466 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams
Fig.15.35
Distribution of direct stress inZ-section beam of Example 15.14.
Thedistributioninthelowerflangemaybededucedfromantisymmetry;thecompletedistributionis
thenasshowninFig.15.35.
15.5 ApplicabilityofBendingTheory...................................................................
Theexpressionsfordirectstressanddisplacementderivedintheabovetheoryarebasedontheassump-
tionsthatthebeamisofuniform,homogeneouscrosssectionandthatplanesectionsremainplaneafter
bending. The latter assumption is strictly true only if the bending momentsMxandMyare constant
alongthebeam.Variationofbendingmomentimpliesthepresenceofshearloads,andtheeffectofthese
istodeformthebeamsectionintoashallow,inverted“s”(seeSection2.6).However,shearstressesin
beamswhosecross-sectionaldimensionsaresmallinrelationtotheirlengthsarecomparativelylowso
thatthebasictheoryofbendingmaybeusedwithreasonableaccuracy.
Inthin-walledsections,shearstressesproducedbyshearloadsarenotsmallandmustbecalculated,
although the direct stresses may still be obtained from the basic theory of bending so long as axial
constraintstressesareabsent.Deflectionsinthin-walledstructuresareassumedtoresultprimarilyfrom
bendingstrains;thecontributionofshearstrainsmaybecalculatedseparatelyifrequired.
15.6 TemperatureEffects.................................................................................
InSection1.15.1,weconsideredtheeffectoftemperaturechangeonstress–strainrelationships,whereas
in Section 5.11, we examined the effect of a simple temperature gradient on a cantilever beam of
rectangularcrosssectionusinganenergyapproach.However,aswehaveseen,beamsectionsinaircraft
structures are generally thin walled and do not necessarily have axes of symmetry. We shall now
investigatehowtheeffectsoftemperatureonsuchsectionsmaybedetermined.