Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

298 CHAPTER 9 Thin Plates

whereEandνareelasticvaluesofYoung’smodulusandPoisson’sratio.Inthelinearlyelasticregion,
η=1, which means that Eq. (9.7) may be applied at all stress levels. The derivation of a general
expression forηis outside the scope of this book, but one [Ref. 1] giving good agreement with
experimentis

η=

1 −ν^2 e

1 −ν^2 p

Es

E

[

1

2

+

1

2

(

1

4

+

3

4

Et

Es

)^12 ]

whereEtandEsarethetangentmodulusandsecantmodulus(stress/strain)oftheplateintheinelastic
regionandνeandνparePoisson’sratiointheelasticandinelasticranges.

9.3 ExperimentalDeterminationofCriticalLoadforaFlatPlate..................................

InSection8.3,wesawthatthecriticalloadforacolumnmaybedeterminedexperimentally,without
actuallycausingthecolumntobuckle,bymeansoftheSouthwellplot.Thecriticalloadforanactual,
rectangular,thinplateisfoundinasimilarmanner.
ThedisplacementofaninitiallycurvedplatefromthezeroloadpositionwasfoundinSection7.5,
tobe

w 1 =

∑∞

m= 1

∑∞

n= 1

Bmnsin

mπx

a

sin

nπy

b

where

Bmn=

AmnNx

π^2 D

a^2

(

m+n

(^2) a 2
mb^2

) 2

−Nx

WeseethatthecoefficientsBmnincreasewithanincreaseofcompressiveloadintensityNx.Itfollows
thatwhenNxapproachesthecriticalvalue,Nx,CR,thetermintheseriescorrespondingtothebuckled
shapeoftheplatebecomesthemostsignificant.Forasquareplate,n=1andm=1giveaminimum
valueofcriticalloadsothatatthecenteroftheplate

w 1 =

A 11 Nx

Nx,CR−Nx

orrearranging

w 1 =Nx,CR

w 1

Nx

−A 11

Thus,agraphofw 1 plottedagainstw 1 /Nxwillhaveaslope,intheregionofthecriticalload,equalto
Nx,CR.