Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.5 Energy Method for the Calculation of Buckling Loads in Columns 271

Fig.8.14

Shortening of a column due to buckling.

oralternatively,sinceEId^2 v/dz^2 =−M,

U=

EI

2

∫l

0

(

d^2 v

dz^2

) 2

dz (8.45)

ThepotentialenergyVofthebucklingloadPCR,referredtothestraightpositionofthecolumnasthe
datum,isthen

V=−PCRδ

whereδistheaxialmovementofPCRcausedbythebendingofthecolumnfromitsinitiallystraight
position.ByreferencetoFig.7.15(b)andEq.(7.41),weseethat

δ=

1

2

∫l

0

(

dv

dz

) 2

dz

giving

V=−

PCR

2

∫l

0

(

dv

dz

) 2

dz (8.46)

Thetotalpotentialenergyofthecolumnintheneutralequilibriumofitsbuckledstateis,therefore,

U+V=

∫l

0

M^2

2 EI

dz−

PCR

2

∫l

0

(

dv

dz

) 2

dz (8.47)

or,usingthealternativeformofUfromEq.(8.45),

U+V=

EI

2

∫l

0

(

d^2 v

dz^2

) 2

dz−

PCR

2

∫l

0

(

dv

dz

) 2

dz (8.48)