Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.6 Energy Method for the Bending of Thin Plates 247

ItfollowsimmediatelythatthepotentialenergyoftheNxyloadsis

Vxy=−

1

2

∫a

0

∫b

0

2 Nxy

∂w

∂x

∂w

∂y

dxdy (7.44)

andforthecompletein-planeloadingsystemwehave,fromEqs.(7.42),(7.43),and(7.44),apotential
energyof

V=−

1

2

∫a

0

∫b

0

[

Nx

(

∂w

∂x

) 2

+Ny

(

∂w

∂y

) 2

+ 2 Nxy

∂w

∂x

∂w

∂y

]

dxdy (7.45)

Wearenowinapositiontosolveawiderangeofthin-plateproblemsprovidedthatthedeflectionsare
small,obtainingexactsolutionsifthedeflectedformisknownorapproximatesolutionsifthedeflected
shapehastobe“guessed.”
ConsideringtherectangularplateofSection7.3,simplysupportedalongallfouredgesandsubjected
toauniformlydistributedtransverseloadofintensityq 0 ,weknowthatitsdeflectedshapeisgivenby
Eq.(7.27),namely,

w=

∑∞

m= 1

∑∞

n= 1

Amnsin

mπx

a

sin

nπy

b

Thetotalpotentialenergyoftheplateis,fromEqs.(7.37)and(7.39),

U+V=

∫a

0

∫b

0

{

D

2

[(

∂^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 }]

−wq 0

}

dxdy

(7.46)

SubstitutinginEq.(7.46)forwandrealizingthat“cross-product”termsintegratetozero,wehave

U+V=

∫a

0

∫b

0

{

D

2

∑∞

m= 1

∑∞

n= 1

A^2 mn

[

π^4

(

m^2

a^2

+

n^2

b^2

) 2

sin^2

mπx

a

sin^2

nπy

b

− 2 ( 1 −ν)

m^2 n^2 π^4

a^2 b^2

(

sin^2

mπx

a

sin^2

nπy

b

−cos^2

mπx

a

cos^2

nπy

b

)]

−q 0

∑∞

m= 1

∑∞

n= 1

Amnsin

mπx

a

sin

nπy

b

}

dxdy