Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

9.1 Buckling of Thin Plates 295

andforanontrivialsolution

Nx,CR=π^2 a^2 D

1

m^2

(

m^2

a^2

+

n^2

b^2

) 2

(9.3)

ExactlythesameresultmayhavebeendeducedfromEq.(ii)ofExample7.3,wherethedisplacement
wwouldbecomeinfiniteforanegative(compressive)valueofNxequaltothatofEq.(9.3).
WeobservefromEq.(9.3)thateachtermintheinfiniteseriesfordisplacementcorresponds,asin
thecaseofacolumn,toadifferentvalueofcriticalload(notetheproblemisaneigenvalueproblem).
Thelowestvalueofcriticalloadevolvesfromsomecriticalcombinationofintegersmandn—thatis,
the number of half-waves in thexandydirections, and the plate dimensions. Clearlyn=1givesa
minimumvaluesothatnomatterwhatthevaluesofm,a,andb,theplatebucklesintoahalfsinewave
intheydirection.Thus,wemaywriteEq.(9.3)as

Nx,CR=π^2 a^2 D

1

m^2

(

m^2

a^2

+

1

b^2

) 2

or

Nx,CR=

kπ^2 D

b^2

(9.4)

wheretheplatebucklingcoefficientkisgivenbytheminimumvalueof

k=

(

mb

a

+

a

mb

) 2

(9.5)

foragivenvalueofa/b.Todeterminetheminimumvalueofkforagivenvalueofa/b,weplotkasa
functionofa/bfordifferentvaluesofmasshownbythedottedcurvesinFig.9.2.Theminimumvalue
ofkisobtainedfromthelowerenvelopeofthecurvesshownsolidinthefigure.
Itcanbeseenthatmvarieswiththeratioa/bandthatkandthebucklingloadareaminimumwhen
k=4atvaluesofa/b=1,2,3,....Asa/bbecomeslarge,kapproaches4sothatlongnarrowplates
tendtobuckleintoaseriesofsquares.
Thetransitionfromonebucklingmodetothenextmaybefoundbyequatingvaluesofkforthem
andm+1curves.Hence,

mb

a

+

a

mb

=

(m+ 1 )b

a

+

a

(m+ 1 )b

giving

a

b

=

√

m(m+ 1 )

Substitutingm=1,wehavea/b=

√

2 =1.414,andform=2,a/b=

√

6 =2.45,andsoon.
For a given value ofa/b, the critical stress,σCR=Nx,CR/t, is found from Eqs. (9.4) and (7.4),
thatis,

σCR=

kπ^2 E

12 ( 1 −ν^2 )

(

t

b

) 2

(9.6)