Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

Problems 291

P.8.12 Atubularcolumnoflengthlistaperedinwall-thicknesssothattheareaandthesecondmomentofarea
ofitscrosssectiondecreaseuniformlyfromA 1 andI 1 atitscenterto0.2A 1 and0.2I 1 atitsends.
Assumingadeflectedcenter-lineofparabolicform,andtakingthemorecorrectformforthebendingmoment,
usetheenergymethodtoestimateitscriticalloadwhentestedbetweenpin-center,intermsoftheprecedingdata
andYoung’smodulusE.Hence,showthatthesavinginweightbyusingsuchacolumninsteadofonehavingthe
sameradiusofgyrationandconstantthicknessisabout15%.

Ans. 7.01EI 1 /l^2.

P.8.13 Auniformcolumn(Fig.P.8.13)oflengthlandbendingstiffnessEIisrigidlybuiltinatendz=0andsimply
supportedatendz=l.Thecolumnisalsoattachedtoanelasticfoundationofconstantstiffnessk/unitlength.

Fig. P.8.13

Representingthedeflectedshapeofthecolumnbyapolynomial

v=

∑p

n= 0

anηn, whereη=z/l

determine the form of this function by choosing a minimum number of termsp, such that all the kinematic
(geometric)andstaticboundaryconditionsaresatisfied,allowingforonearbitraryconstantonly.
Usingtheresultthusobtained,findanapproximationtothelowestflexuralbucklingloadPCRbytheRayleigh–
Ritzmethod.

Ans. PCR=21.05EI/l^2 +0.09kl^2.

P.8.14 FigureP.8.14showsthedoublysymmetricalcrosssectionofathin-walledcolumnwithrigidlyfixedends.
Find an expression, in terms of the section dimensions and Poisson’s ratio, for the column length for which the
purelyflexuralandthepurelytorsionalmodesofinstabilitywouldoccuratthesameaxialload.
In which mode would failure occur if the length were less than the value found? The possibility of local
instabilityistobeignored.

Ans. l=( 2 πb^2 /t)

√

( 1 +ν)/255.Torsion.

P.8.15 Acolumnoflength2lwiththedoublysymmetriccrosssectionshowninFig.P.8.15iscompressedbetween
theparallelplatensofatestingmachinewhichfullypreventstwistingandwarpingoftheends.
Usingthefollowingdata,determinetheaveragecompressivestressatwhichthecolumnfirstbucklesintorsion

l=500mm,b=25.0mm,t=2.5mm,E=70000N/mm^2 ,E/G=2.6

Ans. σCR=282N/mm^2.