Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

240 CHAPTER 7 Bending of Thin Plates

Theseconditionsmaybesatisfiedbytheassumptionofadeflectedformoftheplategivenby

w=

∑∞

m= 1

∑∞

n= 1

Amnsin

mπx

a

sin

nπy

b

SubstitutingthisexpressionintoEq.(i)gives

Amn=

16 q 0

π^6 Dmn

[(

m^2

a^2

+

n^2

b^2

) 2

+

Nxm^2

π^2 Da^2

] foroddmandn

Amn=0 forevenmandn

Therefore,

w=

16 q 0

π^6 D

∑∞

m=1,3,5

∑∞

n=1,3,5

1

mn

[(

m^2

a^2

+

n^2

b^2

) 2

+

Nxm^2

π^2 Da^2

]sin

mπx

a

sin

nπy

b

(ii)

ComparingEq.(ii)withEq.(i)ofExample7.1,weseethat,asaphysicalinspectionwouldindicate,
thepresenceofatensilein-planeforcedecreasesdeflection.Conversely,acompressivein-planeforce
wouldincreasethedeflection.

7.5 BendingofThinPlatesHavingaSmallInitialCurvature.......................................

Supposethatathinplatehasaninitialcurvaturesothatthedeflectionofanypointinitsmiddleplaneis
w 0 .Weassumethatw 0 issmallcomparedwiththethicknessoftheplate.Theapplicationoftransverse
andin-planeloadswillcausetheplatetodeflectafurtheramountw 1 sothatthetotaldeflectionisthen
w=w 0 +w 1 .However,inthederivationofEq.(7.33),wenotethattheleft-handsidewasobtainedfrom
expressionsforbendingmomentswhichthemselvesdependonthechangeofcurvature.Wetherefore
usethedeflectionw 1 ontheleft-handside,notw.Theeffectonbendingofthein-planeforcesdepends
onthetotaldeflectionwsothatwewriteEq.(7.33)

∂^4 w 1

∂x^4

+ 2

∂^4 w 1

∂x^2 ∂y^2

+

∂^4 w 1

∂y^4

=

1

D

[

q+Nx

∂^2 (w 0 +w 1 )

∂x^2

+Ny

∂^2 (w 0 +w 1 )

∂y^2

+ 2 Nxy

∂^2 (w 0 +w 1 )

∂x∂y

]

(7.34)