Settlement is commonly separated into immediate settlement, primary consolidation and secondary compression:
$$ S_{\mathrm{total}}=S_i+S_c+S_s $$Topic guide: For background, method selection and related checks, see Consolidation Testing and Settlement Analysis.
Each component has different parameters, time behaviour and limitations. This page focuses on one-dimensional consolidation calculations.
Primary consolidation of normally consolidated soil
For a layer treated as normally consolidated:
$$ S_c= \frac{C_cH}{1+e_0} \log_{10} \left( \frac{\sigma'_0+\Delta\sigma'}{\sigma'_0} \right) $$| Symbol | Meaning |
|---|---|
| \(S_c\) | Primary consolidation settlement |
| \(C_c\) | Compression index |
| \(H\) | Initial thickness of compressible layer |
| \(e_0\) | Initial void ratio |
| \(\sigma'_0\) | Initial vertical effective stress at the representative point |
| \(\Delta\sigma'\) | Applied vertical effective-stress increase |
The stress increment generally varies with depth. A layer may need to be divided into sublayers rather than represented by one midpoint value.
Overconsolidated soil
If final effective stress remains below preconsolidation pressure \(\sigma'_p\):
$$ S_c= \frac{C_rH}{1+e_0} \log_{10} \left( \frac{\sigma'_0+\Delta\sigma'}{\sigma'_0} \right) $$If the final stress crosses \(\sigma'_p\):
$$ S_c= \frac{C_rH}{1+e_0} \log_{10} \left( \frac{\sigma'_p}{\sigma'_0} \right) + \frac{C_cH}{1+e_0} \log_{10} \left( \frac{\sigma'_0+\Delta\sigma'}{\sigma'_p} \right) $$The notation for recompression or swelling index varies. The software must identify the selected laboratory parameter and convention.
Worked normally consolidated example
Assume:
- \(C_c=0.25\)
- \(H=3.0\) m
- \(e_0=0.90\)
- \(\sigma'_0=100\) kPa
- \(\Delta\sigma'=50\) kPa
The illustrative settlement is approximately 69.5 mm. It depends entirely on the stated one-dimensional assumptions and parameters.
Time rate of consolidation
The dimensionless time factor is:
$$ T_v=\frac{c_vt}{H_{dr}^2} $$Rearranging:
$$ t=\frac{T_vH_{dr}^2}{c_v} $$Where \(H_{dr}\) is the maximum drainage path. For a layer draining at both top and bottom, it may be half the layer thickness. For single drainage, it may equal the full thickness.
Common theoretical reference points for one-dimensional consolidation include approximately:
- \(T_v=0.197\) at 50 percent average consolidation
- \(T_v=0.848\) at 90 percent average consolidation
The drainage boundary, coefficient of consolidation and selected theoretical relationship must be recorded.
Secondary compression
A common one-dimensional expression is:
$$ S_s= \frac{C_\alpha H}{1+e_p} \log_{10}\left(\frac{t_2}{t_1}\right) $$The start time, end time, reference void ratio and laboratory basis must be defined. Secondary compression should not be added mechanically without confirming its relevance.
Required calculation record
- soil layers and sublayers
- initial and final effective stresses
- stress-distribution method
- groundwater model
- \(C_c\), \(C_r\), \(C_\alpha\), \(e_0\), \(e_p\), \(c_v\) and their sources
- preconsolidation pressure
- drainage boundaries
- calculation dates or time interval
- immediate, primary and secondary components
- assumptions and method version
Important limitations
One-dimensional calculations may not represent:
- three-dimensional deformation
- lateral strain
- non-linear stiffness
- construction staging
- changing groundwater
- creep not captured by the selected parameters
- structured, organic or highly variable deposits
- ground improvement
- smear or well resistance around vertical drains
Related resources
Authoritative reference
Settlement predictions require representative parameters, a suitable stress model and review by a suitably qualified geotechnical professional.