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3/21/2013
1
FOUNDATION ANALYSIS AND DESIGN S G
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
SETTLEMENT AND ALLOWABLE BEARING CAPACITYBEARING CAPACITY
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Settlement of Foundation
Theultimatebearingcapacityqu isrealizedatasettlementlevelofSu.
IfthefactorofsafetyagainstbearingcapacityfailureisFS,thentheallowablebearingcapacityisqsafe=qu/FS.Thesettlementcorrespondingtoqsafe isS.
IfSall istheallowablelevelofsettlementfortheproposedfoundationthenqall(S) bethecorrespondingallowable
Inthedesignofanyfoundation,onemustconsiderthesafetyagainstbearingcapacityfailureaswellasagainstexcessivesettlement ofthefoundation.
Loadperunitareaq versussettlementSforafoundation
qall(S) p gbearingcapacity.
ForfoundationswithsmallerwidthsofB,Sall maybemorethanS;however,forlargervaluesofB,Sall
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2
Verticalstressincreaseinasoilmasscausedbyfoundationload
maybedeterminedusing
Vertical Stress Increase due to Applied Load
BoussinesqsAnalysis
WestergaardsAnalysis
NewmarksAnalysis
2:1MethodNewmarks chart
PressurebulbVerticalstressdistributioninhorizontalandverticalplane
Analysis
Approximatemethods
Equivalentpointloadmethod
Equivalentpointloadmethod
BasicAssumptions: Soilmediumishomogeneous,elastic,isotropicandextendstoagreaterdepth
Loadisappliedatthegroundsurface Theloadedareaisflexible
2:1Method
UniformlydistributedloadonrectangularareaTheverticalstressatapointbeneaththecentreofauniformlyloadedrectangularareamaybefound(a)usingtheinfluencevalueforacornerbytheprincipleof
superposition,dividingtherectangleintofourequalpartsbylinesparalleltothesidesandpassingthroughthecentre(Eq.givenbelow)
(b)Equivalentpointloadmethod(c)2:1method
I II
IV IIIA
AST
)( IVIIIIIIZ IIIIq
VerticalstressatPointA
Byprincipleofsuperposition
Vertical Stress Increase due to Applied Load
IM N
O
Q
RP
)( IVIIIIIIZ IIIIq
ZoneI=MQAT,ZoneII=PRATZoneIII=NQAS,ZoneIV=ORAS
SincezoneIVisdeductedtwice,itsinfluencehastobeaddedonce
Flexible and Rigid FoundationsContactpressureandsettlementsforaflexiblefoundations
Elastic/Clayeysoil Granular/Sandysoil
ClayContactpressureandsettlementsforarigidfoundations
Elastic/Clayeysoil Granular/Sandysoil
Clay
Elastic Settlement of Foundation
H ze dzS0
Elasticsettlementofashallowfoundationcanbeestimatedbyusingthetheoryofelasticity
dzE ysxs
H
zs
)(1
0
(FromHookeslaw)Forflexiblefoundation f
s
se IE
qBS
21
For rigid foundation 930 SS
BIE
qS fs
se
21 ;
where,Se = Elastic settlementq = Net applied pressure on the foundation B = Width of the foundationEs = Average modulus of elasticity of soil (measured from Z = 0 to 4B)s= Poissons ratio of the soilIf = Influence factor depends on rigidity and shape of the foundationH = Thickness of the soil layerx, y, z, are the stress increase due to the net applied foundation load in the x, y and z directions resp.
Forrigidfoundation ),()( 93.0 centreflexibleeRigide SS (limitedtoZ=4B)
Harr (1966)fordetailedderivation
Elastic Settlement of FoundationInfluencefactor,If sEs
Usingtriaxial testForNCclay:Es =250cu to500cu
Forsettlementatcornersofloadedarea(s =0.5)s u uForOCclay:Es =750cuto1000cu
Elastic Settlement of FoundationDepthcorrectionfactor byFox(1948)
Forfoundationsplacedatsomedepthbeneaththegroundsurface,adepthcorrectionmaybeappliedtotheelasticsettlementcomputed.
Depthcorrectionfactordependsonthedepthtowidthratioandlengthtowidthratioofthefoundation.
Fox (1948) has developed a chart for depth correction factor I
surfaceat foundationfor settlementCalculatedDdepth at foundationfor settlement Correctedfactor Depth
Fox(1948)hasdevelopedachartfordepthcorrectionfactor,Id
CorrectedsettlementforfoundationatanydepthD dfs
se IIE
qBS
21
3/21/2013
3
qB
Elasticsettlementoffoundationonsaturatedclay(Janbu etal.1956) Janbu etal.(1956)proposedanequationforevaluatingtheaveragesettlementofflexiblefoundationsonsaturatedclaysoils.(undrained Eu ands =0.5)
ChristianandCarrier(1978)modifiedthevaluesofA1 andA2 andpresentedinthechartform
Thismethodcanbeusedwhenthethicknessoftheclaylayerisknown.
Variation of A1 an A2
Elastic Settlement of Foundation
se E
qBAAS 21where,Se = Elastic settlementq = Net applied pressure on the foundation Es = Average modulus of elasticity of soilA1 = correction factor depends on H/B, L/BA2 = correction factor depends on Df/BUse the Christian and Carrier chart to get A1 and A2*Elastic settlement of clay will be determined only when design warranties
Elastic Settlement of Foundation
zEIqqCCS sze )'(21ElasticsettlementofgranularsoilsStraininfluencefactor(Schmertmann andHartman,1978)
(semiempiricalmethod)
where,Se = Elastic settlementq = stress at the level of the foundation, q = overburden pressureEs = Modulus of elasticity of soil (for various depth, z)
''5.011 qq
qCC1 = a correction factor for depth of foundation qq
1.0yearsin timelog2.012CC2 = a correction factor for creep in soil
Iz = Strain Influence factorVariation of Iz can be determined from the chart or can be developed using the following guidelines
For Square and circular footingIz = 0.1 at z = 0Iz = 0.5 at z = z1=0.5BIz = 0 at z = z2 = 2B
For Foundation with L/B 10Iz = 0.2 at z = 0Iz = 0.5 at z = z1=BIz = 0 at z = z2 = 4B
It gives total settlement for sandy soil. This method is primarily used when field test like SPT or SCPT test are done (to get Es at various depth).
Empirical relation based on SPT
TotalsettlementfromSPTNvalue SettlementofafootingofwidthBunderunitintensityofpressureresting(1kg/cm2)ondrycohesionless depositwithknowncorrectedpenetrationvalue(N)canbereadfromthechart.
Thesettlementunderanyotherpressuremaybecomputedbyassumingthatthesettlementisproportionaltotheintensityofpressure.
Ifthewatertableisatashallowdepth,
IS:8009recommendation
Forcohesionless soil
thesettlementreadfromthechartshallbemultipliedbycorrectionfactorwshowninfigure
Consolidationsettlementisatimedependentprocessthatoccursduetotheexpulsionofexcessporewaterpressureinsaturatedclayeysoilsbelowthegroundwatertableandiscreatedbytheincreaseinstresscausedbythefoundationload
Primary Consolidation Settlement
Samplecollectedforconsolidationtest
elog 21 eeC
TheslopeoftheeversuslogplotforthenormallyconsolidatedportionofthesoilisreferredtoascompressionindexCcTheslopeoftheeversuslogplotfortheover consolidated portion of the soil is
'3
'4
43
log eeCc
Recompression Compression
e log
'1
'2log
CsoverconsolidatedportionofthesoilisreferredtoasswellindexCs
)1( oo ee
HHConsolidationsettlementSc = o
o
HeeH
)1( since
Cs foragivensoil=(1/4to1/5)Cc
Cc =0.009(Liquidlimit 10)ForNCclayForremouldedclay Cc =0.007(Liquidlimit 10)
c=preconsolidationpressure
OCR1
''
log1 o
oc
o
sc He
CS UseCs
Foroverconsolidatedclay,o< c 1
'
'
'
'
loglog occccsc HCHCS
MethodI
o
Method2 DividetheclaylayerintoseveralthinlayershavingthicknessesofHc(1),Hc(2)
Findtheinsituaverageeffectivestresses(i.e.theeffectivestressatthemiddleofeachclaylayers)
Findtheaveragestressincreaseatthemiddleofeachlayer
CalculatetotalScasthesummationofalllayers
''g
1g
1 cc
ooc
oc ee
Forstress
3/21/2013
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Permissible Settlement for Shallow FoundationInfluenceofstructuralrigidityondifferentialsettlement
FlexiblestructureIthasaverylittletendencyforloadtransferandthuscouldhavelarger
differentialsettlements
RigidstructureIthasagreatercapacityforloadtransferandthusprovidesmoreresistancetoexcessivedifferentialsettlements
Ifthecommuteddifferentialsettlementismorethanpermissiblevalue,thedesign(i.e.dimensionsoffootings)
needstobemodifiedeventhoughthemagnitudeoftotalsettlementis
acceptable
Permissible Settlement for Shallow FoundationAngulardistortion Bjerrum (1963)
RecommendationsofEuropeanCommitteeforstandardizationon
Skempton andMcDonald(1956)
differentialsettlementparameters
Permissible Settlement for Shallow FoundationIS19041986
Allowable Bearing Capacity (Settlement consideration)StandardPenetrationTest
2
2
)( 23.0)3(35 wcorrsettlementall RdB
BNq
TerzaghiandPeck(1948)
B1.2mModifiedMeyerhofsequation
2)( 20 wcorrsettlementall RdNq
Allthementionedequationsareforthesettlementof25mm.
Allowablebearingcapacity(settlementconsideration)foranysettlementcanbedetermined
InkPa
InkPa
21 BDfd BDw wR 215.02
corrc Nq 4
2
2
)(3.05.12 wcorrsettlementall RdB
BNq
usingthefollowingequation
)()( 25'' settlementallsettlementall q
Sq B>1.2m
StaticConePenetrationTest
Coneresistanceqc inkq/cm2 butqall(settlement)inkPa2)( 7.2 wcsettlementall Rqq
ModifiedMeyerhofsequation
Allowable Bearing Capacity using Plate Load Test
Forc=0soil
Ultimatebearingcapacityandultimatesettlement
p
fpuuf B
Bqq *
2
)3.0()3.0(
fp
pfpf BB
BBss
puuf qq
p
fpf BB
ss
For=0soil
Immediate settlement
Thismethodcanbeusedforthedesignofashallowfoundationforagivensafesettlement.
Minimumtwoplateloadtestsareconductedwithvaryingplatesize(B1,B2)
Theloadcorrespondingtosafesettlements1,s2 ..(i.e Q1,Q2..)areobtainedfromloadsettlementcurvesobtainedfromalltests.
HouselsMethod
PlateconfigurationQ1 = A1m+P1n ; Q2 = A2m+P2nImmediatesettlement
fs
se IE
qBS
21
fs
se IE
mqBS
21
S
qB
fs
s IE
m
21
;
Q1 A1m+P1n;Q2 A2m+P2nFindmandnwhere,A1 andA2 aretheareaofplates;P1 andP2 aretheperimetersofplatesFootingconfigurationSubmandnforfootingandgetQQ=Am+Pn (samesettlementasplate)Q=(Q/s1)*s (forothersettlements)AandPareareaandperimeterofproposedfooting;s1 safesettlementoffooting(sameasplate)
Allowable / Safe Bearing Capacity of Soil (NBC, 1983)
Caution:
Thevaluesarejustguidevalues.Thiscannotsubstitutedetailedgeotechnicalinvestigationofany
proposedprojects
3/21/2013
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Case Studies
Transcona Grain Elevator
http://www.cgs.ca/pdf/110916%20Skaftfeld%20photos%20Transcona%20Grain%20Elevator_MINIMISED.pdf
Leaning tower of PisaCostanzo,D.;Jarniolkowski,M.,Lancellotta,R.,andPepe,M.C.(1994),LeaningTowerofPisa:DescriptionoftheBehavior,Settlement94BanquetLecture,TexasA&MUniversity
SOIL-STRUCTURE INTERACTIONINTERACTION
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
ConventionalDesignofFoundation
Intheconventionalanalysisanddesignoffootings,tipofthecolumnisfixedandassumesthesoilpressureisassumedtobeuniformorlinearlyvarying.Thecentroidofthesoilpressureiscoincidentwiththelineofactionoftheresultantcolumnloads.
StructuralAnalysis&Design
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Generalassumption Assumefoundationasrigid Soilreactionasconstant Designasinvertedbeamsorslabs Noconsiderationtothetypeofsoil
Inreality,footingisneitherperfectlyrigidnorperfectlyflexible.Soilreaction/contactpressuredistributiondependsonsoiltypeandrigidityofthestructure
ContactPressureDistribution
ContactpressureandsettlementprofilesforfoundationonsandTheassumptionofuniformpressuredistributionwillgiveaconservativedesignforrigidfooting
onsandysoilsasthemaximumbendingmomentisoverestimated
Rigidfooting
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Flexiblefooting
ContactPressureDistributionContactpressureandsettlementprofilesforfoundationonclay
Theassumptionofuniformpressuredistributionwillresultinaslightlyunsafedesignforrigidfootingsonclaysasthemaximumbendingmomentatthecentreisunderestimated.
Rigidfooting
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Flexiblefooting
Realisticdistributionofcontactpressureneedstobeconsidered
SoilFoundationStructureInteraction
Structures
Foundation
Response of any structure
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Underlying Soil / rock
y
3/21/2013
6
Soil StructureInteractionStudies
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Soil StructureInteractionStudies
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Theproblemofsoilfoundationstructureinteractionisgenerallysolvedbyincorporatingthereactionfromthefoundation,intotheresponsemechanismofthestructure,byidealizingthefoundationbyasuitablemathematicalmodel.
Soil StructureInteractionAnalysis
Evenifthefoundationmediumhappenstobecomplexinsomeproblems,inamajorityofcases,theresponseofthestructureatthecontactsurfaceisofprimeinterestandhence,itwouldbeofimmensehelpintheanalysis,ifthefoundationcanberepresentedbyasimplemathematicalmodel,withoutforegoingthedesiredaccuracy.
To accomplish this objective, many foundation models have been proposed and a
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Itisgenerallyobservedthatthemodeling ofthesuperstructureandfoundationarerathersimplerandstraightforwardthanthatofthesoilmediumunderneath.
However,soilishavingverycomplexcharacteristics,sinceitisheterogeneous,anisotropicandnonlinearinforcedisplacementcharacteristics.
Thepresenceoffluctuationofwatertablefurtheraddstoitscomplexity.
Toaccomplishthisobjective,manyfoundationmodelshavebeenproposedandacomprehensivereviewpertainingtothesehasbeengivenbymanyauthors.
CriticalAspectsofSSITheproblemofsoilfoundationstructureinteractionisgenerallysolvedbyincorporatingthereactionfromthefoundationintotheresponsemechanismofthestructure,byidealizingthefoundationbyasuitablemathematicalmodel.
SoilCriticalaspectsofSSIistheconceptualmodellingof Foundation
Simplifiedmodels
Canbemodelled asbeam/plate/shell
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Thefootingcanbemodelledasabeamoraplateorashellandclassicalbendingtheoriescanbeusedforrepresentingtheirresponse.
Thesoilreactionhastobeincorporatedintheintegratedanalysisofsoilstructureinteractionequationbymodellingthesoilappropriatelyusingdifferentmodels
Structure Canbeeasilymodelled
Inertia=>baseshearandmoment
CriticalAspectsofSeismicSSI
Relativefoundationrotation
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Relativefoundationdisplacement
TheoryofElasticFoundationTheresponseoffoundationsoilsystemsubjectedtoexternalloadsdependsonthegeometryofthefoundation(beamoraplate)andidealisationofsoil.Mostofthefootingscanbeconsideredaseitherbeams(onedimensional)orplates(twodimensional:rectangular,squares,circular,annularorothershapes).
Plateonanelasticfoundation
q(x)
z (y)
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Beamonanelasticfoundation
q(x, y)z
y
(y)
3/21/2013
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BeamonElasticFoundation
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
BeamonElasticFoundation
R(x)-q(x)
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Aimshouldbetoincluderealisticvalueofsoilreaction/contactpressure[modelidealizedsoilbehavior,i.e.R(x)]asfaraspossibletogettheactual
responseofsoilfoundationinteraction R(x)-q(x)
Idealised SoilBehavior/ModelingSoilMediaThemechanicalresponseofnaturallyoccurringsoilscanbeinfluencedbyavarietyoffactors.
Shape,sizeandmechanicalpropertiesoftheindividualsoilparticles Theconfigurationofthesoilstructure Theintergranularstressesandstresshistory Presenceofsoilmoisture,degreeofsaturationandsoilpermeability
Thesefactorsgenerallycontributetostress strainphenomenonwhichdisplaymarkedly non linear irreversible and time dependent and to soil masses which
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
markedlynonlinear,irreversibleandtimedependent,andtosoilmasseswhichexhibitanisotropicandnonhomogeneousmaterialproperties.
Hence,anyattempttosolveasoilfoundationinteractionproblem,takingintoaccountallsuchmaterialcharacteristics,isclearlyanimpossibletask
Inordertoobtainmeaningfulandrealizableinformationforpracticalproblemsofsoilfoundationinteractionitbecomesnecessarytoidealise thebehaviour ofthesoilbytakingintoaccountspecificaspectsofitsbehaviour.
Idealised SoilBehavior/ModelingSoilMediaThesimplesttypeofidealised soilresponseassumelinearelasticbehaviour ofthesupportingsoilmedium
Assumptionoflinearityandreversibilityofdeformationsimplicitinlinearelasticbehavior,ofcoursenotalwaysrigorouslysatisfiedbynaturallyoccurringsoilmasses
Assuminglinearelasticbehaviour ontheotherhandconsiderablyreducestheanalyticalrigorexpendedinthesolutionofaparticularboundaryvalueproblemandprovidesusefulinformationtomany
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Basedonthecomplexity;soilmodelsareclassifiedas
practicalproblemsofsoilmechanicsandfoundationengineeringwhichwouldotherwisebeintractable.
OneparametermodelTwoparametermodelThreeparametermodelFourparametermodeletc
WinklerModel
Z
Structure
Foundation
Soil
Structure
Foundation
Idealisedspringtorepresentsoil Z
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
TheearliestformulationofthesoilmodelwasduetoWinkler,whoassumedthesoilmodeltoconsistofcloselyspaced
independentlinearsprings
Oneparametermodel
Surfacedisplacementdueto:
Nonuniformload
Concentratedload
Z
X
Uniformflexibleload
WinklerModel
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
x
z
Rigidload
Responseofactualfoundation
3/21/2013
8
WinklerModel
Theloaddeflectionrelationatanypointisgivenby
R(x)=q=kw,where
According to the Winklers idealization, deformation of foundation due to applied load isconfined to loaded regions only. If such a soil model is subjected to a partially distributedsurface loading, q, the springs will not be affected beyond the loaded region. For such asituation, an actual foundation is observed to have the surface deformation.
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
k isthespringconstantandisoftenreferredtoasthefoundationmoduluscanbeequatedtomodulusofsubgradereactionofsoil.
w istheverticaldeflectionofthecontactsurface.
Discrete,independent,linearelasticsprings Simpletouse Soilbehaviourislinearingeneral AnumberofstudiesintheareaofsoilstructureinteractionhavebeenconductedonthebasisofWinklerhypothesisforitssimplicity.
CoefficientofsubgradereactionThecoefficientofsubgradereactionistheunitpressurerequiredtoproduceaunit
settlement.ItcanbeobtainedfromPlateloadtest Inclayeysoils,settlementundertheloadtakesplaceoveralongperiodoftimeandthecoefficientshouldbedeterminedonthebasisofthefinalsettlement.
Onpurelygranularsoils,settlementtakesplaceshortlyaftertheloadapplication.
Assumptionsqks pressure bearing
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Thevalueofks isindependentofthemagnitudeofpressure Thevalueofks hasthesamevalueforeverypointonthesurfaceofthefooting
pws settlement
Boththeassumptionsarestrictlynotaccurate.Thevalueofks decreaseswiththeincreaseofthemagnitudeofthepressureanditisnotsameforeverypointofsurfaceofthefootingasthesettlementoftheflexiblefootingvariesfrompointtopoint.However,themethodissupposedtogiverealisticvaluesforcontactpressuresandissuitableforbeamormatdesignwhenonlyaloworderofsettlementisrequired
Coefficientofsubgradereaction
Plateloadtest Theplateshouldobviouslybeaslargeaspossible,consistentwithbeingabletoexerttheverticalforcesrequired.
Thestandardplateiseitheracircularshapeof760mmdiameterorasquareshape760x760mm,16mmthick,andrequiresstiffeningbymeansofothercircular/squareplatesplacedconcentricallyaboveit.
Invariably a large plate does not settle uniformly The settlement must therefore be
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Invariably,alargeplatedoesnotsettleuniformly.Thesettlementmust,therefore,bemonitoredbymeansofthreeorfourdialgaugesequallyspacedaroundtheperimeterinordertodeterminethemeansettlement.
Supportsforthesedialgaugesshouldbesitedwelloutsidethezoneofinfluenceofthejackingloadwhichismeasuredbyaprovingring.Whenchoosingadiameterofplatetouseforthetest,dueconsiderationshouldalsobegiventothelimitedzoneofinfluenceoftheloadedplate.Typically,thesoilwillonlybeeffectivelystressedtoadepthof1.251.50timesthediameteroftheplate.
Coefficientofsubgradereaction
Thesubgradereactionatanypointalongthebeamisassumedtobedirectlyproportionaltotheverticaldisplacementofthebeamatthatpoint,i.e,soilisassumedtobeelasticandobeysHookes Law
Plateloadtest
ks Coefficientofsubgradereactionorcoefficientofelasticuniformcompressionqbearingpressureatapointalongthebeamw verticaldisplacementofthe beam at that point
wqks settlement
pressure bearing
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Hooke s Law.thebeamatthatpoint
Infoundationdesign,thevalueofks isthesecantmodulusofthegraphovertheestimatedworkingrangeofbearingpressure.
Thevalueofthemodulusofsubgradereaction(ks)obtainedfromthetestvariesaccordingtothesizeofplateused.
Coefficientofsubgradereaction
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
ks dependsnotonlyonthedeformationcharacteristicsofthesoilbutalsoonthesizeofcontactareabetweenplateandsubgrade.
Thevariationofks withplatesizecreatesanobviousdifficultyindecidingwhichplatesizeshouldbeusedasthestandardorreferencefordefiningvaluesofks foranalysis.
Coefficientofsubgradereaction
Forc=0soil
Ultimatebearingcapacityandultimatesettlement
p
fpuuf B
Bqq *
2
)3.0()3.0(
fp
pfpf BB
BBss
puuf qq
p
fpf BB
ss
For=0soil
22
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
ForCohesionless soil Determineksp fromplateloadtestorfromestimate(0.3mdia)
Findksf usingtherelation
22
)3.0()3.0(
)3.0()3.0(
**
p
f
f
p
p
pu
pf
fp
p
f
p
pu
f
uf
BB
BB
sq
BBBB
BB
sq
sq
2
)3.0()3.0(
p
f
f
pspfs B
BBB
kk
ForCohesivesoil Determineksp fromplateloadtestorfromestimate(0.3mdia)
Findksf usingtherelation(forwidth) Findksf usingtherelation(forlength)
L
Lkk spfs 5.1152.0
3.0,5.1
3.0,spfs
kk
f
p
p
pu
f
uf
BB
sq
sq
*f
pspsf BB
kk
Forlargerlength(mat/raft)
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CoefficientofsubgradereactionApplicationofcoefficientofsubgrade
reactiontolargermats/rafts
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Portionofthematthatexperiencemoresettlementproducemorecompressioninthespring
Sumofthesespringsmustequaltheappliedstructuralloadsplustheweightofthemat
NosinglevalueofKstrulyrepresentstheinteractionbetweenthesoilandthemat
WinklerModels
Basic limitations of Winkler hypothesis lies in the fact that this model cannot accountfor the dispersion of the load over a gradually increasing influence area with increasein depth.
Moreover, it considers linear stressstrain behaviour of soil.
The most serious demerit of Winkler model is the one pertaining to the independenceof the springs. So the effect of the externally applied load gets localized to thesubgrade only to the point of its application
LimitationsofWinklers hypothesis
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
subgrade only to the point of its application.
This implies no cohesive bond exists among the particles comprising soil medium.
The deficiency of the Winkler's Model in describing the continuous behavior of realsoil masses and the mathematical complexities of the elastic continuum has lead tothe development of many other simple soil behaviour models. These models possessome of the characteristics features of continuous elastic solids.
The term "Two Parameter signifies that the model is defined by two independentelastic constant.
TwoParameterElasticModels
First approach is to eliminate the discontinuityof Winkler spring by providing mechanicalinteraction between the individual springelements by either elastic membranes, elasticbeams or elastic layers capable of inducingpurely shear deformation (i.e. FilonenkoBorodich, Hetenyi, Pasternak and Kerr).
Twoapproaches
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
The Second approach proceeds from the elasticcontinuum model and introduces constraints orsimplifying assumptions with respect to thedistribution of displacements and stresses(Reissner, Vlazov and Leontiev).
FILONENKO&BORODICHMODEL(1945)
Basic Model
T T
ThemodelproposedbyFilonenkoBorodich (1940,1945)acquirescontinuitybetweentheindividualspringelments intheWinklermodelbyconnectingthemtoathinelasticmembraneunderaconstanttensionT.
Byconsideringtheequilibriumofthemembranespringsystem,itcanbeshownthatfor3Dproblems(eg.Rectangularorringfoundation),thesurfacedeflectionofthesoilmediumduetoapressureqisgivenby
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Z
),(),(),(),( 2 yxwTyxkwyxqyxR 2
2
2
22
yx
Inthecaseof2Dproblems(eg.stripfoundation)
2
2 )()()()(dx
xwdTxkwxqxR
FILONENKO&BORODICHMODEL(1945)Surfacedisplacementofthemodel:
Basic Model
Concentrated load
T T T
Px
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Z Z
FILONENKO&BORODICHMODEL(1945)Surfacedisplacementdueto:
Rigid load Uniform flexible load
T
XT
T
q
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Z
T
Z
3/21/2013
10
HETENYIMODEL(1946)
),(),(),(),( 4 yxwDyxkwyxqyxR Where,
Inthismodel,interactionbetweentheindependentspringelementsisaccomplishedbyincorporatinganelasticplatein3Dproblemsoranelasticbeamin2Dproblems.
3hE
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
,D istheflexuralrigidityoftheplateEp &p areyoungsmodulusandpoisonsratioofplatematerial;hp isthethicknessofplate. 22
4
4
4
4
44 2
yxyx
)1(12 2ppphED
4
4 )()()()(dx
xwdDxkwxqxR For2Dproblems
PASTERNEKMODEL(1954)
Xq (x,y) Shear layer with shear modulus G
Stressesintheshearlayer
ThemodelforsoilbehaviorproposedbyPasternakassumestheexistenceofshearinteractionbetweenthespringelement.Thiscanbeaccomplishedbyconnectingthespringelementstoalayerofincompressibleverticalelementswhichdeformintraverseshearonly.
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Z),(),(),(),(2 yxwGyxkwyxqyxR p
The continuity in this model is characterized by the consideration of the shear layer.
A comparison of this model with that of FilonenkoBorodich implies their physical equivalency(T has been replaced by G).
For3Dcase
3ParametermodelKerr (1964) proposed a 3 parameter soil model where in a shear layer issandwiched between two Winkler layers to represent the substances which can berepresented neither by Winkler foundation nor by isotropic elastic continuum.
A shear layer is introduced in the Winkler foundation and the spring constantsabove and below this layer is assumed to be different as per this formulation.
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
ViscoelasticFoundationModel Theexpressionviscoelasticsignifiesthedualnatureofamaterial:ontheonehanditbehavesinaviscousway,asaliquid,ontheotherhandelastically,asasolid.
Foranidealsolid,Hookes lawholds:thestress,,appliedisproportionaltothedeformation,,andtheproportionalityconstantisthemodulusofelasticityE,so =E. BesidesEalsootherquantitiesplayarole,suchasthe
Responseofanidealspring
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
shearmodulus,G,inashearingdeformationortorsion,whichisrelatedtoE.ForthesakeofsimplicityweshallmainlyuseEasarepresentativequantityfortheelasticstiffnessinanygeometryofloading. AsamodelforE,helicalspringwithstiffnessEisassumed. Theresponseisinstantaneous,withoutanytimedependency,andtherecoveryafterreleaseofthestressisalsoinstantaneousandcomplete
ViscoelasticFoundationModel
ForanidealliquidNewtonslawholds:Thestressisproportionaltotherateofdeformationd /dt ;theproportionalityconstantistheviscosity,so = . d /dt.Asamodelwechooseadashpot;withinacylinderfilledwithafluidapistoncanmovewithsomeclearance.Thereisnoinstantaneousresponse;thedeformationisproportionaltotime,andnorecoverytakesplace.Thedashpotischaracterizedby.
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
Responseofanidealfluid/liquid
ApplicabilityofViscoelasticFoundationModel
Contrarytotheelasticfoundations,viscoelastic foundationbedsexhibittimedependentresponsewhensubjectedtoexternalloading,i.e.suchfoundationshaveanimmediatesettlementdirectlyaftertheloadingisappliedandanadditionalsettlementincreasingnonlinearlywithtimeduetogradualoutflowofporewater.
ModellingoftimedependentresponseofsuchfoundationbedsisachievedbyusingmechanicalelementssuchasWinklersprings,viscousdashpotsandtheir
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
g p g , pcombinations.
Severalresearchershaveusedvarioustypesofviscoelastic lumpedparametermodelstorepresentthebehaviourofviscoelastic soilbeds.
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MaxwellViscoelasticFoundationModel TheMaxwellViscoelastic foundationModel(ME)consistsofaHookean elasticelement(HE)andNewtonianviscouselement(NE)connectedinseries.
ThecorrespondingstructuralformulaisexpressedME=HE NEwhere istheappliednormalstress,tistheelasped time,EisthemodulusofHokean elasticelement, isthecoefficientofnormalviscosityoftheNewtonian viscouselement
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
ItisobservedthattheMaxwellmodelcanreciprocatetheinstantaneousdeformationandtheinstantaneousrecoveryduetotheapplicationandremovalofloadingrespectively.However,theviscousstrainduetoaconstantstressincreaseslinearlywithtime.
Further,afterremovalofstress,theviscousstrainremainsconstantwithtime.Thus,thisparticularmodelisfoundunsuitableforproperrepresentationofthetimedependentbehaviourofthesoil,however,itrightlycapturestheinstantaneousdeformationbehaviour.
Thus,Maxwellmodelissuitableforsimulatingshorttermbehaviourofaviscoelasticsubgrade,butisunsuitableforlongtermbehaviour.
KelvinVoigtViscoelasticFoundationModel TheKelvinVoigtViscoelastic Model(KV)consistsofaHookean elasticelement(HE)andaNewtonianviscouselement(NE)connectedinparallel.
Thismodeliscapableofpredictingthetimedependentsettlementbehaviour,creepphenomenonandthequasiviscousflowofaviscoelastic medium.However,itisnotcapableofpredictingtheinstantaneoussettlementandtheinstantaneouspartialrecoveryofthemediumduetosuddenloadingandunloadingrespectively.
Hence,theKelvinVoigtmodelissuitablefortherepresentationofthelongtermbehaviourofthesoil.
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
ThecorrespondingstructuralformulaisexpressedKV=HE+NE Thefirstpartcorrespondstothepurelyelasticstresswhichis
independentofthestrainrate Thesecondpartcorrespondstotheviscousstresswhichislineraily
proportionaltothestrainrate.Thusforazerostrainrate,apureelasticstateisobtainedwhereasfornonzerostrainrates,theelasticstateisaddeduponbyamagnitudeequaltotheviscouspartofthestress
ResponseofMaxwellandKelvinFoundationModel
Maxwellmodel
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
KelvinVoigtmodel
PoyntingThompsonViscoelasticFoundationModel ThePoyntingThompsonViscoelastic model(PTh)isathreeparametermodelwhichisacombinationofHookean elasticspringattachedinserieswithaviscoelastic KelvinVoigtelement.
Itiscapableofidealizingthestresstimeandthecorrespondingstraintimebehaviourofaviscoelastic subgrade.However,thismodelfailstopredictpermanentdeformationobtainedasaresultofloadingofaviscoelastic subgrade.
ThecorrespondingstructuralformulaisexpressedPTh =HE1(HE2+NE)=HE1KV
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur
BurgerViscoelasticFoundationModel ThefourparameterBurgerModelisacombinationofMaxwellelementconnectedinserieswithaKelvinVoigtelement.
Thismodeliscapableofexpressingthecomplexphenomenaassociatedwithlongtermreversibleandirreversibledeformation,alongwiththemodellingofinstantaneousstrainandrecoveryduetosuddenloadingandunloadingoftheelement.
Burgermodelisthemostgeneralisedmodelandallotherviscoelastic modelscanbeobtainedfromtheburgermodelbyomittingoneoftheelementsfromtheburgerelement.
CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur