Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

236 CHAPTER 7 Bending of Thin Plates

Differentiatingthedeflectionfunctiongives

∂^4 w

∂x^4

= 0

∂^4 w

∂y^4

= 0

∂^4 w

∂x^2 ∂y^2

= 4 A

SubstitutinginEq.(7.20),wehave

0 + 2 × 4 A+ 0 =constant=

q

D

Thedeflectionfunctionisthereforevalidand

A=

q

8 D

ThebendingmomentdistributionsaregivenbyEqs.(7.7)and(7.8);thatis,

Mx=−

q

4

[y^2 −by+ν(x^2 −ax)](i)

My=−

q

4

[x^2 −ax+ν(y^2 −by)](ii)

Fortheedgesx=0andx=a,

Mx=−

q

4

(y^2 −by) My=−

νq

4

(y^2 −by)

Fortheedgesy=0andy=b,

Mx=−

νq

4

(x^2 −ax) My=−

q

4

(x^2 −ax)

7.4 CombinedBendingandIn-PlaneLoadingofaThinRectangularPlate.......................

So far our discussion has been limited to small deflections of thin plates produced by different
formsoftransverseloading.Inthesecases,weassumedthatthemiddleorneutralplaneoftheplate
remainedunstressed.Additionalin-planetensile,compressive,orshearloadswillproducestressesin
the middle plane, and these, if of sufficient magnitude, will affect the bending of the plate. Where
the in-plane stresses are small compared with the critical buckling stresses, it is sufficient to con-
sider the two systems separately; the total stresses are then obtained by superposition. On the other
hand, if the in-plane stresses are not small, then their effect on the bending of the plate must be
considered.
The elevation and plan of a small elementδxδyof the middle plane of a thin deflected plate are
showninFig.7.12.Directandshearforcesperunitlengthproducedbythein-planeloadsaregiventhe
notationNx,Ny,andNxyandareassumedtobeactinginpositivesensesinthedirectionsshown.Since
therearenoresultantforcesinthexorydirectionsfromthetransverseloads(seeFig.7.9),weneed