(1.5.4)-2.4 Statically Indeterminate Str材料力学材料力学

Chapter2AxiallyLoadedMembers

MechanicsofMaterialsStaticallyIndeterminateStructuresWelcometomechanicsofmaterials,inthissection,wearegoingtodiscussstaticallyindeterminatestructres.StaticallyIndeterminateStructuresStaticallyindeterminatebarStaticallydeterminatebarStaticallydeterminatestructures:

Thereactionsandinternalforcescanbedeterminedsolelyfromequationsofequilibrium.Staticallyindeterminatestructures:Thereactionsandinternalforcescannotbefoundbystaticsalone.Thesprings,bars*,andcablescoveredintheprecedingsectionshaveoneimportantfeatureincommon—theirreactionsandinternalforcescanbedeterminedsolelyfromequationsofequilibrium.Structuresofthistype*areclassifiedasstaticallydeterminate.However,forstructures,likethisone*,therearenowtwoverticalreactions,RAandRB,butonlyoneusefulequationofequilibriumintheverticaldirection.Itisnotsufficientforfindingallthereactionssolyfromequationsofequilibrium,stucturesofthiskind*areclassifiedasstaticallyindeterminate.Toanalyzesuchstructures,wemustsupplementtheequilibriumequationswithadditionalequationspertainingtothedisplacementsofthestructure.EquationofcompatibilityEquationofequilibrium:Equationofcompatibility：Force-displacementrelations:AFDADDToseehowastaticallyindeterminatestructureisanalyzed,considerthisexample*.Forthisstructure,onlyoneequationofequilibriumisavailable*.TherearetwounknownsRaandRb.Anadditionalequationisneeded.Basedupontheobservationthatabarwithbothendsfixeddoesnotchangeinlength,thatis*δAB=0,thisequationservesastheadditionalequation.whichiscalledanequationofcompatibility*.Inordertosolvethesetwoequations,introducetheforce-displacementrelations*,andexpressthecompatibilityequationintermsoftheunknownforcesRAandRB.Nowsolvesimultaneously*theequilibriumequation,thecompatibilityequation,andtheforce-displacementrelations,thetwounknownscanbefound*.Withallthereactions*known,internalaxialforces*,stresses*anddisplacementquantities*canbedetermined.

Example:Findexpressionsforallsupportreactionforcesintheplaneframe.MemberACisaflatprismaticbaroflength

L,width

b,andthicknesst.ColumnDBFhasconstant

thicknesstandtaperslinearlyfromwidthb

towidth5/4b.E

isconstant

fortheframe.Unkowns：

Ax,Ay;

Bx,By;

Fx,Fy;

RDDegreeofstaticindeterminacy=1Equationsofequilibrium:ThebeamAC:Thecompatibilityequation:ThecolumnDBF:Now,let’slookatthisexample,tofindallthereactionforcesinthisframe.First,drawthefreebodydiagram*foreachcomponentinthisframe.Ascanbeseen,thereare7*unknownforces.Andthereare6equationsofequilibriumintotal,3foreachcomponent.So,thisplaneframeisone-degree*staticallyIndeterminate.Oneadditionalequation*isneeded.Then,writeallequilibriumequations*basedonthefreebodydiagrams.ForthecolumnDBF,summingmoments*aboutpointF,givesforceBx*.summingforces*alongxdirection,givesFx*.summingforces*alongydirection,gives*.AndforthememberAC,summingforces*alongxdirection,givesAx*,summingmoments*aboutpointA,givesBy*,andsummingforces*alongy,givesAy*.Now,onlytwoforcesareleftasunknows,RdandFy.Next,wearegoingtodeterminethesetwoforcescombiningthecompatibilityequation.Force-dispalcementrelation:Again,introducetheforce-displacementrelation*,inordertoexpressδDintermsofunknownforces.Thecolumncanbeseenthelowerpartandupperpart,therefore*.TheaxialforceforthelowerpartisBy-RD,whichiscompressive,andfortheupperpartispositiveRD.Substituting3PforBy,thisexpressionissimplified