SME
Soil Mechanics · Problem type
Time for a degree of consolidation
Use this for “A 4 m clay layer drains at its top and bottom, with cᵥ = 0.002 cm²/s. How long until it is 50% consolidated?”
Chart
Tᵥ(50) ≈ 0.1963
This is an example. Type your own values to start a new problem.
Hdr (m)
=
H (m)
÷
N
Tᵥ
=
π
÷
4
×
(
%)
2
t (yr)
=
Tᵥ
×
(
Hdr (m)
)
2
÷
cᵥ (cm²/s)
Formulas
Hdr
=
H
N
2
=
4
2
Tᵥ
=
π
4
×
(
U
100
)
2
0.19635
=
π
4
×
(
50
100
)
2
t
=
Tᵥ
×
Hdr
2
cᵥ
3,926,991
=
0.19635
×
2
2
2 ×
10
-7
Formulas are worked in m²/s, s. The step-by-step shows the conversions.
Assumptions
•
Drained at one face (Hdr = H) or both, top and bottom (Hdr = H ÷ 2): the farthest water travels.
•
The degree of consolidation U is the share of the final settlement reached so far.
•
Tᵥ = (π ÷ 4)U² holds for U up to 60%.
Step-by-step
WE KNOW
H = 4 m, N = 2, cᵥ = 0.002 cm²/s, U = 50%
FIND
drainage path (Hdr), time factor (Tᵥ), time (t)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m²/s, s).
cᵥ = 0.002 cm²/s = 2 × 10⁻⁷ m²/s (1 m²/s = 10⁴ cm²/s)
Step 2 · Find drainage path (Hdr)
Use Hdr = H ÷ N
Water leaves by the nearer drained face: half the layer away with two, all of it with one.
Hdr
=
4
2
Hdr = 2 m
Step 3 · Find time factor (Tᵥ)
Use Tᵥ = π ÷ 4 × (U ÷ 100)²
Early on, the time factor grows as the square of the degree of consolidation.
Tᵥ
=
π
4
×
(
50
100
)
2
Tᵥ
=
0.7854
×
0.5
2
Tᵥ = 0.7854 × 0.25
Tᵥ = 0.19635
Step 4 · Find time (t)
Use t = Tᵥ × Hdr² ÷ cᵥ
Time grows with the square of the drainage path and falls as cᵥ rises.
t
=
0.19635
×
2
2
2 ×
10
-7
t
=
0.785398
2 ×
10
-7
t
=
3,926,990.817
s
Step 5 · Convert the answers
Change each answer back to the unit you chose.
t = 3,926,990.817 s = 0.1244 yr (1 yr = 31557600 s)
Check
Put the numbers back in. Both sides should match.
2
=
4
2
✓
0.19635
=
π
4
×
(
50
100
)
2
✓
3,926,991
=
0.19634954
×
2
2
2 ×
10
-7
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Consolidation
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