CHAPTER 16 Shear of Beams..........................................................................
InChapter15,wedevelopedthetheoryforthebendingofbeamsbyconsideringsolidorthickbeam
sections and then extended the theory to the thin-walled beam sections typical of aircraft structural
components.Infact,itisonlyinthecalculationofsectionpropertiesthatthin-walledsectionssubjected
tobendingaredistinguishedfromsolidandthicksections.However,forthin-walledbeamssubjected
to shear, the theory is based on assumptions applicable only to thin-walled sections, so we shall not
consider solid and thick sections; the relevant theory for such sections may be found in any text on
structuralandstressanalysis[Ref.1].Therelationshipsbetweenbendingmoments,shearforces,and
loadintensitiesderivedinSection15.2.5stillapply.
16.1 GENERAL STRESS, STRAIN, AND DISPLACEMENT
RELATIONSHIPS FOR OPEN AND SINGLE CELL
CLOSED SECTION THIN-WALLED BEAMS
In this section, we shall establish the equations of equilibrium and expressions for strain which are
necessary for the analysis of open section beams supporting shear loads and closed section beams
carrying shear and torsional loads. The analysis of open section beams subjected to torsion requires
adifferentapproachandisdiscussedseparatelyinChapter17.Therelationshipsareestablishedfrom
firstprinciplesfortheparticularcaseofthin-walledsectionsinpreferencetotheadaptionofEqs.(1.6),
(1.27),and(1.28),whichrefertodifferentcoordinateaxes;theform,however,willbeseentobethe
same. Generally, in the analysis we assume that axial constraint effects are negligible that the shear
stressesnormaltothebeamsurfacemaybeneglected,sincetheyarezeroateachsurfaceandthewall
is thin, that direct and shear stresses on planes normal to the beam surface are constant across the
thickness,andfinallythatthebeamisofuniformsectionsothatthethicknessmayvarywithdistance
aroundeachsectionbutisconstantalongthebeam.Inaddition,weignoresquaresandhigherpowers
ofthethicknesstinthecalculationofsectionproperties(seeSection15.4.5).
Theparametersintheanalysisisdistancemeasuredaroundthecrosssectionfromsomeconvenient
origin.Anelementδs×δz×tofthebeamwallismaintainedinequilibriumbyasystemofdirectand
shearstressesasshowninFig.16.1(a).Thedirectstressσzisproducedbybendingmomentsorbythe
bending action of shear loads, whereas the shear stresses are due to shear and/or torsion of a closed
sectionbeamorshearofanopensectionbeam.Thehoopstressσsisusuallyzerobutmaybecaused,
Copyright©2010,T.H.G.Megson. PublishedbyElsevierLtd. Allrightsreserved.
DOI:10.1016/B978-1-85617-932-4.00016-6 479