Rak-50 3149 f. l6- Drained Undrained Analysis

Drained and Undrained

Analysis

1

Analysis

Prof. Minna KarstunenUniversity of Strathclyde

Thanks to Dennis Waterman, Antonio Gens, Marcelo Sanchez & Helmut Schweiger

Outline

• Drained / undrained conditions

• Modelling undrained behaviour with Plaxis

• Influence of constitutive model and parameters – Influence of dilatancy

2

– Influence of dilatancy

– Undrained behaviour with Mohr-Coulomb Model

– Undrained behaviour with Hardening Soil Model

• Summary

Nicoll

Highway, Highway,

Singapore

Drained / Undrained Conditions

In undrained conditions, no water movementtakes place and, therefore, excess pore pressuresbuild up

∆u ≠ 0, ∆σ ≠ ∆σ'∆u ≠ 0, ∆σ ≠ ∆σ'

In drained conditions, no excess pore pressuresbuild up

∆u = 0, ∆σ = ∆σ'


• Drained analysis appropriate when

– Permeability is high

– Rate of loading is low

– Short term behaviour is not of interest for – 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 < 0.10 (U < 10%) use undrained analysis

T > 0.40 (U > 70%) use drained analysis

tDγ

EkT

2

w

oed====

k = Permeability

Eoed = Oedometric modulus = 1/mv

γw = Unit weight of water

D = Drainage length

t = Construction time

T = Dimensionless time factor

U = Degree of consolidation

Undrained Behaviour

Implications of undrained soil behaviour:

– Excess pore pressures are generated

– No volume change

In fact small volumetric strains develop because a In fact small volumetric strains develop because a

finite (but high) bulk modulus of water is introduced

in the finite element formulation

– Predicted undrained shear strength depends

on soil model used

– Assumption of dilatancy angle has serious

effect on results

Triaxial test (NC)Typical results from drained (left) and undrained (right) triaxial tests on normally

consolidated soils (from Atkinson & Bransby, 1978)

Triaxial test (OC)Typical results from drained (left) and undrained (right) triaxial tests on overconsolidated soils

Stress Paths in Undrained Triaxial Test – NC / OC

1 3 2

t−

=σ σ

1 3

1 3

2

2

2

's

s

′ ′+

′ ′+

=

=

σ σ

σ σ

Strength Parameters According to MC

� Mohr-Coulomb parameters in terms of effective stress

tancτ σ ϕ′ ′ ′= +ϕ′

τ

c′ σ ′

11 33 sin ; sin 2 tan tan2

c c′ ′ ′ ′ + ′ ′ ′= + = + ′ ′

− σ σϕ

σ

ϕ ϕ

σ

In terms of principal effective stresses

Strength Parameters According to MC

� Mohr-Coulomb parameters in terms of total stresses

� Only undrained conditions!

τ

uC

total stressestancτ σ ϕ′ ′ ′= +

� Soil behaves as if it was cohesive

� : undrained shear strength

� only changes if drainage occurs (no change if undrained conditions prevail)

( )u u

c s=

1 3

2 F

σ σ−

,σ σ′

uC

u-C

1 3

2 F

σ σ′ ′−

Effective stresses

uc

ucτ =

What is the critical case: drained or

undrained?

NC

OC

Loading

Unloading

t

NC

OC

note that for soils in general:

•level of safety against failure is lower for short term (undrained) conditions for loading problems (e.g. embankment)

•level of safety against failure is lower for long term (drained) conditions for unloading problems (e.g. excavations)

however …

Unloading

s, s’

What is the critical case: drained or

undrained?t

NC

OC

Loading

•For very soft NC soil, factor of safety against failure may be lower for short term (undrained) conditions for unloading problems (e.g. excavations)

•For very stiff OC soil, factor of safety against failure may be lower for short term (undrained) conditions for loading problems (e.g. embankment)

s, s’

Unloading

FE Modelling of Undrained

Behaviour

What Plaxis does when specifying

type of material behaviour: undrained ?

The issue:

•Constitutive equations are formulated in terms of σ’•Constitutive equations are formulated in terms of σ’

•We need to compute D

εσ ∆=∆ '' D

D∆σ = ∆ε

FE Modelling of Undrained Behavior

Principle of effective stress �

with

since the strains are the same in each phase,

fσσσ ∆+∆=∆ '

[ ]Twwwf ppp 000∆∆∆=∆σ

εσ ∆=∆ ff D33

33

00

01ef KD =

n

KK

f

e ≅

εσ ∆=∆ '' D

' ' ( ' )f f fD D D D D∆σ + ∆σ = ∆σ = ∆ε = ∆ε + ∆ε = + ∆ε

pore fluid stiffness, related to thebulk modulus of pore fluid (water) Kf

fDDD += '

33 n

� We need D D∆σ = ∆ε

FE Modelling of Undrained Behavior

� Example: linear elastic model + plane strain

∆εDσ∆ ′=′

4 2 2' ' ' 0

3 3 3

2 4 2' ' ' 0

3 3 3

2 2 4' ' ' 0

exx xx

e

yy yy

ezz zz

K G K G K G

K G K G K G

K G K G K G

+ − −

′σ ε ′σ ε− + − = ′ σ ε − − +′σ γ

& &

& &

& &

)1(2)21(3 ν+=

ν−=

EG

EK fDDD += '

'G G=

2 2 4' ' ' 0

3 3 3

0 0 0

exy xy

K G K G K G

G

− − +′σ γ

& &

4 2 20

3 3 3

2 4 20

3 3 3

2 2 40

3 3 3

0 0 0

exx xx

e

yy yy

ezz zz

exy xy

K G K G K G

K G K G K G

K G K G K G

G

+ − −

σ ε σ ε− + − = σ ε − − +σ γ

& &

& &

& &

& &

=∆σ D∆ε

FE Modelling of Undrained Behavior� Example: linear elastic model + plane strain

4 2 2' ' ' 0

3 3 3 02 4 2

0' ' ' 0 3 3 3

0

e e e

e e e

K G K G K G

K K K

K K KK G K G K GD

K K K

+ − −

− + − = +

fDDD += '

02 2 4

' ' ' 0 0 0 0 03 3 3

0 0 0

e e eK K K

K G K G K G

G

− − +

4 2 2' ' ' 0

3 3 3

2 4 2' ' ' 0

3 3 3

2 2 4' ' ' 0

3 3 3

0 0 0

e e e

e e e

e e e

K G K K G K K G K

K G K K G K K G KD

K G K K G K K G K

G

+ + − + − +

− + + + − + = − + − + + +

FE Modeling of Undrained Behavior� Example: linear elastic model + plane strain

fDDD += '

4 2 2' ' ' 0

3 3 3

2 4 2' ' ' 0

' 3 3 3

2 2 4' ' ' 0

3 3 3

0 0 0

e e e

e e e

e

e e e

K G K K G K K G K

K G K K G K K G KD D D

K G K K G K K G K

G

+ + − + − +

− + + + − + = + = − + − + + + 0 0 0 G

4 2 20

3 3 3

2 4 20

3 3 3

2 2 40

3 3 3

0 0 0

K G K G K G

K G K G K GD

K G K G K G

G

+ − −

− + − = − − + ' eK K K= +

4 4'

3 3e

K G K K G+ + = +

FE Modelling of Undrained

Behavior

All the above (which is valid for any soil (or model) for which the principle of effective stress applies) can be easily combined with the FEM

• instead of specifying the components of D, specify D'‚ and Ke

• when calculating stresses,

fDDD += '

εσ ∆=∆ '' D

f e vKσ ε∆ = ∆

fσσσ ∆+∆=∆ '

Undrained Behaviour with PLAXIS

A value must be set for Ke

• real value of Ke = Kw/n ~ 1•106 kPa (for n = 0.5)

• in fact, the pore-fluid is assigned a bulk modulus that is substantially larger than that of the soil skeleton (which ensures that during undrained loading the volumetric strains are very small)are very small)

(((( ))))(((( ))))

(((( ))))u

u

u

uw

total

213

1G2

213

E

n

K'KK

νννν−−−−

νννν++++====

νννν−−−−====++++====

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


uu

(((( ))))(((( ))))(((( ))))'1213

1'EK

u

u

totalνννν++++νννν−−−−

νννν++++==== assuming νu = 0.495

• Note: this procedure gives reasonable results only for ν' < 0.35 !

• Note: in Version 8 B-value can be entered explicitly for undrained materials


Skempton’s B parameter (undrained conditions)

( )w

w

w

KK'

KB

K

K'BBB

+=

+

=σ∆+σ∆+σ∆=∆=∆n

n

ppw ;

1

1 ;

3

1321

therefore, if Kw

is very large (compared to K’), B is very close to 1


Method A (analysis in terms of effective stresses):type of material behaviour: undrainedeffective strength parameters (c', ϕ', ψ‘)effective stiffness parameters (E50', ν‘)

Method B (analysis in terms of effective stresses):type of material behaviour: undrainedtype of material behaviour: undrainedtotal strength parameters (c = cu, ϕ = 0, ψ = 0)effective stiffness parameters (E50', ν‘)

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


(Method A)

• Analysis in terms of effective stress

• Type of material behaviour: undrained

• u changes (excess pore water pressures generated)

• Constitutive equations are formulated in terms of σ’

' 'D∆σ = ∆ε' 'D∆σ = ∆ε

In the case of Mohr Coulomb model:

effective strength parameters c’, ϕ’, ψeffective stiffness parameters E50', ν'

• the total stiffness matrix is computed as: fDDD += '


(Method A)

u

uf

t

ESPTSP

c

– single set of parameters in terms of effective stress (undrained, drained, consolidation analysis consistent)

– realistic prediction of pore pressures (if model is appropriate)

– the undrained analysis can be followed by a consolidation analysis (correct pore pressures, correct drained parameters)

– cu is a consequence of the model, not an input parameter!!

s, s’

ESP

uc

( ) )''('' ; )''(''1

0 3123122 σ∆+σ∆ν=σ∆σ∆+σ∆ν−σ∆′

==ε∆E

For plane strain: the undrained effective stress path rises vertically

• In the case of the Mohr-Coulomb model (in plane strain), it is easy to compute cu analytically

Undrained Strength for Method A

Linear Elasticity

( ) ( ) 0)'1(''3

1 '''

3

1'0' 0

'

'31321 ≈ν+σ∆+σ∆=σ∆+σ∆+σ∆=∆⇒≈∆⇒≈

∆=ε∆ pp

K

pv

( ) 0'''2

1 31 ≈∆=σ∆+σ∆ s

Effective Stress

Path, ESP B’

a’ =

c’c

osφ’

sinφ’( )1 3

2t

σ − σ=

Plane strain: effective stress path rises vertically


uc

A’

a’ =

c’c

os

( ) ( )1 3 1 3' '

' , 2 2

s sσ + σ σ + σ

= =

{ }

{ }0 0

0 0

1'cos ' 'sin ' 'cos ' sin '

2

'cos ' 1 1 sin ' ,

' ' 2

u o vo ho

u ho

v v vo

c c s c

c cK K

′ ′= + = + +

′= + + =

′

φ φ φ σ σ φ

σφφ

σ σ σ

uc

,o o

t s′

t


• The Mohr Coulomb model in terms of effective stresses OVERESTIMATES the undrained shear strength of soft clays!

s, s’

u realc

uMCc





Undrained Behaviour with PLAXIS(Method B)

• analysis in terms of effective stress• type of material behaviour: undrained • u changes

• constitutive equations are formulated in terms of σ’ (but

strength in total stresses!)

εσ ∆=∆ D

total strength parameters c = cu, ϕ = 0, ψ = 0effective stiffness parameters E50', ν'

fDDD += '

' 'D∆σ = ∆ε


3 ; 0.495

2 1u u

EE ν

ν

′= =

′+

Resulting undrained stiffness parameters

t


s, s’

uc

– parameters in terms of total stress and effective stress

– prediction of pore pressures (generally unrealistic)

– the undrained analysis should not be followed by a consolidation analysis (pore pressures unrealistic)

– cu is an input parameter!!

ESP TSP






(Method C)

TSP=ESP

t

uc

– Parameters in terms of total stress

– No prediction of pore pressures (only total stresses are obtained)

– The undrained analysis can not be followed by a consolidation analysis

– cu is an input parameter!!

s, s’


(Method C)

• Analysis in terms of total stress• Type of material behaviour: drained (in spite of modelling an undrained case)

• Porewater pressure does not change (because it is not calculated)

• Constitutive equations are formulated in terms of σ• Constitutive equations are formulated in terms of σ

εσ ∆=∆ D

total strength parameters c = cu, ϕ = 0, ψ = 0total stiffness parameters Eu, νu = 0.495

Undrained Shear Strength from

Advanced Models

� Although it is possible, in a few simple cases, to obtain

an analytical expression for cu, it is advisable to perform

a numerical “laboratory” test to check the value of

undrained shear strength actually supplied by the model

� It is important to perform the numerical “laboratory” test

under the same condition as in the analysis

� Plane strain, triaxial, simple shear

� Correct initial stresses

� Compression, extension, simple shear

� Not all cu values are achievable with a particular model

Influence of Dilatancy on

Undrained Shear Strength

If we set then, negative volumetric plastic deformations occur at failure:

0>ψ

0

e p

v v v∆ε = ∆ε + ∆ε

∆ε ≈

(elastic-plastic behavior)

(undrained conditions)

Therefore, in undrained analysis, dilatancy, , must be set to zero!

0v

∆ε ≈ (undrained conditions)

result: unlimited increase of q (or t), i.e. infinite strength!!

0 0 ' ' 0

At failure: ' 0

sin 0

p e e

v v vp K

q M p q

t s t

∆ < ⇒ ∆ > ⇒∆ = ∆ >

∆ = ∆ ⇒ ∆ >

′ ′∆ = ∆ ⇒ ∆ >

ε ε ε

ϕ

ψ

Comparison MC-HS (influence ψ)

200

225

250

275

300

Simulation of undrained triaxial compression test – MC / HS model - q vs ε1

ε1 [%]

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

q [kN

/m2]

0

25

50

75

100

125

150

175

MC non dil

MC dil

HS_1 non dil

HS_1 dil

Comparison MC-HS (influence ψ)

175

200

225

250

275

300

MC non dil

MC dil

HS_1 non dil

HS_1 dil

total stress path

Simulation of undrained triaxial compression test – MC / HS model - q vs p´

p' [kN/m2]

0.00 25.00 50.00 75.00 100.00 125.00 150.00 175.00 200.00 225.00 250.00

q [

kN

/m2]

0

25

50

75

100

125

150

175

Comparison MC-HS (influence ψ)e

xce

ss p

ore

pre

ssu

re [kN

/m2]

60

70

80

90

100

MC non dil

MC dil

HS_1 non dil

HS_1 dil

Simulation of undrained triaxial compression test – MC / HS model - ∆pw vs ε1

ε1 [%]

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

exce

ss p

ore

pre

ssu

re [kN

/m

-20

-10

0

10

20

30

40

50

Summary

• FEM analysis of undrained conditions can be performed in effectivestresses and with effective stiffness and strength parameters (Method A)

• Method A must be used:

– if consolidation/long term analysis are required

– advanced soil models are adopted

• undrained shear strength is a result of the constitutive model

• care must be taken with the choice of the value for dilatancy angle

• Methods B (and C) provide alternative ways to analyze undrained problems but:

– the constitutive model does not generally represent the true soil

behaviour (before failure)

– potentially useful for stability problems in undrained conditions

(specification of undrained shear strength is straightforward)

Date post:	19-Jan-2016
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Rak-50 3149 f. l6- Drained Undrained Analysis

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