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8/13/2019 4-2 Numerica(Finite Element) for deep excavation
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1. Finite element analysis procedure
Part II: Finite Element Method
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Governing equation
Problem Domain of Interest
Problem Domain
Prescribed Displacement
Prescribed Force
Divide problem domain into many finite elements
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System Static Equilibrium (F vs s)
0bT s
Stress tensor gravitational (body) force vector
Forces and stresses must equilibrium in the system
Governing equation
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Compatibility (u vs e)
uTe
Strain vector Displacement vector
Displacement and strain must agree in the system
Positive for compression
Governing equation
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Stress-Strain Relationships (s vs e)
es D
Stress Tensor Strain Tensor
Constitutive model to characterize material behavior
Material Constitutive (Stiffness) Matrix
Governing equation
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Internal work
The principal of virtual work
q
S
T
V
T
V
T dSqudVbudV
q
se
External work
by body force
External work by
prescribed force
extint FF Check convergence,
If not converged, change(increase) displacement u
Governing equation
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nGn dKR
Gaussian Elimination method or Frontal solver method:
nGn RKd1
ndBe
ndBDD es
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CACULATE
Displacement Increment u
CONVERT to
Strain Increment e
INTEGRATE
Stress Increment s
Iteration Scheme
e.g. Newton-Raphson
START at
a Given Boundary Conditions
e.g. Prescribed force or displacement
CACULATE
External Force Increment Fext
CHECK
Convergence
FextFint
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Calculation
Convergence Criteria
%1
F
FF
ext
extint
3%~5% is more practical
error default tolerance
Calculation until system equilibrium is reached
Convergence
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1. Define Problem Type/Dimension
2. Define Geometry and Material
3. Set Boundary Condition
4. Generate Mesh
5. Generate Initial Condition
6. Simulate Construction Procedure
7. Calculate until Convergence is reached
8. Validate the Finite Element Model
Analysis procedure
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2. Types of analysis
Effective stress analysis:
All of the calculation in the computer program are based on the
effective stress. Soil and water are treated as different material.
Input parameters: nf ,,, Ec
Total stress analysis:All of the calculation in the computer program are based on the
total stress. Only one material, soil-water mixture, exists.
No excess porewater is generated in the program.
Input parameters: uuu Es nf ,,0,
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Effective stress analysis: soil and water are treated as two-phase material.
The stress of water (porewater pressure) and soil (effective stress) are
computed separatelyTotal stress analysis: soil and water are treated as a single material.
The total stress of the single-phase material is computed.
CLAY
hs u
Effective stress analysis
hs
Total stress analysis
satg
0, fsats
g
f,c
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Sand (drained behavior)--------Effective stress analysis
Clay (undrained behavior)-----Effective stress undrained analysis
-----Total stress undrained analysis
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Linear Elastic Model:Hooks Elasticity Law
Nonlinear Elastic Model:
Duncan Hyperbolic Model
Linear Elastic- Perfectly Plastic Model:
Mohr-Coulomb Model
Elasto-Plastic Model:
Hardening Soil Model
Hardening Soil Model w/ Small Strain
Cam-Clay Model
s
e
s linearlyincreases w/ e
s
e
s nonlinearlyincreases w/ e
s
e
s linearly increasesw/ e and reaches toa peak strength
3. Constitutive model
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Hardening
Peak Softening
xx
Residual
Compression
Dilatancy
Constant
Volume
Real Soil Stress-Strain-Volume Relationships
Comparison of constitutive model
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Elastic Model Elasto-Plastic Model
Simple, require less input
parameters (E, u) Complicated, many
parameters
Efficient computation Great computation effort
Less numerical problems
(convergence)
Many numerical problems
(integration, convergence) Results are ok for working
stress conditions
(away from failure)
However, plastic model can
describe more realistically
plastic deformation of soil
(plastic strain, dilatancy,
stress history) Close form solutions are
available
Comparison of constitutive model
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Plastic Deformation:Unloading (e.g., Excavation)
Dilatancy
Dense sand or OC clay underdrained condition
(e.g., Retaining Structure)
Stress History
e=f (s, s)
current stress state, stress increment
s
e
Elastic
Elaso-Plastic
eeep
ev
e
Elastic(No dilation, Volume change is
controlled only by Poissons
ratio)
Elaso-Plastic
(by dilation angle)
Comparison of constitutive model
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Set Boundary Conditions
Prescribed Force: External force
Prescribed Displacement: u 0, v0
Horizontal fixity u=0, v0
Vertical fixity u0, v=0
Total fixity u=0, v=0standard fixity
Prescribed force or displacement
Boundary condition
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Generate Initial Condition
Initial water pressure
Initial effective stress
Phreatic level (for knowing GWT)
Pore pressure distribution
(for excess pore water pressure or seepage conditions)
Ko procedure (for horizontal ground and soil layers) Gravity loading (e.g., slopes)
Initial stress
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(A) Linear elastic elastoplastic model
s
e
E
0
s
e
E
0
E
E
Linear elastic elastoplastic model Linear elastic perfectly plastic model
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Linear elastic elastoplastic model
e
plasticity
s
Test results
Linear elastic
Yielding stress
Simulation of the real soil
E
Defined by c, f
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Disturbed soils
31 ss
1e
3s
1s
Undisturbed soils
Problems for the simulation of real soil
c fEis underestimated may not be affected
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For sand-like material:
502.058.65 NVs
Wave velocity from relationship of Vs and N
(Taipei silty clay)
Shear modulus from wave equation
2sVG (=density of soil)
)1(2 s
sEGn
Elastic theory
3.025.0 snSand
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Youngs modulus of the soil
)1(2 ss GE n
is a parameter which take into account of the high stiffness atsmall strain, =0.3-0.5
For clay-like material (total stress analysis or non-porous mode):
Eu/su can be assumed to be 450-500, or obtained from back
analysis of well-documented case histories
We can establish the relationship between Eand Ndirectly
E=2000N(kPa), for example
mu=0.495 (saturated clay)
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M
p
q
pln
e
1p
ae
cse
ke k
l
Unloading/reloading
Isotropic virgin consolidation line (IV
One dimensional consolidation line
Critical state line(CSL)
(a)(b)
(B). Cam-clay and other high order models (MITE3, MITS1)
q
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F
pMq
Critical state line ( CSL )
Elastic wall
Isotropic virgin consolidation line
Yielding surface
q
e
p
E
E Y
XX
. State boundary surface
Unloading / reloading
X
Y
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The equations for the state boundary:
The yielding equation
)/1(
222
2
/
lk
pqM
M
p
p
e
l
eep ae exp
222
2
0/ pqM
Mpp
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The Cam-clay model requires the followingparameters: , , , and .v M lE k
f
f
sin3
sin6
M f
f
sin3
sin6
M
303.2cC
l303.2sCk
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e
s
Failure stress
E
E
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Yield function-I
Yield function-II
Definitions of cap yield surface :
~
32
(C) Hardening soil model (HS model)effective stress model
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(D) HS small modelevolve from HS model
Typical field measuring result
Typical analysis result using the finite element
method
Wall deformation curve
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Wang (1997):
Numerical program:
FLAC
Soil model:
Creep modelhyperbolic model
Excavation case:
TNEC
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Yielding surfaces for clayey soils
Common material??
3s
1s
Area 1
Area 2
Area 3
Area 4
Initial yielding surface Y3
Y2
Y1
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Variation of normalized secant modulus with strain
=209kPasuc= 63kPa
=430kPasuc= 139kPa=310kPasuc= 102kPa
e1
e
s0
s1s1-s3
Esec=-
Esec
suc
0.001 0.01 0.1 110
100
1000
London clay
Jrdine et al. (1984)
5000
Straine (%)
vs
vsv
s
e1
s1s0
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For HS Small model, two additional parameters are required
0.7
7.0g
7.0g0G
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G0: Initial or very small strain shear; which can be obtainedfrom small strain test, bender element test, cross hole field
test or seismic survey
g0.7: the strain level at which the secant shear modulus is
reduced to 70% of G0; This value can be determined fromthe small strain test, or use the following correlation
]2sin)1()2cos1(2[
9
101
0
7.0 fsfg Kc
G(PLAXIS manual suggest)
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A good case
history, TNEC is
used in evaluation
of the performanceof the above models
GL.-2.8
GL.-4.9
GL.-8.6
GL.-11.8
GL.-15.2
GL.-17.3
GL.-19.7
1FL GL.+0.0
B1F GL.-3.5
B2F GL.-7.1
B3F GL.-10.3
B4F GL.-13.7
B5F GL.-17.1
H300x300x10x15 GL.-2.3
H400x400x13x21 GL.-16.5
unit: meter
GL.-35.0
CL =30 =33%-38%
LL=33-36PI=13-16
Gravel
GP N>100
SM N=4-11 =31
CL
=32%-40%
LL=29-39
PI=9-23
=29
SM N=22-24 =31
CL N=9-11 =29
SM
N=14-37
=32
GL.-35.0
GL.-46.0
GL.+0.0
GL.-2.0
GL.-8.0GL.-5.6
GL.-33.0
GL.-37.5
f
f
f
f
f
fSungshan I
Sungshan II
Sungshan III
Sungshan IV
Sungshan V
Sungshan VIStage 1
Stage 2
Stage 3
Stage 4
Stage 5
Stage 6
Stage 7
4. Evaluation of the commonly used soil models
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Comparison
MCC model
Depth(m)
50
45
40
35
30
25
2015
10
5
0
Displacement (cm)
141210 8 6 4 2 0Distance from the wall (m)
0 10 20 30 40 50 60 70 80 90 100
Settlement(cm)
-8
-6-4
-2
0
excavation stage
Field measurementMCC model
3
4
6
7
5
2
346
7
5
2
1
real soil parameter
Dep
th(m)
50
45
40
35
30
25
20
15
10
5
0
Displacement (cm)
141210 8 6 4 2 0Distance from the wall (m)
0 10 20 30 40 50 60 70 80 90 100
Settlement(cm)
-8
-6
-4
-2
0
excavation stage
Field measurementMCC model
34
67
5
2
76
54 3
2
1
adjusted parameter,k/l = 0.25
real soil parameter
adjusted parameter,k/l = 0.25
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HS model
Depth(m
)
5045
40
35
30
25
20
15
10
5
0
Displacement (cm)
141210 8 6 4 2 0Distance from the wall (m)
0 10 20 30 40 50 60 70 80 90 100
Settle
ment(cm)
-8
-6
-4
-2
0
excavation stage
Field measurementHS model
3
4
67
5
2
76
54 3
2
1
Comparison Real soil parameter
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q
p
Real soil yield surface
Modified Cam-Clay yield surface
Critical state line
K0line
A
B
C
D
Relationship of stress path in modified Cam-Clay yield surface
and real soil yield surface
E
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Comparison
HS small
model
Depth(m)
50
45
40
35
30
25
2015
10
5
0
Displacement (cm)
141210 8 6 4 2 0Distance from the wall (m)
0 10 20 30 40 50 60 70 80 90 100
Settlem
ent(cm)
-8
-6-4
-2
0
excavation stage
Field measurementHS small model
34
6
7
5
2
7
65 4 3
21
Dep
th(m)
50
45
40
35
30
25
20
15
10
5
0
Displacement (cm)
141210 8 6 4 2 0Distance from the wall (m)
0 10 20 30 40 50 60 70 80 90 100
Settlement(cm)
-8
-6
-4
-2
0
excavation stage
Field measurementHS small model
3
4
6
7
5
2
7
6
5 4
32
1
real soil parameter
adjusted parameter,0.7= 10-5
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Comparison Mohr-Coulomb model (f= 0 )
Depth(m
)
50
45
40
35
30
2520
15
10
5
0
Displacement (cm)
141210 8 6 4 2 0Distance from the wall (m)
0 10 20 30 40 50 60 70 80 90 100
Settle
ment(cm)
-8
-6
-4
-2
0
excavation stage
Field measurementHS small model
3
4
6
7
5
2
7
65 4
32
1
back analysis (E/Su=500)
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Comparison Duncan-Chang model
Displacement (cm)
Depth(
m)
Distance from the wall (m)
S
ettlement(cm)
Field observation
34
67
34
6
7
excavation stage
Duncan-Chang model
5
21
5
2
1
141210 8 6 4 2 0
50
45
40
35
30
25
20
15
10
5
00 10 20 30 40 50 60 70 80 90 100
-8
-6
-4
-2
0
back analysis
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q
MCC yield surface
K0-line
K-line
Advanced effective stress model
Gyration
p
This will introduce a gyration rate related parameter
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Wall displacement(cm) Distance from wall(m)
Settlement(cm)
Depth(m)
0 20 40 60 80
-10
-8
-6
-4
-2
0
12 10 8 6 4 2 0
50
40
30
20
10
0
Measurement
Analysis with small strain
Conventional elastoplastic analysis
.
Comparison 3KS model
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Excav. surface
FIGURE 8.25 The finite element meshes used in the analysis of excavation
(a) bad mesh (b) good mesh
Excav. surface
(a)
(b)
Retaining wall
Retaining wall
5. Mesh generation for excavation
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Boundary conditions
(a)
Retaining wall
(b)
Retaining wall
FIGURE 8.26 Boundary conditions of the finite element mesh
(a) boundary outside the excavation zone is allocated with rollers
(b) boundary outside excavation zone is allocated with hinges
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final depth
FIGURE 8.27 Distance of the boundary required for the analysis of wall deflection or ground settlement
ground settlement :
wall deflection :
eH
D
eHD 4
eHD 3
6 Corner effect on deformation behavior
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6. Corner effect on deformation behavior
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10
20
30
40
2 4 60 2 4 60 2 4 60
Lateral wall deformation (cm)
STAGE 4: STAGE 6: STAGE 7:
FEM 2D FEM 3D Measured values
0
D
epth(m)
8
FIGURE 8.29 Comparisons of the wall deflection from plane strain analysis, threedimensional analysis and field measurement respectively in a cornerof the Haihaw Financial Center excavation
struts
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B
B
L
L
FIGURE 6.30 Relationship between the plane strain ratio and the aspect ratio of an excavation
(a) PSR, the length-width ratio, and the distance from the corner
Distance to the corner (m)
0 5 10 15 20 25 30 35 40 45 500.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
0.2
0.30.5 0.6 0.7 0.8
PSR=0.9
(a)
PSR=Plane strain ratioB=Width d=Distance to the cornerL= Length
Section to be
evaluated
Section to be
evaluated
(b)
0.4
d
d
PSR
=0.1