ce5101 lecture 6 - 1d consolidation - terzhagi theory (oct 2013).ppt

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CE 5101 Lecture 6 – 1D Consolidation

Oct 2013

Prof Harry Tan

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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

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Terzaghi 1D Vertical Flow

• Formulation of Theory

• Useful Approximations

• Elastic Solutions

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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 > ∞)

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1D CONSOLIDATION

soft clay layerfully saturated

z

pw = pw, o

´ = ´

rigid impermeable layer

D

initialground surface apply surcharge loadrapidly

rigid impermeable layer

pw = pw, o + pw, t=o

pw, t=o =

´ = ´

t = 0

rigid impermeable layer

pw = pw, o + pw, t

pw, t = t´

´ = ´ + t´

settlement st

0 < t < ∞

consolidation takes place

rigid impermeable layer

pw = pw, o

´ = ´ +

settlement s

t = ∞

consolidation process completed

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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

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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)

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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

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1D CONSOLIDATION

degree of consolidation Ut

permeable

permeable

D

D

Tv

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

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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

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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

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Initial Excess Pore Pressures Distributions

Case 0 Case 0

Case 0

Case 0

Case 1 Case 2

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Initial Excess PP Distributions

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Initial Excess PP Distributions

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Initial Excess PP Distributions

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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

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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

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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

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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)

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By Eye Method for Determining Pc

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Casagrande Method for Determining Pc

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EX Casagrande Method for Determining Pc

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Log-log Method for Determining Pc

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Determine Pc - Janbu

Pc

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Idealized 1D Consolidation e-logP Curve

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Correction to get Field Curve for NC Clays

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Correction to get Field Curve for OC Clays

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Factors Affecting Accuracy of Pc

Sample DisturbanceLoad Increment Ratio

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Factors Affecting Accuracy of Pc

Load Increment Duration

Related to the influence of secondary compression on test results

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Taylor Square root time Method for cvExperimental CurveTheory Curve

Correction ratio =0.9209/0.7976=1.15

Tv90 = 0.848

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Casagrande Log time Method for cv

Correction for U0 based on parabolic relation upto U50

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Example of Use of Sqrt time and log time methods

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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

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Example of Hyperbolic Method

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What is a high quality test?

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

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FEM Theory

• Formulation

• Stress Equilibrium – Deformation Part

• Continuity Equilibrium – Hydraulic Part

• Global Assembly

• Step by step Integration (Implicit Method)

• Output

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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

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FINITE ELEMENT FORMULATION FOR CONSOLIDATION (2)

Mechanical problem: equilibrium equation

Stiffness matrix

Coupling matrix

Incremental load vector

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FINITE ELEMENT FORMULATION FOR CONSOLIDATION (2)

Hydraulic (flow) problem: continuity equation

Flow matrix

Coupling matrix

Water compressibility matrix

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FINITE ELEMENT FORMULATION FOR CONSOLIDATION (3)

Global system of equations

Step-by-step integration procedure

0 < < 1 ; Generally, fully implicit)

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FINITE ELEMENT FORMULATION FOR CONSOLIDATION (4)

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

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FEM compare Terzaghi

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Plaxis Model at 1 day

Load = 100 kPa

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FEM compare Terzaghi

Terzhagi theory

Plaxis Ver 9.0

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FEM compare Terzaghi

Terzhagi theory

Plaxis

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Fully Coupled with Unsaturated Soil Model - Plaxis 2010

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Fully Coupled with Unsaturated Soil Model - Plaxis 2010

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Fully Coupled with Unsaturated Soil Model - Plaxis 2010

Results for Terzaghi’s 1D Consolidation Test

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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

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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

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1D CONSOLIDATION – NUMERICAL SIMULATION

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

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1D CONSOLIDATION – NUMERICAL SIMULATION

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

1D CONSOLIDATION – NUMERICAL SIMULATION

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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 <

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influence of parameters in HS-model

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 <

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influence of parameters in HS-model

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 <

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Consolidation Modeling in a Reclaimed Land

Why a Mohr-Coulomb Model is grossly incorrect ?

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Consider a Reclaimed LandSand Loading in 365 days

10m Reclaim Sand

15m Marine Clay

Sea Bed

Closed consolidation boundaries all round

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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

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Compare Settlements of seabed

MC = 400 mm in 2500 days

HS = 4,350 mm in 12,700 days

Which Model is Correct ?

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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

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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

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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

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

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Definition of Secondary Compression Index

ionconsolidatprimary of end

at timetwhere

tt

log

ee

tlog

eC

p

p

p

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Bjerrum data on Secondary Compression in 1D Oedometer Test

Apparent Pc

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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

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

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Effects of Pre-consolidation Pressure on cv and C

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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

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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

Creep Settlements by Janbu

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Creep Settlements by Janbu

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