Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.2 Inelastic Buckling 261

Also,Pisappliedthroughthecentroidofeachendsectionadistanceefromnnsothat

∫d^1

0

σx(y 1 +e)dA+

∫d^2

0

σv(y 2 −e)dA=−Pv (8.10)

FromFig.8.8(b),

σx=

σ 1

d 1

y 1 σv=

σ 2

d 2

y 2 (8.11)

Theanglebetweentwoclose,initiallyparallel,sectionsofthecolumnisequaltothechangeinslope
d^2 v/dz^2 ofthecolumnbetweenthetwosections.This,inturn,mustbeequaltotheangleδφinthestrain
diagramofFig.8.8(c).Hence,

d^2 v

dz^2

=

σ 1

Ed 1

=

σ 2

Etd 2

(8.12)

andEq.(8.9)becomes,fromEqs.(8.11)and(8.12)

E

d^2 v

dz^2

∫d^1

0

y 1 dA−Et

d^2 v

dz^2

∫d^2

0

y 2 dA= 0 (8.13)

Further,inasimilarmanner,fromEq.(8.10)

d^2 v

dz^2

⎛

⎝E

∫d^1

0

y^21 dA+Et

∫d^2

0

y^22 dA

⎞

⎠+ed

(^2) v
dz^2

⎛

⎝E

∫d^1

0

y 1 dA−Et

∫d^2

0

y 2 dA

⎞

⎠=−Pv (8.14)

Thesecondtermontheleft-handsideofEq.(8.14)iszerofromEq.(8.13).Therefore,wehave

d^2 v

dz^2

(EI 1 +EtI 2 )=−Pv (8.15)

inwhich

I 1 =

∫d^1

0

y^21 dA and I 2 =

∫d^2

0

y^22 dA

the second moments of area aboutnnof the convex and concave sides of the column, respectively.
Putting

ErI=EI 1 +EtI 2

or

Er=E

I 1

I

+Et

I 2

I

, (8.16)