Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

252 CHAPTER 7 Bending of Thin Plates

Endsofminoraxis

σx,max=

± 6 pa^4 b^2

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

, σy,max=

± 6 pb^4 a^2

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

P.7.9 Use the energy method to determine the deflected shape of a rectangular platea×b, simply supported
alongeachedgeandcarryingaconcentratedloadWataposition(ξ,η)referredtoaxesthroughacornerofthe
plate.Thedeflectedshapeoftheplatecanberepresentedbytheseries

w=

∑∞

m= 1

∑∞

n= 1

Amnsin

mπx

a

sin

nπy

b

Ans. Amn=

4 Wsin

mπξ

a

sin

nπη

b

π^4 Dab[(m^2 /a^2 )+(n^2 /b^2 )]^2

P.7.10 If,inadditiontothepointloadW,theplateofproblemP.7.9supportsanin-planecompressiveloadofNx
perunitlengthontheedgesx=0andx=a,calculatetheresultingdeflectedshape.

Ans. Amn=

4 Wsin

mπξ

a

sin

nπη

b

abDπ^4

[(

m^2

a^2

+

n^2

b^2

) 2

−

m^2 Nx

π^2 a^2 D

]

P.7.11 Asquareplateofsideaissimplysupportedalongallfoursidesandissubjectedtoatransverseuniformly
distributedloadofintensityq 0 .ItisproposedtodeterminethedeflectedshapeoftheplatebytheRayleigh–Ritz
methodemployinga“guessed”formforthedeflectionof

w=A 11

(

1 −

4 x^2

a^2

)(

1 −

4 y^2

a^2

)

inwhichtheoriginistakenatthecenteroftheplate.
Commentonthedegreetowhichtheboundaryconditionsaresatisfiedandfindthecentraldeflectionassuming
ν=0.3.

Ans.

0.0389q 0 a^4

Et^3

P.7.12 A rectangular platea×b, simply supported along each edge, possesses a small initial curvature in its
unloadedstategivenby

w 0 =A 11 sin

πx

a

sin

πy

b

Determine,usingtheenergymethod,itsfinaldeflectedshapewhenitissubjectedtoacompressiveloadNxper
unitlengthalongtheedgesx=0,x=a.

Ans. w=

A 11

[

1 −

Nxa^2

π^2 D

/(

1 +

a^2

b^2

) 2 ]sin

πx

a

sin

πy

b