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Civil
Elastic layer settlement from constrained modulus
Settlement of a soil profile split into sublayers, each compressing by its thickness times the representative effective stress increase divided by its one-dimensional constrained modulus M.
- Ref
- CIV-55
- Rev
- v1
- Field
- Civil
- Result
- S [m]
A layered soil is split into sublayers, each treated as one-dimensional: the strain is Δσ′/M, where M is the constrained modulus, the stiffness under confinement with no lateral strain. The settlement is the sum over layers, S = ΣHiΔσ′i/Mi. Use a modulus matched to the stress range of each layer, and a representative stress increase at its middle.
Settlement of a soil profile split into sublayers, each compressing by its thickness times the representative effective stress increase divided by its one-dimensional constrained modulus M. Small strains and no lateral deformation. Use a modulus suited to the stress range of each layer. Up to three layers here; leave unused ones at zero thickness.
Calculator
- Settlement of layer 1, H₁Δσ′₁/M₁0.01 m
- Settlement of layer 2, H₂Δσ′₂/M₂0.006 m
- Settlement of layer 3, H₃Δσ′₃/M₃0 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The total settlement is 16 mm; layer 1 provides 62.5 % of it.
- The softest layer governs, so a thin soft layer can contribute more settlement than a thick stiff one.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| M1 Constrained modulus of layer 1 | 1,000,000 to 1e8 Pa | 0.0072 | 0.126 | 0.1188 | +10% | −5.68 % |
| H1 Thickness of layer 1 | 0 to 20 m | 0.006 | 0.106 | 0.1 | +10% | +6.25 % |
| Δσ′2 Stress increase in layer 2 | 0 to 500,000 Pa | 0.01 | 0.11 | 0.1 | +10% | +3.75 % |
| M2 Constrained modulus of layer 2 | 1,000,000 to 1e8 Pa | 0.0109 | 0.1 | 0.0891 | +10% | −3.41 % |
| Δσ′1 Stress increase in layer 1 | 0 to 500,000 Pa | 0.006 | 0.08933 | 0.08333 | +10% | +6.25 % |
| H2 Thickness of layer 2 | 0 to 20 m | 0.01 | 0.05 | 0.04 | +10% | +3.75 % |
| H3 Thickness of layer 3 | 0 to 20 m | 0.016 | 0.016 | 0 | +10% | 0 % |
| Δσ′3 Stress increase in layer 3 | 0 to 500,000 Pa | 0.016 | 0.016 | 0 | +10% | 0 % |
| M3 Constrained modulus of layer 3 | 1,000,000 to 1e8 Pa | 0.016 | 0.016 | 0 | +10% | 0 % |
Sweep of Stress increase in layer 1
| Stress increase in layer 1 (Pa) | Settlement (m) |
|---|---|
| 0 | 0.006 |
| 62,500 | 0.01642 |
| 125,000 | 0.02683 |
| 187,500 | 0.03725 |
| 250,000 | 0.04767 |
| 312,500 | 0.05808 |
| 375,000 | 0.0685 |
| 437,500 | 0.07892 |
| 500,000 | 0.08933 |
Sweep one variable
Variables and how each one contributes
Settlement
What the equation gives you. Everything else feeds it.
Thickness of layer 1
The thickness of layer 1, in m (zero if absent). Raises the settlement of this layer in direct proportion. Raising it by 10 % (to 2.2) moves the result by +6.25 %, from 0.016 to 0.017 m.
Thickness of layer 2
The thickness of layer 2, in m (zero if absent). Raises the settlement of this layer in direct proportion. Raising it by 10 % (to 3.3) moves the result by +3.75 %, from 0.016 to 0.0166 m.
Thickness of layer 3
The thickness of layer 3, in m (zero if absent). Raises the settlement of this layer in direct proportion. Raising it by 10 % (to 0) moves the result by +0 %, from 0.016 to 0.016 m.
Stress increase in layer 1
The effective stress increase at the middle of layer 1, in Pa. Raises the settlement of this layer in direct proportion. Raising it by 10 % (to 66,000) moves the result by +6.25 %, from 0.016 to 0.017 m.
Stress increase in layer 2
The effective stress increase at the middle of layer 2, in Pa. Raises the settlement of this layer in direct proportion. Raising it by 10 % (to 33,000) moves the result by +3.75 %, from 0.016 to 0.0166 m.
Stress increase in layer 3
The effective stress increase at the middle of layer 3, in Pa. Raises the settlement of this layer in direct proportion. Raising it by 10 % (to 0) moves the result by +0 %, from 0.016 to 0.016 m.
Constrained modulus of layer 1
The constrained modulus of layer 1, in Pa. A stiffer layer settles less, in inverse proportion. Raising it by 10 % (to 1.32×107) moves the result by −5.68 %, from 0.016 to 0.01509 m.
Constrained modulus of layer 2
The constrained modulus of layer 2, in Pa. A stiffer layer settles less, in inverse proportion. Raising it by 10 % (to 1.65×107) moves the result by −3.41 %, from 0.016 to 0.01545 m.
Constrained modulus of layer 3
The constrained modulus of layer 3, in Pa. A stiffer layer settles less, in inverse proportion. Raising it by 10 % (to 1.1×107) moves the result by +0 %, from 0.016 to 0.016 m.
Worked example with the values above
- Layer terms H·Δσ′/M: layer 1 = 2 × 60,000 / 1.2×107 = 0.01; layer 2 = 3 × 30,000 / 1.5×107 = 0.006 m.
- Sum of the terms: 0.016 m.
- S = 0.016 m.
Assumptions and limits
- One-dimensional compression with a representative layer, and effective stresses throughout.
- Primary consolidation by Terzaghi theory with a uniform initial excess pore pressure.
Common mistakes
- Using natural logarithms with a base-10 compression index.
- Using total instead of effective stresses.
- Using the layer thickness instead of the drainage path for the consolidation time.
Statistics and uncertainty
Settlement predictions typically agree with field values only to within a factor of 1.5 to 2, and the time of consolidation can be out by an order of magnitude because the field coefficient of consolidation differs from the laboratory one. Observe the settlement during construction and update the prediction.
Where it is used
- Predicting settlement of embankments and foundations on clay.
- Planning how long a surcharge must stay in place.
Maths you need
- Void ratio: volume of voids divided by volume of solids
- Compression index: void ratio change per log₁₀ cycle of effective stress
- Preconsolidation stress: the largest past effective stress the clay has carried