Simplified Critical-State Soil Mechanics Paul W. Mayne Georgia Institute of Technology Paul W. Mayne Georgia Institute of Technology.

SimplifiedCritical-State Soil Mechanics

SimplifiedCritical-State Soil Mechanics

Paul W. MayneGeorgia Institute of Technology

Paul W. MayneGeorgia Institute of Technology

PROLOGUEPROLOGUE Critical-state soil mechanics is an

effective stress framework describing mechanical soil response

In its simple form here, we consider only shear loading and compression-swelling.

We merely tie together two well-known concepts: (1) one-dimensional consolidation behavior (via e-logsv’

curves); and (2) shear stress-vs. normal stress ( -t sv’) plots from direct

shear (alias Mohr’s circles).

Critical State Soil Mechanics (CSSM)

Critical State Soil Mechanics (CSSM)

Experimental evidence 1936 by Hvorslev (1960, ASCE) Henkel (1960, ASCE Boulder) Parry (1961) Kulhawy & Mayne (1990): Summary of

200+ soils Mathematics presented elsewhere

Schofield & Wroth (1968) Roscoe & Burland (1968) Wood (1990) Jefferies & Been (2006)

Basic form: 3 material constants (f', Cc, Cs) plus initial state parameter (e0, svo', OCR)

Critical State Soil Mechanics (CSSM)

Critical State Soil Mechanics (CSSM)

Constitutive Models in FEM packages: Original Cam-Clay (1968) Modified Cam Clay (1969) NorSand (Jefferies 1993) Bounding Surface (Dafalias) MIT-E3 (Whittle, 1993) MIT-S1 (Pestana, 1999; 2001) Cap Model “Ber-Klay” (Univ. California) others (Adachi, Oka, Ohta, Dafalias, Nova, Wood, Huerkel)

"Undrained" is just one specific stress path Yet !!! CSSM is missing from most textbooks and

undergrad & grad curricula.

One-Dimensional Consolidation One-Dimensional Consolidation Sandy Clay (CL), Surry, VA: Depth = 27 m

0.5

0.6

0.7

0.8

0.9

1.0

1 10 100 1000 10000

Eff ective Vertical Stress, svo' (kPa)

Void

Rati

o,

e

Cc = 0.38

Cr = 0.04

svo'=300 kPa

sp'=900

kPa

Overconsolidation Ratio, OCR = 3

Cs = swelling index (= Cr)

cv = coef. of consolidation

D' = constrained modulus

Cae = coef. secondary compression

k ≈ hydraulic conductivity

sv’

Direct Shear Test ResultsDirect Shear Test Results

sv’t

Direct Shear Box (DSB)

sv’t

Direct Simple Shear (DSS)

t d t

gs

Slow Direct Shear Tests on Triassic Clay,NC

0

20

40

60

80

100

120

140

0 1 2 3 4 5 6 7 8 9 10

Displacement, d (mm)

She

ar S

tres

s, t

(k

Pa) sn'

(kPa)= 214.5

135.0

45.1

Slow Direct Shear Tests on Triassic Clay, Raleigh, NC

0

20

40

60

80

100

120

140

0 50 100 150 200 250

Eff ective Normal Stress, sn' (kPa)

She

ar S

tres

s, t

(k

Pa)

0.491 = tanf '

Strength Parameters:

c' = 0; f ' = 26.1 oPeak

Peak

Peak

CSSM for DummiesCSSM for Dummies

sCSL’ sNC’

Effective stress sv'

She

ar s

tres

s t

Voi

d R

atio

, e

NC

CC

tanf'CSL

Effective stress sv'

Voi

d R

atio

, e

NC

CSL

CSSM Premise:

“All stress paths

fail on the critical

state line (CSL)”

CSL

fc=0

e0e0

sCSL’ ½sNC’

Log sv'

CSSM for DummiesCSSM for Dummies

Log sv'

Effective stress sv'

Sh

ear

str

ess

t

Vo

id R

ati

o, e

Vo

id R

ati

o, e

NCNC

CC

tanf'CSL

CSLCSL

STRESS PATH No.1

NC Drained Soil

Given: e0, svo’, NC

(OCR=1)

e0

svo

svo

Drained Path: Du = 0

tmax = c + s tanf

ef

De

Volume Change is

Contractive: evol =

De/(1+e0) < 0

Effective stress sv'

c’=0

CSSM for DummiesCSSM for Dummies

Log sv'

Effective stress sv'

Sh

ear

str

ess

t

Vo

id R

ati

o, e

Vo

id R

ati

o, e

NCNC

CC

tanf'CSL

CSLCSL

STRESS PATH No.2

NC Undrained Soil

Given: e0, svo’, NC

(OCR=1)

e0

svo

svo

Undrained Path: DV/V0 = 0

+Du = Positive Excess Porewater Pressures

svf

svf

Dutmax = cu=su

Effective stress sv'

CSSM for DummiesCSSM for Dummies

Log sv'

Effective stress sv'

She

ar s

tres

s t

Voi

d R

atio

, e

NC NC

CC

tanf'

CSL

CSLCSL

Note: All NC

undrained

stress paths are

parallel

to each other, thus:

su/svo’ = constant

Effective stress sv'

DSS: su/svo’NC =

½sinf’

Voi

d R

atio

, e

CSSM for DummiesCSSM for Dummies

Log sv'Effective stress sv'

Effective stress sv'

Sh

ear

stre

ss t

Voi

d R

atio

, e

Vo

id R

atio

, e

NC NC

CC

tanf'

CSL

CSLCSL

CS

sp'

sp'

OC

Overconsolidated States:

e0, svo’, and OCR = sp’/svo’

where sp’ = svmax’ = Pc’ =

preconsolidation stress;

OCR = overconsolidation ratio

CSSM for DummiesCSSM for Dummies

Log sv'

Effective stress sv'

Sh

ear

str

ess

t

Vo

id R

ati

o, e

NC NC

CC

tanf'

CSL

CSLCSL

CS

OC

Stress Path No. 3

Undrained OC Soil:

e0, svo’, and OCR

svo'

e0

svo'

Stress Path: DV/V0 = 0

Negative Excess Du

Effective stress sv'svf'

Vo

id R

ati

o, e

Du

CSSM for DummiesCSSM for Dummies

Log sv'

Effective stress sv'

Sh

ear

str

ess

t

Vo

id R

ati

o, e

Vo

id R

ati

o, e

NC NC

CC

tanf'

CSL

CSLCSL

CS

OC

Stress Path No. 4

Drained OC Soil:

e0, svo’, and OCR

Stress Path: Du =

0 Dilatancy: DV/V0 > 0

svo'

e0

svo'

Effective stress sv'

Critical state soil mechanicsCritical state soil mechanics

• Initial state: e0, svo’, and OCR = sp’/svo’

• Soil constants: f’, Cc, and Cs (L = 1-Cs/Cc)

• For NC soil (OCR =1): Undrained (evol = 0): +Du and tmax = su = cu

Drained (Du = 0) and contractive (decrease evol)

• For OC soil: Undrained (evol = 0): -Du and tmax = su = cu

Drained (Du = 0) and dilative (Increase evol)

There’s more ! There’s more ! Semi-drained, Partly undrained, Cyclic response….. Semi-drained, Partly undrained, Cyclic response…..

Equivalent Stress ConceptEquivalent Stress Concept

Log sv'

Stress sv'

Sh

ear

stre

ss t

Vo

id R

atio

, e NC

NC

CC

tanf'

CSL

CSLCSL

CS

OC

1. OC State (eo, svo’, sp’)

svo'

2. Project OC state to NC

line for equivalent stress,

se’

3. se’ = svo’ OCR[1-Cs/Cc]

svo'

e0

Effective stress sv'

Vo

id R

atio

, e

sp' sp'

se'

se'

su

svf'

ep

De = Cs log(sp’/svo’)

De = Cc log(se’/sp’)

De

at se’suOC = suNC

Critical state soil mechanicsCritical state soil mechanics

• Previously: su/svo’ = constant for NC soil

• On the virgin compression line: svo’ = se’

• Thus: su/se’ = constant for all soil (NC &

OC)

• For simple shear: su/se’ = ½sin f’

• Equivalent stress: Normalized Undrained Shear

Strength:

su/svo’ = ½ sinf’ OCR L

where L = (1-Cs/Cc)

se’ = svo’ OCR[1-Cs/Cc]

Undrained Shear Strength from CSSMUndrained Shear Strength from CSSM

0.0

0.1

0.2

0.3

0.4

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

sinf'

s u/s

vo' N

C

(DS

S)

AGS Plastic

Amherst

Ariake

Bootlegger

Bothkennar

Boston Blue

Cowden

Hackensack

James Bay

Mexico City

Onsoy

Porto Tolle

Portsmouth

Rissa

San Francisco

Silty Holocene

Wroth (1984)

su/svo'NC (DSS) =½sinf'

plasticity index, PI (%)

s u/s

vo' N

C

(DS

S)

AGS Plastic AmherstAriake Bootlegger

Bothkennar Boston BlueCowden Hackensack

James Bay Mexico CityOnsoy Porto Tolle

Portsmouth RissaSan Francisco Silty Holocene

Undrained Shear Strength from CSSMUndrained Shear Strength from CSSM

0.1

1

10

1 10 100Overconsolidation Ratio, OCR

DS

S U

ndra

ined

Str

engt

h , s

u/s v

o' Amherst CVVC

Atchafalaya

Bangkok

Bootlegger Cove

Boston Blue

Cowden

Drammen

Hackensack

Haga

Lower Chek Lok

Maine

McManus

Paria

Portland

Portsmouth

Silty Holocene

Upper Chek Lok

40

30

20

f' = 40o

20o 30o

su/svo' = ½ sinf' OCRL

Note: L = 1 - Cs/Cc 0.8

IntactClays

L

Porewater Pressure Response from CSSMPorewater Pressure Response from CSSM

-6

-5

-4

-3

-2

-1

0

1

1 10 100Overconsolidation Ratio, OCR

Nor

mal

ized

Por

ewat

er, D u

/svo'

Amherst CVVC

Atchafalaya

Bangkok

Bootlegger Cove

Boston Blue

Cowden

Drammen

Hackensack

Haga

Lower Chek Lok

Maine

McManus

Paria

Portland

Portsmouth

Silty Holocene

Upper Chek Lok

20

30

40

L = 0.9 0.8 0.7

IntactClays

f' = 20o 30o 40o

Dus/svo' = 1 - ½cosf'OCRL

Yield SurfacesYield Surfaces

Log sv'

Normal stress sv'

Sh

ear

stre

ss t

Vo

id R

atio

, eNC NC

CSL

CSL

CSL

OC

Normal stress sv'

Vo

id R

atio

, e

sp'

sp'

OC

Yield surface represents 3-d preconsolidation

Quasi-elastic behavior within the yield surface

Critical state soil mechanicsCritical state soil mechanics• This powerpoint: geosystems.ce.gatech.edu

• Classic book: Critical -State Soil Mechanics by Schofield & Wroth (1968): http://www.geotechnique.info

• Schofield (2005) Disturbed Soil Properties and Geotechnical Design Thomas Telford

• Wood (1990): Soil Behaviour and CSSM

• Jefferies & Been (2006): Soil liquefaction: a critical-state approach www.informaworld.com

ESA versus TSA• Effective stress analysis (ESA) rules:

c' = effective cohesion intercept (c' = 0 for OCR < 2 and c' ≈ 0.02 sp' for short term loading)

f' = effective stress friction angle t = c' + s' tan f' = Mohr-Coulomb strength

criterion sv' = sv - u0 - Du = effective stress

• Total stress analysis (TSA) is (overly) simplistic for clay with strength represented by a single parameter, i.e. "f = 0" and tmax = c = cu = su = undrained shear strength (implying "Du = 0")

Explaining the myth that "f = 0"

The effective friction angle (f') is usually between 20 to 45 degrees for most soils. However, for clays, we here of "f = 0" analysis which applies to total stress analysis (TSA). In TSA, there is no knowledge of porewater pressures (PWP). Thus, by ignoring PWP (i.e., Du = 0), there is an illusional effect that can be explained by CSSM. See the following slides.

5.1688665.7431846.3813157.09035

7.8781678.7535199.72613210.8068112.0075713.3417514.8241616.4712918.3014320.3349322.5943625.10485

0.5

0.6

0.7

0.8

10 100 1000

Vo

id R

atio

, e

Log Effective stress, sv'

0.5

0.6

0.7

0.8

0 100 200 300 400 500

Vo

id R

ati

o, e

sv' (kPa)

0

100

200

300

0 100 200 300 400 500

t=

Sh

ear

Str

ess

(kP

a)

sv' (kPa)

f' = 30 °Cc = 0.50Cr = Cs = 0.05

(Undrained) Total Stress Analysis - ConsolidatedUndrained Triaxial Tests

Three specimens initially consolidated to svc' = 100, 200, and 400 kPa

(Undrained) Total Stress Analysis

0 100 200 300 400 5000

100

200

300

Effective stress, sv' (kPa)

=

(

)t

Shear

Str

ess

kPa

su100

su200

su400

In TSA, however, Du not known, so plot stress paths for "Du = 0"

Obtains the illusion that " f ≈ 0° "

5.1688665.7431846.3813157.09035

7.8781678.7535199.72613210.8068112.0075713.3417514.8241616.4712918.3014320.3349322.5943625.10485

0.5

0.6

0.7

0.8

0 100 200 300 400 500 600

Vo

id R

ati

o, e

sv' (kPa)

0

100

200

300

0 100 200 300 400 500 600

t=

Sh

ear

Str

ess

(kP

a)

sv' (kPa)

0.5

0.6

0.7

0.8

10 1000

Vo

id R

ati

o, e

sv' (kPa)

VCL

CSL

Cs from Pc' = 400 kPa

Cs from Pc' = 500 kPa

Cs from Pc' = 600 kPa

Another set of undrained Total Stress Analyses (TSA) for UU tests on clays:

UU = Unconsolidated Undrained

(Undrained) Total Stress Analysis

0 100 200 300 400 5000

100

200

300

Effective stress, sv' (kPa)

=

(

)t

Shear

Str

ess

kPa

su

Again, Du not known in TSA, so plot for stress paths for "Du = 0"

Obtains the illusion that " f = 0° "

Explaining the myth that "f = 0"

Effective Stress Analyses (ESA)• Drained Loading (Du = 0)• Undrained Loading (DV/V0 = 0)

Total Stress Analyses (TSA) Drained Loading (Du = 0) Undrained Loading with " f = 0"

analysis: DV/V0 = 0 and "Du = 0"

Cambridge University q-p' spaceCambridge University q-p' space

P' = (s1' + s2' + s3')/3

q =

(s 1

- s

3) TriaxialCompression

CSL

'sin3

'sin6

ff

cMs2' = s3'

s1'

Undrained NCStress Path

Undrained OCStress Path

svo' = P0'

DrainedStress Path3V : 1H

L

22

)'/( 0

OCRMps c

TCu

Port of Anchorage, AlaskaPort of Anchorage, Alaska

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Effective Stress, p'* = (s1'+s2'+s3')/(3sp')

Dev

iato

ric

Str

ess

= q

* =

(s1-s

3)/s

p'

Bootlegger

Cove Clay

Mc = (q/p')f = 1.10

Mc = 6sinf '/(3-sinf ')f' = 27.7o

0.1

1

10

1 10 100

Overconsolidation Ratio, OCR

Str

en

gth

Ra

tio

, su/s

vo'

DSS Data

CIUC Data

MCC Pred CIUC

MCC Pred DSS

Critical State Soil Mechanics(Modified Cam Clay)

f ' = 27.7o

L = 0.75

Cavity Expansion – Critical State Model for Evaluating OCR in Clays from Piezocone Tests

OCRM

q uT b

vo

2

1

1 95 1

1

. '

/

s

L

where M = 6 sinf’/(3-sinf’)

and L = 1 – Cs/Cc 0.8

qc

fs

ub

qT

0

2

4

6

8

10

12

14

16

18

20

0 1 2 3 4 5 6

Overconsolidation Ratio, OCR

Dep

th (

met

ers)

CPTU

CRS

IL Oed

RF

Bothkennar, UK

Cambridge University q-p' spaceCambridge University q-p' space

P' = (s1' + s2' + s3')/3

q =

(s 1

- s

3)

CSL'sin3

'sin6

ff

cM

Yield SurfaceOriginal Cam Clay

Modified Cam Clay

Pc'

Bounding Surface

Cap ModelCap Model

Anisotropic Yield SurfaceAnisotropic Yield Surface

P’ = (s1’ + s2’ + s3’)/3

q =

(s 1

- s

3)Mc = 6sinf’/(3-sinf’)

fctn(K0NC)

Y3 = Limit State

Yield Surface

e0

svo’

K0

G0

Y2

CSL

sp’

Y1

OCOCNCNC

Cambridge University q-p' spaceCambridge University q-p' space

P' = (s1' + s2' + s3')/3

q =

(s 1

- s

3) fctn(K0NC)

Y3 = Limit State

Yield SurfaceCSL

sp’

'sin3

'sin6

ff

cM

Apparent

Mc

MIT q-p' spaceMIT q-p' space

P' = ½(s1' + s3')

q =

½(s

1 - s 3

)

fctn(K0NC)

Yield Surface

sp’

'sintan f c

OCOCDiaz-Rodriguez, Leroueil, and Aleman (1992, JGE)

Diaz-Rodriguez, Leroueil, and Aleman (1992, JGE)

Diaz-Rodriguez, Leroueil, & Aleman

(ASCE Journal Geotechnical

Engineering July 1992)

Yield Surfaces of Natural ClaysYield Surfaces

of Natural Clays

Friction Angle of Clean Quartz Sands

Friction Angle of Clean Quartz Sands

(Bolton, 1986 Geotechnique) (Bolton, 1986 Geotechnique)

State Parameter for Sands, y(Been & Jefferies, 1985; Jefferies & Been 2006)

log p'

voidratio

e

p' = ⅓ (s1'+s2'+s3')

VCLCSL

l10

l10

p0'

y = e0 - ecsl

Dry of Critical (Dilative)

Wet of Critical (Contractive)

e0

ecsl

State Parameter for Sands, y(Simplified Critical State Soil Mechanics)

log p'

voidratio

e

p' = ⅓ (s1'+s2'+s3')

VCLCSL

l10

l10

p0'

y = e0 - ecsl

y = (Cs - Cc )∙log[ ½ cos f' OCR ]

Du = (1 - ½ cos f' OCRL ]∙svo'

e0

ecsl

then CSL = OCR = 2/cosf'

Georgia Tech

State Parameter for Sands, y(Been, Crooks, & Jefferies, 1988)

log OCRp = log2L + Y/( -k l)where OCRp = R = overconsolidation ratio in Cambridge q-p' space, = 1- /L k l, l = Cc/ln(10) = compression index, and k Cs/ln(10) = swelling index

log OCRp = log2L + Y/( -k l)where OCRp = R = overconsolidation ratio in Cambridge q-p' space, = 1- /L k l, l = Cc/ln(10) = compression index, and k Cs/ln(10) = swelling index

MIT Constitutive Models Whittle et al. 1994: JGE Vol. 120 (1)

"Model prediction of anisotropic behavior of Boston Blue Clay"

MIT-E3: 15 parameters for clay Pestana & Whittle (1999) "Formulation of

unified constitutive model for clays and sands" Intl. J. for Analytical & Numerical Methods in Geomechanics, Vol. 23

MIT S1: 13 parameters for clay MIT S1: 14 parameters for sand

MIT E-3 Constitutive Model

Whittle (2005)

MIT S-1 Constitutive ModelPestana and Whittle (1999)

MIT S-1 Constitutive Model

Predictions forBerlin Sands

(Whittle, 2005)

Critical state soil mechanicsCritical state soil mechanics• Initial state: e0, svo’, and OCR = sp’/svo’

• Soil constants: f’, Cc, and Cs

• Link between Consolidation and Shear Tests• CSSM addresses:

NC and OC behavior Undrained vs. Drained (and other paths) Positive vs. negative porewater pressures Volume changes (contractive vs. dilative) su/svo’ = ½ sinf’ OCRL where L = 1-Cs/Cc

• Yield surface represents 3-d preconsolidation• State parameter: y = e0 - ecsl

• Yield surface represents 3-d preconsolidation• State parameter: y = e0 - ecsl

Simplified Critical State Soil Mechanics

Log sv' Effective stress sv'

Effective stress sv'

Sh

ear

str

ess

t

Vo

id R

ati

o, e

Vo

id R

ati

o, eNC

NCCC

CSL

CSLCSL

CS

sp'

sp'

OC

Four Basic Stress

Paths:

1. Drained NC (decrease

DV/Vo)

2. Undrained NC (positive Du)

3. Undrained OC (negative

Du)

4. Drained OC (increase

DV/Vo)

f'

eNC

consolidationswellingeOC

12

3

4

YieldSurface

dilative

contractive

+Du

-Du

sCS½sp

tmax = su NC

tmax = stanf

su OC

tmax = c+stanf c'