CE5101 Lecture 6 - 1D Consolidation - Terzhagi Theory (OCT 2013).ppt

1

CE 5101 Lecture 6 – 1D Consolidation

Oct 2013

Prof Harry Tan

2

Outline

• Terzaghi Theory• Useful Elastic Solutions• Oedometer Tests• FEM Theory• FEM compared with Terzaghi• Consolidation of Realistic Soils• Example of Consolidation in Reclaimed Land• Secondary Compression and Creep

3

Terzaghi 1D Vertical Flow

• Formulation of Theory

• Useful Approximations

• Elastic Solutions

4

1D CONSOLIDATION

Assumptions made:

soil is fully saturated

pore water is incompressible

Darcy's law is valid

isotropic (constant) permeability

linear elastic soil behaviour

load applied instantaneously

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

5

1D CONSOLIDATION

soft clay layerfully saturated

z

pw = pw, o

´ = ´

rigid impermeable layer

D

initialground surface apply surcharge loadrapidly


pw = pw, o + pw, t=o

pw, t=o =

´ = ´

t = 0


pw = pw, o + pw, t

pw, t = t´

´ = ´ + t´

settlement st

0 < t < ∞

consolidation takes place


pw = pw, o

´ = ´ +

settlement s

t = ∞

consolidation process completed

6

1D CONSOLIDATION

2w

2

vw

z

pc

t

p

w

oedv γ

Ekc

0m

TMt

v2

eM

21U

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

with

cv …. coefficient of consolidation

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

pw = 0 at t = ∞ (independent of z)

1m22

1M

7

1D CONSOLIDATION

tDγ

Ek

D

tcT

2w

oed2

v

v

Ut ……… average degree of consolidation

Tv ……… dimensionless time factor

s

s

p

ppU t

0,w

t,wo,wt

NOTE:

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

U .... depends on Tv and boundary conditions

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

8

1D CONSOLIDATION

clay layerfully saturated

z

/ w

t = 0

impermeable

45°

t = t1 t = t2

t = t = t3

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

slope of Isochrones > hydraulic gradient

t1: bottom of layer not yet influenced by consolidation process

D

surcharge load

9

1D CONSOLIDATION

degree of consolidation Ut

permeable

permeable

D

D

Tv

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

10

Terzaghi 1D Vertical Flow Consolidation

5.0..,2.0 vv UeiT

21.0

442

22

18

1v

vTT

v eeU

v

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

2

11

Drainage Boundaries

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

12

Initial Excess Pore Pressures Distributions

Case 0 Case 0

Case 0

Case 0

Case 1 Case 2

13

Initial Excess PP Distributions

14


15


16

Superposition of Elastic Solutions

drained

undrained

= +

Case 0 Case 1

A A0A1

For a given Tv, find U0 and U1

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

17

Superposition of Elastic Solutions

• Let ultimate settlement be SAf

• Then degree of consolidation for A is: • 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:

•

• Therefore,

AfAfAfA S

AS

S

AS

S

ASU

)1()0()(

fAA

fAA S

ASU

S

ASU

11

00

)1(;

)0(

Af

fAA

Af

fAAA S

SU

S

SUU 1

10

0

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

18

1D Consolidation Test (Oedometer Test)

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

19

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)

20

By Eye Method for Determining Pc

21

Casagrande Method for Determining Pc

22

EX Casagrande Method for Determining Pc

23

Log-log Method for Determining Pc

24

Determine Pc - Janbu

Pc

25

Idealized 1D Consolidation e-logP Curve

26

Correction to get Field Curve for NC Clays

27

Correction to get Field Curve for OC Clays

28

Factors Affecting Accuracy of Pc

Sample DisturbanceLoad Increment Ratio

29

Factors Affecting Accuracy of Pc

Load Increment Duration

Related to the influence of secondary compression on test results

30

Taylor Square root time Method for cvExperimental CurveTheory Curve

Correction ratio =0.9209/0.7976=1.15

Tv90 = 0.848

31

Casagrande Log time Method for cv

Correction for U0 based on parabolic relation upto U50

32

Example of Use of Sqrt time and log time methods

33

Rectangular Hyperbolic Method for cvSridharan and Prakash 1981,1985

2972.0B

tfor35.1A

tfor04.2A

where

c

BmHcand

)1A(m

ct

,Therefore

Amt/tcmt

CMT

/t

U/T

90

60

2

v

34

Example of Hyperbolic Method

35

What is a high quality test?

36

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

37

FEM Theory

• Formulation

• Stress Equilibrium – Deformation Part

• Continuity Equilibrium – Hydraulic Part

• Global Assembly

• Step by step Integration (Implicit Method)

• Output

38

FINITE ELEMENT FORMULATION FOR CONSOLIDATION (1)

Effective stresses

Constitutive law

Discretization

In terms of excess pore pressure same shape functions for

displacements and pore pressures

39


Mechanical problem: equilibrium equation

Stiffness matrix

Coupling matrix

Incremental load vector

40


Hydraulic (flow) problem: continuity equation

Flow matrix

Coupling matrix

Water compressibility matrix

41


Global system of equations

Step-by-step integration procedure

0 < < 1 ; Generally, fully implicit)

42


Time step Automatic time stepping is required Critical time step

Consolidation analysis Prescribed time Maximum excess pore pressure

vc

H

80

2

vc

H

40

2

43

FEM compare Terzaghi

44

Plaxis Model at 1 day

Load = 100 kPa

45


Terzhagi theory

Plaxis Ver 9.0

46


Terzhagi theory

Plaxis

47

Fully Coupled with Unsaturated Soil Model - Plaxis 2010

48


49


Results for Terzaghi’s 1D Consolidation Test

50

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

51

1D CONSOLIDATION – NUMERICAL SIMULATION

Investigate influence of:

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

permeability depending on void

ratio

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

applied load = 100 kPasoil layer 2D = 10 mdrainage at top and bottom

52


time [days]

0.01 0.1 1 10 100 1000

sett

lem

ent

[mm

]

0

20

40

60

80

100

reference elasticpore water compressible (B=0.85)permeability e-dependentHardening Soil model

53


time [days]

0.01 0.1 1 10 100 1000

exce

ss p

ore

pre

ssu

re [

kPa]

-100

-80

-60

-40

-20

0

reference elasticpore water compressible (B=0.85)permeability e-dependentHardening Soil model

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

elastic Hardening Soil model


55

influence of parameters in HS-model

time [days]

0.001 0.01 0.1 1 10 100

vert

ical

dis

pla

cem

ents

[m

m]

-120

-100

-80

-60

-40

-20

0

HS_ref B=0.85E50 <

E50 >

Ko_nc >

Ko_nc <

56


time [days]

0.01 0.1 1 10 100

exce

ss p

ore

pre

ssu

re [

kPa]

-100

-80

-60

-40

-20

0

HS_ref B=0.85E50 <

E50 >

Eoed <

Ko_nc >

Ko_nc <

57


time [days]

0.001 0.01 0.1 1 10 100

deg

ree

of

con

soli

dat

ion

[%

]

0

20

40

60

80

100

HS_ref B=0.85E50 <

E50 >

Eoed >

Ko_nc >

Ko_nc <

58

Consolidation Modeling in a Reclaimed Land

Why a Mohr-Coulomb Model is grossly incorrect ?

59

Consider a Reclaimed LandSand Loading in 365 days

10m Reclaim Sand

15m Marine Clay

Sea Bed

Closed consolidation boundaries all round

60

Soil Parameters

Equivalent Oedometer Parameters in HS Model:

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

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

62

Compare Settlements of seabed

MC = 400 mm in 2500 days

HS = 4,350 mm in 12,700 days

Which Model is Correct ?

63

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 amount 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

64

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

65

Conclusions

• MC Model cannot be used for consolidation analysis 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 both rate and amount of consolidation settlements

66

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

67

Definition of Secondary Compression Index

ionconsolidatprimary of end

at timetwhere

tt

log

ee

tlog

eC

p

p

p

68

Bjerrum data on Secondary Compression in 1D Oedometer Test

Apparent Pc

69

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

Secondary compression index is independent of soil thickness for most cases

70

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

71

Effects of Pre-consolidation Pressure on cv and C

72

Typical values for C

NC Clays 0.005-0.02

Organic Clays, highly plastic > 0.03

OCR> 2 <0.001

Values of C/ Cc

Organic Silts 0.035-0.06

Peats 0.035-0.085

Canadian Muskeg 0.09-0.1

Singapore MC 0.04-0.06

SF Baymud 0.04-0.06

Leda Clay 0.03-0.06

Creep Settlements by Janbu

73

Can identify 3 different phases for 3 different mechanisms of settlements:

• Immediate is Elastic Undrained Compression• Consolidation is Drained (elastic plus plastic) Cap Compression • Creep is time-dependent secondary compression at constant effective stress


74


75

CE5101 Lecture 6 - 1D Consolidation - Terzhagi Theory (OCT 2013).ppt

Documents

Transcript of CE5101 Lecture 6 - 1D Consolidation - Terzhagi Theory (OCT 2013).ppt