CHAPTER 7 Bending of Thin Plates..................................................................
Generally, we define a thin plate as a sheet of material whose thickness is small compared with its
other dimensions but which is capable of resisting bending in addition to membrane forces. Such
a plate forms a basic part of an aircraft structure, being, for example, the area of stressed skin
bounded by adjacent stringers and ribs in a wing structure or by adjacent stringers and frames in a
fuselage.
In this chapter, we shall investigate the effect of a variety of loading and support conditions on
the small deflection of rectangular plates. Two approaches are presented: an “exact” theory based
on the solution of a differential equation and an energy method relying on the principle of the sta-
tionary value of the total potential energy of the plate and its applied loading. The latter theory
will subsequently be used in Chapter 9 to determine the buckling loads for unstiffened and stiffened
panels.
7.1 PureBendingofThinPlates........................................................................
The thin rectangular plate of Fig. 7.1 is subjected to pure bending moments of intensityMxandMy
perunitlengthuniformlydistributedalongitsedges.Theformerbendingmomentisappliedalongthe
edgesparalleltotheyaxis,andthelatteralongtheedgesparalleltothexaxis.Weshallassumethat
thesebendingmomentsarepositivewhentheyproducecompressionattheuppersurfaceoftheplate
andtensionatthelower.
Ifwefurtherassumethatthedisplacementoftheplateinadirectionparalleltothezaxisissmall
compared with its thicknesstand that sections which are plane before bending remain plane after
bending, then, as in the case of simple beam theory, the middle plane of the plate does not deform
duringthebendingandisthereforeaneutralplane.Wetaketheneutralplaneasthereferenceplanefor
oursystemofaxes.
Letusconsideranelementoftheplateofsideδxδyandhavingadepthequaltothethicknesstof
theplateasshowninFig.7.2(a).Supposethattheradiiofcurvatureoftheneutralplanenareρxand
ρyinthexzandyzplanes,respectively(Fig.7.2(b)).Positivecurvatureoftheplatecorrespondstothe
positivebendingmoments,whichproducedisplacementsinthepositivedirectionofthezordownward
axis. Again, as in simple beam theory, the direct strainsεxandεycorresponding to direct stresses
Copyright©2010,T.H.G.Megson. PublishedbyElsevierLtd. Allrightsreserved.
DOI:10.1016/B978-1-85617-932-4.00007-5 219