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Soil Mechanics I

6 – Consolidation

1 Definition

2 Theory of one-dimensional consolidation

3 Application: Settlement

4 Coefficient of consolidation; lab determination

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Undrained loading → excess pore pressure Δu

u = u0 + Δu

→ hydraulic gradient

→ seepage

→ volume changes

→ a change in effective stress Δσ'

With time decrease in Δu

u = u0 = end of consolidation

NB: during consolidation the rate of deformation is not linear (which will be shown later in the theory)

Definition

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1 Consolidation ≡ dissipation of excess pore pressure

×

2 Consolidation is a coupled process: seepage + volume change

(εV ≠ 0; ε

V >0, <0)

Definition

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Generally: consolidation in 3D

Special case: large loaded area + relatively thin soil layer → 1D

In-situ:

Lab model:

Definition

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Spring Analogy:

Pressures (σ, u) at different given times t1, t

2...: isochrones

One-dimensional consolidation

[3]

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Spring Analogy (a more realistic one + both-side drainage):

One-dimensional consolidation

[3]

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Degree of consolidation

Degree of consolidation Uz

(= ΔVz(t) / ΔV

z, fin )

σ' - σ1' = (σ

2' - σ

1') – u = Δu – u = u

i – u

Uz = (e – e

1) / (e

2 – e

1) = (σ' - σ

1') / (σ

2' - σ

1') = (u

i - u) / u

i = 1 - u / u

i

[3]

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Average degree of consolidation U (Uavg

) for both-side drainage

(= ΔV(t) / ΔVfin

)

One-dimensional consolidation: U directly linked to settlement

Degree of consolidation

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Assumptions:

1 Saturated soil (S=1)

2 Water and grains incompressible

1 + 2: principle of effective stress valid

3 Darcy's law (v = k i, k = const.)

4 Homogeneous, linearly elastic soils, small deformations

5 Compression + seepage in 1-D

(+ top and bottom drainage)

One-dimensional consolidation – Terzaghi's Theory

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Continuity of flow (Qout

= Qin) + compressibility of the soil (ε

V ≠ 0)

One-dimensional consolidation – Terzaghi's Theory

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Hydraulic gradient in depth z:

iz = ∂(u/γ

w) / ∂z = 1/γ

w ∂u / ∂z

Hydraulic gradient in depth z+dz:

iz+dz

= 1/γw

∂u / ∂z + 1/γw

∂2u / ∂z2 dz

Darcy:

dQ = k i dx dy dt

dQout

= k 1/γw

∂u / ∂z dx dy dt

dQin

= k 1/γw

(∂u / ∂z + ∂2u / ∂z2 dz) dx dy dt

Volume change:

dQout

- dQin = - k / γ

w ∂2u / ∂z2 dz dx dy dt

(dx dy = 1: dQout

- dQin = - k / γ

w ∂2u / ∂z2 dz dt )

One-dimensional consolidation – Terzaghi's Theory

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Volume change - compressibility:

Using coefficient of compressibility (e.g.):

av= - Δe / Δσ' = - de / dσ'

Volume change = settlement

Δdz = - de / (1+e1) dz = a

v dσ' / (1+e

1) dz =

= mv dσ' dz = 1/E

oed dσ' dz

constant total stress (Δσ = 0): Δσ' = - Δu

Δdz = - 1/Eoed

du dz

du = ∂u / ∂t dt

Δdz = - 1/Eoed

∂u / ∂t dt dz

One-dimensional consolidation – Terzaghi's Theory

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dQout

– dQin = - k / γ

w ∂2u / ∂z2 dz dt Δdz = - 1 / E

oed ∂u / ∂t dt dz

∂u / ∂t = k Eoed

/ γw ∂2u / ∂z2

k Eoed

/ γw = c

v ... coefficient of consolidation

∂u / ∂t = cv

∂2u / ∂z2

One-dimensional consolidation – Terzaghi's Theory

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∂u / ∂t = cv

∂2u / ∂z2

Mathematical solution – initial and boundary conditions

t = 0: Δu = u1 = Δσ = Δσ' = σ

2' – σ

1'

z = 0 ; z = 2H: Δu = 0

the initial excess pore pressure is a function of depth z:

the initial excess pore pressure varies linearly with depth z (or constant):

Z = z / H (dimensionless depth)

T = cv t / H2 (dimensionless time = „time factor“)

One-dimensional consolidation – Terzaghi's Theory

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Degree of consolidation

Uz = 1- u / u

1 = 1 - ∑(f

1(Z) × f

2(T))

[1]

One-dimensional consolidation – Degree of consolidation

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A clay layer of h = 6m and cv = 6 ×10-8 m2s-1 is drained from the top and bottom.

What is the degree of consolidation at 1/20, 1/4, 1/2, 3/4 of the clay thickness in one year after an undrained loading.

T = cv t / H2 = 6×10-8 × 365 × 24 × 3600 / 9=

= 0,21 ≈ 0,2

(h = 2H)

Depth ½ h: Z = z / H = H / H = 1 Uz ≈ 0,25 = 25%

Depth ¼ h: Z = z / H = (¼ 2H) / H = 0,5 Uz ≈ 0,45 = 45%

Depth ¾ h: Z = z / H = (¾ 2H) / H = 1,5 Uz ≈ 0,45 = 45%

Depth 1/20 h: Z = z / H = (1/20 2H) / H = 0,1 Uz ≈ 0,9 = 90%

Degree of consolidation U(z)

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U % = 100 (1 - ∑f (T))

Graphically (depending on boundary and initial conditions...)

For U < 60% T ≈ π/4 U2

[1]

Average Degree of Consolidation U (Uavg

)

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U % = 100 (1 - ∑f (T))

graphically (for u constant or linear with depth)

[3]

Average Degree of Consolidation U (Uavg

)

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A clay layer of h = 6m and cv = 6 ×10-8 m2s-1 is drained from the top and bottom.

What is the degree of consolidation at 1/20, 1/4, 1/2, 3/4 of the clay thickness in one year after an undrained loading.

a) What part of the final settlement takes places in one year after loading?b) When does 95% of the final settlement happen?

a) T = 0,2 → U = 50%

U = s(t) / sfinal

1 year: 50% of final settlement

b) T = cv t / H2

t = T H2 /cv

U = 0,95 →T ≈ 1,2

U = 0,95 → t ≈ 1,2 × 32 / (6×10-8) = 1,8×108 s ≈ 6 years

Average Degree of Consolidation U (Uavg

)

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Oedometer: log t method (Casagrande)

Coefficient of Consolidation cv - determination

[1]

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Oedometer: log t method (Casagrande)

....definition of creep by EOP (End Of Primary) compression

Coefficient of Consolidation cv - determination

[3]

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Oedometer: √t method (Taylor)On the theoretical curve U vs √T (right) Taylor noted, that the initial part of the

relationship is linear, and that after decreasing its slope by 15%, the intersection corresponds to T for U=0.9.

It si used in curve fitting method (left)

[1]

Coefficient of Consolidation cv - determination

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Typical values of cv

Clay

n × 10-8 m2s-1

Silt

n × 10-4 m2s-1

(± kaolin in Lab Class No4)

1×10-8 m2s-1 ≈ 0,3 m2 / year

Coefficient of Consolidation cv - determination

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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 for the course Soil Mechanics ILiterature for the course Soil Mechanics I

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[1] Taylor, D. W. (1948) The fundamentals of mechanics of soils. John Wiley & Sons .

[2] Terzaghi, K. (1943) Theoretical soil mechanics. John Wiley & Sons.

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

References – used figures etc.