2D and 3D Consolidation - Biot Theory (OCT2011 Color)

CE5101 Seepage and ConsolidationLecture 7- 2D and 3D Consolidation

Prof Harry TanOCT 2011

1

CE 5101 Lecture 7 – 2D and 3D Consolidation3D Consolidation

OCT 2011

1

Prof Harry Tan

Outline• Biot Theory (2D and 3D Coupled

Consolidation))• FEM compare with Schiffman et. al. 1967

(Mandel-Cryer Effect in 2D)• Undrained, Consolidation and Drained

Response• Case 1 – WCRS Excavation

2

• Case 1 – WCRS Excavation • Compare CRISP with Plaxis• Case 2 –Barcelona Breakwater



2

Biot Theory (2D and 3D Coupled Consolidation)

• Equilibrium Equations

• Compatibility Equations –Strain/Displacements

• Constitutive Equations

• Continuity Equations

3

• Boundary Conditions

• Assume infinitesimal strain conditions

Equilibrium Equations

4



3

5

6



4

7

8



5

9

10



6

Cryer-Mandel Problem

11

Comments on True 3D Consolidation

• True 3D consolidation couples total stresses and continuity of soil water (excess PP)continuity of soil-water (excess PP)

• Psuedo 3D uncouples these two phenomena

• When total stress distribution is constant at all time, the rate of change of excess PP = rate of change of volume at all points in the soil

Thi i t l f 1D lid ti h

12

• This is true only for 1D consolidation where there is a direct relationship between excess PP and volume change



7

Schiffman Strip Footing 1967

• Plane-strain consolidation in 2DSt f l ti th• Stresses from elastic theory are independent of elastic constants

• Total stresses are same at start and end of consolidation

• However they vary with time; they exceed

13

initial value during consolidation• So excess PP first increases before it

starts to dissipate with consolidation

FEM compare with Schiffman et. al. 1967 (Mandel-Cryer Effect in 2D)

Mean total stress

Excess pore pressures

14

Mean effective stress



8

Plaxis Biot ConsolidationSchiffman, Chen and Jordan 1967

ExcPP at x/a=1

15Strip Footing on Elastic Half-space

All round closed boundary condition

0.6

Stress [kN/m2]

Stress at x/a=1

Mean Tota l Stress

Mean Ef fect iv e Stress

Plaxis Biot ConsolidationSchiffman, Chen and Jordan 1967

Mean total stress

0.2

0.3

0.4

0.5

Ex cess Pore Pressure

Mean excess PP

Input:

k=0.001 m/day

Eoed=10000 kPa

cv=k*Eoed/

16

1e-3 1e-2 0.1 1 10 100 1e30

0.1

Time [day]

Strip Footing on Elastic Half-space

Mean effective stress

cv=k Eoed/w

cv=1 m2/day



9

Plaxis Biot ConsolidationRamp Loading – Schiffman 1960

1

Excess PP [kN/m2]

Chart 1

Ver 8.2

To

0.2

0.4

0.6

0.8Ver 7.2

To

17

1e-5 1e-4 1e-3 1e-2 0.1 1 100

Time [day]

ExcPP at Base – Closed consolidation boundary

Ramp Loading with To=0.1


0

Displacement [m]

Chart 2

Ver 8.2

V 7 2

8 3

-6e-3

-4e-3

-2e-3Ver 7.2

18

Settlement at Top – Closed consolidation boundary

Ramp Loading with To=0.1

1e-5 1e-4 1e-3 1e-2 0.1 1 10-0.01

-8e-3

Time [day]



10


1

Excess PP [kN/m2]

At Base

Point B

Point B

To

0.2

0.4

0.6

0.8Point B

To=0.1 To=0.5

19

Excess PP at Base – Closed consolidation boundary

Ramp Loading with To=0.1 and 0.5

1e-5 1e-4 1e-3 1e-2 0.1 1 100

Time [day]

TYPES OF ANALYSIS

Drained Loading/Construction/ excavation: very slow (in relation to the g y (

soil permeability)

Undrained Loading/Construction/ excavation: very fast (in relation to the

soil permeability)

Intermediate cases: consolidation analysisB th h i l d h d li (fl ) bl i t t

20

Both mechanical and hydraulic (flow) problems interact

More complex computations: coupled analysis



11

Undrained, Consolidation and Drained Response

• Definition drained / undrained• Modeling undrained behaviour with Plaxis• Modeling undrained behaviour with Plaxis

• In terms of effective stresses with drained strength parameters

• In terms of effective stresses with undrained strength parameters

• In terms of total stresses with undrained strength parameters

• Example of Underwater Cut Slope

21

• Example of Underwater Cut Slope• Example of Clay Embankment • Summary

DRAINED / UNDRAINED Drained analysis appropriate when

• permeability is high• rate of loading is low• short term behaviour is not of interest for problem considered

Undrained analysis appropriate when• permeability is low and rate of loading is high

• short term behaviour has to be assessed

Suggestion by Vermeer & Meier (1998) for deep excavations:T < 0.1 (U=35%) use undrained conditionsT > 0.40 (U=70%) use drained conditions

22

tDγ

EkT

2w

oedk = permeabilityEoed = oedometer modulusw = unit weight of waterD = drainage lengtht = construction timeTv = dimensionless time factor



12

UNDRAINED BEHAVIOUR WITH PLAXIS

uuw 1G2EK'KK

PLAXIS automatically adds stiffness of water when undrained material type is chosen using the following approximation

u

u

u

uwtotal 213213n

'KK

'1213

1'EK

u

utotal

assuming u = 0.495

Note:

23

Note: - this procedure gives reasonable B-values only for ´ < 0.35 !- real value of Kw/n ~ 1.106 kPa (for = 0.5)

- NB: in Version 8, B-value can be entered explicitly

UNDRAINED BEHAVIOUR WITH PLAXISExample 1:

E´ = 3 000 kPa, ´ = 0.3, u = 0.495

K´ = 2 500 kPa, Ktotal = 115 000 kPa Kw/n = 112 500 kPa

with = 0.978 is reasonable value for saturated soil

Example 2:

wKnK

B'

1

1

24

E´ = 3 000 kPa, ´ = 0.45, u = 0.495

K´ = 10 000 kPa, Ktotal = 103 103 kPa Kw/n = 93 103 kPa

B = 0.903 is poor value for saturated soil



13

UNDRAINED BEHAVIOUR WITH PLAXIS

Method A (analysis in terms of effective stresses):type of material behaviour: undrainedeffective strength parameters c´, ´, ´

ff ti tiff t E ´ ´effective stiffness parameters E50 ,

Method B (analysis in terms of effective stresses):type of material behaviour: undrainedundrained strength parameters c = cu, = 0, = 0effective stiffness parameters E50´, ´

25

Method C (analysis in terms of total stresses):type of material behaviour: drainedtotal strength parameters c = cu, = 0, = 0undrained stiffness parameters Eu, u = 0.495

Notes on different methods:

Method A: recommended soil behaviour is always governed by effective stresses increase of shear strength during consolidation includedg g essential for exploiting features of advanced models such as the

Hardening Soil model, the Soft Soil model and the Soft Soil Creep model

Method B: only when no information on effective strength parameters is

available cannot be used with the Soft Soil model and the Soft Soil Creep

d l

26

model

Method C: NOT recommended no information on excess pore pressure distribution (total stress

analysis)



14

Consider fully undrained isotropic elastic behaviour(Mohr Coulomb in elastic range)

pw = p > p´ = 0

centre of Mohr Circle remains at the same point centre of Mohr Circle remains at the same point

'cos'c'sin2

1c o'

yo'xu

27

Fig.6 Mohr Circle for evaluating undrained shear strength (plane strain)

28



15

Undrained strengths of MC vs Real NC Soils

29

Factor of Safety of Cuts/ExcavationsCritical FS is Long-term unloading condition,

For permanent cuts drained strength is key parameter for safe design

For temporary cuts, need to consider if

30

undrained or partially drained condition



16

Factor of Safety of EmbankmentsCritical FS is Short-term loading condition,

undrained strength isundrained strength is key parameter for safe design

31

Example of Underwater CUT Slope (Unloading Problem)

LIMIT EQUILIBRIUM ANALYSIS OF CUT SLOPESThe figures below show the results of SLOPE/W calculations of FS for a underwater cut slope in the undrained and drained condition by Bishop's Simplified methoddrained condition, by Bishop s Simplified method.Drained and Undrained ParametersThe drained parameters are c'=2 kPa, '=240, =16 kN/m3

The equivalent undrained parameters are obtained from:

0 5924i1'i1K

kPa83.124cos2c clay; of top At

'sin cosφoc'c

0

0u

,mu

32

kPa/m 1.94 24 sin 4.77 'sin

kPa/m 4.77 0.5916/2 K12

0.5924sin -1 'sin1K

0,m

0

,v,

m

00



17

Bishop‘s FS for Drained CUTCut Slope in Clay (Drained)

1.403

Description: ClaySoil Model Mohr Co lomb

Water

Water Level

atio

n (m

)

12

14

16

18

20

22

24

26

33

Soil Model: Mohr-CoulombUnit Weight: 16Cohesion: 2Phi: 24

1:2 Cut

Distance (m)

0 5 10 15 20 25 30 35 40 45 50 55 60

Ele

va

0

2

4

6

8

10

12

Bishop‘s FS for UnDrained CUTCut Slope in Clay (UnDrained)

2.085

Water

Water Level

tion

(m)

12

14

16

18

20

22

24

26

34

Description: ClaySoil Model: S=f(datum)Unit Weight: 16C - Datum: 1.83Rate of Increase: 1.94Datum (elevation): 20

1:2 Cut

Distance (m)

0 5 10 15 20 25 30 35 40 45 50 55 60

Ele

vat

0

2

4

6

8

10

12



18

PLAXIS Analysis Cases Drained Analysis with c’=2 kPa and ’=24o

Method A (analysis in terms of effective stresses):type of material behaviour: undrainedtype of material behaviour: undrainedeffective strength parameters c´, ´, ´effective stiffness parameters E50´, ´ Method B (analysis in terms of effective stresses):type of material behaviour: undrainedundrained strength parameters c = cu, = 0, = 0effective stiffness parameters E50´, ´

35

effective stiffness parameters E50 , Method C (analysis in terms of total stresses):type of material behaviour: drainedtotal strength parameters c = cu, = 0, = 0undrained stiffness parameters Eu, u = 0.495

Drained CUT, Plaxis FS=1.39 cf LE=1.40

Drained Analysis withEffective strength parameters c´=2 kPa, ´=24, ´=0Effective stiffness parameters E50´=15000 kPa, ´=0.2p ,

36

ONLY ONE POSSIBLE METHOD IN DRAINED ANALYSIS ie Drained Strength and Stiffness parameters

Solutions by FEM and LE will agree very well



19

Method A - UnDrained CUT plus Full Consolidation Plaxis c/phi FS=1.37 cf LEM=1.40

Method A (undrained)Effective strength parameters c´=2 kPa, ´=24o, ´=0o

Effective stiffness parameters E ´=15000 kPa ´=0 2Effective stiffness parameters E50 =15000 kPa, =0.2

Slip circle same as Drained Case

37

Method A - UnDrained CUT, Plaxis FS=2.26 cf LE=2.09

Method A (in terms of effective stresses, undrained)Effective strength parameters c´=2 kPa, ´=24o, ´=0o

Effective stiffness parameters E50´=15000 kPa ´=0 2Effective stiffness parameters E50 =15000 kPa, =0.2

38

• Deeper slip surface than Drained Case

• FEM (A) and LE not identical because undrained strength profile in the two cases are slightly different

• But slip surface of FEM (A) and LE are nearly identical



20

Method B - UnDrained CUT, Plaxis FS=2.13 cf LE=2.09

Method B (in terms of effective stresses, undrained)Undrained strength parameters c=1.83 kPa, ∆c=1.94 kPa, =0, =0 0, 0Effective stiffness parameters E50´=15000 kPa, ´=0.2

39


• FEM (A and B) and LE not identical because undrained strength profile in all three cases are slightly different

• But slip surface of FEM (A and B) and LE are nearly identical

Method C - UnDrained CUT, Plaxis FS=2.14 cf LE=2.09

Method C (in terms of total stresses, Drained)Total strength parameters c=1.83 kPa, ∆c=1.94 kPa, =0 =0 0, 0Undrained stiffness parameters E50=18625 kPa, =0.49

40


• FEM (A,B,C) and LE not identical because undrained strength profile in all 4 cases are slightly different; but B and C are nearly identical

• But slip surface of FEM (A,B,C) and LE are nearly identical



21

SUMMARY OF FS FOR CUT SLOPES

Analysis Condition PLAXIS SLOPE/W

Drained (Method A) 1 39 1 40Drained (Method A) 1.39 1.40

A+Consolidation 1.37 1.40

Undrained (A) 2.26 2.09

Undrained (B) 2.13 2.09

Undrained (C) 2.14 2.09

41

( )

• Only Drained analysis is FEM and LE identical

• In Undrained analysis there are differences in strength profiles

• Undrained plus Consolidation is close to Drained Case

Embankment Undrained Analysis (Loading Problem)

Embankment on Clay(Total Undrained Condition) Slope/W FS=1.029

PLAXIS M th d A1.029

Description: fillSoil Model: Mohr-CoulombUnit Weight: 20Cohesion: 0Phi: 33Piezometric Line #: 1

Description: ClaySoil Model: S=f(datum)Unit Weight: 16C - Datum: 1.83Rate of Increase: 1.94Datum (elevation): 0

Water Table

Hei

ght

(m)

-8

-6

-4

-2

0

2

4

6

8

10

12

PLAXIS Method A FS=1.019

42

Datum (elevation): 0Piezometric Line #: 1

Distance (m)

0 10 20 30 40-10

-8

For the same Undrained Strength profile,

Slip surface in FEM and LE are nearly identical



22

Embankment Drained Analysis

2.592

Embankment on Clay (Drained Condition)

Drained FS = 2.52

Description: ClaySoil Model: Mohr-CoulombUnit Weight: 16Cohesion: 2Phi: 24Piezometric Line #: 1

Description: fillSoil Model: Mohr-CoulombUnit Weight: 20Cohesion: 0Phi: 33Piezometric Line #: 1 Water Level

0 10 20 30 40-10

-8

-6

-4

-2

0

2

4

6

8

10

Undrained Method A+ Consolidation FS=2.11

43

Drained Analysis: FEM and LE nearly identical results

Consolidation Analysis is not the exactly same as Drained response

SUMMARY Undrained analysis should be performed in effective stresses

and with effective stiffness and strength parameters Undrained Analysis with Full Consolidation may not agree with y y g

Drained Analysis due to different end state stress states

Note that for NC-soils in general

factor of safety against failure is lower for short term

(undrained) conditions for loading problems (e.g.

embankments)

44

factor of safety against failure is lower for long term (drained)

conditions for unloading problems (e.g. excavations)



23

Case 1 - WCRS Excavation Example of Deep Excavation

45

1D Closed Box Swelling

46



24

Effects of PermeabilityHeave at A (0.5m below Fmn Level)

0.03

Heave at A [m]

Uy-A(0.5...

k=1e-9

0.01

0.015

0.02

0.025UND

DRN

k=1e-7

k=1e-8

47

0 50 100 150 200-5e-3

0

5e-3

Time [day]

Effects of PermeabilityExc PP at D (1.5m below Fmn Level)

250

Excess PP at D(1.5m) [kN/m2]

EPP-D(1...

k=1e 9

100

150

200

k=1e-9

UND

DRN

k=1e-7

k=1e-8

48

0 50 100 150 2000

50

Time [day]



25

Effects of PermeabilityExc PP at E (6.1m below Fmn Level)

250

Excess PP at E(6.1m) [kN/m2]

EPP-E(6...

k 1 9

100

150

200

k=1e-9

UND

DRN

k=1e-7

k=1e-8

49

0 50 100 150 2000

50

Time [day]

Excavate to Formation Levelk=1e-7 m/s k=1e-8 m/s k=1e-9 m/s

50



26

WCRS – CPG Design Mesh

• Compare Drained and Undrained Case

• Cases at k=1e-7m/s, 1e-8m/s, 1e-9m/s

51

WCS Soil Undrained Strength Profile in Plaxis

95

100

105

Cu=Method B or C

Cu-Method A

70

75

80

85

90

Dep

th (

m)

Cu Method A

52

60

65

0 50 100 150 200 250 300 350

Cu (kPa)

'sin)1(2

1'cos' ' vKocCu Method A, Cu is:



27

Drained and Undrained (Method A)Displacements at Formation Level

Undrained Drained

53

Drained and UndrainedBMs at Formation Level

Undrained Drained

54



28

Consolidation Analysisassume: k=1e-7, 1e-8 and 1e-9 m/s

55

Cases of k=1e-7 to 1e-9 m/sDisplacements at Formation Level

56k = 1e-7 m/s k = 1e-8 m/s k = 1e-9 m/s



29

Cases of k=1e-7 to 1e-9 m/sBMs at Formation Level

57k = 1e-7 m/s k = 1e-8 m/s k = 1e-9 m/s

Cases of k=1e-7 to 1e-9 m/sExcess PP at Formation Level

58k = 1e-7 m/s k = 1e-8 m/s k = 1e-9 m/s



30

Wall Deflection at B (15/83.85 – 1.65m above FL)

0.2

Ux at B [m]

Ux at B

DRN

0.1

0.15

UND

k=1e-7

k=1e-8

k=1e-9

59

0 50 100 150 200 2500

0.05

Time [day]

Heave at C(0/78.7 – 3.5m below FL)

0.03

0.035

Heave at C [m]

Uy at C

DRN

5e-3

0.01

0.015

0.02

0.025

0.03UND

k=1e-7

k=1e-8

k=1e-9

60

0 30 60 90 120-5e-3

0

5e 3

Time [day]



31

One North StationWT& E tiWT& Excavation next to INSEAD

Is it Drained or Undrained?

61

Depth Vs. Log (Permeability, m/s) (One North and Fusionpolis Site)

0

-1.00E+01 -9.00E+00 -8.00E+00 -7.00E+00 -6.00E+00 -5.00E+00 -4.00E+00 -3.00E+00 -2.00E+00 -1.00E+00 0.00E+00

Log (Permeability, m/s)

Field Permeability Tests Data from One North/Fusionpolis Site

(Jurong Formation Residual Soils)

5

10

15

20 De

pth

, m

62

25

30

35

Single Packer Test

Falling Head Test

Variable HeadPermeability Test



32

WT-7 next to INSEAD – D1.8m CBP Wall with 30m deep cut,

20m soils, 10m rock excavation

Sandy Clay G/Sa/SC=0/25/75

Sandy Silt G/Sa/SC=0/32/68

Sandy Clayey Silt G/Sa/SC=1/17/82

63

One North - WT7 I19after cast base slab and remove lowest anchor

0.00

5.00

0 20 40 60 80 100 120Wall Deflection (mm)

Wall Response is much closer to Drained Behavior at One North Wall Type 7

10.00

15.00

20.00

25.00

Dep

th (

m)

Drained

64

30.00

35.00

40.00

Drained

Undrained

I19



33

CONCLUSIONSFrom the above study, the following conclusions can be made:• The site at WCS showed fairly consistent thick layers of sandy soils with

GSD consisting of more than 60% sands and gravels.• Limited field permeability tests showed k-values in these soils ranging from

1e-5 to 1e-7 m/s.1e 5 to 1e 7 m/s.• These soils are likely to have k-values > 1e-6 m/s, therefore, they should be

modeled as drained materials.• The consolidation parametric studies showed that with soils of stiffnesses

greater than 20,000 kPa, k=1e-7 m/s will result in drained response over any reasonable construction period of 1.5 to 2 years.

• Experience from the recent One North excavation supports this observation.• Undrained analysis cannot apply to this site.• Consolidation analysis must be done very carefully to reflect the true

65

y y ystiffnesses and permeabilities of the site soils, and this will show results very close to fully drained behavior.

• It is more prudent to design the excavation system using fully drained analysis for this site

Compare CRISP with Plaxis – 1D swelling box experiment

Lower GWT to each top levelExcavate 2m in 30 days

Excavate 3.5m in 30 days

Excavate 4m in 30 days




Lower GWT to each top level

Set PP=0 at each top level

66

Track PP at 1m and 6m below last excavated level



34

Swell at 1m below FML

67

Swell at 6m below FML

68



35

PP in CRISP and Plaxis

• CRISP and Abacus are formulated in terms of active (total) pore pressures i.e.( ) p p

• U = Uss + Uexcess

• Plaxis is formulated in terms of Excess Pore Pressures: U = Uexcess

• Uexcess is produced by Undrained loading or unloading of soil clusters specified as “Undrained” type

69

• Steady PP is obtained from phreatic GWT or Seepage computation by Plaxis or Plaxflow program

• Active PP = Uss + Uexcess

PP in CRISP and Plaxis• In Sage Crisp, the “EXCESS” pore pressure always refer to the

original in-situ definition of PWP, regardless of changes of phreatic levels. Thus, it is not the conventional definition ofphreatic levels. Thus, it is not the conventional definition of “EXCESS” pore pressure in typical soil mechanics sense. Rather, it is a reflection of the “incremental variation” of PWP in reference to the original in-situ PWP.

• While in Plaxis, the “EXCESS” is exactly the same definition of conventional definition in typical soil mechanics sense with the “EXCESS” PWP referring to the PWP in excess of the phreatic line.

• Thus, there is some subtle difference between the two “EXCESS” PWP from the two software and can not be

70

EXCESS PWP from the two software and can not be compared directly. As such, the total PWP at a same point was compared instead which expected to give roughly the same values, and it is a indicator of variation of PWP during various stages of constructions and accompanying consolidations.



36

Active (Total) PP at 1m below FML

71

Active (Total) PP at 6m below FML

72



37

Case 2 - Barcelona Breakwater

73

Barcelona breakwater

y

CaissonRubble

x

A

A

0 1

23 45 6

78

9 10

11

12 13

Soft silty clay

20m

50m

74

Gravel



38

Barcelona breakwater: stages (1)

D d iDredging

Bench construction

75

Consolidation

Barcelona breakwater stages (2)

Caisson constructionconstruction

Consolidation

76

Storm loading



39

Initial pore pressures

Active pore pressures

77Groundwater head

Pore pressures after placing the bench

Active pore pressures

78Excess pore pressures



40

Excess pore pressures during consolidation

Initial

After 30 days

79

Final

Displacements during construction and consolidation

Bench construction

80Consolidation



41

Caisson construction

Displacements

81Incremental shear strains

Excess pore pressures during consolidation (caisson)

Initial

After 30 days

82

Final



42

Failure (factor of safety)

Incremental displacements

Factors of safety

After construction

83Incremental shear strains

FS=1.06

After 30 daysFS=1.60

End of consolidationF=1.74

Documents

2D and 3D Consolidation - Biot Theory (OCT2011 Color)