Soil Mechanics I 4 – Compressibility - cuni.cz

Soil Mechanics I
4 – Compressibility

1. Isotropic compression
2. Nonlinearity
3. Overconsolidation; OC vs. creep
4. One-dimensional compression - parameters
5. Settlement (one-dimensional compressibility)
6. At-rest earth coefficient; Eoed vs E' for elastic material

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Isotropic Compression
Loading - isotropic Δσ1' = Δσ2' = Δσ3' = Δσ' = Δp'; q = const.

- anisotropic Δσ1'; Δσ2'; Δσ3' → Δp'

Δp' ≠ 0 → structural changes
(changes in position of soil grains) = compression

Δp' > 0 → loose soil → dense soil = change in porosity shown by (a) → (b):

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2

Isotropic Compression

For isotropic compression (i.e. for q=const.):
Bulk Modulus K = dp' / dεV ≠ const.

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3

Isotropic Compression
Isotropic compression

(semi)logarithmic plot in working
stress range is linear
loading
Normal Compression Line
NCL:
e = N – λ ln p'
unloading – reloading
e = eκ – κ ln p'

Parameters for isotropic
compression: λ; κ; N
They are constants

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Overconsolidation

MZ1_4 April 24, 2013 Skempton (1964)

5

Overconsolidation

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Overconsolidation

„overconsolidation stress“ = yield stress σY'
determination (from oedometer data)

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Overconsolidation

„Overconsolidation Stress“ vs Creep
→ the meaning of „overconsolidation stress“ - ???

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One Dimensional Compression
One-dimensional compression (oedometer)

εV = - Δh / h0 = - Δe / (1+e0) ≡ s / h0
s – settlement

Settlement calculation:
s = - Δe / (1+e0) h0 = εV h0

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One Dimensional Compression
semi-logarithmic plot of stress vs e:

compressibility index Cc= - Δe / Δlogσv'

unloading: Ccr (or Cs ← 'swelling')

Cc is a real parameter – constant (for working stress
range)

NB:
In determining Cc, NCL must be measured (not the
compressibility at OC...)

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One Dimensional Compression
semi-logarithmic plot of stress vs ε:
modified compressibility index Ccε= Δε / Δlogσv'
unloading Crε (or Csε ← 'swelling')

εV = - Δe / (1+e0) → Ccε= Cc /(1+e0)

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One Dimensional Compression
arithmetic plot of stress vs ε:
oedometer modulus Eoed= Δσv' / Δε
coefficient of volume change mv = 1 / Eoed

Not a real parameter – not a constant – depends on
stress level

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One Dimensional Compression
arithmetic plot of stress vs e:

coefficient of compressibility av = - Δe / Δσv'

Not a real parameter – not a constant – depends on
stress level

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One Dimensional Compression
Settlement computation

s = - Δe / (1+e0) h0 = εV h0
summing over layers: s = ∑ si

using different „parameters“:

s = Cc h0 / (1+e0) log ((σv0' + Δσv') / σv0')

s = Ccε h0 log ((σv0' + Δσv') / σv0')

s = mv h0 Δσv' (mv not a constant)

s = h0 Δσv' / Eoed (Eoed not a constant)

Δσv' change in effective stress

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One Dimensional Compression
Settlement computation – OC soil

for example with Ccε

s = Crε h0 log (σp' / σv0') + Ccε h0 log ((σv0' + Δσv') / σp')

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One Dimensional Compression

One-dimensional compression – „at rest“ coefficient K0

εh ≡ 0
εh = 1 / E' (σh' – ν' (σh' + σv')) = 0
σh' – ν' σh' – ν' σv' = 0
σh' / σv' = ν' / (1 – ν' ) = K0 = at rest coefficient

NB:
1 The above definition of K0 is valid for elastic material only (Hooke's law
used);
With soils: ν' not a constant

2 Effective stress ratio K0 constant for NC soil only (while NC soil is not
elastic at all)

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K0 vs Young's Modulus in Elasticity

Elastic relationship between Oedometer a Young's Moduli
Oedometer (Δσv; σh' = σv' K0; εh =0):
εv = σv' / Eoed'
Hooke's Law for a specimen in the oedometer:
εv = 1 / E' (σv' – ν' (σh' + σh')) = σv' / E' (1 – ν' (2ν' / (1 – ν'))
εv = σv' / E' (1 – 2ν'2 / (1 – ν'))

→ 1 / Eoed' = 1 / E' (1 – 2ν'2 / (1 – ν'))

→ Eoed' = E' / (1 – 2ν'2 / (1 – ν')) for an elastic material only

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One Dimensional Compression – Lab Class No 7
Kaolin Clay - Lab class No 7

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One Dimensional Compression

Typical values of „parameters“

Cc; Ccr≈ Cc / 5 Cc = 0,2 – 0,5
clay Cc = 0,15 – 0,3
silt, silty clay Cc > 1; Cc = 7-10
sensitive clay Cc = 10 – 15
peat

Eoed

fine-grained soil Eoed = 1 – 30 MPa

kaolin – Lab Class Eoed = 3 – 8 MPa

sand Eoed = 5 – 100 MPa

gravel Eoed = 20 – 500 MPa

Oedometer modulus is stress dependent – not a parameter

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LLitieterraatuturreefoforrththeeccoouurrsseeSSooililMMeecchhaannicicssII

http://labmz1.natur.cuni.cz/~bhc/s/sm1/
Atkinson, J.H. (2007) The mechanics of soils and foundations. 2nd ed. Taylor & Francis.

Further reading:

Wood, D.M. (1990) Soil behaviour and critical state soil mechanics. Cambridge
Univ.Press.

Mitchell, J.K. and Soga, K (2005) Fundamentals of soil behaviour. J Wiley.

Atkinson, J.H: and Bransby, P.L. (1978) The mechanics of soils. McGraw-Hill, ISBN 0-
07-084077-2.

Bolton, M. (1979) A guide to soil mechanics. Macmillan Press, ISBN 0-33318931-0.

Craig, R.F. (2004) Soil mechanics. Spon Press.

Holtz, R.D. and Kovacs, E.D. (1981) An introduction to geotechnical engineering,
Prentice-Hall, ISBN 0-13-484394-0

Feda, J. (1982) Mechanics of particulate materials, Academia-Elsevier.)

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References – used figures etc.
[1] Atkinson, J.H. (2007) The mechanics of soils and foundations. 2nd ed. Taylor & Francis.

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