Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.6 Energy Method for the Bending of Thin Plates 249

Example 7.4
Arectangularplatea×b,issimplysupportedalongeachedgeandcarriesauniformlydistributedload
ofintensityq 0 .Assumingadeflectedshapegivenby

w=A 11 sin

πx

a

sin

πy

b

determinethevalueofthecoefficientA 11 and,hence,findthemaximumvalueofdeflection.

The expression satisfies the boundary conditions of zero deflection and zero curvature (i.e., zero
bendingmoment)alongeachedgeoftheplate.SubstitutingforwinEq.(7.46),wehave

U+V=

∫a

0

∫b

0

[

DA^211

2

{

π^4

(a^2 b^2 )^2

(a^2 +b^2 )^2 sin^2

πx

a

sin^2

πy

b

− 2 ( 1 −ν)

×

[

π^4

a^2 b^2

sin^2

πx

a

sin^2

πy

b

−

π^4

a^2 b^2

cos^2

πx

a

cos^2

πy

b

]}

−q 0 A 11 sin

πx

a

sin

πy

b

]

dxdy

fromwhich

U+V=

DA^211

2

π^4

4 a^3 b^3

(a^2 +b^2 )^2 −q 0 A 11

4 ab

π^2

sothat

∂(U+V)

∂A 11

=

DA 11 π^4

4 a^3 b^3

(a^2 +b^2 )^2 −q 0

4 ab

π^2

= 0

and

A 11 =

16 q 0 a^4 b^4

π^6 D(a^2 +b^2 )^2

giving

w=

16 q 0 a^4 b^4

π^6 D(a^2 +b^2 )^2

sin

πx

a

sin

πy

b

Atthecenteroftheplate,wisamaximumand

wmax=

16 q 0 a^4 b^4

π^6 D(a^2 +b^2 )^2

Forasquareplateandassumingν=0.3,

wmax=0.0455q 0

a^4

Et^3

whichcomparesfavorablywiththeresultofExample7.1.