Settlement of foundation

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3/21/2013 1 FOUNDATION ANALYSIS AND DESIGN CE632N Dr. Rajesh Sathiyamoort hy , IIT Kanpur SETTLEMENT AND ALLOWABLE CE632N Dr. Rajesh Sathiyamoorthy, IIT Kanpur Settlement of Foundation  The ultimate bearing capacity q u  is realized at a settlement  level  of  S u .  If  the factor of  safety against bearing capacity failure is FS, then the allowable bearing capacity is q safe = q u /FS. The settlement  corresponding to q safe  is S.  If  S all  is the allowable level  of  settlement  for the proposed foundation then q all(S)  be the corresponding allowable In the design of  any foundation, one must consider the safety against bearing capacity failure as well as against excessive settlement  of  the foundation. Load per unit area q versus settlement  S for a foundation bearing capacity.   For foundations with smaller widths of  B, S all  may be more than S’; however, for larger values of  B, S all  < S. Hence,  for smaller foundation widths,  the bearing capacity controls;  for larger foundation widths,  the allowable settlement  controls.   However, it is always necessary to estimate  the total settlement  under structural load and then arrive at the allowable bearing capacity. Settlement of Foundation  During settlement  the soil transitions  from the current body (or self weight) stress state to a new one under the additional  applied load.  The stress change Δq from this added load produces a timedependent  accumulation  of  particle rolling,  sliding,  crushing,  and elastic distortions  in a limited influence zone  Foundation settlements  must be estimated  with great care for buildings,  bridges,  towers, power plants,  and similar highcost structures .  For structures such as fills,  earth dams,  levees,  braced sheeting,  and retaining walls a greater margin of  error in the settlements  can usually be tolerated. enea  e oa e  area.  The statistical  accumulation  of  movements in the direction of  interest is the settlement.  In the vertical  direction  the settlement  will be defined as H.  The principal  components  of  H are particle rolling and sliding, which produce a change in the void ratio, and grain crushing, which alters the material slightly.   Only a very small fraction of  H is due to elastic deformation of  the soil grains. As a consequence, if  the applied stress is removed,  very little of  the settlement  ΔH is recovered.  Even though H has only a very small elastic component,  it is convenient  to treat the soil as a pseudoelastic material  with "elastic" parameters E s , G', μ and k s to estimate settlements . Major Problems with Soil Settlement Analys is Obtaining a reliable stress profile from the applied load Obtaining reliable values of  the elastic parameters (E s  and μ)  Correlations are commonl  Undisturbed soil?  Improper estimate of insitu test.. Anisotropy is neglected Theory of  Elasticity  equations  are usually  Stress distribution?  Effective depth of influence?  used, particularly for preliminary design studies  , the influence depth H below the loaded area taken from H = 0 to ∞ (but more correctly from 0 to about 4B or 5B). Estimated values are then used in an equation of  the general form  H dH  H 0   Where ε= strain = Δq/Es  ; but Δq = f(H, load),  E s  = f(H, soil variation),  and H is the estimated depth of  stress change caused by the foundation load. Settlement of Foundation Total Settlement S = S e +S c +S s Elastic/ Immediate Settlement S e Consolidation  Settlement S cs  = S c +S s  Immediate Settlement:  Occurs immediately after the construction.  This is computed using elasticity theory (Important  for Granular soil)  Primary Consolidation:  Due to gradual  dissipation of  pore pressure induced by external loading and consequently expulsion of  water from the soil mass, hence volume change. (Important  for Inorganic clays) Primary Consolidation  Settlement S c Secondary  Consolidation  Settlement  S s  Secondary Consolidation:  Occurs at constant  effective stress with volume change due to rearrangement  of  particles. (Important  for Organic soils) Settl ement Cla yey soil Sandy soil Shortterm/ Immediate Di st or ti on D is to rt io n and consolidation Long  –term / delayed Pri. and sec. consolidation negligible For the calculation of  foundation settlement,  it is required to estimate the vertical  stress increase in the soil mass due to the net load applied on the foundation.

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Transcript of 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

    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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    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

    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

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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

    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

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    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)

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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

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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.

    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)

  • 3/21/2013

    9

    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.

  • 3/21/2013

    11

    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