Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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)isofstandardformandis

v=Acosλz+Bsinλz+

kvc

2 P

z

TheconstantsAandBarefoundusingtheboundaryconditionsofthecolumnwhichare:v=0,when
z=0;v=vc,whenz=L/2;and(dv/dz)=0,whenz=L/2.
Fromthefirstofthese,A=0,whilefromthesecond

B=

vc

sin(λL/ 2 )

(

1 −

kλ

4 P

)

Thethirdboundaryconditiongives,sincevc
=0,therequiredequation;thatis,
(
1 −
kL
4 P

)

cos

λL

2

+

k

2 Pλ

sin

λL

2

= 0

Rearranging

P=

kL

4

(

1 −

tan(λL/ 2 )

λL/ 2

)

IfP(bucklingload)= 4 π^2 EI/L^2 ,thenλL/ 2 =πsothatk= 4 P/L.Finally,ifk→∞

tan

λL

2

=

λL

2

(ii)