Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

222 CHAPTER 7 Bending of Thin Plates

Ifwisthedeflectionofanypointontheplateinthezdirection,thenwemayrelatewtothecurvature
oftheplateinthesamemannerasthewell-knownexpressionforbeamcurvature.Hence

1

ρx

=−

∂^2 w

∂x^2

1

ρy

=−

∂^2 w

∂y^2

thenegativesignsresultingfromthefactthatthecentersofcurvatureoccurabovetheplateinwhich
regionzisnegative.Equations(7.5)and(7.6)thenbecome

Mx=−D

(

∂^2 w

∂x^2

+ν

∂^2 w

∂y^2

)

(7.7)

My=−D

(

∂^2 w

∂y^2

+ν

∂^2 w

∂x^2

)

(7.8)

Equations(7.7)and(7.8)definethedeflectedshapeoftheplateprovidedthatMxandMyareknown.If
eitherMxorMyiszero,then

∂^2 w

∂x^2

=−ν

∂^2 w

∂y^2

or

∂^2 w

∂y^2

=−ν

∂^2 w

∂x^2

andtheplatehascurvaturesofoppositesigns.ThecaseofMy=0isillustratedinFig.7.3.Asurface
possessingtwocurvaturesofoppositesignisknownasananticlasticsurface,asopposedtoasynclastic
surface,whichhascurvaturesofthesamesign.Further,ifMx=My=M,thenfromEqs.(7.5)and(7.6)

1

ρx

=

1

ρy

=

1

ρ

Therefore,thedeformedshapeoftheplateissphericalandofcurvature

1

ρ

=

M

D( 1 +ν)

(7.9)

Fig.7.3

Anticlastic bending.