Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.5 Energy Method for the Calculation of Buckling Loads in Columns 273

stiffer,andoffersgreaterresistancetobuckling,asweobservefromthehighervaluesofcriticalload.
Suchbucklingmodes,asstatedinSection8.1,areunstableandaregenerallyofacademicinterestonly.
Ifthedeflectedshapeofthecolumnisknown,itisimmaterialwhichofEqs.(8.47)or(8.48)isused
for the total potential energy. However, when only an approximate solution is possible, Eq. (8.47) is
preferable,sincetheintegralinvolvingbendingmomentdependsontheaccuracyoftheassumedform
ofv,whereasthecorrespondingterminEq.(8.48)dependsontheaccuracyofd^2 v/dz^2 .Generally,for
anassumeddeflectioncurve,visobtainedmuchmoreaccuratelythand^2 v/dz^2.
Supposethatthedeflectioncurveofaparticularcolumnisunknownorextremelycomplicated.We
thenassumeareasonableshapewhichsatisfies,asfaraspossible,theendconditionsofthecolumnand
thepatternofthedeflectedshape(Rayleigh–Ritzmethod).Generally,theassumedshapeisintheform
ofafiniteseriesinvolvingaseriesofunknownconstantsandassumedfunctionsofz.Letussuppose
thatvisgivenby

v=A 1 f 1 (z)+A 2 f 2 (z)+A 3 f 3 (z)

SubstitutioninEq.(8.47)resultsinanexpressionfortotalpotentialenergyintermsofthecriticalload
andthecoefficientsA 1 ,A 2 ,andA 3 astheunknowns.Assigningstationaryvaluestothetotalpotential
energywithrespecttoA 1 ,A 2 ,andA 3 inturnproducesthreesimultaneousequationsfromwhichtheratios
A 1 /A 2 ,A 1 /A 3 ,andthecriticalloadaredetermined.Absolutevaluesofthecoefficientsareunobtainable
sincethedeflectionsofthecolumninitsbuckledstateofneutralequilibriumareindeterminate.
Asasimpleillustration,considerthecolumnshowninitsbuckledstateinFig.8.15.Anapproximate
shapemaybededucedfromthedeflectedshapeofatip-loadedcantilever.Thus,

v=

v 0 z^2

2 l^3

( 3 l−z)

This expression satisfies the end-conditions of deflection—that is,v=0atz=0andv=v 0 atz=l.
In addition, it satisfies the conditions that the slope of the column is zero at the built-in end and
that the bending moment—d^2 v/dz^2 —is zero at the free end. The bending moment at any section is
M=PCR(v 0 −v)sothatsubstitutionforMandvinEq.(8.47)gives

U+V=

PCR^2 v^20

2 EI

∫l

0

(

1 −

3 z^2

2 l^2

+

z^3

2 l^3

) 2

dz−

PCR

2

∫l

0

(

3 v 0

2 l^3

) 3

z^2 ( 2 l−z)^2 dz

Fig.8.15

Buckling load for a built-in column by the energy method.