Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.3 Plates Subjected to a Distributed Transverse Load 233

inwhichmrepresentsthenumberofhalfwavesinthexdirectionandnrepresentsthecorresponding
number in theydirection. Further,Amnare unknown coefficients, which must satisfy the preceding
differentialequationandmaybedeterminedasfollows.
Wemayalsorepresenttheloadq(x,y)byaFourierseries;thus,

q(x,y)=

∑∞

m= 1

∑∞

n= 1

amnsin

mπx

a

sin

nπy

b

(7.28)

Aparticularcoefficientam′n′iscalculatedbyfirstmultiplyingbothsidesofEq.(7.28)bysin(m′πx/a)
sin(n′πy/b)andintegratingwithrespecttoxfrom0toaandwithrespecttoyfrom0tob.Thus,

∫a

0

∫b

0

q(x,y)sin

m′πx

a

sin

n′πy

b

dxdy

=

∑∞

m= 1

∑∞

n= 1

∫a

0

∫b

0

amnsin

mπx

a

sin

m′πx

a

sin

nπy

b

sin

n′πy

b

dxdy

=

ab

4

am′n′

since

∫a

0

sin

mπx

a

sin

m′πx

a

dx=0whenm

=m′

=

a

2

when m=m′

and

∫b

0

sin

nπy

b

sin

n′πy

b

dy=0whenn

=n′

=

b

2

when n=n′

Itfollowsthat

am′n′=

4

ab

∫a

0

∫b

0

q(x,y)sin

m′πx

a

sin

n′πy

b

dxdy (7.29)

Substitutingnowforwandq(x,y)fromEqs.(7.27)and(7.28)intothedifferentialequationforw,we
have

∑∞

m= 1

∑∞

n= 1

{

Amn

[(

mπ

a

) 4

+ 2

(mπ

a

) 2 (nπ

b

) 2

+

(nπ

b

) 4 ]

−

amn

D

}

sin

mπx

a

sin

nπy

b

= 0