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SM1_6 January 9, 2018 1
Soil Mechanics I
6 – Consolidation
1 Definition
2 Theory of one-dimensional consolidation
3 Application: Settlement
4 Coefficient of consolidation; lab determination
SM1_6 January 9, 2018 2
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
SM1_6 January 9, 2018 3
1 Consolidation ≡ dissipation of excess pore pressure
×
2 Consolidation is a coupled process: seepage + volume change
(εV ≠ 0; ε
V >0, <0)
Definition
SM1_6 January 9, 2018 4
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]
SM1_6 January 9, 2018 6
Spring Analogy (a more realistic one + both-side drainage):
One-dimensional consolidation
[3]
SM1_6 January 9, 2018 7
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]
SM1_6 January 9, 2018 8
Average degree of consolidation U (Uavg
) for both-side drainage
(= ΔV(t) / ΔVfin
)
One-dimensional consolidation: U directly linked to settlement
Degree of consolidation
SM1_6 January 9, 2018 9
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
SM1_6 January 9, 2018 12
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
SM1_6 January 9, 2018 13
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
SM1_6 January 9, 2018 14
∂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
SM1_6 January 9, 2018 15
Degree of consolidation
Uz = 1- u / u
1 = 1 - ∑(f
1(Z) × f
2(T))
[1]
One-dimensional consolidation – Degree of consolidation
SM1_6 January 9, 2018 16
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)
SM1_6 January 9, 2018 17
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
)
SM1_6 January 9, 2018 18
U % = 100 (1 - ∑f (T))
graphically (for u constant or linear with depth)
[3]
Average Degree of Consolidation U (Uavg
)
SM1_6 January 9, 2018 19
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
)
SM1_6 January 9, 2018 20
Oedometer: log t method (Casagrande)
Coefficient of Consolidation cv - determination
[1]
SM1_6 January 9, 2018 21
Oedometer: log t method (Casagrande)
....definition of creep by EOP (End Of Primary) compression
Coefficient of Consolidation cv - determination
[3]
SM1_6 January 9, 2018 22
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
SM1_6 January 9, 2018 23
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
SM1_6 January 9, 2018 24
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
SM1_6 January 9, 2018 25
[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.