光波导系统 - 轴向负载

AxialLoad4.1Saint-Venant’sPrincipleLocalizedDeformation

*Deformationislocalized.*Farfromtheload,thestressisuniform,

=P/A.

*Basedonexperiments,ageneralruletodeterminethedistance:

D=largestdimensionofthecrosssectionSaint-Venant’sPrincipleIfloadisreplacedbyastaticallyequivalentload,onlylocalizedregionisaffected.AtadistancelargerthanD,thestressandstraindistributionsremainthesame.Fortwodifferentloadings,thestressesatsectionc-carethesame.Arubbermembraneisstretched.Thelocalizeddistortionisonlyatthegrips.4.2ElasticDeformationofanAxiallyLoadedMemberGeneralFormulusConsidergeneralloadingsandchangingcross-sectionAtlocationx,cutasmallsegmentdx.TheinternalforceisP(x),andtheextensionisd

.Totalextension:*Procedure:Externalforcesinternalforcestressstraindeformation*CompressiveinternalforceP(x)<0,

,

,

<0.ConstantLoadandCross-SectionAreaPiecewiseConstantLoadandCross-SectionAreaSignConvention*>0:tensilestress;

<0:compressivestress*>0:elongation;

<0:contractionDividethebarintothreesegments:AB,BC,CDCutatmiddleofeachsegment,drawFBD,andfindinternalforce.

Internalforcediagram4.3PrincipleofSuperpositionSeveralexternalloads:P1,P2,……,Pn.Applyeachloadindividually;

P1

1,

1,d1

P2

2,

2,d2

…………… Pn

n,

n,dn

Thetotalsolutionifallloadsareapplied:

stress:

=

1+

2+……+

n

strain:

=

1+

2+……+

n

deformation:d

=d1+d2+……+dn

*SuperpositionprincipleisapplicableonlyforcasesinwhichthedeformationissmallandHooke’slawcanbeapplied.*Forlargedeformation,superpositionprincipleisnotapplicable.4.4StaticallyIndeterminateAxiallyLoadedMemberConceptofStaticallyIndeterminacy*Equilibriumequation:FA+FB=P*1equation<2unknownsFAandFBcannotbesolved.*Staticallyindeterminacy:#ofequations<#ofunknowns*Staticallydeterminacy:#ofequations=#ofunknownsCompatibility(Kinematic)Condition*UppersegmentAC:internalforceFA,deformation

AC

*LowersegmentCB:internalforceFB,deformation

CB

*Compatibilitycondition:FAandFBarethensolvedas*Compatibilityconditions:fromload-deformationrelationship*Solveindeterminateproblem:

equilibriumequations compatibilityconditions4.5TheForceMethodofAnalysisforAxiallyLoadedMember(skip)4.6ThermalStress*Materialsexpandwhentemperatureincreases.Materialscontractwhentemperaturedecreases.*Forahomogeneousandisotropicmaterial,

T=LT

T-changeinlength,L–originallength

T–temperaturechange

-linearcoefficientofthermalexpansionunit:SIsystem1/C°,FPSorIPS:1/F°4.7StressConcentrationStressConcentrationStressconcentrationoccursat(i)locationwherecrosssectionchanges,or(ii)locationwhereloadisapplied.ReductionofStressConcentration*Avoidabruptand/or