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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

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critical state soil mechanics
Critical-State Soil Mechanics

Paul W. Mayne, PhD, P.E.Civil & Environmental EngineeringGeorgia Institute of Technology

prologue
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.
critical state soil mechanics cssm
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.
one dimensional consolidation

svo\'=300 kPa

sp\'=900 kPa

Cr = 0.04

Cc = 0.38

One-Dimensional Consolidation

Overconsolidation Ratio, OCR = 3

Cs = swelling index (= Cr)

cv = coef. of consolidation

D\' = constrained modulus

Cae = coef. secondary compression

k ≈ hydraulic conductivity

sv’

slide5

sv’

t

t

t

t

d

gs

Direct Simple Shear (DSS)

Direct Shear Test Results

sv’

Direct Shear Box (DSB)

slide6

Void Ratio, e

NC

CSL

CSL

Effective stress sv\'

CSL

tanf\'

CSSM

CC

Void Ratio, e

NC

Log sv\'

CSSM Premise:

“All stress paths fail on the critical state line (CSL)”

Shear stress t

f

c=0

Effective stress sv\'

slide7

De

ef

CSL

CSSM

CC

Void Ratio, e

Void Ratio, e

e0

NC

NC

CSL

CSL

svo

Effective stress sv\'

Log sv\'

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\'

slide8

e0

svf

svo

CSL

svf

svo

CSSM

CC

Void Ratio, e

Void Ratio, e

NC

NC

CSL

CSL

Effective stress sv\'

Log sv\'

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\'

slide9

CSL

CSSM

CC

Void Ratio, e

Void Ratio, e

NC

NC

CSL

CSL

Effective stress sv\'

Log sv\'

tanf\'

Note: All NC undrained

stress paths are parallel

to each other, thus:

su/svo’ = constant

Shear stress t

DSS: su/svo’NC = ½sinf’

Effective stress sv\'

slide10
CSSM

CC

Void Ratio, e

OC

Void Ratio, e

CS

NC

NC

CSL

CSL

Effective stress sv\'

sp\'

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\'

slide11

e0

svo\'

svf\'

Du

CSSM

CC

Void Ratio, e

Void Ratio, e

OC

CS

NC

NC

CSL

CSL

Effective stress sv\'

Log sv\'

CSL

Stress Path No. 3

Undrained OC Soil:

e0, svo’, and OCR

tanf\'

Shear stress t

Stress Path: DV/V0 = 0

Negative Excess Du

svo\'

Effective stress sv\'

slide12

e0

svo\'

svo\'

CSSM

CC

Void Ratio, e

Void Ratio, e

OC

CS

NC

NC

CSL

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\'

critical state soil mechanics1
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 (decreaseevol)
  • For OC soil:
    • Undrained (evol = 0): -Du and tmax = su = cu
    • Drained (Du = 0) and dilative (Increaseevol)

There’s more !

Semi-drained, Partly undrained, Cyclic…..

equivalent stress concept

e0

De

ep

svo\'

se\'

svf\'

Equivalent Stress Concept

CC

NC

Void Ratio, e

Void Ratio, e

NC

OC

CS

sp\'

sp\'

CSL

CSL

Effective stress sv\'

Log sv\'

CSL

1. OC State (eo, svo’, sp’)

tanf\'

2. Project OC state to NC line for equivalent stress, se’

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\'

critical state soil mechanics2
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)

yield surfaces

Yield Surface

Yield Surfaces

NC

NC

CSL

OC

OC

Void Ratio, e

Void Ratio, e

sp\'

CSL

sp\'

Normal stress sv\'

Log sv\'

CSL

  • Yield surface represents 3-d preconsolidation
  • Quasi-elastic behavior within the yield surface

Shear stress t

Normal stress sv\'

cavity expansion critical state model for evaluating ocr in clays from piezocone tests

fs

ub

qc

 qT

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

critical state soil mechanics3
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
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