Mat Foundation analysis and design Coduto10.pdf

Mat Foundation analysis and design Coduto10.pdf

1. I. 8M.ioo.- .I
10
Mats
Chapter10Mats
1
Columns
/
T
353
Themereformulationofaproblemisfarmoreoftenessentialthanits
solution,whichmaybemerelyamatterofmathematicalorexperimental
skill.Toraisenewquestions,newpossibilities,toregardoldproblems
fromanewanglerequirescreativeimaginationandmarksreal
advancesinscience.
AlbertEinstein
Thesecondtypeofshallowfoundationisthematfoundation,asshowninFigureIO.!.A
matisessentiallyaverylargespreadfootingthatusuallyencompassestheentirefootprint
ofthestructure.Theyarealsoknownasraftfoundations.
Foundationengineersoftenconsidermatswhendealingwithanyofthefollowing
conditions:
•Thestructuralloadsaresohighorthesoilconditionssopoorthatspreadfootings
wouldbeexceptionallylarge.Asageneralruleofthumb,ifspreadfootingswould
covermorethanaboutone-thirdofthebuildingfootprintarea,amatorsometype
ofdeepfoundationwillprobablybemoreeconomical.
•Thesoilisveryerraticandpronetoexcessivedifferentialsettlements.Thestruc­
turalcontinuityandflexuralstrengthofamatwillbridgeovertheseirregularities.
Thesameistrueofmatsonhighlyexpansivesoilspronetodifferentialheaves.
•Thestructuralloadsareerratic,andthusincreasethelikelihoodofexcessivediffer­
entialsettlements.Again,thestructuralcontinuityandflexuralstrengthofthemat
willabsorbtheseirregularities.
352
Figure10.1Amatfoundationsupporteddirectlyonsoil.
Lateralloadsarenotuniformlydistributedthroughthestructureandthusmaycause
differentialhorizontalmovementsinspreadfootingsorpilecaps.Thecontinuityof
amatwillresistsuchmovements.
r
•Theupliftloadsarelargerthanspreadfootingscanaccommodate.Thegreater
weightandcontinuityofamatmayprovidesufficientresistance.
•Thebottomofthestructureislocatedbelowthegroundwatertable,sowaterproof­
ingisanimportantconcern.Becausematsaremonolithic,theyaremucheasierto
waterproof.Theweightofthematalsohelpsresisthydrostaticupliftforcesfrom
thegroundwater.
Manybuildingsaresupportedonmatfoundations,asaresilos,chimneys,andother
typesoftowerstructures.Matsarealsousedtosupportstoragetanksandlargemachines.
Typically,thethickness,T,is1to2m(3-6ft),somatsaremassivestructuralelements.
TheseventyfivestoryTexasCommerceTowerinHoustonisoneofthelargestmat­
supportedstructuresintheworld.Itsmatis3m(9ft9in)thickandisbottomed19.2m
(63ft)belowthestreetlevel.
Althoughmostmatfoundationsaredirectlysupportedonsoil,sometimesengineers
usepile-orshaft-supportedmats,asshowninFigure10.2.Thesefoundationsareoften
calledpiledrafts.andtheyarehybridfoundationsthatcombinefeaturesofbothmatand
deepfoundations.Pile-andshaft-supportedmatsarediscussedinSection1!.9.

2. 354 Chapter10Mats
Building
ttt
Mat
10.1RigidMethods
p
~
M
/""P"..
~
355
Figure10.2Apile-orshaft-supportedmatfoundation.Thedeepfoundationsarenot
necessarilydistributedevenlyacrossthemat.
Variousmethodshavebeenusedtodesignmatfoundations.Wewilldividethem
intotwocategories:Rigidmethodsandnonrigidmethods.
10.1RIGIDMETHODS
Figure10.3Bearingpressuredistributionforrigidmethod.
Althoughthistypeofanalysisisappropriateforspreadfootings,itdoesnotaccu­
ratelymodelmatfoundationsbecausethewidth-to-thicknessratioismuchgreaterin
mats,andtheassumptionofrigidityisnolongervalid.Portionsofamatbeneathcolumns
andbearingwallssettlemorethantheportionswithlessload,whichmeansthebearing
pressurewillbegreaterbeneaththeheavily-loadedzones,asshowninFigure10.4.This
redistributionofbearingpressureismostpronouncedwhenthegroundisstiffcompared
tothemat,asshowninFigure10.5,butispresenttosomedegreeinallsoils.
Becausetherigidmethoddoesnotconsiderthisredistributionofbearingpressure,it
doesnotproducereliableestimatesoftheshears,moments,anddeformationsinthemat.
Inaddition,evenifthematwasperfectlyrigid,thesimplifiedbearingpressuredistribu­
tionsinFigure10.3arenotcorrect-inreality,thebearingpressureisgreaterontheedges
'-c
andsmallerinthecenterthanshowninthisfigure.
Figure10.4Therigidmethodassumestherearenoflexuraldeflectionsinthemat.so
thedistributionofsoilbearingpressureissimpletodefine.However.thesedetlectionsare
importantbecausetheyinfluencethebearingpressuredistribution.
Thesimplestapproachtostructuraldesignofmatsistherigidmethod(alsoknownasthe
conventionalmethodortheconventionalmethodofstaticequilibrium)(Teng,1962).This
methodassumesthematismuchmorerigidthantheunderlyingsoils,whichmeansany
distortionsinthemataretoosmalltosignificantlyimpactthedistributionofbearingpres­
sure.Therefore,themagnitudeanddistributionofbearingpressuredependsonlyonthe
appliedloadsandtheweightofthemat,andiseitheruniformacrossthebottomofthemat
(ifthenormalloadactsthroughthecentroidandnomomentloadispresent)orvarieslin­
earlyacrossthemat(ifeccentricormomentloadsarepresent)asshowninFigure10.3.
Thisisthesamesimplifyingassumptionusedintheanalysisofspreadfootings,asshown
inFigure5.10e.
Thissimpledistributionmakesiteasytocomputetheflexuralstressesanddeflec­
tions(differentialsettlements)inthemat.Foranalysispurposes,thematbecomesanin­
vertedandsimplyloadedtwo-wayslab,whichmeanstheshears,moments,and
deflectionsmaybeeasilycomputedusingtheprinciplesofstructuralmechanics.Theen­
gineercanthenselecttheappropriatematthicknessandreinforcement.
,
I
I
--1
rm
RigidMat
Soil
Bearing
Pressure
tJI1JJI1IJID
NonrigidMat
(Exaggerated)

3. 356 Chapter10Mats 10.2NonrigidMethods 357
Rock
Where:
k,=coefficientofsubgradereaction
q
=bearingpressure
l)=settlement
(a)
Stiff
Soil
Thecoefficientk,hasunitsofforceperlengthcubed.Althoughweusethesameunitsto
expressunitweight,
k,isnotthesameastheunitweightandtheyarenotnumerically
equal.
Theinteractionbetweenthematandtheunderlyingsoilmaythenberepresentedas
a"bedofsprings,"eachwithastiffnessk,perunitarea,asshowninFigure10.6.Portions
ofthematthatexperiencemoresettlementproducemorecompressioninthe"springs,"
whichrepresentsthehigherbearingpressure,whereasportionsthatsettlelessdonotcom­
pressthespringsasfarandthushavelessbearingpressure.Thesumofthesespring
forcesmustequaltheappliedstructuralloadsplustheweightofthemat:
(b)
LP+Wr-Uf)=JqdA=Jl)k,dA (10.2)
Figure10.5Distributionofbearingpres­
sureunderamatfoundation;(a)onbedrock
orveryhardsoil;(b)onstiffsoil;(c)onsoft
soil(AdaptedfromTeng,
1962).
~///////}///////////W-
~iUl.llii111l-J
~: .
(c)
Where:
LP=sumofstructuralloadsactingonthemat
Wr=weightofthemat
CoefficientofSubgradeReaction
Becausenonrigidmethodsconsidertheeffectsoflocalmatdeformationsonthedistribu­
tionofbearingpressure,weneedtodefinetherelationshipbetweensettlementandbear­
ingpressure.Thisisusuallydoneusingthecoefficientofsubgradereaction,
ks(also
knownasthemodulusofsubgradereaction,orthesubgrademodulus):
Weovercometheinaccuraciesoftherigidmethodbyusinganalysesthatconsiderdefor­
mationsinthematandtheirinfluenceonthebearingpressuredistribution.Theseare
callednonrigidmethods,andproducemoreaccuratevaluesofmatdeformationsand
stresses.Unfortunately,nonrigidanalysesalsoaremoredifficulttoimplementbecause
theyrequireconsiderationofsoil-structureinteractionandbecausethebearingpressure
distributionisnotassimple.
Figure10.6Thecoefficientofsubgradereactionformsthebasisfora"bedofsprings"
analogytomodelthesoil-structureinteractioninmatfoundations.(10.1)
Ik,=~I
10.2NONRIGIDMETHODS

4. ThenextstepupfromaWinkleranalysisistouseacoupledmethod,whichusesaddi­
tionalspringsasshowninFigure10.9.Thiswaytheverticalspringsnolongeractinde­
pendently,andtheuniformlyloadedmatofFigure10.8exhibitsthedesireddishshape.In
principle,thisapproachismoreaccuratethantheWinklermethod,butitisnotclearhow
toselectthek,valuesforthecouplingsprings,anditmaybenecessarytodevelopcustom
softwaretoimplementthisanalysis.
~~~~~~~
~
358 Chapter10Mats
Un=porewaterpressurealongbaseofthemat
q=bearingpressurebetweenmatandsoil
A=mat-soilcontactarea
o=settlementatapointonthemat
Thismethodofdescribingbearingpressureiscalledasoil-structureinteraction
analysisbecausethebearingpressuredependsonthematdeformations,andthematde­
formationsdependonthebearingpressure.
WinklerMethod
The"bedofsprings"modelisusedtocomputetheshears,moments,anddeformationsin
themat,whichthenbecomethebasisfordevelopingastructuraldesign.Theearliestuse
ofthese"springs"torepresenttheinteractionbetweensoilandfoundationshasbeenat­
tributedtoWinkler(1867),sotheanalyticalmodelissometimescalledaWinklerfounda­
tionortheWinklermethod.Italsoisknownasabeamonelasticfoundationanalysis.
InitsclassicalformtheWinklermethodassumeseach"spring"islinearandactsin­
dependentlyfromtheothers,andthatallofthespringshavethesame
k,.Thisrepresenta­
tionhasthedesiredeffectofincreasingthebearingpressurebeneaththecolumns,and
thusisasignificantimprovementovertherigidmethod.However,itisstillonlyacoarse
representationofthetrueinteractionbetweenmatsandsoil(HainandLee,1974;Hor­
vath,1983),andsuffersfrommanyproblems,includingthefollowing:
1.Theload-settlementbehaviorofsoilisnonlinear,sothek,valuemustrepresent
someequivalentlinearfunction,asshowninFigure10.7.
2.Accordingtothisanalysis,auniformlyloadedmatunderlainbyaperfectlyuniform
soil,asshowninFigure10.8,willsettleuniformlyintothesoil(i.e.,therewillbeno
differentialsettlement)andallofthe"springs"willbeequallycompressed.Inreal­
ity,thesettlementatthecenterofsuchamatwillbegreaterthanthatalongthe
edges,asdiscussedinChapter7.Thisisbecausethe
6.CT:valuesinthesoilare
greaterbeneaththecenter.
3.The"springs"shouldnotactindependently.Inreality,thebearingpressureinduced
atonepointonthematinfluencesmorethanjustthenearestspring.
4.Primarilybecauseofitems2and3,thereisnosinglevalueofk,thattrulyrepre­
sentstheinteractionbetweensoilandamat.
10.2NonrigidMethods
~
ri
~
iJ
0::
.§
"
"
::c
.~
p
(]
Fj~ure10.7The'1-8relalionshipisnonlin­
car.soksmustrepresentsome"equivalent""
linearfunction.
CoupledMethod
ttttttt
-,_---....1
Idealized
Linear~ ,/Function-----.../
,
,
jk,
1/I
,,,
Seulement.(5
TrueBchavior
359
Figureto.8Settlementofauniformly-loadedmatonauniformsoil:(a)perWinkler
analysis.(b)actual.
Items2and3aretheprimarysourcesoferror,andthiserrorispotentiallyunconser­
vative(i.e.,theshears,moments,anddeflectionsinthematmaybegreaterthanthosepre­
dictedbyWinkler).Theheartoftheseproblemsistheuseofindependentspringsinthe
Winklermodel.Inreality,aloadatonepointonthematinducessettlementbothatthat
pointandintheadjacentpartsofthemat,whichiswhyauniformlymatexhibitsdish­
shapedsettlement,nottheuniformsettlementpre!iictedbyWinkler.
J
PerWinkler
Actual

5. 360
Chapter10Mats
Mal
CouplingSpring
FiKure10.9ModeJingofsoil-struclUreinleractionusingcoupledsprings.
10.2NonrigidMethods 361
Pseudo-CoupledMethod
Thepseudo-coupledmethod(Liao,1991;Horvath,1993)isanattempttoovercomethe
lackofcouplingintheWinklermethodwhileavoidingthedifficultiesofthecoupled
method.Itdoessobyusing"springs"thatactindependently,buthavedifferentk,values
dependingontheirlocationonthemat.Toproperlymodeltherealresponseofauniform
soil,the"springs"alongtheperimeterofthematshouldbestifferthanthoseinthecenter,
thusproducingthedesireddish-shapeddeformationinauniformly-loadedmat.Ifconcen­
tratedloads,suchasthosefromcolumns,alsoarepresent,theresultingmatdeformations
areautomaticallysuperimposedonthedish-shape.
Modelstudiesindicalethatreasonableresultsareobtainedwhen
k,valuesalongthe
perimeterofthematareabouttwicethoseinthecenter(ACI,1993).Wecanimplement
thisinavarietyofways,includingthefollowing:
1.Dividethematintotwoormoreconcentriczones,asshowninFigure10.10.The
innermostzoneshouldbeabouthalfaswideandhalfaslongasthemat.
2.Assignak,valuetoeachzone.Thesevaluesshouldprogressivelyincreasefromthe
centersuchthattheoutermostzonehasak,abouttwiceaslargeattheinnermost
zone.Example10.1illustratesthistechnique.
3.Evaluatetheshears,moments,anddeformationsinthematusingtheWinkler"bed
ofsprings"analysis,asdiscussedlaterinthischapter.
4.Adjustthematthicknessandreinforcementasneededtosatisfystrengthandser­
viceabilityrequirements.
ACI(1993)foundthepseudo-coupledmethodproducedcomputedmoments18to
25percenthigherthanthosedeterminedfromtheWinklermethod,whichisanindication
ofhowunconservativeWinkercanbe.
MostcommercialmatdesignsoftwareusestheWinklermethodtorepresentthe
soil-structureinteraction,andthesesoftwarepackagesusuallycanaccommodatethe
I
J
ZoneA
k,=50Ib/in'
ZoneBk,=75Iblin'
ZoneCk,=100Ib/in'
FiKureto.IOAlypicalmaldividedinlozonesforapseudo-coupledanalysis.Tbecoef­
ficientofsuhgradcreal.'tioll.k,.progressivelyincreasesfromtheinnermostzonetothe
outermostlone.
pseudo-coupledmethod.Giventhecurrentstateoftechnologyandsoftwareavailability,
thisisprobablythemostpracticalapproachtodesigningmostmatfoundations.
Multiple-ParameterMethod
Anotherwayofrepresentingsoil-structureinteractionistousethemultipleparameter
method(Horvath,1993).Thismethodreplacestheindependently-actinglinearspringsof
theWinklermethod(asingle-parametermodel)withspringsandothermechanicalele­
ments(amultiple-parametermodel).Theseadditionalelementsdefinethecouplingeffects.
Themultiple-parametermethodbypassestheguessworkinvolvedindistributingthe
k,valuesinthepsuedo-coupledme!hodbecausecouplingeffectsareinherentlyincorpo­
ratedintothemodel,andthusshouldbemoreaccurate.However,ithasnotyetbeenim­
plementedintoreadily-availablesoftwarepackages.Therefore,thismethodisnotyet
readytobeusedonroutineprojects.
FiniteElementMethod
Allofthemethodsdiscussedthusfarattempttomodelathree-dimensionalsoilbyaseries
ofone-dimensionalsprings.Theydosoinordertomaketheproblemsimpleenoughto
performthestructuralanalysis.Analternativemethodwouldbetouseathree-dimensional

6. 362 Chapter10Mats 10.3DeterminingtheCoefficientofSubgradeReaction 363
mathematicalmodelofboththematandthesoil,orperhapsthemat,soil,andsuperstruc­
ture.Thiscanbeaccomplishedusingtheftniteelementmethod.
Thisanalysismethoddividesthesoilintoanetworkofsmallelements,eachwith
definedengineeringpropertiesandeachconnectedtotheadjacentelementsinaspecified
way.Thestructuralandgravitationalloadsarethenappliedandtheelementsarestressed
anddeformedaccordingly.This.inprinciple.shouldbeanaccuraterepresentationofthe
mat,andshouldfacilitateapreciseandeconomicaldesign.
Unfortunately,suchanalysesarenotyetpracticalforroutinedesignproblemsbe­
cause:
1.Athree-dimensionalfiniteelementmodelrequirestensofthousandsorperhaps
hundredsofthousandsofelements,andthusplacecorrespondingdemandsoncom­
puterresources.fewengineershaveaccesstocomputersthatcanaccommodate
suchintensiveanalyses.
2.Itisdifficulttodeterminetherequiredsoilpropertieswithenoughprecision,espe­
ciallyatsiteswherethesoilsarehighlyvariable.Inotherwords,theanalysis
methodfaroutweighsourabilitytoinputaccurateparameters.
Nevertheless,thisapproachmaybecomemoreusableinthefuture,especiallyasincreas­
inglypowerfulcomputersbecomemorewidelyavailable.
Thismethodshouldnotbeconfusedwithstructuralanalysismethodsthatusetwo­
dimensionalfiniteelementstomodelthematandWinkleI'springstomodelthesoil.Such
methodsrequirefarlesscomputationalresources,andarewidelyused.Wewilldiscuss
thisuseoffiniteelementanalysesinSection
lOA.
10.3DETERMINING THECOEFFICIENT OFSUBGRADE REACTION
MostmatfoundationdesignsarecurrentlydevelopedusingeithertheWinklermethodor
thepseudo-coupledmethod,bothofwhichdependonourabilitytodefinethecoefficient
ofsubgradereaction,
k,.Unfortunately,thistaskisnotassimpleasitmightfirstappear
becausek,isnotafundamentalsoilproperty.Itsmagnitudealsodependsonmanyother
factors,includingthefollowing:
•Thewidthoftheloadedarea-Awidematwillsettlemorethananarrowone
withthesameqbecauseitmobilizesthesoiltoagreaterdepthasshowninFig­
ure8.2.Therefore,eachhasadifferentk,.
•Theshapeoftheloadedarea-Thestressesbelowlongnarrowloadedareasare
differentfromthosebelowsquareloadedareasasshowninFigure7.2.Therefore,
ks
willdiffer.
•Thedepthoftheloadedareabelowthegroundsurface-Atgreaterdepths,the
changeinstressinthesoilduetoqisasmallerpercentageoftheinitialstress,so
thesettlementisalsosmallerandk,isgreater.
1
•Thepositiononthemat-Tomodelthesoilaccurately,k,needstobelargernear
theedgesofthematandsmallernearthecenter.
•Time-Muchofthesettlementofmatsondeepcompressiblesoilswillbedueto
consolidationandthusmayoccuroveraperiodofseveralyears.Therefore,itmay
benecessarytoconsiderbothshort-termandlong-termcases.
Actually,thereisnosingle
k,value,evenifwecoulddefinethesefactorsbecause
the
q-Srelationshipisnonlinearandbecauseneithermethodaccountsforinteractionbe­
tweenthesprings.
Engineershavetriedvarioustechniquesofmeasuringorcomputing
k,.Somerely
onplateloadteststomeasurek,insitu.However,thetestresultsmustbeadjustedtocom­
pensateforthedifferencesinwidth,shape,anddepthoftheplateandthemat.Terzaghi
(1955)proposedaseriesofcorrectionfactors,buttheextrapolationfromasmallplatetoa
matissogreatthatthesefactorsarenotveryreliable.Plateloadtestsalsoincludethedu­
biousassumptionthatthesoilswithintheshallowzoneofinfluencebelowtheplateare
comparabletothoseinthemuchdeeperzonebelowthemat.Therefore,plateloadtests
generallydonotprovidegoodestimatesofk,formatfoundationdesign.
Othershaveusedderivedrelationshipsbetweenk,andthesoil'smodulusofelasti­
city,E(VesicandSaxena,1970;Scott,1981).Althoughtheserelationshipsprovidesome
insight,theytooarelimited.
Anothermethodconsistsofcomputingtheaveragematsettlementusingthetech­
niquesdescribedinChapter7andexpressingtheresultsintheformofk,usingEquation
10.1.Ifusingthepseudo-coupledmethod,usek,valuesinthecenterofthematthatare
lessthanthecomputedvalue,andk,valuesalongtheperimeterthataregreater.This
shouldbedoneinsuchawaythattheperimetervaluesaretwicethecentralvalues,and
theintegralofallthevaluesovertheareaofthematisthesameastheproduceofthe
originalk,andthematarea.Example10.1describesthismethodology.
Example10.1
Astructureistobesupportedona30-mwide,50-mlongmatfoundation.Theaveragebear­
ingpressureis120kPa.Accordingtoasettlementanalysisconductedusingthetechniques
describedinChapter7,theaveragesettlement,ll,willbe30mm.Determinethedesignval­
uesof
k,tobeusedinapseduo-coupledanalysis.
Solution
Computeaverage k,usingEquation10.1:
q120kPa=4000kN/m-'
(k.)",.•=5=0.030ID
Dividethematintothreezones,asshowninFigurelO.l!,with (k,lc= 2(k,.)Aand
(k,)B=1.5(k,lA

7. 364 Chapter10Mats
50m
10.4StructuralDesign
10.4STRUCTURAL DESIGN
365
Figure10.11MatfoundationforExample10.1.
Computetheareaofeachzone:
A,=(25m)(15m) =375m2
A{j=(37.5m)(22.5m)-(25m)(15m) =469m2
Ac=(50m)(30m)-(37.5m)(22.5m) =656m2
37.5m
25m
ZoncA
15m22.5In
ZoneB
ZoneC
30
m
GeneralMethodology
Thestructuraldesignofmatfoundationsmustsatisfybothstrengthandserviceabilityre­
quirements.Thisrequirestwoseparateanalyses,asfollows:
StepI:Evaluatethestrengthrequirementsusingthefactoredloads(Equations
2.7­
2.15)andLRFDdesignmethods(whichACIcallsultimatestrengthdesign).
Thematmusthaveasufficientthickness,T,andreinforcementtosafelyresist
theseloads.Aswithspreadfootings,Tshouldbelargeenoughthatnoshearre­
inforcementisneeded.
Step2:Evaluatematdeformations(whichistheprimaryserviceabilityrequirement)
usingtheunfactoredloads(Equations2.1-2.4).Thesedeformationsarethere­
sultofconcentratedloadingatthecolumnlocations,possiblenon-uniformities
inthemat,andvariationsinthesoilstiffness.Ineffect,thesedeformationsare
theequivalentofdifferentialsettlement.Iftheyareexcessive,thenthematmust
bemadestifferbyincreasingitsthickness.
Closed-FormSolutions
Computethedesignk,values:
A,(k,)A+An(k,){j+Ac(k,)c=(A.+ AB+Ac)(k,),,,,
375(k,)A
+469(1.5)(k,),+656(2)(k,)A=1500(k,L"
2390(k,)A
=1500(k,),,,,
(k,)A=0.627(k,),,,g
Becauseitissodifficulttodevelopaccurate k,.values,itmaybeappropriatetocon­
ductaparametricstudiestoevaluateitseffectonthematdesign.ACI(1993)suggests
varyingk,fromone-halfthecomputedvaluetofiveortentimesthecomputedvalue,and
basingthestructuraldesignontheworstcasecondition.
Thiswiderangein
k,.valueswillproduceproportionalchangesinthecomputed
totalsettlement.However,weignorethesetotalsettlementcomputationsbecausetheyare
notreliableanyway,andcomputeitusingthemethodsdescribedinChapter7.These
changesin
k,.havemuchlessimpactontheshears,moments,anddeflectionsinthemat,
andthushaveonlyasmallimpactonthestructuraldesign.
(k,)A=(0.627)(4000kN/m-') =2510kN/m3
(k,)n=(1.,5)(0.627)(4000kN/m-') =3765kN/m3
(k,)c
=(2)(0.627)(4000kN/m') =5020kN/m3
0(=Answer
0(=Answer
0(=Answer
---1
WhentheWinklermethodisused(i.e.,whenall"springs"havethesame k,.)andthe
geometryoftheproblemcanberepresentedintwo-dimensions,itispossibletodevelop
closed-formsolutionsusingtheprinciplesofstructuralmechanics(Scott,1981;Hetenyi,
1974).Thesesolutionsproducevaluesofshear,moment,anddeflectionatallpointsin
theidealizedfoundation.Whentheloadingiscomplex,theprincipleofsuperpositionmay
beusedtodividetheproblemintomultiplesimplerproblems.
Theseclosed-formsolutionswereonceverypopular,becausetheyweretheonly
practicalmeansofsolvingthisproblem.However,theadventandwidespreadavailability
ofpowerfulcomputersandtheassociatedsoftwarenowallowsustouseothermethods
thataremorepreciseandmoreflexible.
FiniteElementMethod
Today,mostmatfoundationsaredesignedwiththeaidofacomputerusingthefiniteele­
mentmethod(FEM).Thismethoddividesthematintohundredsorperhapsthousandsof
elements,asshowninFigure10.12.Eachelementhascertaindefineddimensions,aspec­
ifiedstiffnessandstrength(whichmaybedefinedintermsofconcreteandsteelproper­
ties)andisconnectedtotheadjacentelementsinaspecifiedway.
Thematelementsareconnectedtothegroundthroughaseriesof"springs,"which
aredefinedusingthecoefficientofsubgradereaction.Typically,onespringislocatedat
eachcornerofeachelement.
Theloadsonthematincludetheexternallyappliedcolumnloads,appliedlineloads,
appliedarealoads,andtheweightofthematitself.Theseloadspressthematdownward,

8. 366 Chapter10Mats
10.6BearingCapacity 367
T.
-tYPlcalElement
10.6BEARINGCAPACITY
Becauseoftheirlargewidth,matfoundationsonsandsandgravelsdonothavebearing
capacityproblems.However,bearingcapacitymightbeimportantinsiltsandclays,espe­
ciallyifundrainedconditionsprevail.TheFargoGrainSilofailuredescribedinChapter6
isanotabl6exampleofabearingcapacityfailureinasaturatedclay.
WecanevaluatebearingcapacityusingtheanalysistechniquesdescribedinChap­
ter6.Itisgoodpracticetodesignthematsothebearingpressureatallpointsislessthan
theallowablebearingcapacity.
Plan
111ii111i1i1i"1'11
Prolile
Figure10.12UseofthefiniteelemeIllmethodtoanalyzematfoundations.Thematis
dividedintoaseriesofelementswhicharcconnectedinaspecified
IJ./ay.Theelements
areconnectedtothegroundthrougha"bedofsprings."
andthisdownwardmovementisresistedbythesoil"springs."Theseopposingforces,
alongwiththestiffnessofthemat,canbeevaluatedsimultaneouslyusingmatrixalgebra,
whichallowsustocomputethestresses,strains,anddistortionsinthemat.Iftheresultsof
theanalysisarenotacceptable,thedesignismodifiedaccordinglyandreanalyzed.
Thistypeoffiniteelementanalysisdoesnotconsiderthestiffnessofthesuperstruc­
ture.Inotherwords,itassumesthesuperstructureisperfectlyflexibleandoffersnoresis­
tancetodeformationsinthemat.Thisisconservative.
Thefiniteelementanalysiscanbeextendedtoincludethesuperstructure,themat,
andtheunderlyingsoilinasinglethree-dimensionalfiniteelementmodel.Thismethod
would,inprinciple,beamoreaccuratemodelofthesoil-structuresystem,andthusmay
produceamoreeconomicaldesign.However,suchanalysesaresubstantiallymorecom­
plexandtime-consuming,anditisverydifficulttodevelopaccuratesoilpropertiesforsuch
models.Therefore,theseextendedfiniteelementanalysesarerarelyperformedinpractice.
10.5TOTALSETTLEMENT
Thebedofspringsanalysesproduceacomputedtotalsettlement.However,thisvalueis
unreliableandshouldnotbeusedfordesign.Theseanalysesareusefulonlyforcomput­
ingshears,moments,anddeformations(differentialsettlements)inthemat.Totalsettle­
mentshouldbecomputedusingthemethodsdescribedinChapter7.
----.-.-tIii...
SUMMARY
MajorPoints
1.Matfoundationsareessentiallylargespreadfootingsthatusuallyencompasstheen­
tirefootprintofastructure.Theyareoftenanappropriatechoiceforstructuresthat
aretooheavyforspreadfootings.
2.Theanalysisanddesignofmatsmustincludeanevaluationoftheflexuralstresses
andmustprovidesufficientflexuralstrengthtoresistthesestresses.
3.Theoldestandsimplestmethodofanalyzingmatsistherigidmethod.Itassumes
thatthematismuchmorerigidthantheunderlyingsoil.whichmeansthemagnitude
anddistributionofbearingpressureiseasytodetermine.Thismeanstheshears,mo­
ment,anddeformationsinthematareeasilydetermined.However,thismethodis
notanaccuraterepresentationbecausetheassumptionofrigidityisnotcorrect.
4.Nonrigidanalysesaresuperiorbecausetheyconsidertheflexuraldeflectionsinthe
matandthecorrespondingredistributionofthesoilbearingpressure.
S.Nonrigidmethodsmustincludeadefinitionofsoil-structureinteraction.Thisis
usuallydoneusinga"bedofsprings"analogy,witheachspringhavingalinear
force-displacementfunctionasdefinedbythecoefficientofsubgradereaction,
k,.
6.ThesimplestandoldestnonrigidmethodistheWinklermethod,whichusesinde­
pendentsprings,allofwhichhavethesame
k,.Thismethodisanimprovementover
rigidanalyses,butstilldoesnotaccuratelymodelsoil-structureinteraction,primar­
ilybecauseitdoesnotconsidercouplingeffects.
7.ThecoupledmethodisanextensionoftheWinklermethodthatconsiderscoupling
betweenthesprings.
8.Thepseudo-coupledmethodusesindependentsprings,butadjuststhe
k,valuesto
implicitlyaccountforcouplingeffects.
9.Themultipleparameterandfiniteelementmethodsaremoreadvancedwaysofde­
scribingsoil-structureinteraction.
10.Thecoefficientofsubgradereactionisdifficulttodetermine.Fortunately,themat
designisoftennotoverlysensitivetoglobalchangesin
k,.Parametricstudiesare
oftenappropriate.

9. 368 Chapter10Mats
10.6BearingCapacity
369
11.
IftheWinklermethodisusedtodescribesoil-structureinteraction.andthemat
geometryisnottoocomplex,thestructuralanalysismaybeperformedusing
closed-formsolutions.However,thesemethodsaregenerallyconsideredobsolete.
12.Moststructuralanalysesareperformedusingnumericalmethods,especiallythefi­
niteelementmethod.Thismethodusesfiniteelementstomodelthemat,andde­
finessoil-structureinteractionusingtheWinkerorpseudo-coupledmodels.In
principle,italsocouldusethemultipleparametermodel.
13.Adesigncouldbebasedentirelyonathree-dimensionalfiniteelementanalysisthat
includesthesoil,mat,andsuperstructure.However,suchanalysesarebeyondcur­
rentpractices,mostlybecausetheyaredifficulttosetupandrequireespecially
powerfulcomputers.
14.ThetotalsettlementisbestdeterminedusingthemethodsdescribedinChapter7.
Donotusethecoefficientofsubgradereactiontodeterminetotalsettlement.
15.Bearingcapacityisnotaproblemwithsandsandgravels,butcanbeimportantin
siltsandclays.ItshouldbecheckedusingthemethodsdescribedinChapter6.
Vocabulary
10.5A25-mdiametercylindricalwaterstoragetankistobesupportedonamatfoundation.The
weightofthetankanditscontentswillbe50,000kNandtheweightofthematwillbe
12.000kN.Accordingtoasettlementanalysisconductedusingthetechniquesdescribedin
Chapter7.thetotalsettlementwillbe40mm.Thegroundwatertableisatadepthof5m
belowthebottomofthemat.Usingthepseudo-coupledmethod.dividethematintozonesand
computekforeachZOlle.Thenindicatethehigh-endandlow-endvaluesofk.thatshouldbe
usedintheanalysis.
10.6Anofficebuildingistobesupportedon150-ftx300-ftmatfoundation.Thesumofthecol­
umnloadsplustheweightofthematwillbe90.000k.Accordingtoasettlementanalysiscon­
ductedusingthetechniquesdescribedinChapter7.thetotalsettlementwillbe1.8inches.The
groundwatertableisatadepthof10ftbelowthebottomofthemal.Usingthepseudo­
coupledmethod.dividethematintozonesandcompositeeachzone.Thenindicatethehigh­
endandlow-endvalvesofk,thatshouldbeusedintheanalysis.
Beamonelasticfoundation
Bedofsprings
Coefficientofsubgrade
reaction
Coupledmethod
Finiteelementmethod
Matfoundation
Multipleparametermethod
Nonrigidmethod
Pile-supportedmat
Pseudo-coupledmethod
Raftfoundation
Rigidmethod
Shaft-supportedmat
Soil-structureinteraction
Winklermethod
COMPREHENSIVE QUESTIONS ANDPRACTICEPROBLEMS
10.1ExplainthereasoningbehindthestatementinSection10.6:"Becauseoftheirlargewidth,mat
foundationsonsandsandgravelsdonothavebearingcapacityproblems."
10.2Howhasthedevelopmentofpowerfulandinexpensivedigitalcomputersaffectedtheanalysis
anddesignofmatfoundations?Whatchangesdoyouexpectinthefutureasthistrendcon­
tinues?
10.3Amatfoundationsupportsfortyrwocolumnsforabuilding.Thesecolumnsarespacedona
uniformgridpattern.Howwouldthemomentsanddifferentialsettlementschangeifweused
anonrigidanalysiswithaconstantk,inlieuofarigidanalysis?
10.4AccordingtoasettlementanalysisconductedusingthetechniquesdescribedinChapter7,a
certainmatwillhaveatotalsettlementof2.1inchesiftheaveragebearingpressureis
5500lb/ft'.Computetheaveragek,andexpressyouranswerinunitsofIb/in.1.
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