Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.6 Flexural–Torsional Buckling of Thin-Walled Columns 281

thatis,

I 0 =3.52× 105 +0.22× 105 =3.74× 105 mm^4

ThetorsionconstantJisobtainedusingEq.(18.11)whichgives

J= 2 ×37.5×2.5^3 / 3 + 75 ×2.5^3 / 3 =781.3mm^4

Finally,isfoundtobe (seeRef.3)

=2.5×37.5^3 × 752 / 24 =30.9× 106 mm^6

SubstitutingtheprecedingvaluesinEqs.(8.77),weobtain

PCR(xx)=6.5× 104 N PCR(yy)=0.41× 104 N PCR(θ)=2.22× 104 N

The column will, therefore, buckle in bending about the Cyaxis when subjected to an axial load of
0.41× 104 N.
Equation(8.73)forthecolumnwhosebuckledshapeisdefinedbyEqs.(8.71)mayberewrittenin
termsofthethreeseparatebucklingloadsgivenbyEqs.(8.77).Thus,
∣ ∣ ∣ ∣ ∣ ∣
0 P−PCR(xx) −PxS
P−PCR(yy) 0 PyS
PyS −PxS I 0 (P−PCR(θ ))/A

∣ ∣ ∣ ∣ ∣ ∣

= 0 (8.78)

Ifthecolumnhas,say,Cxasanaxisofsymmetry,thentheshearcenterliesonthisaxis,andyS=0.
Equation(8.78)therebyreducesto
∣
∣
∣
∣

P−PCR(xx) −PxS

−PxS I 0 (P−PCR(θ ))/A

∣

∣

∣

∣=^0 (8.79)

TherootsofthequadraticequationformedbyexpandingEq.(8.79)arethevaluesofaxialload,which
willproduceflexural–torsionalbucklingaboutthelongitudinalandxaxes.IfPCR(yy)islessthanthe
smallestoftheseroots,thecolumnwillbuckleinpurebendingabouttheyaxis.

Example 8.4
Acolumnoflength1mhasthecrosssectionshowninFig.8.18.Iftheendsofthecolumnarepinned
andfreetowarp,calculateitsbucklingload;E=70000N/mm^2 ,G=30000N/mm^2.

Inthiscase,theshearcenterSispositionedontheCxaxissothatyS=0andEq.(8.79)applies.The
distancex ̄ofthecentroidofareaCfromthewebofthesectionisfoundbytakingfirstmomentsofarea
abouttheweb.Thus,

2 ( 100 + 100 + 100 ) ̄x= 2 × 2 × 100 × 50

whichgives

x ̄=33.3mm