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'