Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.4 Stability of Beams under Transverse and Axial Loads 267

Fig.8.11

Bending of a uniformly loaded beam-column.

giving

d^2 v

dz^2

+

P

EI

v=

w

2 EI

(z^2 −lz) (8.29)

ThestandardsolutionofEq.(8.29)is

v=Acosλz+Bsinλz+

w

2 P

(

z^2 −lz−

2

λ^2

)

,

whereAandBare unknown constants andλ^2 =P/EI. Substituting the boundary conditionsv=0at
z=0andlgives

A=

w

λ^2 P

B=

w

λ^2 Psinλl

(l−cosλl)

sothatthedeflectionisdeterminateforanyvalueofwandPandisgivenby

v=

w

λ^2 P

[

cosλz+

(

1 −cosλl

sinλl

)

sinλz

]

+

w

2 P

(

z^2 −lz−

2

λ^2

)

(8.30)

Inbeamcolumns,asinbeams,weareprimarilyinterestedinmaximumvaluesofstressanddeflection.
For this particular case, the maximum deflection occurs at the center of the beam and is, after some
transformationofEq.(8.30),

vmax=

w

λ^2 P

(

sec

λl

2

− 1

)

−

wl^2

8 P

(8.31)

Thecorrespondingmaximumbendingmomentis

Mmax=−Pvmax−

wl^2

8

or,fromEq.(8.31)

Mmax=

w

λ^2

(

1 −sec

λl

2

)

(8.32)