Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

160 CHAPTER 5 Energy Methods

(a) Deriveaformulaforthecentraldeflectionofthebeam,duetoP,whensimplysupportedateachendofthe

span.

(b) Ifbothendsofthespanareencastré,determinethemagnitudeofthefixedendmoments.

Ans. 3 PL^3 / 128 EI,5PL/48(hogging).

P.5.5 ThetubularsteelpostshowninFig.P.5.5supportsaloadof250NatthefreeendC.Theoutsidediameterof
thetubeis100mm,andthewallthicknessis3mm.Neglectingtheweightofthetubefindthehorizontaldeflection
atC.Themodulusofelasticityis206000N/mm^2.

Ans. 53.3mm.

Fig. P.5.5

P.5.6 AsimplysupportedbeamABofspanLanduniformsectioncarriesadistributedloadofintensityvarying
fromzeroatAtow 0 /unitlengthatBaccordingtothelaw

w=

2 w 0 z

L

(

1 −

z

2 L

)

perunitlength.Ifthedeflectedshapeofthebeamisgivenapproximatelybytheexpression

v=a 1 sin

πz

L

+a 2 sin
2 πz
L
evaluatethecoefficientsa 1 anda 2 andfindthedeflectionofthebeamatmidspan.

Ans. a 1 = 2 w 0 L^4 (π^2 + 4 )/EIπ^7 ,a 2 =−w 0 L^4 / 16 EIπ^5 , 0.00918w 0 L^4 /EI.

P.5.7 Auniformsimplysupportedbeam,spanL,carriesadistributedloadingwhichvariesaccordingtoaparabolic
lawacrossthespan.Theloadintensityiszeroatbothendsofthebeamandw 0 atitsmidpoint.Theloadingisnormal
toaprincipalaxisofthebeamcrosssection,andtherelevantflexuralrigidityisEI.Assumingthatthedeflected
shapeofthebeamcanberepresentedbytheseries

v=

∑∞

i= 1

aisin

iπz

L

findthecoefficientsaiandthedeflectionatthemidspanofthebeamusingthefirsttermonlyintheaboveseries.

Ans. ai= 32 w 0 L^4 /EIπ^7 i^7 (iodd),w 0 L^4 /94.4EI.

P.5.8 FigureP.5.8showsaplanepin-jointedframeworkpinnedtoarigidfoundation.Allitsmembersaremade
ofthesamematerialandhaveequalcross-sectionalareaA,exceptmember12whichhasareaA

√

2.