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Critical-State Soil Mechanics. Paul W. Mayne, PhD, P.E. Civil & Environmental Engineering Georgia Institute of Technology. PROLOGUE. Critical-state soil mechanics is an effective stress framework describing mechanical soil response - PowerPoint PPT Presentation
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Critical-State Soil Mechanics
Paul W. Mayne, PhD, P.E.
Civil & Environmental EngineeringGeorgia Institute of Technology
PROLOGUE Critical-state soil mechanics is an effective stress
framework describing mechanical soil response
In its simplest form here, we consider only shear-induced loading.
We merely tie together two well-known concepts: (1) one-dimensional consolidation behavior, represented by e-logv’ curves; and (2) shear stress-vs. normal stress (v’)
from direct shear box or simple shearing.
Herein, only the bare essence of CSSM concepts are presented, sufficient to describe strength & compressibility response.
Critical State Soil Mechanics (CSSM)
Experimental evidence provided by Hvorslev (1936; 1960, ASCE); Henkel (1960, ASCE Boulder) Henkel & Sowa (1961, ASTM STP 361)
Mathematics presented elsewhere, including: Schofield & Wroth (1968); Burland (1968); Wood (1990).
In basic form: 3 material constants (', Cc, Cs) plus initial state (e0,
vo', OCR)
Constitutive Models, include: Original Cam-Clay, Modified Cam Clay, NorSand, Bounding Surface, MIT-E3 (Whittle, 1993) & MIT-S1 (Pestana) and others (Adachi, Oka, Ohta, Dafalias)
"Undrained" is just one specific stress path
Yet !!! CSSM is missing from most textbooks and undergrad & grad curricula in the USA.
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
vo'=300 kPa
p'=900
kPa
Overconsolidation Ratio, OCR = 3
Cs = swelling index (= Cr)
cv = coef. of consolidation
D' = constrained modulus
Ce = coef. secondary compression
k ≈ hydraulic conductivity
v’
Direct Shear Test Results
v’
Direct Shear Box (DSB)
v’
Direct Simple Shear (DSS)
s
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, (mm)
She
ar S
tres
s,
(k
Pa) n'
(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, n' (kPa)
She
ar S
tres
s,
(k
Pa)
0.491 = tan '
Strength Parameters:
c' = 0; ' = 26.1 oPeak
Peak
Peak
CSSM
Log v'
Effective stress v'
She
ar s
tres
s
Voi
d R
atio
, e
NC
CC
tan'
CSLEffective stress v'
Voi
d R
atio
, e
NC
CSL
CSSM Premise:
“All stress paths
fail on the critical
state line (CSL)”
CSL
c=0
CSSM
Log v'
Effective stress v'
Sh
ear
stre
ss
Vo
id R
atio
, e
Vo
id R
atio
, e
NCNC
CC
tan'
CSL
CSLCSL
STRESS PATH No.1
NC Drained Soil
Given: e0, vo’, NC
(OCR=1)
e0
vo
vo
Drained Path: u = 0
max = c + tan
ef
e
Volume Change is
Contractive: vol =
e/(1+e0) < 0
Effective stress v'
c’=0
CSSM
Log v'
Effective stress v'
Sh
ear
stre
ss
Vo
id R
atio
, e
Vo
id R
atio
, e
NCNC
CC
tan'
CSL
CSLCSL
STRESS PATH No.2
NC Undrained Soil
Given: e0, vo’, NC
(OCR=1)
e0
vo
vo
Undrained Path: V/V0 = 0
+u = Positive Excess Porewater Pressures
vf
vf
umaxcu=su
Effective stress v'
CSSM
Log v'
Effective stress v'
She
ar s
tres
s
Voi
d R
atio
, e
NC NC
CC
tan'
CSL
CSLCSL
Note: All NC undrained
stress paths are
parallel
to each other, thus:
su/vo’ = constant
Effective stress v'
DSS: su/vo’NC = ½sin’
Voi
d R
atio
, e
CSSM
Log v'Effective stress v'
Effective stress v'
Sh
ear
stre
ss
Voi
d R
atio
, e
Vo
id R
atio
, e
NC NC
CC
tan'
CSL
CSLCSL
CS
p'
p'
OC
Overconsolidated States:
e0, vo’, and OCR = p’/vo’
where p’ = vmax’ = Pc’ =
preconsolidation stress;
OCR = overconsolidation ratio
CSSM
Log v'
Effective stress v'
Sh
ear
stre
ss
Vo
id R
atio
, e
NC NC
CC
tan'
CSL
CSLCSL
CS
OC
Stress Path No. 3
Undrained OC Soil:
e0, vo’, and OCR
vo'
e0
vo'
Stress Path: V/V0 = 0
Negative Excess u
Effective stress v'vf'
Vo
id R
atio
, eu
CSSM
Log v'
Effective stress v'
Sh
ear
stre
ss
Vo
id R
atio
, e
Vo
id R
atio
, e
NC NC
CC
tan'
CSL
CSLCSL
CS
OC
Stress Path No. 4
Drained OC Soil:
e0, vo’, and OCR
Stress Path: u =
0 Dilatancy: V/V0 > 0
vo'
e0
vo'
Effective stress v'
Critical state soil mechanics
• Initial state: e0, vo’, and OCR = p’/vo’
• Soil constants: ’, Cc, and Cs ( = 1-Cs/Cc)
• For NC soil (OCR =1): Undrained (vol = 0): +u and max = su = cu
Drained (u = 0) and contractive (decrease vol)
• For OC soil:
Undrained (vol = 0): -u and max = su = cu
Drained (u = 0) and dilative (Increase vol)
There’s more ! There’s more ! Semi-drained, Partly undrained, Cyclic….. Semi-drained, Partly undrained, Cyclic…..
Equivalent Stress Concept
Log v'
Stress v'
Sh
ear
stre
ss
Vo
id R
atio
, e NC
NC
CC
tan'
CSL
CSLCSL
CS
OC
1. OC State (eo, vo’, p’)
vo'
2. Project OC state to NC
line for equivalent stress,
e’
3. e’ = vo’ OCR[1-Cs/Cc]
vo'
e0
Effective stress v'
Vo
id R
atio
, e
p' p'
e'
e'
su
vf'
ep
e = Cs log(p’/vo’)
e = Cc log(e’/p’)
e
at e’suOC = suNC
Critical state soil mechanics
• Previously: su/vo’ = constant for NC soil
• On the virgin compression line: vo’ = e’
• Thus: su/e’ = constant for all soil (NC &
OC)
• For simple shear: su/e’ = ½sin ’
• Equivalent stress: Normalized Undrained Shear
Strength:
su/vo’ = ½ sin’ OCR
where = (1-Cs/Cc)
e’ = vo’ OCR[1-Cs/Cc]
Undrained 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
sin'
s u/
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/vo'NC (DSS) =½sin'
plasticity index, PI (%)
s u/
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 CSSM
0.1
1
10
1 10 100
Overconsolidation Ratio, OCR
DS
S U
ndra
ined
Str
engt
h, s
u/
vo' 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
' = 40o
20o 30o
su/vo' = ½ sin' OCR
Note: = 1 - Cs/Cc 0.8
IntactClays
Porewater Pressure Response from CSSM
-6
-5
-4
-3
-2
-1
0
1
1 10 100Overconsolidation Ratio, OCR
Nor
mal
ized
Por
ewat
er P
ress
ure,
u
/vo
' 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
= 0.9 0.8 0.7
Intact
' = 20o 30o 40o
us/vo' = 1 - ½cos'OCR
Yield Surfaces
Log v'
Normal stress v'
Sh
ear
stre
ss
Vo
id R
atio
, eNC NC
CSL
CSL
CSL
OC
Normal stress v'
Vo
id R
atio
, e
p'
p'
OC
Yield surface represents 3-d preconsolidation
Quasi-elastic behavior within the yield surface
Port 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'* = (1'+2'+
3')/(3p')
Dev
iato
ric
Str
ess
= q
* =
(1-
3)/
p'
Bootlegger
Cove Clay
Mc = (q/p')f = 1.10
Mc = 6sin '/(3-sin ')' = 27.7o
0.1
1
10
1 10 100
Overconsolidation Ratio, OCR
Str
en
gth
Ra
tio
, su/
vo'
DSS Data
CIUC Data
MCC Pred CIUC
MCC Pred DSS
Critical State Soil Mechanics(Modified Cam Clay)
' = 27.7o
= 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
. '
/
where M = 6 sin’/(3-sin’)
and = 1 – Cs/Cc 0.8
qc
fs
ub
qqTT
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
Critical state soil mechanics
• Initial state: e0, vo’, and OCR = p’/vo’
• Soil constants: ’, Cc, and Cs ( = 1-Cs/Cc)
• Using effective stresses, CSSM addresses: NC and OC behavior
Undrained vs. Drained (and other paths)
Positive vs. negative porewater pressures
Volume changes (contractive vs. dilative)
su/vo’ = ½ sin’ OCR where = 1-Cs/Cc
Yield surface represents 3-d preconsolidation
