Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

238 CHAPTER 7 Bending of Thin Plates

Fig.7.13

Component of shear loads in thezdirection.

The determination of the contribution of the shear loads to the equilibrium of the element in the
zdirection is complicated by the fact that the element possesses curvature in bothxzandyzplanes.
Therefore,fromFig.7.13,thecomponentinthezdirectionduetotheNxyshearloadsonlyis

(

Nxy+

∂Nxy

∂x

δx

)

δy

(

∂w

∂y

+

∂^2 w

∂x∂y

δx

)

−Nxyδy

∂w

∂y

or

Nxy

∂^2 w

∂x∂y

δxδy+

∂Nxy

∂x

∂w

∂y

δxδy

neglectingtermsofalowerorder.Similarly,thecontributionofNyxis

Nyx

∂^2 w

∂x∂y

δxδy+

∂Nyx

∂y

∂w

∂x

δxδy

ThecomponentsarisingfromthedirectforcesperunitlengtharereadilyobtainedfromFig.7.12,
namely,
(
Nx+

∂Nx

∂x

δx

)

δy

(

∂w

∂x

+

∂^2 w

∂x^2

δx

)

−Nxδy

∂w

∂x

or

Nx

∂^2 w

∂x^2

δxδy+

∂Nx

∂x

∂w

∂x

δxδy

andsimilarly

Ny

∂^2 w

∂y^2

δxδy+

∂Ny

∂y

∂w

∂y

δxδy