Basic Concepts of Plasticity and Mohr Coulomb ModelMohr Coulomb Model

Prof. Minna Karstunen

University of Strathclyde

Basic Concepts of Plasticity

Example of elasto-plastic behaviour: traction test (1D) in metals

YieldingF

AF∆L

A

C

A

Y

B

A

DO

B

A

DO

Plastic Behaviour of Soils

B

A

DO

∆

0

L

L

Y

L0

Elastic behaviour

CC

e

(logscale )σ ′sp ~Y

Idealization of elasto-plastic behaviour

Yσ

σP

Yield point, the stress

Plastic behaviour: unrestricted plastic

F

A

B

A

D

C

O

∆

0

L

L

Plastic Behaviour of Soils

εeεpεε

Yield point, the stress cannot be higher than

this value

Elastic behaviour

flow takes place at this stress level.

ε = ε e + ε p

� Idealization of elasto-plastic behaviour, different models

σ σ σ σYield limit depends on (effective) stresses

Plastic Behaviour of Soils

{ { {to ta l e la s t ic p la s tic

ε ε ε= +e pd d d

Rigid PerfectlyPlastic

Elasto-plastic perfect plasticity

Elasto-plastic hardening

Elasto-plasticsoftening

ε ε ε ε

On softening models

Y0 = yield stress

YF = failure stress

� Plastic models allow

� to determine in a direct way the ultimate states and failure

� to model irrecoverable strains

� to model changes in material behaviour

Some Basic Concepts

� to model changes in material behaviour

� to model a more proper way the behaviour of fragile or

quasi-fragile materials

Some Basic Concepts

• Strains (ε)– Total strains– Elastic strains (recoverable on unloading) – Plastic strains (not recoverable on unloading)

Total strains = Elastic strains + Plastic strains

• Stresses (σ)

Total stresses = Effective stresses + Pore Pressures

Some Basic Concepts

• Stresses are related to elastic strains even in nonlinear theories

• Stresses are stresses - there is nothing like elastic stress and plastic stress.elastic stress and plastic stress.

• We talk mainly in terms of effective stress.

Elasto-Plastic Models

• An elastic law

• A criterion for yielding (Yield function/surface)

Ideal plastic

Stress

Strain

• The direction of plastic flow (Flow rule)

• Does the yield function change due to plastic flow? If yes, how? (Hardening/Softening rule)

Stress

Strain

Strain hardening

Strain

� Used to delimit the

elastic domain

� It is a

PLASTICOn the surface

IMPOSSIBLE STATEOutside

1σ

Yield Surface

generalization of

the 1D case

� Yield limit (1-D) �

Yield surface (2D-

3D)

ELASTICInside yield

surface ( , ) 0ij i

F σ ξ =

2σ

3σ

F(σ’,hi)=0

Yield Surface

{ } { }( ) 0F dhh,d =++ σσ{ }( ) 0, =hF σ

� Fixed yield surface F (σij) = 0

� Perfect plasticity

� Expanding yield surface – F(σij,hi) = 0

� Hardening plasticity

� Contractive yield surface – F(σij,hi) = 0

Yield Surface

ij i

� Softening plasticity

� The expansion or contraction of the YS is controlled by the hardening (or softening) parameters hi

� The stress state must be either inside the surface or on the surface (stress states outside the surface are not allowed).

� Stress inside the surface F(σij,hi) < 0 � elastic strain only

� Stress on the surface F(σij,hi) = 0 � elastic and plastic strain

� The YS is often expressed in term of the stresses or stress invariants.

� p',q are typical stress variables used to describe soil behaviour and, also, to

define the YS

� Therefore typical expression of the YS are as follows:

( ), , 0f p q p′ ′ =

Yield Surface

( ), 0f hσ =

� where is a typical hardening parameter (h) used in geotechnical models.

The hardening parameter(s) control the expansion or contraction of the YS.

( )0, , 0f p q p′ ′ =

0p′

( ), 0ijf hσ =

Flow Rule

• In one-dimensional problem, it is clear that plastic strains take place along the direction of applied stress

• In 2D or 3D we need to make a

σ’1, ε1p

εp

• In 2D or 3D we need to make a hypothesis regarding the direction of plastic flow (relative magnitude of plastic strain increments)

σ’3, ε3p

� To evaluate plastic deformations the existence of a plastic potential (g or G ) is

assumed.

� The plastic potential provides the direction of the plastic strain:

0),,( =′ ζqpg

� Plastic Deformations

Plastic Potential and Flow Rule

'( , ) 0ij

g σ ζ =

� where is the parameter that control the size of the plastic potential

� It is also necessary to define the flow rule

ζ

0),,( =′ ζqpg

;p p

p q

g gd d d d

p qε λ ε λ

∂ ∂= =

′∂ ∂

( , ) 0ij

g σ ζ =

� In general, p

ij

ij

gd dε λ

σ

∂=

∂

control the magnitude of plastic deformation

control the “direction” of the plastic deformations: the “vector”

of the plastic deformations is normal to the g = constant

surfaces

� Plastic Deformations

Plastic Potential and Flow Rule

( , , ) 0g p q ζ′ =

� Yield Surface (f) and Plastic Potential (g) are generally different functions

� If f ≡ g => associated plasticity

� The components of the plastic deformations are related, i.e. there is a

coupling, which is defined by the flow rule

� The plastic deformations depend on the stress state rather than the

Plastic Potential and Flow Rule

� The plastic deformations depend on the stress state rather than the

increment of the stresses applied

Plastic Potential and Flow Rule

FLOW RULE – NON ASSOCIATEDFLOW RULE ASSOCIATED

� The flow rule defines direction of plastic strain increment

� So, we know the plastic-strain direction, but how we can determine the

magnitude?

� It is necessary to provide a description of the variation of the size and/or

position of the yield surface during plastic deformations (i.e. how the YS

evolve during yielding)

0 0 ( , )p p

p qp p ε ε′ ′=

Hardening rule

q

YS

0 00

p p

p qp p

p q

p pdp d dε ε

ε ε

′ ′∂ ∂′ = +

∂ ∂

p'po’

YS

� The plastic state is reached when the stress state is on the surface:

� It is assumed that once yield occurs (i.e. f = 0), the stresses must remain on the

( )0, , 0f p q p′ ′ =

Consistency condition

� It is assumed that once yield occurs (i.e. f = 0), the stresses must remain on the

yield surface during plastic deformation.

� This constraint is enforced by the consistency condition as follows:

0df =

0

0 0 0 0

0 0

0

0

0

p p p p

p q p q

p p

p

p

p

q

f f fdf dp dq

p q p

p p p p

p pf f f g gdf dp dq d d

p q p p q

gd d

gd d

p q

dp

dpε ε ε ε

λ λε ε

λ λε ε

∂ ∂ ∂′= + +

′ ′∂ ∂ ∂

′ ′ ′ ′∂ ∂ ∂ ∂= + = +

∂ ∂ ∂ ∂

′ ′∂ ∂∂ ∂ ∂ ∂ ∂′= + + + =

′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂

∂

′∂

′

′∂

∂

Consistency condition

0

0

0

0

0p p

p q

dp

df dp dq d dp q p p q

f fdp dq

p qd

pf

p

λ λε ε

λ

′

′= + + + = ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂′ +

′∂ ∂=

′∂∂−

′∂ ∂

14444244443

0

p p

p q

pg g

p qε ε

′∂∂ ∂+

′∂ ∂ ∂

( )0, , 0f p q p′ ′ =

Now we can determine the magnitude of the plastic strain

{ }

∂

∂=

'σλε

gdd p

0 0 0 0

0 0

1

p p p

p

q

p

p

p

p

q p q

p

p

p

f f f fdp dq dp dq

p q p q

p p p pf g g f g g

p p q

gd

p p q

f

d

d p pf g g

g

qpd

ε ε ε ε

ε

ε

εε

∂ ∂ ∂ ∂′ ′+ +

′ ′∂ ∂ ∂ ∂

′ ′ ′ ′∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂− + − +

′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂

−= ′ ′∂ ∂∂ ∂ ∂

∂ ∂=

∂∂=

′

g f g

dpp p q p

f g f g dq

∂ ∂ ∂ ′′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

Plastics Deformations

0 0

0

q

p p

p q

d p pf g g

p p q

ε

εε

′ ′∂ ∂∂ ∂ ∂ + ′ ′∂ ∂ ∂ ∂ ∂

f g f g dq

p q q q

∂ ∂ ∂ ∂ ′∂ ∂ ∂ ∂

Plastic modulus, H

H = 0 → perfect plasticity – and Eq. above not valid!

H > 0 → plasticity with hardening

H < 0 → plasticity with softening

1. Elastic Strain

2. Yield surface

′

′

′

′=

qd

pd

G

K

d

de

q

e

p

310

01

ε

ε

0),,( 0 =′′ pqpf

Theory of Plasticity

3. Plastic Potential

4. Flow rule

0),,( =′ ζqpg

;p p

p q

g gd d d d

p qε λ ε λ

∂ ∂= =

′∂ ∂

5. Hardening law

6. Plastic deformations

p

qp

q

p

pp

p

p

q

p

p

dp

dp

pd

pp

εε

εε

εε

∂

′∂+

∂

′∂=′

′=′

000

00 ),(

� Summary

Theory of Plasticity

7. Total deformations

′

∂

∂

∂

∂

∂

∂

′∂

∂′∂

∂

∂

∂

′∂

∂

′∂

∂

∂

∂

∂

′∂+

′∂

∂

∂

′∂

′∂

∂

−=

dq

pd

q

g

q

f

q

g

p

f

p

g

q

f

p

g

p

f

q

gp

p

gp

p

fd

d

p

q

p

p

p

q

p

p

εε

ε

ε

00

0

1

e pp p p

e pq q q

d d d

d d d

ε ε ε

ε ε ε

= +

Elastoplastic Matrix

{ } [ ] { }εσ dDdep

⋅=

[ ] [ ] { } { }

{ } [ ] { }eT

T

gf

eep

aDaH

bbDD

⋅⋅+=

⋅⋅−=1

β

β

{ } [ ] { }

{ }{ }

{ }{ }

{ }{ }

{ } [ ] { } { } [ ] { }f

e

fg

e

g

fg

g

T

g

eT

f

aDbaDb

fa

ga

ah

h

FH

aDaH

⋅=⋅=

∂

∂=

∂

∂=

⋅

∂

∂⋅

∂

∂−=

⋅⋅+=

;

';

' σσ

ε

β

Elastoplastic Matrix

0=∂

∂

h

F { } [ ] { }q

eT

f aDa ⋅⋅=β

1. ß is a scalar quantity, for perfect plasticity

H = 0 >

2. Dep depends on state of stress / strain and represents a

[ ] [ ] [ ] [ ]∫=ν

dVBDBKepT

2. Dep depends on state of stress / strain and represents a tangential stiffness

3. Vectors {b}f and {b}g contain derivatives of the yield function and the plastical potential function > Dep is symmetric only for associated flow rule

4. Finite element stiffness matrix is also nonsymmetric for non-associated flow rule

Mohr Coulomb Model

MohrMohr--Coulomb Idealisation of Coulomb Idealisation of GeomaterialsGeomaterials

σ’1 –σ’3

σ’3

σ’2=σ’3

σ’

ε1

σ’3

σ3

E’

Mohr Coulomb Yield/Failure ConditionMohr Coulomb Yield/Failure Condition

Yielding (and failure) takes place in the soil mass when mobilised (actual) shear stress at any plane (ττττm )becomes equal to shear strength (ττττf ) which is given by:

ττττm = c’+ σσσσ’n tanφφφφ’ = ττττf

where c’ and φφφφ’ are strength parameters.

f(σσσσ’ )= ττττ - σσσσ’n tanφφφφ ’– c’= 0

τ

Failure criterion

nσ ′

1σ ′

3σ ′

2σ ′

1 2 3σ σ σ′ ′ ′> >

Note that the value of intermediate stress (σ2’) does not influence failure

σ ′

φ ′

τ

τf

1σ ′3σ ′

AA BB

CC

DD

σ ′

9 0 φ ′− 2θ

tanf n

cτ σ φ′ ′ ′= +

c’

nσ ′AA BB

nfσ ′

4 5 2φ ′−

1 3σ σ′ ′>

4 5 2φ ′−

1σ ′

3σ ′3σ ′

1σ ′

FAILUREFAILURE

PLANESPLANES

1 3 1 3( ) ( ) sin 2 cos 0F cσ σ σ σ φ φ′ ′ ′ ′ ′ ′ ′= − − + − =

Mohr-Coulomb in Principal Stress Space

• Mohr – Coulomb failure surface is a irregular hexagon in the principal stress space

σ’3 σ’1=σ’2= σ’3

stress space

σ’1

σ’2

Mohr-Coulomb in Principal Stress Space

• It has corners that may sometimes create problems in

Mohr-Coulomb

pdεr

σ’1

problems in computations

σ’2σ’3

Flow Rule for Mohr Coulomb

For Mohr-Coulomb flow rule is defined through the ‘dilatancy angle’ of the soil.

τYield function

φ’Plastic potential function

G(σσσσ’)= ττττ - σσσσ’n tanψψψψ’ – const.= 0

where ψψψψ’ is the dilatancy angle and ψψψψ’≤≤≤≤ φφφφ’.

nσ ′

1σ ′

3σ ′

1 2 3σ σ σ′ ′ ′> >

ψ’

, τ γ&

Associated and Non-Associated Flow Rule

τ, dγ p F=0

G=0

, p

n nσ ε′ &σ’n, dεn

p

G=0

How to understand dilatancyi.e., why do we get volume changes when applying shear stresses?

ϕ = ψ + ϕi

The apparent externally mobilized angle of friction on horizontal planes (ϕ) is larger than the angle of friction resisting sliding on the inclined planes (ϕi)

strength = friction + dilatancy

How to understand dilatancy

• When dense sands or overconsolidated clays are sheared they dilate

• Larger the particle size, greater • Larger the particle size, greater the dilation

• Mohr-Coulomb idealisation implies dilation at a constant rate when soil is sheared. This is unrealistic.

MC model p'- q- space

2'3 qp +

31 '2'3' σσ −= p q−= 13 '' σσ

1 3 1 3( ) ( )sin 2 cos 0F cσ σ σ σ φ φ′ ′ ′ ′ ′ ′ ′= − − + − =

( ) 3131 '''2'3

1' σσσσ −=+= qp 3

2'32'2'3' 11

qpqp

+=+−= σσ

3

'3'2'3' 33

qpqp

−=−−= σσ

3

'6'' 31

qp +=+ σσ

MC model p'- q- space

6sin 2 cos

3

3 6 sin sin 6 cos

p qor q c

q p q c

φ φ

φ φ φ

′ + = +

′= + +

′6sin 6 coscφ φ′ ′

*

*6sin 6 cos,

3 sin 3 sin

q p c

cwhere c

η

φ φη

φ φ

′= +

′ ′= =

′ ′− −

* 0F q p cη ′= − − =MC - Model formulated in p' - q

6sin 6 cos

3 sin 3 sin

cq p

φ φ

φ φ

′ ′′= +

′ ′− −

MC model p'-q- space

Assuming associated flow rule and ideal plasticity

∆

∆

=

∆

∆e

q

e

v

G

K

q

p

ε

ε

'30

0''

( )

( )

*, 0

0 :

F p q q p c

F Fdp dq consistency condition

p q

η′ ′= − − =

∂ ∂′+ =

′∂ ∂consistency condition

Formulation of Dep for MC

( )

( )3

p

v v

p

q q

dp K d d Hooks law

dq G d d

ε ε

ε ε

′ = −

= −

:p

v

Fd d flowrule

pε λ

∂=

′∂

Substituting into consistency condition leads to:

p

q

p

Fd d

qε λ

′∂

∂=

∂

3 3 0v q

F F F F F FKd Kd Gd Gd

p p p Q q qε λ ε λ

∂ ∂ ∂ ∂ ∂ ∂− + − =

′ ′ ′∂ ∂ ∂ ∂ ∂ ∂

Formulation of Dep for MC

0,3

0 3

v

v qq

dKF FF FKd Gd

dp q Gp Qd

F F F F

εε ε

ελ

∂ ∂ ∂ ∂+ ′∂ ∂′∂ ∂ = =

∂ ∂ ∂ ∂ ∂3

0,

0 3

dF F F F F

K Gp p q q K pF F

p q G F

q

λ = =∂ ∂ ∂ ∂ ∂ +

′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ′∂ ∂ ∂ ∂

Formulation of Dep for MC

2

3, 1,

3

,

v q

p p

v q

T

Kd GdF Fd

p q K G

d d d d

η ε εη λ

η

ε λη ε λ

− +∂ ∂= − = =

′∂ ∂ +

= − =

dp ε ε−′

* 0F q p cη ′= − − =

{ } { }{ } { }[ ]{ }2

0

0 3

0 0

0 3 0 3 1

0 0 3

0 3 0 3 1 3

p

v v

p

q q

v

q

v v

q q

d dK

G d d

dK Kd

dG G

d dK K K G

d dG G K G

dp

dq

ε ε

ε ε

ε ηλ

ε

ε εη η

ε εη

−=

−

−= −

− −= −

+

′

Formulation of Dep for MC

{ }{ }

2

2

2 2

0 0 31

0 3 0 33 3

0 31

v

q

v

dK K K G

dG GK G K G

dK K GK

εη η

εη η

εη η

−= −

+ −

−= −

{ }

{ }

2 2

2 2

2 2

2

2 2

0 3 3 3 9

3

3 3

3 93

3 3

q

v

q

dG K G GK G

K GKK

dK G K G

dGK GG

K G K G

εη η

η η

εη η

εη

η η

= −+ −

−+ +

=

−+ +

Formulation of Dep for MC

2 2

2 2

2

2 2

3

3 3

3 93

3 3

ep

K GKK

K G K GD

GK GG

K G K G

η η

η η

η

η η

− + +

=

− + + 3 3K G K Gη η+ + det 0ep

D = perfect plasticity

T

Te e e

ep

T

e

F FD D D

DF F

D

σ σ

σ σ

∂ ∂ − ∂ ∂ =

∂ ∂ ∂ ∂

% %

% %

In general form :

for non associated flow:

σ∂

∂Q

MC model for element tests

Limitations of MC model (1)

Limitations of MC model (2)

Warning for dense sands

Tips for fine-grained soils

Drawbacks of MC

• Mohr-Coulomb failure criterion is well proven through experiments for most geomaterials, but data for clays is still contradictory!

• An associated flow rule implies continuous dilation at a • An associated flow rule implies continuous dilation at a constant rate upon shearing; this is unrealistic and leads to negative pore pressures in undrained conditions. In an non-associated flow rule with ψ’ < φ’, the rate of dilation is less, but it is still constant. If ψ’ = 0 then the rate of dilation is zero.

• Care must be taken in applying the model for undrained loading. This will be discussed further as part of lecture on Drained and Undrained Analysis.

Drawbacks of MC

• Soils on shearing exhibit variable volume changecharacteristics depending on pre-consolidation pressurewhich cannot be accounted for with MC

• In soft soils volumetric plastic strains on shearing are compressive (negative dilation) whilst Mohr-Coulomb model

will predict continuous dilation

To summarize the limitations of MC are:

• bi-linearity (const. E’)

• unlimited dilation

• isotropy

• elastic response far from the limit state

• …

� more advanced approximation of soil behavior:

Hardening Soil Model (sand), Soft Soil Model (clay)

Other elastic-perfectly plastic modelsmodels

Stress invariants

( ) ( )zyxM σσσσσσσ ++=++=3

1

3

1321

2J=σ

Mean effective stress

Deviatoric stress (in general form)

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]2

3

2

2

2

1

2

13

2

32

2

212

2

1

6

1

MMM

J

σσσσσσ

σσσσσσ

−+−+−

=−+−+−=

( )( )( )MMMJ σσσσσσ −−−= 3213

−= −

3

31

2

33sin

3

1

σθ

JLode angle -30°< θ < 30°

Meaning of stress invariants

σ⋅=⋅= 22 2JPN

( ) ( ) ( )

2

2

3

2

2

2

1

222

2J

ONOPPN

MMM

=

=−+−+−=

=−=

σσσσσσMON σ⋅= 3

Von Mises

03 0 =−⋅= YF σY0 ….. yield stress for axial compression or tension

> not dependent on Lode angle

03

2Yr ⋅=

Von Mises criterion in spacem −− σσ Von Mises criterion in ππππ - plane

Von Mises in 3D

03 0 =−⋅= YF σY0 ….. yield stress for axial compression or tension

> not dependent on Lode angle

Tresca

0031 =−−= YF σσ

in invariant formulation

0cos2 0 =−⋅⋅= YF θσ

> dependent on Lode angle

Tresca in 3D

0cos2 0 =−⋅⋅= YF θσ > dependent on Lode angle

Tresca – Applications in geotechnics

ucY ⋅= 20

Total stress analysis for undrained behaviour (ϕϕϕϕ = 0)

cu ….. undrained shear strength

No volume change > νννν = 0.5 (for numerical reasons 0.49)

Plastic volumetric strains ?

022 =⋅−⋅= ucF σ

0=

∂

∂=

M

PM

Fdd

σλε

λσ

λε dF

dd P 2=

∂

∂=

0== PM

Pvol dd εε

Plastic volumetric strains ?

assume Lode angle = 0

Assume associated flow rule

Tresca – Applications in geotechnics

0== PM

Pvol dd εε

MC – Invariant formulation

0cossinsin3

1cossin =⋅−⋅⋅−⋅+⋅= ϕθϕσθσϕσ cF M

Drucker-Prager

Drucker-Prager vs MC

Drucker-Prager and Mohr-Coulomb criteria in ππππ - plane

Nonlinear FE and solution techniquestechniques

(as in PLAXIS)

� Local level

- nodes

- stress points or integration points

- elastic point, plastic point, apex point, tension point

- constitutive stress and equilibrium stress

- local error

Outline

- local error

� Global level

- initial stress method

- extrapolation

- arc-length control

- global error

� Load advancement

- ultimate level

- number of steps

Main Topics on Non-linear Analysis

• Calculation

• Basic Concepts and Algorithms

• Local Level

• Global Level

• Load Advancement

Calculation

Initial situation

• Geometry (mesh, loads, boundary conditions)

• Material models and parameters• Material models and parameters

• Initial stresses and pore pressures

• Initial values of state variables

Calculation

Calculation phases

• Calculation types

– Plastic– Plastic

– Consolidation

– Phi-c reduction (limit state analysis)

Calculation (continues)

Calculation phases (continues)

• Loading input

– Staged construction

• Switch on/off parts of geometry• Switch on/off parts of geometry

• Switch on/off structural elements (beams, anchors)

• Switch on/off loads (change input values)

• Change pore pressures

– Total Multipliers (L.A. Ultimate level)

– Incremental Multipliers (L.A. Number of steps)

Calculation (continues)

Output

• Displacements, stresses, forces etc. per step/phase

– Displacements and pore pressures are nodal – Displacements and pore pressures are nodal values

– Stresses, strains and state variables are Gauss point level values

• Load-displacement curves

PHASE

• Find equilibrium in final situation

•Subdivide in load steps

Basic Concepts

•Subdivide in load steps

LOAD STEP

• Find equilibrium for load increment

Basic Algorithm per Step

1. Additional displacements δu = K-1 δP

2. Total increments ∆u = ∆u +δu

3. Strain increment ∆ε = B∆u

4. Determine stresses σ c = σ0 + De(∆ε −∆εp)

5. Internal reaction P in =∫∫∫∫Bt σ c dV

6. Equilibrium?? P in ≈ P ex

no: next iteration δu = K-1 (P ex - P in)

Local Level

6-node triangle 15-node triangle

Nodes and Stress Points

6-node triangle 15-node triangle

•Plane strain

•Working load

•Failure loads

•Phi-c reduction

•Axisymmetry

exact integration for plane strainno reduced integration (often used for quadrilateral elements)

Elastic & Plastic Stress Points for Mohr Coulomb Model

τ

c’φ’

Elastic point f < 0

dεp=0

τ

σ’

cφ’

σ’

c’φ’

σ1σ3

σ’3 σ’1

Plastic point f = 0

dεp≠ 0

σ’3 σ’1

Apex Stress Points and Tension Stress Points

standard setting: tension cut off is activated !

Local Error

Constitutive stress σc:

Stress that follows from the constitutive model (e.g. Mohr Coulomb)

Equilibrium stress σeq:

Stress that is in equilibrium with the external load (calculated from stiffness matrix)

Inaccurate point:

Local Error > Tolerated Error

Standard setting:

Tolerated error = 0.01

c

eqc

ErrorLocalσ

σ−σ=

Local Error

For MC model

maxTErrorLocal

eqc σσ −=

( ) ( ) ( ) ( ) ( ) ( )222222 c

zx

e

zx

c

yz

e

yz

c

xy

e

xy

c

z

e

z

c

y

e

y

c

x

e

x

ce ττττττσσσσσσσσ −+−+−+−+−+−=−

max

( )'cos'),(max 1321

max ϕσσ cT −=

Global Calculation: Loads

• Applied load = Load multiplier x Input load

• Standard setting: Input load = 1 stress unit

• Actual load is specified before each calculation

– Staged construction: Input load

– Total incr. multipliers: Load multipliers– Total incr. multipliers: Load multipliers

Loads (continues)

Most calculations: Staged construction

– Total load in last calculation step

Some calculations: Total multipliers

– Total load in last calculation step– Total load in last calculation step

– ΣMdisp, ΣMloadA, ΣMloadB, ΣMweight, ΣMsf

Very few calculations: Incremental multipliers

– Additional load in one calculation step

– Mdisp, MloadA, MloadB, Mweight, Msf

Nonlinear Analysis

Initial Stress Method orElastic Stiffness Method

Initial Stress Method

initial stress method + extrapolation

cannot be controlled by user

Initial Stress Method +Over Relaxation

Standard setting: 1.2

Absolute maximum: 2.0Absolute maximum: 2.0

For low friction angles (<20˚): 1.5 acceptable

is scaling of out-of-balance forces

can be controlled by user

Initial Stress Method +Arc Length Control

in very sensitive cases > consider switching off

Global Error

load

unbalanceErrorGlobal =

ce qqUnbalance −=

qe= “external” load (including gravity loads)

qc= internal reaction forces (integral of σc)

Convergence requirement:

Global error ≤ Tolerated error

Standard setting:

Tolerated error = 0.01

Convergence Requirements

Accuracy reached if:1. Global error ≤ Tolerated Error

2. Inaccurate stress points ≤ 3 + (pl. soil points)/10

3. Inaccurate interface points ≤ 3 + (pl. intf. points)/10

empirical

3. Inaccurate interface points ≤ 3 + (pl. intf. points)/10

c

eqc

ErrorLocalσ

σ−σ=

load

unbalanceErrorGlobal =

Automatic Load Advancement

Converged within desired

minimum number of iterations

→ Scaling up by factor 2

Not converged within desired minimum number of iterations:

→ Scaling down by factor 2

is influenced by manual settings !

See p. 4.10-4.11 in PLAXIS Reference manual

Automatic Load Advancement

Ultimate level procedure:• Staged construction• Plastic calculation (Total multipliers)

qq

uu

Reaching prescribed ultimate level

Failure before reaching prescribed ultimate level

possible due to arc length control

Automatic Load AdvancementNumber of steps procedure:

• Plastic calculation (Incremental multipliers)• Phi- c reduction (safety analysis)

q

u

Reaching final step

typically used for phi-c-reduction