4-2 Numerica(Finite Element) for deep excavation

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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3

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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4

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



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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)



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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)


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)


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)


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)


0 10 20 30 40 50 60 70 80 90 100

Settlement(cm)

-8

-6

-4

-2

0

excavation stage


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)


0 10 20 30 40 50 60 70 80 90 100

Settle

ment(cm)

-8

-6

-4

-2

0

excavation stage


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

4-2 Numerica(Finite Element) for deep excavation

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