Transcript Academic-Practitioner Forum 16th ICSMGE Osaka

Simplified
Critical-State Soil Mechanics
Paul W. Mayne
Georgia Institute of Technology
PROLOGUE



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)
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Experimental evidence
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Mathematics presented elsewhere
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1936 by Hvorslev (1960, ASCE)
Henkel (1960, ASCE Boulder)
Parry (1961)
Kulhawy & Mayne (1990): Summary of 200+ soils
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)
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Constitutive Models in FEM packages:
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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
Sandy Clay (CL), Surry, VA: Depth = 27 m
svo'=300 kPa
1.0
Cr = 0.04
0.9
Void Ratio, e
sp'=900
kPa
Overconsolidation Ratio, OCR = 3
0.8
Cs = swelling index (= Cr)
cv = coef. of consolidation
D' = constrained modulus
Cae = coef. secondary compression
k ≈ hydraulic conductivity
0.7
0.6
Cc = 0.38
0.5
1
10
100
1000
Effective Vertical Stress, s
svo'
v’ (kPa)
10000
Direct Shear Test Results
Slow Direct Shear Tests on Triassic Clay,NC
(kPa)
sn'
120
(kPa)=
214.5
Peak
Strength Parameters:
c' = 0; f ' = 26.1 o
120
100
t
100
80
Shear Stress,
Shear Stress, t (kPa)
140
Slow Direct Shear Tests on Triassic Clay, Raleigh, NC
140
Peak
135.0
60
40
Peak
20
45.1
0
80
60
0.491 = tan f '
40
20
0
0
1
2
3
4
5
Displacement,
t t
sv’
6
d
7
8
9
10
0
50
(mm)
d
100
150
Effective Normal Stress,
t
t
200
sn'
sv’
gs
Direct Shear Box (DSB)
Direct Simple Shear (DSS)
(kPa)
250
CSSM for Dummies
e0
NC
CSL
sCSL’ sNC’
Log sv'
CSSM Premise:
“All stress paths fail
on the critical state
line (CSL)”
Void Ratio, e
sCSL’  ½sNC’
Shear stress t
Void Ratio, e
CC
c=0
NC
CSL
Effective stress sv'
CSL
tanf'
f
Effective stress sv'
CC
e0
De
NC
ef
Void Ratio, e
Void Ratio, e
CSSM for Dummies
NC
CSL
CSL
STRESS PATH No.1
NC Drained Soil
Given: e0, svo’, NC (OCR=1)
Drained Path: Du = 0
Volume Change is Contractive:
evol = De/(1+e0) < 0
Shear stress t
Log sv'
svo
c’=0
Effective stress sv'
CSL
tmax = c + s tanf
tanf'
svo
Effective stress sv'
CC
e0
NC
Void Ratio, e
Void Ratio, e
CSSM for Dummies
NC
CSL
CSL
svf
svo
Effective stress sv'
CSL
STRESS PATH No.2
NC Undrained Soil
Given: e0, svo’, NC (OCR=1)
Undrained Path: DV/V0 = 0
+Du = Positive Excess
Porewater Pressures
Shear stress t
Log sv'
tanf'
Du
tmax = cu=su
svf
svo
Effective stress sv'
CSSM for Dummies
NC
Void Ratio, e
Void Ratio, e
CC
NC
CSL
CSL
Effective stress sv'
Note: All NC undrained
stress paths are parallel
to each other, thus:
su/svo’ = constant
DSS: su/svo’NC = ½sinf’
Shear stress t
Log sv'
CSL
tanf'
Effective stress sv'
CS
NC
Void Ratio, e
CC
OC
NC
CSL
CSL
Log sv'
Effective stress sv'
s p'
Overconsolidated States:
e0, svo’, and OCR = sp’/svo’
where sp’ = svmax’ = Pc’ =
preconsolidation stress;
OCR = overconsolidation ratio
CSL
Shear stress t
Void Ratio, e
CSSM for Dummies
tanf'
Effective stress sv'
s p'
Void Ratio, e
CC
e0
CS
NC
Void Ratio, e
CSSM for Dummies
OC
NC
CSL
CSL
Effective stress sv'
Log sv'
Stress Path No. 3
Undrained OC Soil:
e0, svo’, and OCR
Stress Path: DV/V0 = 0
Negative Excess Du
Shear stress t
svo' svf'
CSL
tanf'
Du
svo'
Effective stress sv'
Void Ratio, e
CC
e0
CS
NC
Void Ratio, e
CSSM for Dummies
OC
NC
CSL
CSL
Effective stress sv'
Log sv'
Stress Path No. 4
Drained OC Soil:
e0, svo’, and OCR
Stress Path: Du = 0
Dilatancy: DV/V0 > 0
CSL
Shear stress t
svo'
tanf'
svo'
Effective stress sv'
Critical 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 !
Semi-drained, Partly undrained, Cyclic response…
Equivalent Stress Concept
NC
e0
De
ep
CS
sp'
Void Ratio, e
Void Ratio, e
CC
NC
OC
CSL
CSL
2. Project OC state to NC
line for equivalent stress, se’
De = Cs log(sp’/svo’)
De = Cc log(se’/sp’)
3. se’ = svo’ OCR[1-Cs/Cc]
Shear stress t
svo' svf' se' Log sv'
1. OC State (eo, svo’, sp’)
sp'
Effective stress sv'
CSL
tanf'
su
at se’
suOC = suNC
svo'
s e'
Stress sv'
Critical 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: se’ = svo’ OCR[1-Cs/Cc]
Normalized Undrained Shear Strength:
su/svo’ = ½ sinf’ OCRL
where L = (1-Cs/Cc)
Undrained Shear Strength from CSSM
su/svo' NC (DSS)
0.4
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)
0.3
0.2
0.1
su/svo'NC (DSS) =½sinf'
0.0
0.0
0.1
0.2
0.3
0.4
sinf'
0.5
0.6
0.7
0.8
DSS Undrained Strength, su/svo'
Undrained Shear Strength from CSSM
10
o
Intact
Clays
f' = 40
L
30o
20o
1
su/svo' = ½ sinf' OCRL
Note: L = 1 - Cs/Cc  0.8
0.1
1
10
Overconsolidation Ratio, OCR
100
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
Porewater Pressure Response from CSSM
1
Normalized Porewater, Du/svo'
Dus/svo' = 1 - ½cosf'OCR
Amherst CVVC
L
Atchafalaya
Bangkok
0
Bootlegger Cove
Boston Blue
-1
Cowden
Drammen
Hackensack
-2
Haga
Lower Chek Lok
-3
Maine
Intact
Clays
-4
McManus
Paria
f' = 20
o
o
30
40
Portland
o
Portsmouth
-5
Silty Holocene
L = 0.9
0.8
Upper Chek Lok
0.7
20
-6
1
10
Overconsolidation Ratio, OCR
100
30
40
Yield Surfaces
NC
NC
Void Ratio, e
Void Ratio, e
CSL
OC
OC
sp'
CSL
sp'
Yield surface represents
3-d preconsolidation

Quasi-elastic behavior
within the yield surface

Normal stress sv'
CSL
Shear stress t
Log sv'
Normal stress sv'
Critical 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.
0.8
f' = 30 °
Cc = 0.50
Cr = Cs = 0.05
0.7
Void Ratio, e
Void Ratio, e
0.8
0.6
0.5
0.7
0.6
0.5
10
100
1000
0
100
Log Effective stress, s v'
200
300
400
500
400
500
sv' (kPa)
(Undrained) Total Stress
Analysis - Consolidated
Undrained Triaxial Tests
Three specimens initially consolidated
to svc' = 100, 200, and 400 kPa
t = Shear Stress (kPa)
300
200
100
0
0
100
200
300
sv ' (kPa)
(Undrained) Total Stress Analysis
In TSA, however, Du not known, so plot stress paths for "Du = 0"
Obtains the illusion that " f ≈ 0° "
t = Shear Stress (kPa)
300
200
su400
su200
su100
100
0
0
100
200
300
Effective stress, sv' (kPa)
400
500
0.8
Void Ratio, e
Void Ratio, e
0.8
0.7
VCL
0.6
CSL
0.7
0.6
Cs from Pc' = 400 kPa
Cs from Pc' = 500 kPa
Cs from Pc' = 600 kPa
0.5
0.5
10
1000
0
100
200
sv' (kPa)
UU = Unconsolidated Undrained
400
500
600
400
500
600
sv' (kPa)
t = Shear Stress (kPa)
Another set of undrained Total
Stress Analyses (TSA) for UU tests
on clays:
300
300
200
100
0
0
100
200
300
s v ' (kPa)
(Undrained) Total Stress Analysis
Again, Du not known in TSA, so plot for stress paths for "Du = 0"
Obtains the illusion that " f = 0° "
t = Shear Stress (kPa)
300
200
su
100
0
0
100
200
300
Effective stress, sv' (kPa)
400
500
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' space
q = (s1 - s3)
s 1'
s 2' = s 3'
Triaxial
Compression
6 sinf '
Mc =
3 - sinf '
Undrained OC
Stress Path
Undrained NC
Stress Path
( su / p0 ' )TC
M c  OCR 
=


2  2 
L
Drained
Stress Path
3V : 1H
svo' = P0'
P' = (s1' + s2' + s3')/3
Port of Anchorage, Alaska
10
Critical State Soil Mechanics
(Modified Cam Clay)
f ' = 27.7o
L = 0.75
0.7
Bootlegger
0.6
Cove Clay
Strength Ratio, su/s vo'
Deviatoric Stress = q* = (s 1-s 3)/s p'
0.8
0.5
0.4
M c = (q/p')f = 1.10
0.3
M c = 6sinf '/(3-sinf ')
0.2
f ' = 27.7o
1
DSS Data
CIUC Data
MCC Pred CIUC
0.1
MCC Pred DSS
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Effective Stress, p'* = (s 1'+s 2'+s 3')/(3s p')
0.8
0.1
1
10
Overconsolidation Ratio, OCR
100
Cavity Expansion – Critical State Model for Evaluating
OCR in Clays from Piezocone Tests

 qT - ub  
1


OCR = 2 
. M  1  s vo '  
 195
where M = 6 sinf’/(3-sinf’)
L = 1 – Cs/Cc 
Overconsolidation Ratio, OCR
0
0.8
1
2
3
4
5
0
2
4
Depth (meters)
and
1/ L
fs
ub
6
8
10
12
CPTU
CRS
14
16
qc  qT
Bothkennar, UK
18
20
IL Oed
RF
6
q = (s1 - s3)
Cambridge University q-p' space
Mc =
Yield Surface
Original Cam Clay
6 sin f '
3 - sin f '
Modified Cam Clay
Bounding Surface
Cap Model
Pc'
P' = (s1' + s2' + s3')/3
Anisotropic Yield Surface
q = (s1 - s3)
Mc = 6sinf’/(3-sinf’)
Yield Surface
e0
svo’
K0
fctn(K0NC)
OC
Y1
Y2
sp’
NC
Y3 = Limit State
G0
P’ = (s1’ + s2’ + s3’)/3
Cambridge University q-p' space
Mc =
6 sin f '
3 - sin f '
q = (s1 - s3)
Yield Surface
Apparent
fctn(K0NC)
Mc
sp’
Y3 = Limit State
P' = (s1' + s2' + s3')/3
MIT q-p' space
q = ½(s1 - s3)
tanac = sinf '
fctn(K0NC)
Yield Surface
OC
sp’
Diaz-Rodriguez, Leroueil,
and Aleman (1992, JGE)
P' = ½(s1' + s3')
Yield Surfaces
of Natural Clays
Diaz-Rodriguez,
Leroueil, & Aleman
(ASCE Journal
Geotechnical
Engineering July 1992)
Friction Angle of Clean Quartz Sands
(Bolton, 1986 Geotechnique)
State Parameter for Sands, y
(Been & Jefferies, 1985; Jefferies & Been 2006)
l10
void l10
ecsl
ratio
y = e0 - ecsl
e
e0
Wet of Critical
(Contractive)
Dry of Critical
(Dilative)
VCL
CSL
p0'
log p'
p' = ⅓ (s1'+s2'+s3')
State Parameter for Sands, y
(Simplified Critical State Soil Mechanics)
y = (Cs - Cc )∙log[ ½ cos f' OCR ]
l10
void
ratio
e
Du = (1 - ½ cos f' OCRL ]∙svo'
l10
ecsl
then CSL = OCR = 2/cosf'
y = e0 - ecsl
VCL
CSL
e0
p0'
p' = ⅓ (s1'+s2'+s3')
log p'
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, L = 1-k/l, l = Cc/ln(10) = compression
index, and k  Cs/ln(10) = swelling index
Georgia Tech
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 Model
Pestana and Whittle (1999)
MIT S-1 Constitutive Model
Predictions for
Berlin Sands
(Whittle, 2005)
Critical 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
Simplified Critical State Soil Mechanics
NC
Void Ratio, e
eOC
CS
eNC
dilative
-Du
OC
+Du
contractive
Log sv'
s p'
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'
Effective stress sv'
tmax = stanf
Shear stress t
Void Ratio, e
CC
tmax = su NC
su OC
3
4
tmax = c+stanf
1
2
c'
Yield
Surface
sCS½sp
sp'
Effective stress sv'