## What is settlement?

The vertical displacement of the structural foundation due to the loading from the superstructure is known as settlement. It is due to the reduction of air voids present in the soil. The amount of vertical displacement depends on the bearing capacity of the soil on which the foundation is laid. The effect of the settlement depends on the magnitude of settlement, uniformity, and the structural design of the structure. The maximum allowable settlement for residential and office buildings should be less than 18 mm, for industrial buildings, it is 38 mm and for commercial buildings, it is 25 mm.

## Components of settlement

Of a foundation, the total settlement is a combination of immediate settlement, consolidation settlement, and secondary consolidation settlement/ secondary compression.

Total settlement, S_{t} = S_{i} + S_{c} + S_{s}

where, S_{i}= Immediate settlement

S_{c} = Consolidation settlement

S_{s}= Secondary compression settlement

## Types of settlement

### Immediate settlement

It is also known as elastic settlement that occurs immediately or within a week of application of the load. In partially saturated soil, the expulsion of pore air leads to the elastic settlement of soil solids. In saturated soils, the elastic compression of soil solids and pore water leads to elastic settlement.

**Calculation of immediate settlement**

A linear elastic model is commonly used to calculate immediate settlement in saturated or nearly saturated cohesive soils and for granular soils.

The Schmertmann theory is used for the calculation of immediate settlement in clay. The formula is

${S}_{i}=\frac{qB\left(1-{\mu}^{2}\right){I}_{f}}{E}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}where,\phantom{\rule{0ex}{0ex}}q=Netpressureonfoundation\phantom{\rule{0ex}{0ex}}B=widthoffoundation\phantom{\rule{0ex}{0ex}}\mu =Poisson\text{'}sratio\phantom{\rule{0ex}{0ex}}E=Modulusofelasticity\phantom{\rule{0ex}{0ex}}{I}_{f}=Influencefactor$

The Young's modulus can be measured from the slope of the undrained stress-strain curve of the field conditions and this elastic modulus can also be calculated using the undrained shear strength using the Bjerrum method.

The value of the influence factor depends on the shape of the footing (square, circular and rectangular) and the type of footing, either flexible or rigid. Isolated footings are examples of flexible footing and mat footing is an example of rigid footing.

The values of the influence factors for rigid and flexible footing are given in the table.

**Immediate settlement in sand**

It can be computed from the following equation,

### Primary consolidation settlement

This type of settlement occurs after the immediate settlement and is a time taking process. Consolidation is the process by which the volume of the soil is reduced due to the evacuation of water from the pores. Excess pore water pressure is dissipated, increasing in effective stress and volumetric strain. The consolidation is a quick process in granular soils due to its high permeability and is a slow process in cohesive soil. This process stops once after all the pore water is expelled.

**Calculation of primary consolidation settlement**

**If the change in void ratio is given**

The basic equation for computing the consolidation for a compressible layer is given as

$\Delta H=\left(\frac{\Delta e}{1+{e}_{0}}\right){H}_{0}\phantom{\rule{0ex}{0ex}}where,\phantom{\rule{0ex}{0ex}}\Delta H=Consolidationsettlement\phantom{\rule{0ex}{0ex}}\Delta e=Changeinvoidratio\phantom{\rule{0ex}{0ex}}{e}_{0}=Initialvoidratioofsoil\phantom{\rule{0ex}{0ex}}{H}_{0}=Initialheightofthecompressiblelayer\phantom{\rule{0ex}{0ex}}$

Sometimes, the primary consolidation settlement is computed using H_{0} for the entire consolidating layer. In it vertical stress conditions act at mid-depth. This may cause an error in the calculation as the void ratio changes with different strata. So, for proper measurement, the consolidating stratum should be divided into sub-layers for the analysis. The above equation is then applied to each layer and the cumulative value is taken.

**If the compression index of soil is given**

${C}_{c}=\frac{\Delta e}{{\mathrm{log}}_{10}\left({\displaystyle \frac{{\overline{\sigma}}_{0}+\Delta \overline{\sigma}}{\overline{{\sigma}_{0}}}}\right)}\phantom{\rule{0ex}{0ex}}{C}_{c}=Compressionindex\phantom{\rule{0ex}{0ex}}\Delta e={e}_{0}-e=Changeinvoidratio\phantom{\rule{0ex}{0ex}}{\overline{\sigma}}_{0}=Effectivestresscorrespondingto{e}_{0}\phantom{\rule{0ex}{0ex}}\Delta \overline{\sigma}=Changeineffectivestress\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Delta H=\frac{{C}_{c}{H}_{0}{\mathrm{log}}_{10}\left({\overline{\sigma}}_{0}+\Delta \overline{\sigma}\right)}{1+{e}_{0}}\phantom{\rule{0ex}{0ex}}where,\Delta H=Consolidationsettlement\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

**If the coefficient of volume change is given**

$\Delta H={H}_{0}\xb7\Delta \overline{\sigma}\xb7{m}_{v}\phantom{\rule{0ex}{0ex}}where,\phantom{\rule{0ex}{0ex}}\Delta H=Consolidationsettlement\phantom{\rule{0ex}{0ex}}\Delta \overline{\sigma}=Changeineffectivestress\phantom{\rule{0ex}{0ex}}{m}_{v}=Coefficientofvolumechange\phantom{\rule{0ex}{0ex}}$

### Secondary compression settlement

This takes place after the primary consolidation settlement. After the surplus pore pressure has disappeared after primary consolidation, the compression of organic soils continues due to plastic adjustment of the soil and compression of the bonds between individual clay particles under persistent loading. This is called secondary compression or creep and it takes place at constant effective stress. In coarse-grained soil (sand and gravel) secondary consolidation is insignificant, but in highly plastic clays secondary consolidation can range from 10 to 20% of the total settlement.

**Calculation of secondary compression settlement**

The following equation is used to compute the secondary compression,

${S}_{s}=\frac{H\xb7{C}_{\alpha}}{1+{e}_{f}}{\mathrm{log}}_{10}\left(\frac{{t}_{2}}{{t}_{1}}\right)\phantom{\rule{0ex}{0ex}}where,\phantom{\rule{0ex}{0ex}}H=Heightofsoilstrataattheendofprimaryconsolidation\phantom{\rule{0ex}{0ex}}{e}_{f}=finalvoidratioatendofprimaryconsolidation\phantom{\rule{0ex}{0ex}}{t}_{1}=timetakentocompleteprimaryconsolidation\phantom{\rule{0ex}{0ex}}{t}_{2}=timetakentocompletesecondaryconsolidation\phantom{\rule{0ex}{0ex}}{C}_{\alpha}=\frac{\Delta e}{\mathrm{log}\left({t}_{2}\right)-\mathrm{log}\left({t}_{1}\right)}=Coefficientofsecondarycompression$

The stress history of the soil layer becomes important to evaluate its settlement qualities. It is expressed by a term called as Over consolidation ratio (OCR). OCR is a ratio of the pre-consolidation pressure to initial vertical effective stress. For a normally consolidated soil, the value of OCR is 1. For overconsolidated and pre-consolidated soil, the value of OCR is greater than 1.

## Settlement of piles

The total settlement of piles due to the application of load is a combination of settlement of pile shaft, settlement of pile caused by the applied load at pile point, and settlement of the pile caused by the applied load transmitted along the pile shaft.

**Settlement of pile shaft is given by the following equation**,

${S}_{1}=\frac{\left({Q}_{wp}+\xi {Q}_{ws}\right)L}{{A}_{p}{E}_{p}}\phantom{\rule{0ex}{0ex}}where,\phantom{\rule{0ex}{0ex}}{Q}_{wp}=Loadcarriedunderpilepointunderworkingload\phantom{\rule{0ex}{0ex}}{Q}_{ws}=Loadcarriedbyfrictionalresis\mathrm{tan}ceunderworkingload\phantom{\rule{0ex}{0ex}}{A}_{p}=Areaofpilecrosssection\phantom{\rule{0ex}{0ex}}L=Lengthofpile\phantom{\rule{0ex}{0ex}}{E}_{p}=Modulusofelasticity$

**Settlement of pile caused by the applied load at pile point is given by the equation,**

${S}_{2}=\frac{{q}_{wp}D\left(1-{{\mu}_{s}}^{2}\right){I}_{wp}}{{E}_{s}}\phantom{\rule{0ex}{0ex}}D=Widthordiameterofthepile\phantom{\rule{0ex}{0ex}}{q}_{wp}=\frac{{Q}_{wp}}{{A}_{p}}=Pointloadperunitareaatthepilepoint\phantom{\rule{0ex}{0ex}}{E}_{s}=Modulusofelasticityatpilepoint\phantom{\rule{0ex}{0ex}}{\mu}_{s}=Poisson\text{'}sratioofsoil\phantom{\rule{0ex}{0ex}}{I}_{wp}=Influencefactor=0.85$

** Settlement of the pile caused by the applied load transmitted along the pile shaft,**

${S}_{3}=\left(\frac{{Q}_{wp}}{pL}\right)\frac{D}{{E}_{s}}\left(1-{{\mu}_{s}}^{2}\right){I}_{ws}\phantom{\rule{0ex}{0ex}}where,\phantom{\rule{0ex}{0ex}}p=Perimeterofthepile\phantom{\rule{0ex}{0ex}}L=Embeddedlengthofpile\phantom{\rule{0ex}{0ex}}{I}_{ws}=Influencefactor\phantom{\rule{0ex}{0ex}}{E}_{s}=Modulusofelasticityofsoilatorbelowthepile\phantom{\rule{0ex}{0ex}}{\mu}_{s}=Poisson\text{'}sratio$

Influence factor is given by,

${I}_{ws}=2+0.35\sqrt{\frac{L}{D}}$

Where L and D are the dimensions of the pile shaft.

The total settlement of the pile is S = S_{1} +S_{2}+ S_{3}

## Context and Applications

This topic is a vital part of structural analysis and construction engineering and is taught in the following engineering disciplines

- Bachelors of Technology (Civil Engineering)
- Masters of Technology (Geotechnical engineering)

## Practice Problems

Q1) Who is called the father of geotechnical engineering?

- Karl von Terzaghi
- Schmertmann
- Holtz
- Ralph B Peck

**Answer:** Option a

**Explanation:** Karl von Terzaghi is known as the father of geotechnical engineering. He invented various theories (on consolidation, bearing capacity, lateral earth pressures, and stability) in the field of geotechnical engineering.

Q2) Which of the following settlement in geotechnical engineering is used to compute the theory of elasticity?

- Immediate settlement
- Secondary compression
- Settlement
- Subgrade

**Answer:** Option a

**Explanation:** Immediate settlement is calculated based on the theory of elasticity. For saturated or partially saturated cohesive soils, a linear elastic model is generally used for the calculation of immediate settlement.

Q3) What is a raised structure of compacted soil that is used to hold back water called?

- Deformation
- Vertical stress
- Embankment
- Coefficient of volume change

**Answer:** Option c

**Explanation:** An embankment is a structure used for retaining the water in a water storage structure.

Q4) A long, slender column used as a foundation under the soil subsurface is called,

- Raft foundation
- Pile foundation
- Strap footing
- Combined footing

**Answer:** Option b

**Explanation: **Pile foundation is a deep footing that used long, slender columns to transmit the load at a certain depth from the surface. Such type of footings is used either when the bearing capacity of the soil is less or when large structures are to be constructed.

Q5) The ratio of the volume of voids to the volume of solids is,

- Void ratio
- Permeability
- Bearing capacity
- Rigidity factor/Modulus of rigidity

**Answer:** Option a

**Explanation:** Void is the space occupied by air. Void ratio is the ratio of space occupied by the voids to the volume of solids.

### Want more help with your civil engineering homework?

*Response times may vary by subject and question complexity. Median response time is 34 minutes for paid subscribers and may be longer for promotional offers.

### Search. Solve. Succeed!

Study smarter access to millions of step-by step textbook solutions, our Q&A library, and AI powered Math Solver. Plus, you get 30 questions to ask an expert each month.

# Calculation of Settlement Homework Questions from Fellow Students

Browse our recently answered Calculation of Settlement homework questions.

### Search. Solve. Succeed!

Study smarter access to millions of step-by step textbook solutions, our Q&A library, and AI powered Math Solver. Plus, you get 30 questions to ask an expert each month.