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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

- 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.

- 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

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'

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

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

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

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'

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

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'

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'

- 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'

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'

- 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 Surface

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'

fs

ub

qc

qT

where M = 6 sinf’/(3-sinf’)

and L= 1 – Cs/Cc 0.8

- 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