Critical state soil mechanics
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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

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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-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)

  • 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.


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’


sv’

t

t

t

t

d

gs

Direct Simple Shear (DSS)

Direct Shear Test Results

sv’

Direct Shear Box (DSB)


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'


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'


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'


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'


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'


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'


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 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…..


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 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


Undrained Shear Strength from CSSM


Porewater Pressure Response from CSSM


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'


Port of Anchorage, Alaska


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 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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