Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

226 CHAPTER 7 Bending of Thin Plates

Fig.7.7

Determination of shear strainγxy.

∂w/∂yinthexzandyzplanes,respectively.Consideringtherotationofsuchanelementinthexzplane,
asshowninFig.7.7,weseethatthedisplacementuinthexdirectionofapointadistancezbelowthe
neutralplaneis

u=−

∂w

∂x

z

Similarly,thedisplacementvintheydirectionis

v=−

∂w

∂y

z

Hence,substitutingforuandvintheexpressionforγxy,wehave

γxy=− 2 z

∂^2 w

∂x∂y

(7.13)

fromwhichEq.(7.12)

Mxy=G

∫t/^2

−t/ 2

2 z^2

∂^2 w

∂x∂y

dz

or

Mxy=

Gt^3

6

∂^2 w

∂x∂y

ReplacingGbytheexpressionE/ 2 ( 1 +ν)establishedinEq.(1.50)gives

Mxy=

Et^3

12 ( 1 +ν)

∂^2 w

∂x∂y