Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

234 CHAPTER 7 Bending of Thin Plates

Thisequationisvalidforallvaluesofxandysothat

Amn

[(

mπ

a

) 4

+ 2

(mπ

a

) 2 (nπ

b

) 2

+

(nπ

b

) 4 ]

−

amn

D

= 0

orinalternativeform

Amnπ^4

(

m^2

a^2

+

n^2

b^2

) 2

−

amn

D

= 0

giving

Amn=

1

π^4 D

amn

[(m^2 /a^2 )+(n^2 /b^2 )]^2

Hence,

w=

1

π^4 D

∑∞

m= 1

∑∞

n= 1

amn

[(m^2 /a^2 )+(n^2 /b^2 )]^2

sin

mπx

a

sin

nπy

b

(7.30)

inwhichamnisobtainedfromEq.(7.29).Equation(7.30)isthegeneralsolutionforathinrectangular
plateunderatransverseloadq(x,y).

Example 7.1
Athinrectangularplatea×bissimplysupportedalongitsedgesandcarriesauniformlydistributedload
ofintensityq 0 .Determinethedeflectedformoftheplateandthedistributionofbendingmoment.

Sinceq(x,y)=q 0 ,wefindfromEq.(7.29)that

amn=

4 q 0

ab

∫a

0

∫b

0

sin

mπx

a

sin

nπy

b

dxdy=

16 q 0

π^2 mn

,

wheremandnareoddintegers.Formorneven,amn=0.Hence,fromEq.(7.30)

w=

16 q 0

π^6 D

∑∞

m=1,3,5

∑∞

n=1,3,5

sin(mπx/a)sin(nπy/b)

mn[(m^2 /a^2 )+(n^2 /b^2 )]^2

(i)

Themaximumdeflectionoccursatthecenteroftheplate,wherex=a/2,y=b/2.Thus,

wmax=

16 q 0

π^6 D

∑∞

m=1,3,5

∑∞

n=1,3,5

sin(mπ/ 2 )sin(nπ/ 2 )

mn[(m^2 /a^2 )+(n^2 /b^2 )]^2

(ii)

Thisseriesisfoundtoconvergerapidly,thefirstfewtermsgivingasatisfactoryanswer.Forasquare
plate,takingν=0.3,summationofthefirstfourtermsoftheseriesgives

wmax=0.0443q 0

a^4

Et^3