Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

248 CHAPTER 7 Bending of Thin Plates

Thetermmultipliedby2( 1 −ν)integratestozero,andthemeanvalueofsin^2 orcos^2 overacomplete
numberofhalfwavesis^12 ;thus,integrationoftheprecedingexpressionyields

U+V=

D

2

∑∞

m=1,3,5

∑∞

n=1,3,5

A^2 mn

π^4 ab

4

(

m^2

a^2

+

n^2

b^2

) 2

(7.47)

−q 0

∑∞

m=1,3,5

∑∞

n=1,3,5

Amn

4 ab

π^2 mn

Fromtheprincipleofthestationaryvalueofthetotalpotentialenergy,wehave

∂(U+V)

∂Amn

=

D

2

2 Amn

π^4 ab

4

(

m^2

a^2

+

n^2

b^2

) 2

−q 0

4 ab

π^2 mn

= 0

sothat

Amn=

16 q 0

π^6 Dmn[(m^2 /a^2 )+(n^2 /b^2 )]^2

givingadeflectedform

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

whichistheresultobtainedinEq.(i)ofExample7.1.
Theprecedingsolutionisexactsinceweknowthetruedeflectedshapeoftheplateintheformofan
infiniteseriesforw.Frequently,theappropriateinfiniteseriesisnotknownsothatonlyanapproximate
solutionmaybeobtained.Themethodofsolution,knownastheRayleigh–Ritzmethod,involvesthe
selectionofaseriesforwcontainingafinitenumberoffunctionsofxandy.Thesefunctionsarechosen
tosatisfytheboundaryconditionsoftheproblemasfaraspossibleandalsotogivethetypeofdeflection
patternexpected.Naturally,themorerepresentativethe“guessed”functionsare,themoreaccuratethe
solutionbecomes.
Supposethatthe“guessed”seriesforwinaparticularproblemcontainsthreedifferentfunctionsof
xandy.Thus,

w=A 1 f 1 (x,y)+A 2 f 2 (x,y)+A 3 f 3 (x,y),

whereA 1 ,A 2 ,andA 3 areunknowncoefficients.Wenowsubstituteforwintheappropriateexpression
forthetotalpotentialenergyofthesystemandassignstationaryvalueswithrespecttoA 1 ,A 2 ,andA 3
inturn.Thus,

∂(U+V)

∂A 1

= 0

∂(U+V)

∂A 2

= 0

∂(U+V)

∂A 3

= 0

givingthreeequations,whicharesolvedforA 1 ,A 2 ,andA 3.