Cam Clay Presentation

8/13/2019 Cam Clay Presentation

1/23

Constitutive Modeling of ClaysUsing Cam Clay and Modified

Cam Clay Models

A PRESENTATIONBY

AMIT PRASHANTGRADUATE STUDENT

CLARKSON UNIVERSITY


2/23

Discussion on -1. Important definitions

2. Original Cam-clay model parameters3. Stable State Boundary Surface

4. Cam:clay Flow Rule5. Plastic potentials and Normalitycondition

6. Modified Cam-clay Model7. Stress:Strain relationship

8. Examples


3/23

State of Sample During Triaxial test

Mean effective stress, p= ( 1+ 2+ 3)/3 = ( 1+ 2+ 3)/3 uShear stress, q= [{( 1- 2)2+( 2- 3)2+( 3- 1)2}/2] 1/2

Specific volume, v=1+e

Strains Corresponding to p and q:- Volumetric strain, v = 1+ 2+ 3Shear strain, = 1/3[2{( 1-2)2+( 2-3)2+( 3-1)2}]1/2

p'

v p' p'


4/23

DEFINITIONS Yield point:- This is the state of stress at which the

sample starts to deform with plastic deformations.

Failure point:- This is state of stress at which theapplied shear stress is maximum while shearing the sample.

Critical state:- This isthe state of stress (for monotonicloading) at which further sheardeformation can occur without

further change in effective stressand void ratio.

Yield pointFailure point

Critical statefor both the cases

1

q

Yield point may besomewhere in this range


5/23

Soil Strength ParametersSoil constants:-

Slope of Iso-NCL in v-ln p plane, Slope of URL in v-ln p plane, Specific volume on CSL at unit pressure in v-ln p

plane, Slope of CSL in p-q plane, MShear modulus, G

Two tests required to determine parameters:-Triaxial Compression testIsotropic Consolidation test


6/23

Soil Strength Parameters

CSL

Iso-NCL

URL

p'=1

p'

v

vv

Modified Cam-clay

Original Cam-clay

p'

q

CSLYield Surface

1

M


7/23

Volume and Pressure Relationships

NCL :- v=v - ln(p)URL :- v=v - ln(p)CSL :- v= - ln(p)

q=Mp

NOTE:-Specific volume and mean effective stress at critical state, found inthe triaxial compression test may be used to determine the valueas the CSL has the same slope as NCL.


8/23

Stable State Boundary Surface

q

p'

v

CSL

SSBS

q

v

p'CSL

q

pc' pc'/2.72


9/23

Stable State Boundary Surface

Equation of SSBS :-

q=Mp( + v ln(p))/( )OR

V= +( )(1 /M)

Where = Current value of q/p

Equation of Yield Surface in q-p plane:-q=M.p.ln(p c/p)

Where pc = Isotropic pre-consolidation pressure


10/23

Cam-clay Flow ruleEnergy dissipated on yielding:-

p vp+q

p=Mp

p

Flow Rule:- (rearranging above equation)

vp / p = M M > gives positive volumetric strain, i.e. the sample iscompressive in nature. The stress state is called on the dry side ofcritical state line.

M < gives negative volumetric strain, i.e. the sample isdilatative in nature. Stress state is called on the wet side of criticalstate line.


11/23

Plastic Potential Surface

Plastic Strain Increment Vector:- Plastic volumetricstrain and plastic shear strain take the same direction as p and qrespectively. The vector sum of increments of these strains at anystress state is called as plastic strain increment vector.

Plastic Potentials:-Plastic potentials form the familyof curves to which the plastic

strain increment vectors areorthogonal.

p'

q p

p p


12/23

Normality Condition

When the yield loci and plastic potentials coincide each

other, normality condition or associative flow rule exists.This means the plastic strain vectors are orthogonal to theyield loci itself.Cam-clay model follows this normality condition, finding itto be reasonably good assumption from the simplificationpoint of view.In original cam-clay model, theyield curve shows a kink at theIsotropic pre-consolidationpressure and that shows twoplastic strain vectors at one point.

p' p c'

q Strain vectors at pre-consolidation pressure


13/23

Modified Cam-clay ModelThis was proposed with elliptical yield surface, having nocontradiction to normality condition for strain vectors atIsotropic pre-consolidation pressure.Equation of SSBS :-

V= +( ){ln(2) ln(1+( /M) 2)}

Where = Current value of q/p

Equation of Yield locus in q-p plane:-q2+M2p 2=M2.p.p c

Where p c = Isotropic pre-consolidation pressure

pc'

q

p'

CSL YieldSurace

1M M.p'/2

p'/2


14/23

Modified Cam-clay Flow Rule

Energy dissipated on yielding:-

p vp

+q p

=p[ vp2

+(M p

)2

]1/2

Flow Rule:- (rearranging above equation)vp / p = (M 2 2)/2

This again explains deformation the same way as original Cam-clayModel.

M > gives positive volumetric strain, i.e. the sample iscompressive in nature.

M < gives negative volumetric strain, i.e. the sample isdilatative in nature.


15/23

Elastic and Plastic stress-strain response

Elastic stress:strain response:-

Plastic stress:strain response:-

Pre-yield (elastic) andPost-yield (elastic-plastic)deformations.

p p

q p = q

p'

2

()

vp' (M + )2 2(M - )22 2

/(M - )2 242 q

p' p'cA cB p'

= p0q

e1/3G q

/vp'e

0 p'


16/23


17/23

Undrained Test Normally Consolidated Sample

B

A p'

u = p - p'

G

H

A

CSL

v NCL

q

cH p'

A p'

cA

H

cG p'

u p'

G

CSLB

ESP

TSPq

H

G

B

cB p'

AH

GB

uAH

uAG

uf

f u


18/23

Undrained Test Lightly Over-consolidated Sample

CSL

p' p'

v

p'

CSL NCL u

u = p - p'

ESP

TSP

cA p'cC cG p'A

B

G

C

A

B

GC

A

B

G

C

CG B A

uAB

uAGuf

uf


19/23

Undrained Test Highly Over-consolidated Sample

B

TSP

ESP

CSL

A

A

p'

NCLvCSL

u

A

DB C

u = p - p'

q q

p'

C D

B

C

D

uBD

uAB

ABu


20/23

Drained Test Normally Consolidated or Lightly Over-consolidated Sample

A

A

v NCLCSL

p'

e

TSP & ESP H

A

B

CCSL

p'

GB

H

G

A BGH

C

BG

H

C

C


21/23

Drained Test Highly Over-consolidated Sample

CC

p'

A

A

TSP & ESP

v

CSL NCL

q

p'

AB

A

v

B

C

CSL q

B

B

C

e

ACt

e

ABe


22/23

Comments-This model explains most of the trends observed instress:strain behavior of clay, Hardening; Softening

dilatation etc.This model uses Extended von-misses criteria as its failureenvelop and that doesnt include the variation of shear

strength due to changing intermediate principal stress.This is probably due to the unavailability of threedimensional test data during the development of thesemodels.Predictions of undrained tests always show no variation inmean effective stress before yielding, which is not alwaystrue.


23/23

Comments-

The p-q plane superimposed

on triaxial plane shows thatthe strength of highly over-consolidated clay is

over predicted as the claymay not bear this much oftension in principal directions

before failure.

p'

q

2. '3

1 '

Tension

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Cam Clay Presentation