Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.3 Plates Subjected to a Distributed Transverse Load 235

SubstitutionforwfromEq.(i)intotheexpressionsforbendingmoment,Eqs.(7.7)and(7.8),yields

Mx=

16 q 0

π^4

∑∞

m=1,3,5

∑∞

n=1,3,5

[(m^2 /a^2 )+ν(n^2 /b^2 )]

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

sin

mπx

a

sin

nπy

b

(iii)

My=

16 q 0

π^4

∑∞

m=1,3,5

∑∞

n=1,3,5

[ν(m^2 /a^2 )+(n^2 /b^2 )]

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

sin

mπx

a

sin

nπy

b

(iv)

Maximumvaluesoccuratthecenteroftheplate.Forasquareplatea=b,andthefirstfivetermsgive

Mx,max=My,max=0.0479q 0 a^2

ComparingEqs.(7.3)withEqs.(7.5)and(7.6),weobservethat

σx=

12 Mxz

t^3

σy=

12 Myz

t^3

Again,themaximumvaluesofthesestressesoccuratthecenteroftheplateatz=±t/2sothat

σx,max=

6 Mx

t^2

σy,max=

6 My

t^2

Forthesquareplate,

σx,max=σy,max=0.287q 0

a^2

t^2

Thetwistingmomentandshearstressdistributionsfollowinasimilarmanner.
The infinite series (Eq. (7.27)) assumed for the deflected shape of a plate gives an exact solution
fordisplacementsandstresses.However,amorerapid,butapproximate,solutionmaybeobtainedby
assumingadisplacementfunctionintheformofapolynomial.Thepolynomialmust,ofcourse,satisfy
thegoverningdifferentialequation(Eq.(7.20))andtheboundaryconditionsofthespecificproblem.
The “guessed” form of the deflected shape of a plate is the basis for the energy method of solution
describedinSection7.6.

Example 7.2
Showthatthedeflectionfunction

w=A(x^2 y^2 −bx^2 y−axy^2 +abxy)

isvalidforarectangularplateofsidesaandb,builtinonallfouredgesandsubjectedtoauniformly
distributedloadofintensityq.IfthematerialoftheplatehasaYoung’smodulusEandisofthickness
t,determinethedistributionsofbendingmomentalongtheedgesoftheplate.