9.4 Local Instability 2999.4 LocalInstability......................................................................................
WedistinguishedintheintroductoryremarkstoChapter8betweenprimaryandsecondary(orlocal)
instability.Thelatterformofbucklingusuallyoccursintheflangesandwebsofthin-walledcolumns
havinganeffectiveslendernessratio,le/r<20.Forle/r>80,thistypeofcolumnissusceptibletopri-
maryinstability.Intheintermediaterangeofle/rbetween20and80,bucklingoccursbyacombination
ofbothprimaryandsecondarymodes.
Thin-walledcolumnsareencounteredinaircraftstructuresintheshapeoflongitudinalstiffeners,
which are normally fabricated by extrusion processes or by forming from a flat sheet. A variety of
cross sections are used, although each is usually composed of flat plate elements arranged to form
angle,channel,Z-,or“tophat”sections,asshowninFig.9.4.Weseethattheplateelementsfallinto
twodistinctcategories:flangeswhichhaveafreeunloadededgeandwebswhicharesupportedbythe
adjacentplateelementsonbothunloadededges.
In local instability, the flanges and webs buckle like plates, with a resulting change in the cross
sectionofthecolumn.Thewavelengthofthebuckleisoftheorderofthewidthsoftheplateelements,
andthecorrespondingcriticalstressisgenerallyindependentofthelengthofthecolumnwhenthelength
isequaltoorgreaterthanthreetimesthewidthofthelargestplateelementinthecolumncrosssection.
Bucklingoccurswhentheweakestplateelement,usuallyaflange,reachesitscriticalstress,although
insomecasesalltheelementsreachtheircriticalstressessimultaneously.Whenthisoccurs,therotational
restraintprovidedbyadjacentelementstooneanotherdisappears,andtheelementsbehaveasthough
theyaresimplysupportedalongtheircommonedges.Thesecasesarethesimplesttoanalyzeandare
foundwherethecrosssectionofthecolumnisanequal-leggedangle,T-,cruciform,orasquaretube
ofconstantthickness.Valuesoflocalcriticalstressforcolumnspossessingthesetypesofsectionmay
befoundusingEq.(9.7)andanappropriatevalueofk.Forexample,kforacruciformsectioncolumn
isobtainedfromFig.9.3(a)foraplatewhichissimplysupportedonthreesideswithoneedgefreeand
hasa/b>3.Hence,k=0.43,andifthesectionbuckleselastically,thenη=1and
σCR=0.388E(
t
b) 2
(ν=0.3)Itmustbeappreciatedthatthecalculationoflocalbucklingstressesisgenerallycomplicatedwithno
particularmethodgaininguniversalacceptance,muchoftheinformationavailablebeingexperimental.
A detailed investigation of the topic is therefore beyond the scope of this book. Further information
maybeobtainedfromallthereferenceslistedattheendofthischapter.
Fig.9.4
(a) Extruded angle; (b) formed channel; (c) extrudedZ;(d) formed “top hat.”