Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

252 CHAPTER 7 Bending of Thin Plates Endsofminoraxis σx,max= ± 6 pa^4 b^2 t^2 ( 3 b^4 + 2 a^2 b^2 + 3 a^4 ) , σy,max= ± 6 pb^4 a ...

CHAPTER 8 Columns................................................................................... A large proportion of an ai ...

254 CHAPTER 8 Columns IftheperfectcolumnofFig.8.1issubjectedtoacompressiveloadP,onlyshorteningofthecolumn occursnomatterwhatthev ...

8.1 Euler Buckling of Columns 255 sothatthedifferentialequationofbendingofthecolumnis d^2 v dz^2 + PCR EI v= 0 (8.2) Thewell-kno ...

256 CHAPTER 8 Columns Fig.8.3 Buckling loads for different buckling modes of a pin-ended column. gyrationofthecrosssectionsincet ...

8.1 Euler Buckling of Columns 257 Fig.8.4 Behavior of a perfect pin-ended column. However,aswehaveseen,forvaluesofPbelowPCRtheco ...

258 CHAPTER 8 Columns Fig.8.5 Column of Example 8.1. giving d^2 v dz^2 +λ^2 v= kvc 2 EI z (i) ThesolutionofEq.(i)isofstandardfor ...

8.2 Inelastic Buckling 259 NotethatEq.(ii)isthetranscendentalequation,whichwouldbederivedwhendeterminingthebuckling loadofacolum ...

260 CHAPTER 8 Columns Fig.8.7 Elastic moduli for a material stressed above the elastic limit. Fig.8.8 Determination of reduced e ...

8.2 Inelastic Buckling 261 Also,Pisappliedthroughthecentroidofeachendsectionadistanceefromnnsothat ∫d^1 0 σx(y 1 +e)dA+ ∫d^2 0 σ ...

262 CHAPTER 8 Columns whereErisknownasthereducedmodulus,gives ErI d^2 v dz^2 +Pv= 0 ComparingthiswithEq.(8.2),weseethatifPisthec ...

8.3 Effect of Initial Imperfections 263 Extensiveexperimentscarriedoutonaluminiumalloycolumnsbytheaircraftindustryinthe1940s sho ...

264 CHAPTER 8 Columns whereλ^2 =P/EI.Thefinaldeflectedshape,v,ofthecolumndependsontheformofitsunloadedshape, v 0 .Assumingthat v ...

8.3 Effect of Initial Imperfections 265 whichgivesonrearranging δ=PCR δ P −A 1 (8.28) andweseethatagraphofδplottedagainstδ/Phasa ...

266 CHAPTER 8 Columns ThesolutionofEq.(i)isofstandardformandis v=Acosμz+Bsinμz−e Theboundaryconditionsare:v=0,whenz=0and(dv/dz)= ...

8.4 Stability of Beams under Transverse and Axial Loads 267 Fig.8.11 Bending of a uniformly loaded beam-column. giving d^2 v dz^ ...

268 CHAPTER 8 Columns WemayrewriteEq.(8.32)intermsoftheEulerbucklingloadPCR=π^2 EI/l^2 forapin-endedcolumn. Hence, Mmax= wl^2 π^ ...

8.4 Stability of Beams under Transverse and Axial Loads 269 Similarly,thegeneralsolutionofEq.(8.35)is v=Ccosλz+Dsinλz− W Pl (l−a ...

270 CHAPTER 8 Columns The deflection curve is then obtained from Eq. (8.38) by substitutingλafor sinλa(sinceλais now verysmall)a ...

8.5 Energy Method for the Calculation of Buckling Loads in Columns 271 Fig.8.14 Shortening of a column due to buckling. oraltern ...