Boussinesq'S theory

BOUSSINESQ THEORY
VERTICAL STRESS DUE TO POINT LOAD
TABLE FOR VALUES OF BOUSSINESQ’S COEFFICIENT (퐼_퐵)
SOME POINTS FOR USING THE BOUSSINESQ’S EQUATION.
LIMITATIONS OF BOUSSINESQ’S SOLUTION.

1. BOUSSINESQ THEORY
FOR CONCENTRATED LOAD
PREPARED
BY
ER. SANJEEV SINGH
[email protected]

2. CONTENTS
BOUSSINESQTHEORY
VERTICALSTRESSDUETOPOINTLOAD
TABLEFORVALUESOFBOUSSINESQ’SCOEFFICIENT(??????
??????)
SOMEPOINTSFORUSINGTHEBOUSSINESQ’SEQUATION.
LIMITATIONSOFBOUSSINESQ’SSOLUTION.
LIKESHAREANDSUBSCRIBE

3. BOUSSINESQ THEORY
Boussinesq(1885)developedatheorytocalculatestressesanddeformationdue
topointloadappliedatthesurfaceofsemiinfinitesoilmass.
Assumptions:
1.Soilmassiselasticcontinuumhavingaconstantvalueofmodulusofelasticity
(E)i.eitobeysHook’slaw.
2.Thesoilishomogeneousi.eithasidenticalelasticpropertiesitallpointsin
identicaldirection.
3.Thesoilisisotropic,i.eithasidenticalelasticpropertiesinalldirectionata
point.
4.Thesoilmassissemi-infinitei.eitextendstoinfinityinthedownwarddirection
andlateraldirctions.Inotherwords,itislimitedonitstopbutahorizontal
planeandextendstoinfinityinallotherdirections

4. 5.Theselfweightofthesoilisignored.
6.Thesoilisinitiallyunstressed,i.eitisfreefromresidualstressesbefore
theapplicationoftheload.
7.Thetopsurfaceofthemediumisfreeofshearstressesandissubjected
toonlythepointloadataspecifiedlocation.
8.Continuityofstressisconsideredtoexistinthemedium.
9.Thestressesaredistributedsymmetricallywithrespecttozaxis(vertical
axis)

5. VERTICAL STRESS DUE TO POINT LOAD
From fig shows a horizontal surface of an
elastic continuum subject to a point load
Q at a point O. Using logarithmic stress
function for solution of elasticity
problem. Bossinesqproved that the polar
stress σ
�at point P is given by
σ
�=
3
2π
�co�β
�
2
……….(a)
Where R= polar distance between the
origin O and point P.
β= angle which the line OP makes
with the vertical
Or R = ??????
2
+??????
2
Where ??????
2
= �
2
+�
2

6. sinβ=
�
�
and cosβ=
??????
�
NOW, the vertical stress (σ
??????) at point P is given by
σ
??????=σ
�????????????�
2
β……..(a1)
σ
??????=
3
2π
�co�β
�
2
????????????�
2
β
σ
??????=
3�
2π
co�
3
β
�
2
σ
??????=
3�
2π
??????
??????
3
�
2
σ
??????=
3�
2π
??????
3
�
5
………(a2)

7. σ
??????=
3�
2π
??????
2
??????
2
??????
3
�
5
σ
??????=
3�
2π
??????
5
??????
2
�
5
σ
??????=
3�
2π
1
??????
2
??????
5
(�
2
+??????
2
)
5
2
σ
??????=
3�
2π
1
??????
2
1
(1+
??????
??????
2
)
5
2
…….(a3)
Or, σ
??????=
??????
??????�
??????
2
…………….(a4) Where ??????
??????=
3�
2π
1
(1+
??????
??????
2
)
5
2
……..(a5)

8. Thecoefficient??????
??????isknownastheBoussinesqinfluenceforthevertical
stress.Thevalueof??????
??????canbedeterminedforthegivenvalueofr\Zfrom
equation(a5).
σ
�=
�
2π
(
3�
2
�
�
5−1−2??????(
�
2
−�
2
��
2
(�+??????)
+
�
2
�
�
3
�
2
)…….(a6)
σ
�=
�
2π
(
3�
2
�
�
5−1−2??????(
�
2
−�
2
�??????
2
(�+??????)
+
�
2
�
�
3
??????
2
)…….(a7)
Ifv=0.5
σ
�=
3�
2π
�
2
�
�
5 σ
�=
3�
2π
�
2
�
�
5

9. TABLE FOR VALUES OF BOUSSINESQ’S
COEFFICIENT (??????
??????)

10. Thefollowingpointsaretobenotedwhileusingeqs(a4).
1.Theverticalstressdoesnotdependuponthemodulusofelasticity(E)and
thePoissonratio(v).Butthesolutionhasbeenderivedassumingthatthe
soilislinearlyelastic.
2.Theintensityofverticalstressjustbelowtheloadpointisgivenby
σ
�=0.4775
�
??????
2
(becausewhen,r\Z=0.??????
??????=0.4775)
3.Atthesurface(z=0).theverticalstressjustbelowtheloadistheoretically
infinite.However,inanactualcase,thesoilundertheloadyieldsdueto
veryhighstresses.Thepointloadspreadsoverasmallbutfiniteareaand,
therefore,onlyfinitestressesdevelop.
4.Theverticalstress(σ
�)decreasesrapidlywithanincreaseinr\Zratio.
Theoretically,verticalstresswouldbezeroonlyataninfinitedistancefrom
theloadpoint.Actually,atr\Z=5ormore,theverticalstressbecome
extremelysmallandisneglected.

11. 5.Inactualpractice,foundationloadsarenotapplieddirectlyonthe
groundsurface.However,ithasbenestablishedthattheBoussinesq’s
solutioncanbeappliedconservativelytofieldproblemconcerningloads
atshallowdepth,providedthedistance‘z’ismeasuredfromthepointof
applicationoftheload.
6.Boussinesq’ssolutioncanevenbeusedfornegative(upward)loads.For
example,iftheverticalstressdecreaseduetoanexcavationisrequired,
thenegativeloadisequaltotheweigthofthesoilremoved.However,as
thesoilisnotfullyelasticthestressesdeterminedare(necessarily)
approximate.
7.Thefieldmeasurementsindicatesthattheactualstressesaregenerally
smallerthanthetheoreticalvaluesgivenbyBoussinesq’ssolution
especiallyatshallowdepths.Thus,theBoussinesq’ssolutiongives
conservativevaluesandiscommomlyusedinsoilengineeringproblems.

12. LIMITATIONS
1.Thesolutionwasderivedassumingthesoilasanelasticmedium,but
thesoildoesnotbehavesasanelasticmaterial.
2.Whenthestressdecreaseoccursinsoil,therelationbetweenthestress
andthestrainisnotlinearasassumed,therefore,thesolutionisnot
strictlyapplicable.
3.Indeepsanddeposits,themodulusofelasticityincreasewithan
increaseindepthandtherefore,Boussinesq’ssolutionwillnotgive
satisfactoryresults.
4.Thepointloadsappliedbelowgroundsurfacecausessomewhatsmaller
stressesthanarecausedbysurfaceloads,andtherefore,thesolutionis
notstrictlyapplicable.