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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
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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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