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

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’


Critical state soil mechanics

sv’

t

t

t

t

d

gs

Direct Simple Shear (DSS)

Direct Shear Test Results

sv’

Direct Shear Box (DSB)


Critical state soil mechanics

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'


Critical state soil mechanics

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'


Critical state soil mechanics

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'


Critical state soil mechanics

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'


Critical state soil mechanics

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'


Critical state soil mechanics

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'


Critical state soil mechanics

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)


Undrained shear strength from cssm

Undrained Shear Strength from CSSM


Undrained shear strength from cssm1

Undrained Shear Strength from CSSM


Porewater pressure response from cssm

Porewater Pressure Response from CSSM


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'


Port of anchorage alaska

Port of Anchorage, Alaska


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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