Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

266 CHAPTER 8 Columns

ThesolutionofEq.(i)isofstandardformandis

v=Acosμz+Bsinμz−e

Theboundaryconditionsare:v=0,whenz=0and(dv/dz)=0,whenz=L/2.
FromthefirstoftheseA=e,whilefromthesecond

B=etan

μL

2

Theequationforthedeflectedshapeofthecolumnisthen

v=e

[

cosμ(z−L/ 2 )

cosμL/ 2

− 1

]

Themaximumvalueofvoccursatmidspan,wherez=L/2;thatis,

vmax=e

(

sec

μL

2

− 1

)

Themaximumbendingmomentisgivenby

M(max)=Pe+Pvmax

sothat

M(max)=Pesec

μL

2

8.4 StabilityofBeamsunderTransverseandAxialLoads..........................................

Stressesanddeflectionsinalinearlyelasticbeamsubjectedtotransverseloadsaspredictedbysimple
beamtheoryaredirectlyproportionaltotheappliedloads.Thisrelationshipisvalidifthedeflections
are small such that the slight change in geometry produced in the loaded beam has an insignificant
effectontheloadsthemselves.Thissituationchangesdrasticallywhenaxialloadsactsimultaneously
with the transverse loads. The internal moments, shear forces, stresses, and deflections then become
dependentonthemagnitudeofthedeflectionsaswellasthemagnitudeoftheexternalloads.Theyare
alsosensitive,asweobservedintheprevioussection,tobeamimperfectionssuchasinitialcurvature
andeccentricityofaxialload.Beamssupportingbothaxialandtransverseloadsaresometimesknown
asbeam-columnsorsimplyastransverselyloadedcolumns.
First,weconsiderthecaseofapin-endedbeamcarryingauniformlydistributedloadofintensity
wperunitlengthandanaxialloadPasshowninFig.8.11.Thebendingmomentatanysectionofthe
beamis

M=Pv+

wlz

2

−

wz^2

2

=−EI

d^2 v

dz^2