The document presents critical-state soil mechanics (CSSM) as a framework for understanding the mechanical behavior of soils under shear-induced loading, emphasizing its relevance in understanding strength and compressibility. It outlines key concepts, experimental evidence, and various constitutive models related to CSSM, along with the importance of this framework in both theoretical and practical applications. The document further highlights the gaps in educational curricula regarding CSSM and its foundational role in geotechnical engineering.

1. Critical-State Soil Mechanics
For Dummies
Prof. Samirsinh Parmar
Asst. Professor, Dept. of Civil Engg.
Dharmasinh Desai University, Nadiad, Gujarat, INDIA
Mail: samirddu@gmail.com

2. CONTENT OUTLINE
 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-logsv’ curves; and (2) shear
stress-vs. normal stress (t-sv’) from direct shear
box or simple shearing.
 Herein, only the bare essence of CSSM concepts
are presented, sufficient to describe strength &
compressibility response.

3. 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 (f', Cc, Cs) plus initial
state (e0, svo', 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.

4. 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
Effective Vertical Stress, svo' (kPa)
Void
Ratio,
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’

5. Direct 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)
Shear
Stress,
t
(kPa)
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
Effective Normal Stress, sn' (kPa)
Shear
Stress,
t
(kPa)
0.491 = tan f '
Strength Parameters:
c' = 0; f ' = 26.1 o
Peak
Peak
Peak

6. CSSM for Dummies
Log sv'
Effective stress sv'
Shear
stress
t
Void
Ratio,
e
NC
CC
tanf'
CSL
Effective stress sv'
Void
Ratio,
e
NC
CSL
CSSM Premise:
“All stress paths fail
on the critical state
line (CSL)”
CSL
f
c=0

7. CSSM for Dummies
Log sv'
Effective stress sv'
Shear
stress
t
Void
Ratio,
e
Void
Ratio,
e
NC
NC
CC
tanf'
CSL
CSL
CSL
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

8. CSSM for Dummies
Log sv'
Effective stress sv'
Shear
stress
t
Void
Ratio,
e
Void
Ratio,
e
NC
NC
CC
tanf'
CSL
CSL
CSL
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
Du
tmax = cu=su
Effective stress sv'

9. CSSM for Dummies
Log sv'
Effective stress sv'
Shear
stress
t
Void
Ratio,
e
NC NC
CC
tanf'
CSL
CSL
CSL
Note: All NC undrained
stress paths are parallel
to each other, thus:
su/svo’ = constant
Effective stress sv'
DSS: su/svo’NC = ½sinf’
Void
Ratio,
e

10. CSSM for Dummies
Log sv' Effective stress sv'
Effective stress sv'
Shear
stress
t
Void
Ratio,
e
Void
Ratio,
e
NC NC
CC
tanf'
CSL
CSL
CSL
CS
sp'
sp'
OC
Overconsolidated States:
e0, svo’, and OCR = sp’/svo’
where sp’ = svmax’ = Pc’ =
preconsolidation stress;
OCR = overconsolidation ratio

11. CSSM for Dummies
Log sv'
Effective stress sv'
Shear
stress
t
Void
Ratio,
e
NC NC
CC
tanf'
CSL
CSL
CSL
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'
Void
Ratio,
e
Du

12. CSSM for Dummies
Log sv'
Effective stress sv'
Shear
stress
t
Void
Ratio,
e
Void
Ratio,
e
NC NC
CC
tanf'
CSL
CSL
CSL
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'

13. 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…..

14. Equivalent Stress Concept
Log sv'
Stress sv'
Shear
stress
t
Void
Ratio,
e
NC
NC
CC
tanf'
CSL
CSL
CSL
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'
Void
Ratio,
e
sp' sp'
se'
se'
su
svf'
ep
De = Cs log(sp’/svo’)
De = Cc log(se’/sp’)
De
at se’
suOC = suNC

15. 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:
Normalized Undrained Shear Strength:
su/svo’ = ½ sinf’ OCRL
where L = (1-Cs/Cc)
se’ = svo’ OCR[1-Cs/Cc]

16. 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
sinf'
s
u
/
s
vo
'
NC
(DSS)
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'

17. Undrained Shear Strength from CSSM
0.1
1
10
1 10 100
Overconsolidation Ratio, OCR
DSS
Undrained
Strength
,
s
u
/
s
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
f' = 40o
20o
30o
su/svo' = ½ sinf' OCRL
Note: L = 1 - Cs/Cc  0.8
Intact
Clays

18. Porewater Pressure Response from CSSM
-6
-5
-4
-3
-2
-1
0
1
1 10 100
Overconsolidation Ratio, OCR
Normalized
Porewater
Pressure,
Du
/
s
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
L = 0.9 0.8 0.7
Intact
f' = 20o
30o
40o
Dus/svo' = 1 - ½cosf'OCRL

19. Yield Surfaces
Log sv'
Normal stress sv'
Shear
stress
t
Void
Ratio,
e
NC NC
CSL
CSL
CSL
OC
Normal stress sv'
Void
Ratio,
e
sp'
sp'
OC
 Yield surface represents
3-d preconsolidation
 Quasi-elastic behavior
within the yield surface

20. 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'* = (s1'+s2'+s3')/(3sp')
Deviatoric
Stress
=
q*
=
(
s
1
-
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
Strength
Ratio,
s
u
/
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

21. Cavity Expansion – Critical State Model for Evaluating
OCR in Clays from Piezocone Tests
OCR
M
q u
T b
vo
=

-












2
1
195 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
Depth
(meters)
CPTU
CRS
IL Oed
RF
Bothkennar, UK

22. Critical state soil mechanics
• Initial state: e0, svo’, and OCR = sp’/svo’
• Soil constants: f’, Cc, and Cs (L = 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/svo’ = ½ sinf’ OCRL
where L = 1-Cs/Cc
 Yield surface represents 3-d preconsolidation

23. Thank You