Mohr-Coulomb model and soil stiffness

Mohr-Coulomb model

The Mohr-Coulomb model Dennis Waterman Plaxis bv

Mohr-Coulomb model and soil stiffness Objectives: •

To indicate features of soil behaviour

•

To formulate Hooke’s law of isotropic linear elasticity

•

To formulate the Mohr-Coulomb criterion in a plasticity framework

•

To identify the parameters in the LEPP Mohr-Coulomb model

•

To indicate the possibilities and limitations of the MC model

CG1 - Santiago, Chile - Octubre de 2012

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Mohr-Coulomb model

Features of soil behaviour •

Elasticity (reversible deformation; limited)

> stiffness

•

Plasticity (irreversible deformation)

> stiffness, strength

•

Failure (ultimate limit state or critical state)

> strength

•

Presence and role of pore water

•

Undrained behaviour and consolidation

•

Stress dependency of stiffness

•

Strain dependency stiffness

•

Time dependent behaviour (creep, relaxation)

•

Compaction en dilatancy

•

Memory of pre-consolidation pressure

•

Anisotropy (directional strength and/or stiffness)

σ yy σ yz σ zy

Concepts of soil modelling

σ zz

σ zx

σ yx σ xy σ xz

σ xx

•

Relationship between stresses (stress rates) and strains (strain rates)

•

Elasticity (reversible deformations) –

•

dσ σ=f (dεε,σ σ,h)

Plasticity (irreversible deformations) –

•

dσ σ=f (dεε)

Example: Hooke’s law

Perfect plasticity, strain hardening, strain softening

–

Yielding, yield function, plastic potential, hardening/softening rule

–

Example: Mohr-Coulomb yielding

Time dependent behaviour (time dependent deformations) –

Biot’s (coupled) consolidation

–

Creep, stress relaxation

–

Visco elasticity, visco plasticity


dσ σ=f (dεε,σ σ,t)

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Mohr-Coulomb model

Types of stress-strain behaviour σ

Linear-elastic

Non-linear elastic

σ

ε

σ

Elastoplastic

ε

Lin. elast. perfectly-plast.

ε

EP strain-hardening

σ

σ

EP strain-softening

σ

ε

ε

ε

Stress definitions • •

In general, soil cannot sustain tension, only compression PLAXIS adopts the general mechanics definition of stress and strain: Tension/extension is positive; Pressure/compression is negative

σyy σxx

σxx σyy

• •

σyy σxx

σxx σyy

In general, soil deformation is based on stress changes in the grain skeleton (effective stresses) According to Terzaghi’s principle: σ’ = σ - pw


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Mohr-Coulomb model

Elasticity: Hooke’s law 1 − ν  σ xx   ν σ    yy   ν  σ zz  E  σ  = (1 + ν )(1 − 2 ν )  0   xy   0  σ yz      σ zx   0

Inverse:  ε xx  ε   yy   ε zz  1 γ  = E xy    γ yz     γ zx 

 1 − ν  − ν  0   0   0

ν 1−ν

ν 0 0 0

ν ν 1−ν 0 0 0

0 0 0 −ν 0 0

1 2

−ν 1 −ν

−ν −ν 1

0 0 0

0 0 0

0 0 0

0 0 0

2 + 2ν 0 0

0 2 + 2ν 0

1 2

    0   0   2 + 2ν  0 0 0

0 0 0 0 −ν 0

1 2

0  0   0  0   0   −ν

 ε xx  ε   yy   ε zz  γ   xy   γ yz     γ zx 

 σ xx  σ   yy   σ zz  σ   xy   σ yz     σ zx 

Elasticity: Hooke’s law In principal stress / strain components:

ν ν  σ 1  1 −ν E σ  =  ν 1 −ν ν   2  (1 + ν )(1 − 2ν )  σ 3   ν ν 1 −ν 

 ε1  ε   2 ε 3 

In isotropic and deviatoric stress / strain components:

 p K q =  0   

0  ε v  3G  ε s 


(σ 1 + σ 2 + σ 3 )

p=

1 3

q=

1 (σ 1 − σ 2 ) 2 + (σ 2 − σ 3 ) 2 + (σ 3 − σ 1 ) 2 2

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Mohr-Coulomb model

Model parameters in Hooke’s law: dσ1 Two parameters:

- dε1 ⇓

- Young’s modulus E - Poisson’s ratio ν

dε3 ⇐

- σ1 Meaning (axial compr.):

E

dσ 1 d ε1

E = ν=−

1 ν

dε3 dε1

- ε1 1

ε3

Alternative parameters in Hooke’s law: Shear modulus:

G =

d σ xy dγ

xy

=

⇒ dγxy

E 2 (1 + ν )

Bulk modulus:

dp

dp E K= = dε v 3(1 − 2ν ) Oedometer modulus:

Eoed

E (1 − ν ) dσ = 1= dε1 (1 + ν )(1 − 2ν )


dσxy

dεv - dσ1 ⇓ - dε1

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Mohr-Coulomb model

Hooke’s law for effective stress rates The modeling of non-linear soil behaviour requires a relationship between effective stress rates (dσ’ ) and strain rates (dε)

ν' 1 −ν ' ν '  dσ 'xx   ν ' 1 −ν ' ν '  dσ '  yy     ν'  dσ 'zz  ν ' 1 −ν ' E'   = 0 0  dσ 'xy  (1 +ν ')(1 − 2ν ')  0  0  dσ ' yz  0 0    0 0  dσ 'zx   0 Symbolic:

dσ ' = D d ε

0

1 2

0

0

0

0

−ν ' 0 0

0 1 2

−ν ' 0

( )

dε = D

⇔

e

0

0  0  0   0  0   1 −ν ' 2

e −1

 d ε xx  dε   yy   d ε zz     d γ xy   d γ yz     d γ zx 

dσ '

Plasticity Basic principle of elasto-plasticity:

ε ij = ε ije + ε ijp dε ij = dε ije + dε ijp

(total strains) (strain rates)

Elastic strains: Hooke’s law Plastic strains: 3 questions 1. Does plasticity occur? 2. If so, in what direction? 3. How much plasticity?


-> yield function -> potential function -> magnitude dλ

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Mohr-Coulomb model

Plasticity – does plasticity occur? Determination based on yield function f = f (σ’,ε) • • •

If f < 0 If f = 0 and df < 0 If f = 0 and df = 0

Pure elastic behaviour Unloading from plastic state (= elastic behaviour) Elastoplastic behaviour

Plasticity – does plasticity occur? Yield function f is (a.o.) a function of the stress state → f=0 can be represented as a border in the stress space (yield contour) Within the yield contour: On the yield contour: Outside the yield contour:

f<0 f=0 f>0

(impossible stress state)

f=0 Condition: Yield contour must be convex

f<0 f>0


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Mohr-Coulomb model

Plasticity – in what direction? Determination based on potential function g = g (σ’,ε) The direction of plastic strain is determined by the vector Perpendicular to the plastic potential function Metals (a.o): Soils:

f = g (associated flow) f ≠ g (non-associated flow)

q,εs

q,εs

g f=g

f p,εv

p,εv

Plasticity – how much? Determination based on magnitude scalar dλ λ The magnitude of plastic strain can be found with the so-called consistency condition, stating that for plasticity the stress state should remain on the yield surface:

df =

∂f ∂f dσ + dε = 0 ∂σ ∂ε


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Mohr-Coulomb model

Plasticity Basic principle of elasto-plasticity:

ε ij = ε ije + ε ijp dε ij = dε ije + dε ijp

(total strains) (strain rates)

Elastic strain rates:

dε ije = (D e )ijkl dσ 'kl −1

Plastic strain rates:

dε ijp = dλ

∂g ∂σ 'ij

dλ = scalar; magnitude of plastic strains dg/dσ = vector; direction of plastic strains g = plastic potential function

The Mohr-Coulomb failure criterion Origin:

σ’n

F T

τ

T ≤ A + F tanϕ

Coulomb:

τ ≤ c’ - σ’n tanϕ’ τ

T ϕ

ϕ’

A

c’ F


σ’n

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Mohr-Coulomb model

The Mohr-Coulomb failure criterion σ’n In general:

τ σ’3 θ

σ’1 The condition τ ≤ c’ - σ’n tanϕ’ must hold for arbitrary angle θ

The Mohr-Coulomb failure criterion τ c cosϕ -s* sinϕ

MC criterion: t*≤ c cosϕ - s* sinϕ

t* ϕ c

-σ3

-σ1

-σn

-s*


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Mohr-Coulomb model

The Mohr-Coulomb failure criterion MC criterion:

t*≤c’ cosϕ’ - s* sinϕ’ t* = ½(σ’3 - σ’1) s* = ½(σ’3+σ’1)

1 2

(σ '3 −σ '1 ) ≤ c' cos ϕ ' − 12 (σ '3 +σ '1 )sin ϕ ' − σ '1 ≤

2c' cos ϕ ' 1 + sin ϕ ' − σ '3 1 − sin ϕ ' 1 − sin ϕ '

Note: Compression is negative and σ’1≤ σ’2≤ σ’3

Visualisation of the M-C failure criterion τ ϕ’ c’ σ’n -σ’1

a=

2c' cos ϕ ' 1 − sin ϕ '

b=

1 + sin ϕ ' 1 − sin ϕ '

b 1 a -σ’3


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Mohr-Coulomb model

Full Mohr-Coulomb criterion σ1

(σ '3 −σ '2 ) ≤ c' cos ϕ ' − 12 (σ '3 +σ '2 )sin ϕ ' 1 1 2 (σ '2 −σ '3 ) ≤ c ' cos ϕ ' − 2 (σ '2 +σ '3 ) sin ϕ ' 1 1 2 (σ '3 −σ '1 ) ≤ c ' cos ϕ ' − 2 (σ '3 +σ '1 ) sin ϕ ' 1 1 2 (σ '1 −σ '3 ) ≤ c ' cos ϕ ' − 2 (σ '1 +σ '3 ) sin ϕ ' 1 1 2 (σ '2 −σ '1 ) ≤ c ' cos ϕ ' − 2 (σ ' 2 +σ '1 ) sin ϕ ' 1 1 2 (σ '1 −σ '2 ) ≤ c ' cos ϕ ' − 2 (σ '1 +σ '2 ) sin ϕ ' 1 2

σ3

σ2

Reformulation into yield functions 1 2

(σ '3 −σ '1 ) ≤ c' cos ϕ ' − 12 (σ '3 +σ '1 )sin ϕ '

f 2b = 12 (σ '3 −σ '1 ) + 12 (σ '3 +σ '1 )sin ϕ '−c' cos ϕ '


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Mohr-Coulomb model

Reformulation into yield functions f1a = 12 (σ '3 −σ '2 ) + 12 (σ '3 +σ '2 ) sin ϕ '−c' cos ϕ '

σ1

f1b = 12 (σ '2 −σ '3 ) + 12 (σ '2 +σ '3 ) sin ϕ '−c' cos ϕ '

f 2 a = 12 (σ '1 −σ '3 ) + 12 (σ '1 +σ '3 )sin ϕ '−c' cos ϕ ' f 2b = 12 (σ '3 −σ '1 ) + 12 (σ '3 +σ '1 )sin ϕ '−c' cos ϕ '

σ2

σ3

f 3a = 12 (σ '2 −σ '1 ) + 12 (σ '2 +σ '1 )sin ϕ '−c' cos ϕ ' f 3b = 12 (σ '1 −σ '2 ) + 12 (σ '1 +σ '2 )sin ϕ '−c' cos ϕ '

Parameters: Effective cohesion (c’) and effective friction angle (ϕ’)

Plastic potentials of the M-C model g1a = 12 (σ '3 −σ '2 ) + 12 (σ '3 +σ '2 ) sinψ − c' cosψ g1b = 12 (σ '2 −σ '3 ) + 12 (σ '2 +σ '3 ) sinψ − c' cosψ

g 2 a = 12 (σ '1 −σ '3 ) + 12 (σ '1 +σ '3 ) sinψ − c' cosψ g 2b = 12 (σ '3 −σ '1 ) + 12 (σ '3 +σ '1 ) sinψ − c' cosψ

g 3a = 12 (σ '2 −σ '1 ) + 12 (σ '2 +σ '1 )sinψ − c' cosψ g 3b = 12 (σ '1 −σ '2 ) + 12 (σ '1 +σ '2 ) sinψ − c' cosψ Dilatancy angle ψ instead of friction angle ϕ Motivation based on simple shear test


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Mohr-Coulomb model

Failure in a simple shear test: dε yy dγ xy

=

dε yyp dγ xyp

= tanψ

σxy

γxy εyy

ψ

dilatancy

γxy

The LEPP Mohr-Coulomb model Linear-elastic perfectly-plastic stress-strain relationship - Elasticity: - Plasticity:

Hooke’s law Mohr-Coulomb failure criterion

The LEPP model with Mohr-Coulomb failure contour is in PLAXIS called the Mohr-Coulomb model For this model: Plasticity = Failure This does NOT apply to all models!!!


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Mohr-Coulomb model

The LEPP Mohr-Coulomb model Model parameters: -

Young’s modulus (stiffness) Poisson’s ratio Cohesion Friction angle Dilatancy angle

E ν

c

ϕ ψ

Model parameters must be determined such that real soil behaviour is approximated in the best possible way

Parameter determination Parameter determination from: •

Laboratory tests (triaxial test (CD, CU), oedometer test or CRS, simple shear test, …)

•

Field tests (SPT, CPT, pressure meter (Menard, CPM, SBP), dilatometer, …)

•

Correlations with qc , PI , RD and other index parameters

•

Rules-of-thumb, norms, charts, tables

•

Engineering judgement


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Mohr-Coulomb model

MC approximation of a CD triax. test σ1-σ3

σ’3 = confining pressure

E ’50

2c 'cos φ '− 2σ '3 sin φ ' 1 − sin φ '

-ε1

εv 2 sinψ 1 − sinψ

-ε1

1-2ν’

MC approximation of a compression test -σ1

Eoed -ε1

Eoed =

(1 +ν )(1 − 2ν ) E (1 −ν )


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Mohr-Coulomb model

Possibilities and limitations of the LEPP MohrCoulomb model Possibilities and advantages – – – – – –

Simple and clear model First order approach of soil behaviour in general Suitable for many practical applications Limited number and clear parameters Good representation of failure behaviour (drained) Dilatancy can be included

σ1

σ2

σ3

Possibilities and limitations of the LEPP Mohr-Coulomb model Limitations and disadvantages – – – – – – –

Isotropic and homogeneous behaviour Until failure linear elastic behaviour No stress/stress-path/strain-dependent stiffness No distinction between primary loading and unloading or reloading Dilatancy continues for ever (no critical state) Be careful with undrained behaviour No time-dependency (creep)

σ1

σ2


σ3

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Mohr-Coulomb model and soil stiffness

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