Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.2Plates Subjected to Bending and Twisting 225

for whichMt=0, leaving normal moments of intensityMnon two mutually perpendicular planes.
Thesemomentsaretermedprincipalmoments,andtheircorrespondingcurvaturesarecalledprincipal
curvatures.ForaplatesubjectedtopurebendingandtwistinginwhichMx,My,andMxyareinvariable
throughouttheplate,theprincipalmomentsarethealgebraicallygreatestandleastmomentsintheplate.
Itfollowsthattherearenoshearstressesontheseplanesandthatthecorrespondingdirectstresses,for
agivenvalueofzandmomentintensity,arethealgebraicallygreatestandleastvaluesofdirectstress
intheplate.
Let us now return to theloaded plateof Fig. 7.5(a). Wehaveestablished, in Eqs. (7.7) and (7.8),
therelationshipsbetweenthebendingmomentintensitiesMxandMyandthedeflectionwoftheplate.
ThenextstepistorelatethetwistingmomentMxytow.Fromtheprincipleofsuperposition,wemay
considerMxyactingseparatelyfromMxandMy.Asstatedpreviously,Mxyisresistedbyasystemof
horizontalcomplementaryshearstressesontheverticalfacesofsectionstakenthroughoutthethickness
oftheplateparalleltothexandyaxes.Consideranelementoftheplateformedbysuchsections,as
shown in Fig. 7.6. The complementary shear stresses on a lamina of the element a distancezbelow
theneutralplaneare,inaccordancewiththesignconventionofSection1.2,τxy.Therefore,ontheface
ABCD

Mxyδy=−

∫t/^2

−t/ 2

τxyδyzdz

andonthefaceADFE

Mxyδx=−

∫t/^2

−t/ 2

τxyδxzdz

giving

Mxy=−

∫t/^2

−t/ 2

τxyzdz

orintermsoftheshearstrainγxyandmodulusofrigidityG

Mxy=−G

∫t/^2

−t/ 2

γxyzdz (7.12)

ReferringtoEqs.(1.20),theshearstrainγxyisgivenby

γxy=

∂v

∂x

+

∂u
∂y
Werequire,ofcourse,toexpressγxyintermsofthedeflectionwoftheplate;thismaybeaccomplished
asfollows.Anelementtakenthroughthethicknessoftheplatewillsufferrotationsequalto∂w/∂xand