1. Critical-State Soil Mechanics For Dummies
Paul W. Mayne, PhD, P.E. Civil & Environmental Engineering Georgia Institute of Technology Atlanta, GA
2006

2. 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-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
svo'=300 kPa
sp'=900 kPa
Cr = 0.04
Overconsolidation Ratio, OCR = 3
Cs = swelling index (= Cr)
cv = coef. of consolidation
D' = constrained modulus
Cae = coef. secondary compression
k ≈ hydraulic conductivity
Cc = 0.38
sv’

5. sv’ sv’ t t t t Direct Shear Test Results gs d Direct Shear Box (DSB)
Direct Simple Shear (DSS) t t t gs d
Direct Shear Box (DSB)

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

7. Volume Change is Contractive: evol = De/(1+e0) < 0
CSSM for Dummies CC
Void Ratio, e
Void Ratio, e e0 NC ef De NC
CSL
CSL
svo
Log sv'
Effective stress sv'
CSL
tmax = c + s tanf
tanf'
STRESS PATH No.1
NC Drained Soil
Shear stress t
Given: e0, svo’, NC (OCR=1)
Drained Path: Du = 0
Volume Change is Contractive: evol = De/(1+e0) < 0
c’=0
svo
Effective stress sv'

8. +Du = Positive Excess Porewater Pressures
CSSM for Dummies CC
Void Ratio, e
Void Ratio, e e0
svo
svo NC
svf
svf NC
CSL
CSL
Effective stress sv'
Log sv'
CSL
tanf'
STRESS PATH No.2
NC Undrained Soil Du
tmax = cu=su
Shear stress t
Given: e0, svo’, NC (OCR=1)
Undrained Path: DV/V0 = 0
+Du = Positive Excess Porewater Pressures
Effective stress sv'

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

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

11. CSSM for Dummies svo' svf' Log sv' svo' Void Ratio, e Void Ratio, e OCCC
svo' e0
Void Ratio, e
Void Ratio, e OC
svf' CS NC NC
CSL
CSL
Effective stress sv'
Log sv'
CSL
Stress Path No. 3
Undrained OC Soil:
e0, svo’, and OCR
tanf' Du
Shear stress t
Stress Path: DV/V0 = 0
Negative Excess Du
svo'
Effective stress sv'

12. CSSM for Dummies svo' Log sv' Void Ratio, e Void Ratio, e OC NC NC CSL
Effective stress sv'
Log sv'
CSL
Stress Path No. 4
Drained OC Soil:
e0, svo’, and OCR
tanf'
Shear stress t
Stress Path: Du = 0
Dilatancy: DV/V0 > 0
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 ConceptCC NC
svo' e0
Void Ratio, e
svf'
se'
Void Ratio, e De OC NC CS
sp'
sp' ep
CSL
CSL
Log sv'
Effective stress sv'
1. OC State (eo, svo’, sp’)
CSL
2. Project OC state to NC line for equivalent stress, se’
tanf'
Shear stress t su
at se’
suOC = suNC
De = Cs log(sp’/svo’)
De = Cc log(se’/sp’)
3. se’ = svo’ OCR[1-Cs/Cc]
svo'
se'
Stress sv'

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

16. Undrained Shear Strength from CSSM

17. Undrained Shear Strength from CSSM

18. Porewater Pressure Response from CSSM

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

20. Port of Anchorage, Alaska

21. Cavity Expansion – Critical State Model for Evaluating OCR in Clays from Piezocone Tests
where M = 6 sinf’/(3-sinf’)
and L = 1 – Cs/Cc  0.8 qc fs ub
 qT

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