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Soil Mechanics I

5 – Soil Strength

1. Mohr-Coulomb envelope

2. Critical state; critical strength

3. Residual strength

4. CSL in 3D

5. Undrained strength

6. Peak strength

7. Strength of unsaturated soils

8. Interpretation of strength data

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Mohr - Coulomb envelope

τf = c' + σ

f' tgφ' (subscript f means 'failure')

τf = ½ (σ

1' – σ

3') sin 2θ

σf' = ½ (σ

1' + σ

3') + ½ (σ

1' – σ

3') cos 2θ

θ = 45º + ½ φ'

Strength

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Idealized shear test(Drained, u = const):

Strength of soils

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...during shearing after sufficient strain the soil reaches a uniquely defined state - ideal plasticity - critical state

Definition of critical state

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Critical State (shear) + Compressibility:

Wet side of critical × Dry side of critical

Definition of critical state

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Critical State (shear) + Compressibility:

q = M p'e

Γ = e + C

c log p'

(Γ = v + λ ln p')

Definition of critical state

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CSL in 3D v:q:p' (e:q:p'; w:q:p'...)

[1]

Definition of critical state

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Detail: during shear a stochastic movement of grains ... even decay of grains (crushing)

Neglecting detail – continuous process – yielding – dissipation of energy - FRICTION – described by q=Mp'.

Detail: during shear a stochastic change of distance between grains

Neglecting detail - change of v = VOLUME CHANGE – COMPRESSION – described by Γ=v+λlnp'.

Concept of critical state describes at „phenomenological“ (macro) level the microstructural processes.

Developed for „paste“ = „reconstituted“ soil

CSSM however is a general theoretical framework for soil behaviour

Definition of critical state

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S = 1; undrained event ↔ w = const (i.e., n = const; e = const; ...)

Undrained Strength

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S = 1; undrained event ↔ w = const (i.e., n = const; e = const; ...)

Undrained Strength

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NC (loose) vs OC (dense) soil:

...dependence on e → normalization

Peak Strength

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Series of 3 undrained triaxial tests

NC at p'=a; 2a; 3a

[1]

Normalization (Peak Strength)

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Series of 3 undrained triaxial tests

Stress-strain diagrams are similar; at a suitable non-dimensional plot they merge/coincide → normalization

(If normalization successful, the behaviour is identical...(so called 'physical isomorphism'))

[1]

Normalization (Peak Strength)

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Normalization

by „equivalent“/“Hvorslev“ stress at NCL, or

by stress at CSL

logσc' = (e

Γ -e

a) / C

c

[2]

Normalization (Peak Strength)

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Despite normalizing the straight line approximations cannot describe the peak strength envelope successfully:

cp' and φ

p' depend on stress level (on the interval of stress during the experiments)

Peak Strength → normalization for MC strength envelope

[2]

Peak Strength

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Peak Strength - “Hvorslev's M-C peak strength envelope“

τp'=c

p'(e) + σ

p' tgφ

p'

φp' constant - not changing with e

cp'(e) increasing with decreasing e – not

a parameter

[2]

Peak Strength

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M-C equation for “Hvorslev's M-C peak strength envelope“

τp'=c

p'(e) + σ

p' tgφ

p'

cp'(e) is relevant for Peak Strength for both fine-grained soils (clay)

and coarse-grained soils (sand) → not linked to the forces between grains/particles

cp'(e) is the value on the vertical axis depending on e; v; w; n →

not a parameter

Peak Strength

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Normalization by σc': c

p' = c

p'(e) / σ

c'

From the figure: cp' = tgφ

c'- tgφ

p' (i.e. c

p' and φ

p' are not independent)

τp'/σ

c'=c

p' + σ

p'/σ

c' tgφ

p' = (tgφ

c'- tgφ

p') + (σ

p'/σ

c') tgφ

p'

If the normalized strength envelope is linear → cp', φ

p' not depending on e

→ cp', φ

p' material parameters

Peak Strength → normalization for “Hvorslev's M-C peak strength envelope“

[2]

Peak Strength

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Peak envelope is curved since uncemented soils exhibit c' = 0 pro σ'=0

τp' = a σ

p'b

log τp' = log a + b logσ

p'

a and b are parameters depending on the state (w, e, etc., i.e., on stress again...)

Peak Strength → power law strength envelope (rock mechanics: „Hoek-Brown strength“)

[2]

Peak Strength

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Peak Strength → power law strength envelope (rock mechanics: „Hoek-Brown strength“)

Normalising:

τp'/σ

c' = A (σ

p'/σ

c')B = tg φ

c' (σ

p'/σ

c')B

log (τp'/σ

c') = log(tg φ

c') + B (σ

p'/σ

c')

B is a (real) parameter depending on grains/mineralogy only, not on the state

[2]

Peak Strength

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Peak Strength → “secant peak friction angle“

Effect of Dilation (dilatancy angle)

τ'/σ' = tg (φc' + ψ

p)

[2]

Peak Strength

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In Summary:

Three ways to interpret the peak strength data

In practice the least reasonable is used

At least a stress range must be given if the linear M-C peak strength envelope is used

Peak Strength - Summary

[2]

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Capillary Suction considered a stress parameter in unsaturated soils

Strength of Unsaturated Soils

τf = c’ + (σ - ua)f tgφ’ + (ua - uw)f tgφb

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„UU“ triaxial tests on fine-grained soil:MV(F7), I

P=39, w

L=75, S

r=0,98

[...from a site investigation report]

….Interpretation of strength tests

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„UU“ test on unsaturated soilB<1→Δu < Δσ; different Δ(u

a-u

w) for different σ

3→ different initial states for A-A

3 originally

identical specimens; in shearing further Δu→bigger Mohr's circle for higher net stress (←φb<φ')

[3]

….Interpretation of strength tests

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Unsaturated soils, or cavitation in shearing OC saturated soils → the initial part not relevant...

„UU“ tests – cont.

[3]

….Interpretation of strength tests

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„UU“ tests – cont.

„φu c

u“ strength envelope („total parameters“ in practice) from UU or CU triaxial tests

dependence on state (w; e...)dependence on loading rate

→ φu and c

u are no parameters for strength (have no meaning!)

→ cannot be used in practice→ just an incorrect interpretation of the test results („empiricism“)

….Interpretation of strength tests

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Summary and Typical Values

φcr' = critical strength = real parameter (a constant for given soil)

Quartzy Sand φcr' = 32 – 35º

Bohemian loess (silty clay) φcr' = ca 32º

clay - depending on mineralogy:London Clay φ

cr' = ca 22º

Brno (Vienna) Tegel φcr' = ca 25º

North Bohemian clays φcr' = ca 25-26º

Kaolin Clay φcr' = ca 25-27º

φr' = residual strength = real parameter (a constant for given soil)

φr' ≈ 1/2φ

cr'

su = undrained strength – dependence on w; e;... (not a constant (material parameter))

at wL: s

u ≈ 2-3 kPa; at w

P: s

u ≈ 200-300kPa

soft clay su ≈ 20 kPa, firm clay s

u > 50 kPa...

NOT TO BE TABLED (codes, standards....)!

„φu ≠0; c

u strength envelope“ („total parameters“) from UU or CU triaxial tests - no meaning,

no use

φp' c

p' = peak strength – dependence on w; e;... (not a constant (material parameter))

NOT TO BE TABLED (codes, standards....)!

Strength

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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.)

Literature

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[1] Atkinson, J.H. and Bransby, P.L. (1978) The mechanics of soils. An introduction to critical state soil mechanics. McGraw-Hill, Maidenhead (UK).

[2] Atkinson, J.H. (2007) The mechanics of soils and foundations. 2nd ed. Taylor & Francis.

[3] Fredlund, D.G. and Rahardjo, H. (1993) Soil mechanics for unsaturated soils. J Wiley&Sons.

References – used figures etc.