Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.6 Energy Method for the Bending of Thin Plates 241

Theeffectofaninitialcurvatureondeflectionisthereforeequivalenttotheapplicationofatransverse
loadofintensity

Nx

∂^2 w 0

∂x^2

+Ny

∂^2 w 0

∂y^2

+ 2 Nxy

∂^2 w 0

∂x∂y

Thus,in-planeloadsaloneproducebending,providedthereisaninitialcurvature.
Assumingthattheinitialformofthedeflectedplateis

w 0 =

∑∞

m= 1

∑∞

n= 1

Amnsin

mπx

a

sin

nπy

b

(7.35)

thenbysubstitutioninEq.(7.34),wefindthatifNxiscompressiveandNy=Nxy=0,

w 1 =

∑∞

m= 1

∑∞

n= 1

Bmnsin

mπx

a

sin

nπy

b

(7.36)

where

Bmn=

AmnNx

(π^2 D/a^2 )[m+(n^2 a^2 /mb^2 )]^2 −Nx

We shall return to the consideration of initially curved plates in the discussion of the experimental
determinationofbucklingloadsofflatplatesinChapter9.

7.6 EnergyMethodfortheBendingofThinPlates..................................................

Twotypesofsolutionareobtainableforthinplatebendingproblemsbytheapplicationoftheprinciple
of the stationary value of the total potential energy of the plate and its external loading. The first, in
whichtheformofthedeflectedshapeoftheplateisknown,producesanexactsolution;thesecond,the
Rayleigh–Ritzmethod,assumesanapproximatedeflectedshapeintheformofaserieshavingafinite
numberoftermschosentosatisfytheboundaryconditionsoftheproblemandalsotogivethekindof
deflectionpatternexpected.
InChapter5,wesawthatthetotalpotentialenergyofastructuralsystemcomprisedtheinternalor
strainenergyofthestructuralmember,plusthepotentialenergyoftheappliedloading.Wenowproceed
toderiveexpressionsforthesequantitiesfortheloadingcasesconsideredintheprecedingsections.

7.6.1 Strain Energy Produced by Bending and Twisting

Inthinplateanalysis,weareconcernedwithdeflectionsnormaltotheloadedsurfaceoftheplate.These,
asinthecaseofslenderbeams,areassumedtobeprimarilyduetobendingactionsothattheeffects
ofshearstrainandshorteningorstretchingofthemiddleplaneoftheplateareignored.Therefore,it
issufficientforustocalculatethestrainenergyproducedbybendingandtwistingonlyasthiswillbe
applicable,forthereasonoftheprecedingassumption,toallloadingcases.Itmustberememberedthat