Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

Problems 251

whereDistheflexuralrigidity,νisPoisson’sratio,andAisaconstant. CalculatethevalueofAandhencethe
centraldeflectionoftheplate.

Ans. A=a^4 ( 5 − 3 ν)/4,Cen.def.=qa^4 ( 5 − 3 ν)/ 384 D( 1 −ν)

P.7.6 Thedeflectionofasquareplateofsidea,whichsupportsalateralloadrepresentedbythefunctionq(x,y)
isgivenby

w(x,y)=w 0 cos

πx

a

cos

3 πy

a

,

wherexandyarereferredtoaxeswhoseorigincoincideswiththecenteroftheplateandw 0 isthedeflectionatthe
center.
IftheflexuralrigidityoftheplateisDandPoisson’sratioisν,determinetheloadingfunctionq,thesupport
conditionsoftheplate,thereactionsattheplatecorners,andthebendingmomentsatthecenteroftheplate.

Ans. q(x,y)=w 0 D 100

π^4

a^4

cos

πx

a

cos

3 πy

a

Theplateissimplysupportedonalledges.

Reactions:− 6 w 0 D

(π

a

) 2

( 1 −ν)

Mx=w 0 D

(π

a

) 2

( 1 + 9 ν),My=w 0 D

(π

a

) 2

( 9 +ν).

P.7.7 Asimplysupportedsquareplatea×acarriesadistributedloadaccordingtotheformula

q(x,y)=q 0

x

a

,

whereq 0 isitsintensityattheedgex=a.Determinethedeflectedshapeoftheplate.

Ans. w=

8 q 0 a^4

π^6 D

∑∞

m=1,2,3

∑∞

n=1,3,5

(− 1 )m+^1

mn(m^2 +n^2 )^2

sin

mπx

a

sin

nπy

a

P.7.8 Anellipticplateofmajorandminoraxes2aand2bandofsmallthicknesstisclampedalongitsboundary
andissubjectedtoauniformpressuredifferencepbetweenthetwofaces.Showthattheusualdifferentialequation
fornormaldisplacementsofathinflatplatesubjecttolateralloadingissatisfiedbythesolution

w=w 0

(

1 −

x^2

a^2

−

y^2

b^2

) 2

,

wherew 0 isthedeflectionatthecenterwhichistakenastheorigin.
Determinew 0 in terms ofpand the relevant material properties of the plate and hence expressions for the
greateststressesduetobendingatthecenterandattheendsoftheminoraxis.

Ans. w 0 =

3 p( 1 −ν^2 )

2 Et^3

(

3

a^4

+

2

a^2 b^2

+

3

b^4

)

Center, σx,max=

± 3 pa^2 b^2 (b^2 +νa^2 )

t^2 ( 3 b^4 + 2 a^2 b^2 + 3 a^4 )

, σy,max=

± 3 pa^2 b^2 (a^2 +νb^2 )

t^2 ( 3 b^4 + 2 a^2 b^2 + 3 a^4 )