Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.6 Flexural–Torsional Buckling of Thin-Walled Columns 285

Theboundaryconditionsareθ=0whenz=0andz=L,andsincethewarpingissuppressedattheends
ofthebeam,

dθ

dz

=0whenz=0andz=L (seeEq.(17.19))

Puttingθ=0atz=0inEq.(ii)

0 =A+D

or

A=−D

Also,

dθ

dz

=−μAsinμz+μBcosμz+C

andsince(dθ/dz)=0atz=0,

C=−μB

Whenz=L,θ=0sothat,fromEq.(ii),

0 =AcosμL+BsinμL+CL+D

whichmayberewritten

0 =B(sinμL−μL)+A(cosμL− 1 ) (iii)

Thenfor(dθ/dz)=0atz=L,

0 =μBcosμL−μAsinμL−μB

or

0 =B(cosμL− 1 )−AsinμL (iv)

EliminatingAfromEqs.(iii)and(iv)

0 =B[2( 1 −cosμL)−μLsinμL](v)

Similarly,intermsoftheconstantC

0 =−C[2( 1 −cosμL)−μLsinμL] (vi)

or

B=−C

ButB=−C/μsothattosatisfybothequationsB=C=0and

θ=Acosμz−A=A(cosμz− 1 ) (vii)