Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

242 CHAPTER 7 Bending of Thin Plates

Fig.7.14

(a) Strain energy of element due to bending; (b) strain energy due to twisting.

weareonlyneglectingthecontributionsofshearanddirectstrainsonthedeflectionoftheplate;the
stressesproducingthemmustnotbeignored.
Considertheelementδx×δyofathinplatea×bshowninelevationinthexzplaneinFig.7.14(a).
BendingmomentsMxperunitlengthappliedtoitsδyedgeproduceachangeinslopebetweenitsends
equalto(∂^2 w/∂x^2 )δx.However,sinceweregardthemomentsMxaspositiveinthesenseshown,then
thischangeinslope,orrelativerotation,oftheendsoftheelementisnegativeastheslopedecreases
withincreasingx.ThebendingstrainenergyduetoMxisthen

1

2

Mxδy

(

−

∂^2 w

∂x^2

δx

)

Similarly,intheyzplanethecontributionofMytothebendingstrainenergyis

1

2

Myδx

(

−

∂^2 w

∂y^2

δy

)

The strain energy due to the twisting moment per unit length,Mxy,appliedtotheδyedges of the
element,isobtainedfromFig.7.14(b).Therelativerotationoftheδyedgesis(∂^2 w/∂x∂y)δxsothatthe
correspondingstrainenergyis

1

2

Mxyδy

∂^2 w

∂x∂y

δx

Finally,thecontributionofthetwistingmomentMxyontheδxedgesis,inasimilarfashion,

1

2

Mxyδx

∂^2 w

∂x∂y

δy