424 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams
Fig.15.1
Bending of a rubber eraser.
15.1 SymmetricalBending................................................................................
Althoughsymmetricalbendingisaspecialcaseofthebendingofbeamsofarbitrarycrosssection,we
shallinvestigatetheformerfirstsothatthemorecomplexgeneralcasemaybemoreeasilyunderstood.
Symmetricalbendingarisesinbeamswhichhaveeithersinglyordoublysymmetricalcrosssections;
examplesofbothtypesareshowninFig.15.2.Supposethatalengthofbeam,ofrectangularcrosssection,
say,issubjectedtoapure,saggingbendingmoment,M,appliedinaverticalplane.Weshalldefinethis
laterasanegativebendingmoment.ThelengthofbeamwillbendintotheshapeshowninFig.15.3(a)
inwhichtheuppersurfaceisconcaveandthelowerconvex.Itcanbeseenthattheupperlongitudinal
fibersofthebeamarecompressed,whilethelowerfibersarestretched.Itfollowsthat,asinthecaseof
theeraser,betweenthesetwoextremestherearefibersthatremainunchangedinlength.
Thedirectstressthereforevariesthroughthedepthofthebeamfromcompressionintheupperfibers
totensioninthelower.Clearly,thedirectstressiszeroforthefibersthatdonotchangeinlength;we
havecalledtheplanecontainingthesefiberstheneutralplane.Thelineofintersectionoftheneutral
planeandanycrosssectionofthebeamistermedtheneutralaxis(Fig.15.3(b)).
Theproblem,therefore,istodeterminethevariationofdirectstressthroughthedepthofthebeam,
thevaluesofthestresses,andsubsequentlytofindthecorrespondingbeamdeflection.
15.1.1 Assumptions
Theprimaryassumptionmadeindeterminingthedirectstressdistributionproducedbypurebending
isthatplanecrosssectionsofthebeamremainplaneandnormaltothelongitudinalfibersofthebeam
afterbending.Again,wesawthisfromthelinesonthesideoftheeraser.Weshallalsoassumethatthe
materialofthebeamislinearlyelastic—thatis,itobeysHooke’slawandthatthematerialofthebeam
ishomogeneous.