolidation

Oct 2011

Prof Harry Tan

1

Outline

• Terzaghi Theory

U f l El ti S l ti• Useful Elastic Solutions

• Oedometer Tests

• FEM Theory

• FEM compared with Terzaghi

• Consolidation of Realistic Soils

2

• Example of Consolidation in Reclaimed Land

• Secondary Compression and Creep



2

Terzaghi 1D Vertical Flow

• Formulation of Theory

• Useful Approximations

• Elastic Solutions

3

1D CONSOLIDATION

Assumptions made:

soil is fully saturated

pore water is incompressible

Darcy's law is valid

isotropic (constant) permeability

linear elastic soil behaviour

4

load applied instantaneously

one-dimensional problem (length of applied load > ∞)



3

1D CONSOLIDATION

soft clay layerfully saturated

p = p

initialground surface

apply surcharge load rapidly

p = p + p t = 0

z

pw = pw, o

´ = ´

rigid impermeable layer

D


pw = pw, o + pw, t=o

pw, t=o =

´ = ´

t = 0

settlement st

consolidation takes place

settlement s

consolidation process completed

5rigid impermeable layer

pw = pw, o + pw, t

pw, t = t´

´ = ´ + t´

0 < t < ∞


pw = pw, o

´ = ´ +

t = ∞

1D CONSOLIDATION

2 Ek

the change in pore pressure (pw) with time and position within the layer can be expressed by the partial differential equation

2w

2

vw

z

pc

t

p

w

oedv γ

Ekc with

cv …. coefficient of consolidation

with boundary conditions:pw = 0 at the top of layer (independent of t)no flow at bottom of layer

6

0m

TMt

v2

eM

21U

ypw = at t = 0 (independent of z)pw = 0 at t = ∞ (independent of z)

1m22

1M



4

1D CONSOLIDATIONUt ……… average degree of consolidation

Tv ……… dimensionless time factor

tDγ

Ek

D

tcT

2w

oed2

vv

s

s

p

ppU t

0,w

t,wo,wt

NOTE:

D .... drainage path, NOT thickness of layer !

7

g p , y

U .... depends on Tv and boundary conditions

Tv ... depends on problem (pw, o - distribution)

1D CONSOLIDATIONt1: bottom of layer not yet influenced by consolidation processsurcharge load

clay layerfully saturated

/ w

t = 0 t = t1 t = t2

t = t = t3

horizontal tangent > dv/dz = 0 (no flow) at bottom boundary

slope of Isochrones > hydraulic gradient

D

8

z

impermeable

45°



5

1D CONSOLIDATIONpermeable

D

Tv

D

D

9

degree of consolidation Ut

permeable

Isochrones: lines of excess pore pressures (pw, t) at a given time

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

10

21.0

442

22

18

1v

vTT

v eeU

Then

wvm



6

Drainage Boundaries

11

When k is 2 orders smaller it behaves like an impermeable boundary eg

k=1e-8 m/s is an impermeable boundary to sand of k=1e-6 m/s

Initial Excess Pore Pressures Distributions

Case 0 Case 0Case 0

12

Case 0 Case 1 Case 2



7

Initial Excess PP Distributions

13


14



8


15

Superposition of Elastic Solutions

drained Case 0 Case 1

undrained

= +A A0

A1

For a given Tv, find U0 and U1

Combined U = U0(A0/A) + U1(A1/A)

16

Combined U = U0(A0/A) + U1(A1/A)

What may produce this initial Excess PP??

Reclaimed Clay Fill self weight combined with

Imposed Sand Capping weight above reclaimed clay fill



9

Superposition of Elastic Solutions

• Let ultimate settlement be SAf

• Then degree of consolidation for A is: A S

AS

S

AS

S

ASU

)1()0()(

• By definition:

• Therefore:

• Now the amount of settlement is proportional to the area under the pore pressure isochrones. Thus the ultimate settlement is proportional to the area of the initial excess PP isochrones:

AfAfAf SSS

fAA

fAA S

ASU

S

ASU

11

00

)1(;

)0(

Af

fAA

Af

fAAA S

SU

S

SUU 1

10

0

17

•

• Therefore,

A

A

Af

fA

A

A

Af

fA

A

A

S

S

A

A

S

S1100 ;

A

AA

A

AAA A

AU

A

AUU 1

10

0

1D Consolidation Test (Oedometer Test)

18

Void ratio corresponding to full consolidation for each load increment is calculated backwards from final water content and final thickness readings



10

e vs P curve depends on stress historydeposition gives normal curve (Normally Consolidated Soils)unloading by erosion or removal of soil load gives swelling curve (Over-consolidated Soils)

19

By Eye Method for Determining Pc

20



11

Casagrande Method for Determining Pc

21

EX Casagrande Method for Determining Pc

22



12

Log-log Method for Determining Pc

23

Determine Pc - Janbu

PcPc

24



13

Idealized 1D Consolidation e-logP Curve

25

Correction to get Field Curve for NC Clays

26



14

Correction to get Field Curve for OC Clays

27

Factors Affecting Accuracy of Pc

28

Sample DisturbanceLoad Increment Ratio



15

Factors Affecting Accuracy of Pc

Load Increment Duration

Related to the influence of secondary compression on test results

29

Taylor Square root time Method for cvExperimental CurveTheory Curve

T 0 848

Correction ratio =0.9209/0.7976=1.15

Tv90 = 0.848

30



16

Casagrande Log time Method for cv

Correction for U0

based on parabolic relation upto U50

31

Example ofExample of Use of Sqrt time and log time methods

32



17

Rectangular Hyperbolic Method for cvSridharan and Prakash 1981,1985

cmt

CMT

/t

U/T

tfor35.1A

tfor04.2A

where

c

BmHcand

)1A(m

ct

,Therefore

Amt/tcmt/t

90

60

2

v

33

2972.0B

Example of Hyperbolic Method

34



18

What is a high quality test?

35

Cv is one order larger in OC state compare to NC state

36



19

FEM Theory

• Formulation

• Stress Equilibrium – Deformation Part

• Continuity Equilibrium – Hydraulic Part

• Global Assembly

• Step by step Integration (Implicit Method)

37

• Output

FINITE ELEMENT FORMULATION FOR CONSOLIDATION (1)

Effective stresses

Constitutive law

Discretization

38

In terms of excess pore pressure

same shape functions for displacements and pore pressures



20


Mechanical problem: equilibrium equation

Stiffness matrix

39

Stiffness matrix

Coupling matrix

Incremental load vector


Hydraulic (flow) problem: continuity equation

Flow matrix

40

Flow matrix

Coupling matrix

Water compressibility matrix



21


Global system of equations

Step-by-step integration procedure

410 < < 1 ; Generally, fully implicit)


Time step

Automatic time stepping is required

Critical time step

H 2

Consolidation analysis

Prescribed time

vc80

vc

H

40

2

42

Maximum excess pore pressure



22

FEM compare Terzaghi

43

Plaxis Model at 1 day

Load = 100 kPa

44



23


Terzhagi theory

Plaxis Ver 9.0

45


Terzhagi theory

Plaxis

46



24

Fully Coupled with Unsaturated Soil Model - Plaxis 2010

47


48



25


Results for Terzaghi’s 1D Consolidation Test

49

Real Soils Consolidation

• Cv is not constant with consolidation process

• Both kv and mv (or Eoed) are varied as consolidation progress

• Cv is one order larger for OC state compared to NC state

50

compared to NC state



26

1D CONSOLIDATION – NUMERICAL SIMULATION

applied load = 100 kPa

Investigate influence of:

compressibility of pore water (by means of B-value)

soil layer 2D = 10 mdrainage at top and bottom

51

permeability depending on void ratio

elastic-plastic soil behaviour(by means of changing constitutive model)


0

20

reference elasticpore water compressible (B=0.85)

sett

lem

ent

[mm

]

20

40

60

80

( )permeability e-dependentHardening Soil model

52

time [days]

0.01 0.1 1 10 100 1000

80

100



27


kPa]

-100

80

ss p

ore

pre

ssu

re [

k -80

-60

-40

reference elasticpore water compressible

53time [days]

0.01 0.1 1 10 100 1000

exce

s

-20

0

p p(B=0.85)permeability e-dependentHardening Soil model


54distribution of excess pore pressures at 50% consolidation along centre line

elastic Hardening Soil model



28

influence of parameters in HS-model

m]

0

al d

isp

lace

men

ts [

m

-80

-60

-40

-20

HS_ref B=0.85E50 <

E50 >

55

time [days]

0.001 0.01 0.1 1 10 100

vert

ica

-120

-100Ko_nc >

Ko_nc <


a]

-100

s p

ore

pre

ssu

re [

kP -80

-60

-40

HS_ref B=0.85E50 <

E50 >

56time [days]

0.01 0.1 1 10 100

exce

ss

-20

0

E50

Eoed <

Ko_nc >

Ko_nc <



29


%]

0

e o

f co

nso

lid

atio

n [

%

20

40

60

HS_ref B=0.85E50 <

E50 >

57time [days]

0.001 0.01 0.1 1 10 100

deg

ree

80

100

50

Eoed >

Ko_nc >

Ko_nc <

Consolidation Modeling in a Reclaimed LandReclaimed Land

Why a Mohr-Coulomb Model is grossly incorrect ?

58



30

Consider a Reclaimed LandSand Loading in 365 days

10m Reclaim Sand

15m Marine Clay

Sea Bed

59

Closed consolidation boundaries all round

Soil Parameters

60

Equivalent Oedometer Parameters in HS Model:

Cc=1.0 Cs=0.1 eo=2.0 and m=1.0 for logarithmic compression response



31

61HS Model can produce results very close to Oedometer Test Data

Compare Settlements of seabed

MC = 400 mm in 2500 days

HS = 4 350 mm in 12 700 days

Which Model is Correct ?

62

HS 4,350 mm in 12,700 days



32

Amount of Settlement

Single layer 1-D compression Estimate:

Cc=1.0, eo=2.0, Ho=15mPo = 7.5m*5 = 37.5 kPaP_inc = 10m*18 = 180 kPaPf = Po+P_inc = 217.5 kPaSett = Ho*Cc/(1+eo)*log(Pf/Po) = 15000*0.254 = 3,817 mm

• This is a single layer computation and it grossly under-estimate

63

g y p g yamount of settlements; but 3,817 mm >> 400 mm by MC Model, and is much closer to 4,330 mm by HS Model

• Thus HS Model gave realistic answer and MC Model is grossly incorrect

Compare with Program UniSettle Using same oedometer parameters of Cc=1.0, eo=2.0;;

UniSettle = 4428 mm

HS = 4350 mm

UniSettle 15-layer computation gave same results as Plaxis HS model

64



33

Conclusions

• MC Model cannot be used for consolidation analysis of soft soilsanalysis of soft soils

• The linear elastic model in MC cannot predict both the rate and amount of consolidation settlements of highly nonlinear soft clays

• The HS Model with equivalent oedometer parameters will give very good predictions of

65

parameters will give very good predictions of both rate and amount of consolidation settlements

Secondary Compression - Creep Effects, continued settlements under constant effective stress

66



34

Definition of Secondary Compression Index

ionconsolidatprimary of end

at timetwhere

t

tlog

ee

tlog

eC

p

p

p

67

Bjerrum data on Secondary Compression in 1D Oedometer Test

Apparent Pc

68



35

Relation between Instantaneous and delayed compression (a) for different thickness (b) for given thickness

69

Secondary compression index is independent of soil thickness for most cases

Effect of Magnitude of Stress Increment: ratio of secondary to primary compression is largest when stress increment to initial stress is small

70



36

Effects of Pre-consolidation Pressure on cv

and C

71

Typical values for C

NC Clays 0 005 0 02 Values of C/ CcNC Clays 0.005-0.02

Organic Clays, highly plastic >0.03

OCR>2 <0.001

c

Organic Silts 0.035-0.06

Peats 0.035-0.085

Canadian Muskeg 0.09-0.1

Singapore MC 0.04-0.06

72

SF Baymud 0.04-0.06

Leda Clay 0.03-0.06



37

Creep Settlements by JanbuCan identify 3 different phases for 3 different mechanisms of settlements:

• Immediate is Elastic Undrained Compression• Consolidation is Drained (elastic plus plastic) Cap Compression

73

p p ) p p• Creep is time-dependent secondary compression at constant effective stress

Creep Settlements by Janbu

74



38

Creep Settlements by Janbu

75

Documents

1D Consolidation - Terzhagi Theory (SEP 2011)