8.2 Inelastic Buckling 259NotethatEq.(ii)isthetranscendentalequation,whichwouldbederivedwhendeterminingthebuckling
loadofacolumnoflengthL/2,builtinatoneendandpinnedattheother.
8.2 InelasticBuckling....................................................................................
Wehaveshownthatthecriticalstress,Eq.(8.8),dependsonlyontheelasticmodulusofthematerial
of the column and the slenderness ratiol/r. For a given material, the critical stress increases as the
slendernessratiodecreases—inotherwords,asthecolumnbecomesshorterandthicker.Apointisthen
reached when the critical stress is greater than the yield stress of the material so that Eq. (8.8) is no
longerapplicable.Formildsteel,thispointoccursataslendernessratioofapproximately100,asshown
inFig.8.6.Wethereforerequiresomealternativemeansofpredictingcolumnbehavioratlowvalues
ofslendernessratio.
It was assumed in the derivation of Eq. (8.8) that the stresses in the column remained within the
elasticrangeofthematerialsothatthemodulusofelasticityE(=dσ/dε)wasconstant.Abovetheelastic
limitdσ/dεdependsonthevalueofstressandwhetherthestressisincreasingordecreasing.Thus,in
Fig.8.7,theelasticmodulusatthepointAisthetangentmodulusEtifthestressisincreasingbutEif
thestressisdecreasing.
ConsideracolumnhavingaplaneofsymmetryandsubjectedtoacompressiveloadPsuchthatthe
directstressinthecolumnP/Aisabovetheelasticlimit.Ifthecolumnisgivenasmalldeflection,v,in
itsplaneofsymmetry,thenthestressontheconcavesideincreases,whereasthestressontheconvex
side decreases. Thus, in the cross section of the column shown in Fig. 8.8(a), the compressive stress
decreasesintheareaA 1 andincreasesintheareaA 2 ,whereasthestressonthelinennisunchanged.
Sincethesechangestakeplaceoutsidetheelasticlimitofthematerial,wesee,fromourremarksinthe
previousparagraph,thatthemodulusofelasticityofthematerialintheareaA 1 isE,whilethatinA 2
isEt.Thehomogeneouscolumnnowbehavesasifitwerenonhomogeneous,withtheresultthatthe
Fig.8.6
Critical stress–slenderness ratio for a column.