Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.4 Stability of Beams under Transverse and Axial Loads 269

Similarly,thegeneralsolutionofEq.(8.35)is

v=Ccosλz+Dsinλz−

W

Pl

(l−a)(l−z) (8.37)

whereA,B,C,andDareconstantswhicharefoundfromtheboundaryconditionsasfollows.
Whenz=0,v=0,sofromEq.(8.36)A=0.At z=l,v=0giving,fromEq.(8.37),C=−Dtanλl.
Atthepointofapplicationoftheload,thedeflectionandslopeofthebeamgivenbyEqs.(8.36)and
(8.37)mustbethesame.Hence,equatingdeflections

Bsinλ(l−a)−

Wa

Pl

(l−a)=D[sinλ(l−a)−tanλlcosλ(l−a)]−

Wa

Pl

(l−a)

andequatingslopes

Bλcosλ(l−a)−

Wa

Pl

=Dλ[cosλ(l−a)−tanλlsinλ(l−a)]+

W

Pl

(l−a)

Solving the preceding equations forBandDand substituting forA,B,C,andDin Eqs. (8.36) and
(8.37),wehave

v=

Wsinλa

Pλsinλl

sinλz−

Wa

Pl

z forz≤l−a (8.38)

v=

Wsinλ(l−a)

Pλsinλl

sinλ(l−z)−

W

Pl

(l−a)(l−z) forz≥l−a (8.39)

These equations for the beam-column deflection enable the bending moment and resulting bending
stressestobefoundatallsections.
Aparticularcaseariseswhentheloadisappliedatthecenterofthespan.Thedeflectioncurveis
thensymmetricalwithamaximumdeflectionundertheloadof

vmax=

W

2 Pλ

tan

λl

2

−

Wl
4 p
Finally,weconsiderabeam-columnsubjectedtoendmomentsMAandMBinadditiontoanaxial
loadP(Fig. 8.13). The deflected form of the beam-column may be found by using the principle of
superpositionandtheresultsofthepreviouscase.First,weimaginethatMBactsalonewiththeaxial
loadP.IfweassumethatthepointloadWmovestowardBandsimultaneouslyincreasessothatthe
productWa=constant=MB,then,inthelimitasatendstozero,wehavethemomentMBappliedatB.

Fig.8.13

Beam-column supporting end moments.