Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.4 Combined Bending and In-Plane Loading of a Thin Rectangular Plate 237

Fig.7.12

In-plane forces on plate element.

onlytoincludethein-planeloadsshowninFig.7.12whenconsideringtheequilibriumoftheelement
inthesedirections.ForequilibriumparalleltoOx,
(
Nx+

∂Nx

∂x

δx

)

δycos

(

∂w

∂x

+

∂^2 w

∂x^2

δx

)

−Nxδycos

∂w

∂x

+

(

Nyx+

∂Nyx

∂y

δy

)

δx−Nyxδx= 0

Forsmalldeflections,∂w/∂xand(∂w/∂x)+(∂^2 w/∂x^2 )δxaresmall,andthecosinesoftheseangles
arethereforeapproximatelyequaltoone.Theequilibriumequationthussimplifiesto

∂Nx

∂x

+

∂Nyx

∂y

= 0 (7.31)

Similarly,forequilibriumintheydirection,wehave

∂Ny

∂y

+

∂Nxy

∂x

= 0 (7.32)

Notethatthecomponentsofthein-planeshearloadsperunitlengthare,toafirstorderofapproximation,
thevalueoftheshearloadmultipliedbytheprojectionoftheelementontherelevantaxis.