CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams...................................
InChapter11,wediscussedthevarioustypesofstructuralcomponentsfoundinaircraftconstruction
andthevariousloadstheysupport.Wesawthatanaircraftisbasicallyanassemblyofstiffenedshell
structuresrangingfromthesingle-cellclosedsectionfuselagetomulticellularwingsandtailsurfaces,
each subjected to bending, shear, torsional, and axial loads. Other, smaller portions of the structure
consist of thin-walled channel, T-, Z-, “top-hat”-, or I-sections, which are used to stiffen the thin
skinsofthecellularcomponentsandprovidesupportforinternalloadsfromfloors,enginemountings,
andsoforth.Structuralmemberssuchastheseareknownasopensectionbeams,whereasthecellular
componentsaretermedclosedsectionbeams;clearly,bothtypesofbeamaresubjectedtoaxial,bending,
shear,andtorsionalloads.Inthischapter,weshallinvestigatethestressesanddisplacementsinthin-
walledopenandsingle-cellclosedsectionbeamsproducedbybendingloads.
InChapter1,wesawthatanaxialloadappliedtoamemberproducesauniformdirectstressacross
the cross section of the member. A different situation arises when the applied loads cause a beam to
bend which, if the loads are vertical, will take up a sagging () or hogging shape (). This means
thatforloadswhichcauseabeamtosagtheuppersurfaceofthebeammustbeshorterthanthelower
surface,astheuppersurfacebecomesconcaveandtheloweroneconvex;thereverseistrueforloads
which cause hogging. The strains in the upper regions of the beam will, therefore, be different from
thoseinthelowerregions,andsincewehaveestablishedthatstressisdirectlyproportionaltostrain
(Eq.(1.40)),itfollowsthatthestresswillvarythroughthedepthofthebeam.
Thetruthofthiscanbedemonstratedbyasimpleexperiment.Takeareasonablylongrectangular
rubbereraseranddrawthreeorfourlinesonitslongerfacesasshowninFig.15.1(a);thereasonfor
thiswillbecomeclearalittlelater.Nowholdtheeraserbetweenthethumbandforefingerateachend
andapplypressureasshownbythedirectionofthearrowsinFig.15.1(b).Theeraserbendsintothe
shapeshown,andthelinesonthesideoftheeraserremainstraightbutarenowfartherapartatthetop
thanatthebottom.
Since,inFig.15.1(b),theupperfibershavebeenstretchedandthelowerfiberscompressed,there
willbefiberssomewhereinbetweenwhichareneitherstretchednorcompressed;theplanecontaining
thesefibersiscalledtheneutralplane.
Now rotate the eraser so that its shorter sides are vertical and apply the same pressure with your
fingers. The eraser again bends but now requires much less effort. It follows that the geometry and
orientationofabeamsectionmustaffectitsbendingstiffness.Thisismorereadilydemonstratedwith
aplasticruler.Whenflatitrequireshardlyanyefforttobendit,butwhenheldwithitswidthvertical,
itbecomesalmostimpossibletobend.
Copyright©2010,T.H.G.Megson. PublishedbyElsevierLtd. Allrightsreserved.
DOI:10.1016/B978-1-85617-932-4.00015-4 423