Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

286 CHAPTER 8 Columns

Sinceθ=0atz=l,

cosμL= 1

or

μL= 2 nπ

Therefore,

μ^2 L^2 = 4 n^2 π^2

or

σI 0 −GJ

E

=

4 n^2 π^2

L^2

Thelowestvalueoftorsionalbucklingloadcorrespondston=1sothat,rearrangingthepreceding,

σ=

1

I 0

(

GJ+

4 π^2 E

L^2

)

(viii)

ThepolarsecondmomentofareaI 0 isgivenby

I 0 =Ixx+Iyy (seeRef.2)

thatis,

I 0 = 2

(

tdd^2 +

td

3

3

)

+

3 td^3

12

+ 2 td

d^2

4

whichgives

I 0 =

4 ltd^3

12

SubstitutingforI 0 ,J,andinEq.(viii)

σ=

4

4 ld^3

(

sgt^2 +

13 π^2 Ed^4

L^2

)

References

[1] Timoshenko,S.P.,andGere,J.M.,TheoryofElasticStability,2ndedition,McGraw-Hill,1961.
[2] Megson,T.H.G.,StructuralandStressAnalysis,2ndedition,Elsevier,2005.
[3] Megson,T.H.G.,AircraftStructuresforEngineeringStudents,4thedition,Elsevier,2007.