Settlement of foundation

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FOUNDATION ANALYSIS AND DESIGN S G

CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur

SETTLEMENT AND ALLOWABLE BEARING CAPACITYBEARING CAPACITY


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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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

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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


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


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

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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

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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


ConventionalDesignofFoundation

Intheconventionalanalysisanddesignoffootings,tipofthecolumnisfixedandassumesthesoilpressureisassumedtobeuniformorlinearlyvarying.Thecentroidofthesoilpressureiscoincidentwiththelineofactionoftheresultantcolumnloads.

StructuralAnalysis&Design


Generalassumption Assumefoundationasrigid Soilreactionasconstant Designasinvertedbeamsorslabs Noconsiderationtothetypeofsoil

Inreality,footingisneitherperfectlyrigidnorperfectlyflexible.Soilreaction/contactpressuredistributiondependsonsoiltypeandrigidityofthestructure

ContactPressureDistribution

ContactpressureandsettlementprofilesforfoundationonsandTheassumptionofuniformpressuredistributionwillgiveaconservativedesignforrigidfooting

onsandysoilsasthemaximumbendingmomentisoverestimated

Rigidfooting


Flexiblefooting

ContactPressureDistributionContactpressureandsettlementprofilesforfoundationonclay

Theassumptionofuniformpressuredistributionwillresultinaslightlyunsafedesignforrigidfootingsonclaysasthemaximumbendingmomentatthecentreisunderestimated.

Rigidfooting


Flexiblefooting

Realisticdistributionofcontactpressureneedstobeconsidered

SoilFoundationStructureInteraction

Structures

Foundation

Response of any structure


Underlying Soil / rock

y

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Soil StructureInteractionStudies


Soil StructureInteractionStudies


Theproblemofsoilfoundationstructureinteractionisgenerallysolvedbyincorporatingthereactionfromthefoundation,intotheresponsemechanismofthestructure,byidealizingthefoundationbyasuitablemathematicalmodel.

Soil StructureInteractionAnalysis

Evenifthefoundationmediumhappenstobecomplexinsomeproblems,inamajorityofcases,theresponseofthestructureatthecontactsurfaceisofprimeinterestandhence,itwouldbeofimmensehelpintheanalysis,ifthefoundationcanberepresentedbyasimplemathematicalmodel,withoutforegoingthedesiredaccuracy.

To accomplish this objective, many foundation models have been proposed and a


Itisgenerallyobservedthatthemodeling ofthesuperstructureandfoundationarerathersimplerandstraightforwardthanthatofthesoilmediumunderneath.

However,soilishavingverycomplexcharacteristics,sinceitisheterogeneous,anisotropicandnonlinearinforcedisplacementcharacteristics.

Thepresenceoffluctuationofwatertablefurtheraddstoitscomplexity.

Toaccomplishthisobjective,manyfoundationmodelshavebeenproposedandacomprehensivereviewpertainingtothesehasbeengivenbymanyauthors.

CriticalAspectsofSSITheproblemofsoilfoundationstructureinteractionisgenerallysolvedbyincorporatingthereactionfromthefoundationintotheresponsemechanismofthestructure,byidealizingthefoundationbyasuitablemathematicalmodel.

SoilCriticalaspectsofSSIistheconceptualmodellingof Foundation

Simplifiedmodels

Canbemodelled asbeam/plate/shell


Thefootingcanbemodelledasabeamoraplateorashellandclassicalbendingtheoriescanbeusedforrepresentingtheirresponse.

Thesoilreactionhastobeincorporatedintheintegratedanalysisofsoilstructureinteractionequationbymodellingthesoilappropriatelyusingdifferentmodels

Structure Canbeeasilymodelled

Inertia=>baseshearandmoment

CriticalAspectsofSeismicSSI

Relativefoundationrotation


Relativefoundationdisplacement

TheoryofElasticFoundationTheresponseoffoundationsoilsystemsubjectedtoexternalloadsdependsonthegeometryofthefoundation(beamoraplate)andidealisationofsoil.Mostofthefootingscanbeconsideredaseitherbeams(onedimensional)orplates(twodimensional:rectangular,squares,circular,annularorothershapes).

Plateonanelasticfoundation

q(x)

z (y)


Beamonanelasticfoundation

q(x, y)z

y

(y)

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BeamonElasticFoundation


BeamonElasticFoundation

R(x)-q(x)


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


markedlynonlinear,irreversibleandtimedependent,andtosoilmasseswhichexhibitanisotropicandnonhomogeneousmaterialproperties.

Hence,anyattempttosolveasoilfoundationinteractionproblem,takingintoaccountallsuchmaterialcharacteristics,isclearlyanimpossibletask

Inordertoobtainmeaningfulandrealizableinformationforpracticalproblemsofsoilfoundationinteractionitbecomesnecessarytoidealise thebehaviour ofthesoilbytakingintoaccountspecificaspectsofitsbehaviour.

Idealised SoilBehavior/ModelingSoilMediaThesimplesttypeofidealised soilresponseassumelinearelasticbehaviour ofthesupportingsoilmedium

Assumptionoflinearityandreversibilityofdeformationsimplicitinlinearelasticbehavior,ofcoursenotalwaysrigorouslysatisfiedbynaturallyoccurringsoilmasses

Assuminglinearelasticbehaviour ontheotherhandconsiderablyreducestheanalyticalrigorexpendedinthesolutionofaparticularboundaryvalueproblemandprovidesusefulinformationtomany


Basedonthecomplexity;soilmodelsareclassifiedas

practicalproblemsofsoilmechanicsandfoundationengineeringwhichwouldotherwisebeintractable.

OneparametermodelTwoparametermodelThreeparametermodelFourparametermodeletc

WinklerModel

Z

Structure

Foundation

Soil

Structure

Foundation

Idealisedspringtorepresentsoil Z


TheearliestformulationofthesoilmodelwasduetoWinkler,whoassumedthesoilmodeltoconsistofcloselyspaced

independentlinearsprings

Oneparametermodel

Surfacedisplacementdueto:

Nonuniformload

Concentratedload

Z

X

Uniformflexibleload

WinklerModel


x

z

Rigidload

Responseofactualfoundation

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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.


k isthespringconstantandisoftenreferredtoasthefoundationmoduluscanbeequatedtomodulusofsubgradereactionofsoil.

w istheverticaldeflectionofthecontactsurface.

Discrete,independent,linearelasticsprings Simpletouse Soilbehaviourislinearingeneral AnumberofstudiesintheareaofsoilstructureinteractionhavebeenconductedonthebasisofWinklerhypothesisforitssimplicity.

CoefficientofsubgradereactionThecoefficientofsubgradereactionistheunitpressurerequiredtoproduceaunit

settlement.ItcanbeobtainedfromPlateloadtest Inclayeysoils,settlementundertheloadtakesplaceoveralongperiodoftimeandthecoefficientshouldbedeterminedonthebasisofthefinalsettlement.

Onpurelygranularsoils,settlementtakesplaceshortlyaftertheloadapplication.

Assumptionsqks pressure bearing


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


Invariably,alargeplatedoesnotsettleuniformly.Thesettlementmust,therefore,bemonitoredbymeansofthreeorfourdialgaugesequallyspacedaroundtheperimeterinordertodeterminethemeansettlement.

Supportsforthesedialgaugesshouldbesitedwelloutsidethezoneofinfluenceofthejackingloadwhichismeasuredbyaprovingring.Whenchoosingadiameterofplatetouseforthetest,dueconsiderationshouldalsobegiventothelimitedzoneofinfluenceoftheloadedplate.Typically,thesoilwillonlybeeffectivelystressedtoadepthof1.251.50timesthediameteroftheplate.


Thesubgradereactionatanypointalongthebeamisassumedtobedirectlyproportionaltotheverticaldisplacementofthebeamatthatpoint,i.e,soilisassumedtobeelasticandobeysHookes Law

Plateloadtest

ks Coefficientofsubgradereactionorcoefficientofelasticuniformcompressionqbearingpressureatapointalongthebeamw verticaldisplacementofthe beam at that point

wqks settlement

pressure bearing


Hooke s Law.thebeamatthatpoint

Infoundationdesign,thevalueofks isthesecantmodulusofthegraphovertheestimatedworkingrangeofbearingpressure.

Thevalueofthemodulusofsubgradereaction(ks)obtainedfromthetestvariesaccordingtothesizeofplateused.



ks dependsnotonlyonthedeformationcharacteristicsofthesoilbutalsoonthesizeofcontactareabetweenplateandsubgrade.

Thevariationofks withplatesizecreatesanobviousdifficultyindecidingwhichplatesizeshouldbeusedasthestandardorreferencefordefiningvaluesofks foranalysis.


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


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


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


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


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


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


Z Z

FILONENKO&BORODICHMODEL(1945)Surfacedisplacementdueto:

Rigid load Uniform flexible load

T

XT

T

q


Z

T

Z

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HETENYIMODEL(1946)

),(),(),(),( 4 yxwDyxkwyxqyxR Where,

Inthismodel,interactionbetweentheindependentspringelementsisaccomplishedbyincorporatinganelasticplatein3Dproblemsoranelasticbeamin2Dproblems.

3hE


,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.


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.


ViscoelasticFoundationModel Theexpressionviscoelasticsignifiesthedualnatureofamaterial:ontheonehanditbehavesinaviscousway,asaliquid,ontheotherhandelastically,asasolid.

Foranidealsolid,Hookes lawholds:thestress,,appliedisproportionaltothedeformation,,andtheproportionalityconstantisthemodulusofelasticityE,so =E. BesidesEalsootherquantitiesplayarole,suchasthe

Responseofanidealspring


shearmodulus,G,inashearingdeformationortorsion,whichisrelatedtoE.ForthesakeofsimplicityweshallmainlyuseEasarepresentativequantityfortheelasticstiffnessinanygeometryofloading. AsamodelforE,helicalspringwithstiffnessEisassumed. Theresponseisinstantaneous,withoutanytimedependency,andtherecoveryafterreleaseofthestressisalsoinstantaneousandcomplete

ViscoelasticFoundationModel

ForanidealliquidNewtonslawholds:Thestressisproportionaltotherateofdeformationd /dt ;theproportionalityconstantistheviscosity,so = . d /dt.Asamodelwechooseadashpot;withinacylinderfilledwithafluidapistoncanmovewithsomeclearance.Thereisnoinstantaneousresponse;thedeformationisproportionaltotime,andnorecoverytakesplace.Thedashpotischaracterizedby.


Responseofanidealfluid/liquid

ApplicabilityofViscoelasticFoundationModel

Contrarytotheelasticfoundations,viscoelastic foundationbedsexhibittimedependentresponsewhensubjectedtoexternalloading,i.e.suchfoundationshaveanimmediatesettlementdirectlyaftertheloadingisappliedandanadditionalsettlementincreasingnonlinearlywithtimeduetogradualoutflowofporewater.

ModellingoftimedependentresponseofsuchfoundationbedsisachievedbyusingmechanicalelementssuchasWinklersprings,viscousdashpotsandtheir


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


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.


ThecorrespondingstructuralformulaisexpressedKV=HE+NE Thefirstpartcorrespondstothepurelyelasticstresswhichis

independentofthestrainrate Thesecondpartcorrespondstotheviscousstresswhichislineraily

proportionaltothestrainrate.Thusforazerostrainrate,apureelasticstateisobtainedwhereasfornonzerostrainrates,theelasticstateisaddeduponbyamagnitudeequaltotheviscouspartofthestress

ResponseofMaxwellandKelvinFoundationModel

Maxwellmodel


KelvinVoigtmodel

PoyntingThompsonViscoelasticFoundationModel ThePoyntingThompsonViscoelastic model(PTh)isathreeparametermodelwhichisacombinationofHookean elasticspringattachedinserieswithaviscoelastic KelvinVoigtelement.

Itiscapableofidealizingthestresstimeandthecorrespondingstraintimebehaviourofaviscoelastic subgrade.However,thismodelfailstopredictpermanentdeformationobtainedasaresultofloadingofaviscoelastic subgrade.

ThecorrespondingstructuralformulaisexpressedPTh =HE1(HE2+NE)=HE1KV


BurgerViscoelasticFoundationModel ThefourparameterBurgerModelisacombinationofMaxwellelementconnectedinserieswithaKelvinVoigtelement.

Thismodeliscapableofexpressingthecomplexphenomenaassociatedwithlongtermreversibleandirreversibledeformation,alongwiththemodellingofinstantaneousstrainandrecoveryduetosuddenloadingandunloadingoftheelement.

Burgermodelisthemostgeneralisedmodelandallotherviscoelastic modelscanbeobtainedfromtheburgermodelbyomittingoneoftheelementsfromtheburgerelement.


Settlement of foundation

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