Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

288 CHAPTER 8 Columns

P.8.4 A uniform, pin-ended column of lengthland bending stiffnessEIhas an initial curvature such that the
lateraldisplacementatanypointbetweenthecolumnandthestraightlinejoiningitsendsisgivenby

v 0 =a

4 z

l^2

(l−z) (seeFig.P.8.4)

ShowthatthemaximumbendingmomentduetoacompressiveendloadPisgivenby

Mmax=−

8 aP

(λl)^2

(

sec

λl

2

− 1

)

where

λ^2 =P/EI

Fig. P.8.4

P.8.5 Theuniformpin-endedcolumnshowninFig.P.8.5isbentatthecentersothatitseccentricitythereisδ.If
thetwohalvesofthecolumnareotherwisestraightandhaveaflexuralstiffnessEI,findthevalueofthemaximum
bendingmomentwhenthecolumncarriesacompressionloadP.

Ans. −P

2 δ

l

√

EI

P

tan

√

P

EI

l

2

.

Fig. P.8.5

P.8.6 A straight uniform column of lengthland bending stiffnessEIis subjected to uniform lateral loading
w/unitlength.Theendattachmentsdonotrestrictrotationofthecolumnends.Thelongitudinalcompressiveforce
Phaseccentricityefromthecentroidsoftheendsectionsandisplacedsoastoopposethebendingeffectofthe
lateralloading,asshowninFig.P.8.6.Theeccentricityecanbevariedandistobeadjustedtothevaluewhich,for
givenvaluesofPandw,willresultintheleastmaximumbendingmomentonthecolumn.Showthat

e=(w/Pμ^2 )tan^2 μl/ 4

where

μ^2 =P/EI

DeducetheendmomentwhichwillgivetheoptimumconditionwhenPtendstozero.

Ans. wl^2 /16.