Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

278 CHAPTER 8 Columns

ExpandingEq.(8.66)andnotingthatσisconstantoverthecrosssection,weobtain

T(z)=−σ

d^2 u

dz^2

yS

∫

Sect

tds+σ

d^2 u

dz^2

∫

Sect

tyBds−σ

d^2 θ

dz^2

y^2 S

∫

Sect

tds

+σ

d^2 θ

dz^2

2 yS

∫

Sect

tyBds−σ

d^2 θ

dz^2

∫

Sect

ty^2 Bds+σ

d^2 v

dz^2

xS

∫

Sect

tds

−σ

d^2 v

dz^2

∫

Sect

txBds−σ

d^2 θ

dz^2

x^2 S

∫

Sect

tds+σ

d^2 θ

dz^2

2 xS

∫

Sect

txBds

−σ

d^2 θ

dz^2

∫

Sect

tx^2 Bds

(8.67)

Equation(8.67)mayberewritten

T(z)=P

(

xS

d^2 v

dz^2

−yS

d^2 u

dz^2

)

−

P

A

d^2 θ

dz^2

(Ay^2 S+Ixx+AxS^2 +Iyy) (8.68)

InEq.(8.68),thetermIxx+Iyy+A(x^2 S+y^2 S)isthepolarsecondmomentofareaI 0 ofthecolumnabout
theshearcenterS.Thus,Eq.(8.68)becomes

T(z)=P

(

xS

d^2 v

dz^2

−yS

d^2 u

dz^2

)

−I 0

P

A

d^2 θ

dz^2

(8.69)

SubstitutingforT(z)fromEq(8.69)inthegeneralequationforthetorsionofathin-walledbeam(see
Ref.3)wehave

E

d^4 θ

dz^4

−

(

GJ−I 0

P

A

)

d^2 θ

dz^2

−PxS

d^2 v

dz^2

+PyS

d^2 u

dz^2

= 0 (8.70)

Equations(8.61),(8.62),and(8.70)formthreesimultaneousequationswhichmaybesolvedtodetermine
theflexural–torsionalbucklingloads.
Asanexample,considerthecaseofacolumnoflengthLinwhichtheendsarerestrainedagainst
rotationaboutthezaxisandagainstdeflectioninthexandydirections;theendsarealsofreetorotate
aboutthexandyaxesandarefreetowarp.Thus,u=v=θ=0atz=0andz=L.Also,sincethecolumn
isfreetorotateaboutthexandyaxesatitsends,Mx=My=0atz=0andz=L,andfromEqs.(8.61)
and(8.62)

d^2 v

dz^2

=

d^2 u

dz^2

=0atz=0andz=L

Further,theendsofthecolumnarefreetowarpsothat

d^2 θ

dz^2

=0atz=0andz=L