Identidade Corporativa e a Relação Generativa entre o Ator e o Sistema

3.2 Da Natureza do Self em RI

3.2.3 Identidade Corporativa e a Relação Generativa entre o Ator e o Sistema

presented by Tan S A

National University of Singapore

Helmut F. Schweiger

Institute for Soil Mechanics and Foundation Engineering Graz University of Technology, Austria

CONTENTS

 Definition drained / undrained

 Drained / undrained soil behaviour

• Typical results from drained and undrained triaxial tests

• Skempton‘s parameters A and B

 Modelling 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

 Influence of constitutive model and parameters

• Influence of dilatancy

• Undrained behaviour with Mohr-Coulomb Model

• Undrained behaviour with Hardening Soil Model

 Summary

 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 use undrained conditions T > 0.40 use drained conditions

D t γ

E T k ₂

= oed

k = permeability E_oed= oedometer modulus γ_w = unit weight of water D = drainage length t = construction time T_v = dimensionless time factor

UNDRAINED BEHAVIOUR

 Implications of undrained soil behaviour

• excess pore pressures are generated

• no volume change

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 effects on results

Fig.1 Results from undrained triaxial tests using the Mohr-Coulomb and Hardening Soil Model

Fig.2 Typical results from drained (a) and undrained (b) triaxial tests on normally consolidated soils (from Atkinson & Bransby, 1978)

a) b)

TRIAXIAL TEST (OC) – DRAINED / UNDRAINED

Fig.3 Typical results from drained (a) and undrained (b) triaxial tests on overconsolidated soils (from Atkinson & Bransby, 1978)

a) b)

Fig.4 Typical results from undrained triaxial tests on (a) normally consolidated and (b) overconsolidated clay (from Ortigao, 1995)

∆p_w

∆pw

PORE PRESSURE PARAMETERS A AND B

water pore , vol skeleton

vol = ∆ε

∆

' K

' p

skeleton , vol

= ∆ ε

∆ 3

(

1 2 ´

)

´ ' E

K = − ν

Skempton 1954: ^∆^p^w⁼^B

[

^∆^σ₃ ⁺ ^A

(

^∆^σ₁ ⁻^∆^σ₃

) ]

- fully saturated soil

- no inflow / outflow of pore water - bulk modulus of soil grains >>>

- isotropic linear elastic material behaviour (Hooke´s law)

w w water

pore ,

vol K

p n∆

= ε

∆

assuming triaxial compression:

( )

leading to

with

PORE PRESSURE PARAMETERS A AND B

notes on parameters A and B:

for K_wlarge compared to K´, parameter B ~ 1.0 (corresponds to ∆p_w= ∆p > ∆p´ = 0)

small amount of drapped air reduces parameter B significantly (Figure 4)

parameter A depends on stress path, even for elastic material behaviour

parameter A cannot be determined a priori for complex elastic-plastic constitutive models > is a result of the model behaviour for the stress path followed

Fig.5 Dependence of pore pressure parameter B on degree of saturation

UNDRAINED BEHAVIOUR WITH PLAXIS

( ) ( )

(

)

u u w

total 31 2

1 G 2 2

1 3

E n

' K K

K − ν

= + ν

= − +

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

( )

(

1 2

) (

1 '

)

3 1 ' K E

u u

total − ν +ν

= + _{assuming ν}_u_{= 0.495}

Note:

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

- NB: in Version 8 B-value can be entered explicitely

Example 1:

E´ = 3 000 kPa, ν´ = 0.3, ν_u= 0.495

→ K´ = 2 500 kPa, K_total= 115 000 kPa → K_w/n = 112 500 kPa

' 1 nK B 1

with = = 0.978 > reasonable value for saturated soil

Example 2:

E´ = 3 000 kPa, ν´ = 0.45, ν_u= 0.495

→ K´ = 10 000 kPa, K_total= 103 103 kPa → K_w/n = 93 103 kPa B = 0.903 > poor value for saturated soil

UNDRAINED BEHAVIOUR WITH PLAXIS

Method A (analysis in terms of effective stresses):

type of material behaviour: undrained effective strength parameters c´, ϕ´, ψ´

effective stiffness parameters E₅₀´, ν´

Method B (analysis in terms of effective stresses):

type of material behaviour: undrained

undrained strength parameters c = c_u, ϕ = 0, ψ = 0 effective stiffness parameters E₅₀´, ν´

Method C (analysis in terms of total stresses):

type of material behaviour: drained

total strength parameters c = c_u, ϕ = 0, ψ = 0 undrained stiffness parameters E_u, ν_u= 0.495

Notes on different methods:

Method A:

recommended

 soil behaviour is always governed by effective stresses

increase of shear strength during consolidation included

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 model

Method C:

NOT recommended

no information on excess pore pressure distribution (total stress analysis)

UNDRAINED STRENGTH FROM MOHR CIRCLE

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

∆p_w= ∆p > ∆p´ = 0

→ centre of Mohr Circle remains at the same point

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

( )

sin ' c'cos ' 2

c_u =1 σ^'_x^o +σ^'_y^o ϕ + ϕ

Model Number E50 ref Eur

ref Eoed

ref φ Ψ c υur p^ref m K0

nc Rf

kN/m² kN/m² kN/m² ° ° kN/m² - kN/m² - - -

HS_1 30 000 90 000 30 000 35 0 / 10 0.0 0.2 100 0.75 0.426 0.9

HS_2 50 000 150 000 50 000 35 0 0.0 0.2 100 0.75 0.426 0.9

HS_3 15 000 45 000 15 000 35 0 0.0 0.2 100 0.75 0.426 0.9

HS_4 30 000 90 000 40 000 35 0 0.0 0.2 100 0.75 0.426 0.9

HS_5 30 000 90 000 15 000 35 0 0.0 0.2 100 0.75 0.426 0.9

HS_6 50 000 150 000 30 000 35 0 0.0 0.2 100 0.75 0.426 0.9

Table 1 Parameter sets for Hardening Soil model Parameters for MC Model

E = 30 000 kN/m² υur = 0.2 φ = 35°

Ψ = 0° and 10° see also Schweiger (2002)

COMPARISON MC – HS / INFLUENCE ψ

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 75 100 125 150 175 200 225 250 275 300

MC non dil MC dil HS_1 non dil HS_1 dil

Fig.7 Simulation of undrained triaxial compression test – MC / HS model - q vs ε₁

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 75 100 125 150 175 200 225 250 275 300

MC non dil MC dil HS_1 non dil HS_1 dil total stress path

Fig.8 Simulation of undrained triaxial compression test – MC / HS model - q vs p´

COMPARISON MC – HS / INFLUENCE ψ

Fig.9 Simulation of undrained triaxial compression test – MC / HS model -∆p_wvs 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 excess pore pressure [kN/m2 ]

-20 -10 0 10 20 30 40 50 60 70 80 90 100

MC non dil MC dil HS_1 non dil HS_1 dil

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

parameter A

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

MC non dil MC dil HS_1 non dil HS_1 dil

Fig.10 Simulation of undrained triaxial compression test – MC / HS model - A vs ε₁

PARAMETER VARIATION – HARDENING SOIL

p' [kN/m²]

0.00 25.00 50.00 75.00 100.00 125.00 150.00

q [kN/m2 ]

0 25 50 75 100 125 150

HS_1 HS_2 HS_3 HS_4 HS_5 HS_6 total stress path

Fig.11 Simulation of undrained triaxial compression test – HS model - q vs p´

ε₁ [%]

0.00 1.00 2.00 3.00 4.00 5.00 6.00

q [kN/m2 ]

0 25 50 75 100 125 150

HS_1 HS_2 HS_3 HS_4 HS_5 HS_6

Fig.12 Simulation of undrained triaxial compression test – HS model - q vs ε₁

PARAMETER VARIATION – HARDENING SOIL

ε₁ [%]

0.00 1.00 2.00 3.00 4.00 5.00 6.00

excess pore pressure [kN/m2]

0 10 20 30 40 50 60 70 80

HS_1 HS_2 HS_3 HS_4 HS_5 HS_6

Fig.13 Simulation of undrained triaxial compression test – HS model -∆p_wvs ε₁

ε₁ [%]

0.00 1.00 2.00 3.00 4.00 5.00 6.00

parameter A

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

HS_1 HS_2 HS_3 HS_4 HS_5 HS_6

Fig.14 Simulation of undrained triaxial compression test – HS model - A vs ε₁

In document IDENTIDADE, AUTODETERMINAÇÃO E RELAÇÕES INTERNACIONAIS: O CASO DO SAARA OCIDENTAL (página 109-116)