Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

294 CHAPTER 9 Thin Plates

Fig.9.1

Buckling of a thin flat plate.

andthatthesearesmallcomparedwiththethicknessoftheplate.Theserestrictionsthereforeapplyin
thesubsequenttheory.
First, we consider the relatively simple case of the thin plate of Fig. 9.1, loaded as shown, but
simplysupportedalongallfouredges.WehaveseeninChapter7thatitstruedeflectedshapemaybe
representedbytheinfinitedoubletrigonometricalseries

w=

∑∞

m= 1

∑∞

n= 1

Amnsin

mπx

a

sin

nπy

b

Also,thetotalpotentialenergyoftheplateis,fromEqs.(7.37)and(7.45),

U+V=

1

2

∫a

0

∫b

0

[

D

{(

∂^2 w

∂x^2

+

∂^2 w

∂y^2

) 2

− 2 ( 1 −ν)

[

∂^2 w

∂x^2

∂^2 w

∂y^2

−

(

∂^2 w

∂x∂y

) 2 ]}

−Nx

(

∂w

∂x

) 2 ]

dxdy

(9.1)

TheintegrationofEq.(9.1)onsubstitutingforwissimilartothoseintegrationscarriedoutinChapter7.
Thus,bycomparingwithEq.(7.47),

U+V=

π^4 abD

8

∑∞

m= 1

∑∞

n= 1

A^2 mn

(

m^2

a^2

+

n^2

b^2

)

−

π^2 b

8 a

Nx

∑∞

m= 1

∑∞

n= 1

m^2 A^2 mn (9.2)

Thetotalpotentialenergyoftheplatehasastationaryvalueintheneutralequilibriumofitsbuckled
state(i.e.,Nx=Nx,CR).Therefore,differentiatingEq.(9.2)withrespecttoeachunknowncoefficient
Amn,wehave

∂(U+V)

∂Amn

=

π^4 abD

4

Amn

(

m^2

a^2

+

n^2

b^2

) 2

−

π^2 b

4 a

Nx,CRm^2 Amn= 0