312 CHAPTER 9 Thin Plates
Theratioτ/τCRisknownastheloadingratioorbucklingstressratio.ThebucklingstressτCRmaybe
calculatedfromtheformula
τCR,elastic=kssE(
t
b) 2 [
Rd+1
2
(Rb−Rd)(
b
d) 3 ]
(9.33)
wherekssisthecoefficientforaplatewithsimplysupportededges,andRdandRbareempiricalrestraint
coefficientsfortheverticalandhorizontaledgesofthewebpanel,respectively.Graphsgivingkss,Rd,
andRbarereproducedinthestudyofKuhn[Ref.13].
The stress equations (9.27) and (9.28) are modified in the light of these assumptions and may be
rewrittenintermsoftheappliedshearstressτas
σF=kτcotα
( 2 AF/td)+0.5( 1 −k)(9.34)
σS=kτtanα
(AS/tb)+0.5( 1 −k)(9.35)
Further,thewebstressσtgivenbyEq.(9.15)becomestwodirectstresses:σ 1 alongthedirectionofα
givenby
σ 1 =2 kτ
sin2α+τ( 1 −k)sin2α (9.36)andσ 2 perpendiculartothisdirectiongivenby
σ 2 =−τ( 1 −k)sin2α (9.37)ThesecondarybendingmomentofEq.(9.25)ismultipliedbythefactork,whiletheeffectivelengths
forthecalculationofstiffenerbucklingloadsbecome(seeEqs.(9.24))
or
le=ds/√
1 +k^2 ( 3 − 2 b/ds) forb<1.5d
le=ds forb>1.5dwheredsistheactualstiffenerdepth,asopposedtotheeffectivedepthdoftheweb,takenbetweenthe
web/flangeconnections,asshowninFig.9.13.WeobservethatEqs.(9.34)through(9.37)areapplicable
toeitherincompleteorcompletediagonaltensionfieldbeams,since,forthelattercase,k=1giving
theresultsofEqs.(9.27),(9.28),and(9.15).
Insomecases,beamstaperalongtheirlengths,inwhichcasetheflangeloadsarenolongerhorizontal
buthaveverticalcomponentswhichreducetheshearloadcarriedbytheweb.Thus,inFig.9.14,where
disthedepthofthebeamatthesectionconsidered,wehave,resolvingforcesvertically,
W−(FT+FB)sinβ−σt(dcosα)sinα= 0 (9.38)Forhorizontalequilibrium,
(FT−FB)cosβ−σttdcos^2 α= 0 (9.39)