Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

534 CHAPTER 18 Combined Open and Closed Section Beams

Thetorsionalrigidityofthecompletesectionisthen

GJ= 5000 × 107 + 6 × 107 = 5006 × 107 Nmm^2

Inallunrestrainedtorsionproblems,thetorqueisrelatedtotherateoftwistbytheexpression

T=GJ

dθ

dz

Theangleoftwistperunitlengthisthereforegivenby

dθ

dz

=

T

GJ

=

10 × 106

5006 × 107

=0.0002rad/mm

SubstitutingforTinEq.(17.1)fromEq.(17.4),weobtaintheshearflowintheclosedsection.Thus,

qcl=

(GJ)cl

2 A

dθ

dz

=

5000 × 107

2 × 20000

×0.0002

fromwhich

qcl=250N/mm

Themaximumshearstressintheclosedsectionisthen250/1.5=166.7N/mm^2.
Intheopenportionofthesection,themaximumshearstressisobtaineddirectlyfromEq.(17.10)
andis

τmax,op= 25000 × 2 ×0.0002=10N/mm^2

Itcanbeseenfromtheabovethatintermsofstrengthandstiffness,theclosedportionofthewingsection
dominates.Thisdominancemaybeusedtodeterminethewarpingdistribution.Havingfirstfoundthe
positionofthecenteroftwist(theshearcenter),thewarpingoftheclosedportioniscalculatedusing
the method described in Section 17.1. The warping in the walls 13 and 34 is then determined using
Eq.(17.19),inwhichtheoriginforthesweptareaARistakenatthepoint1andthevalueofwarping
isthatpreviouslycalculatedfortheclosedportionat1.

Problems..............................................................................................

P.18.1 ThebeamsectionofExample18.1(seeFig.18.2)issubjectedtoabendingmomentinaverticalplaneof
20kNm.Calculatethemaximumdirectstressinthecrosssectionofthebeam.

Ans. 172.5N/mm^2.

P.18.2 AwingboxhasthecrosssectionshowndiagrammaticallyinFig.P.18.2andsupportsashearloadof100
kNinitsverticalplaneofsymmetry.Calculatetheshearstressatthemidpointoftheweb36ifthethicknessofall
wallsis2mm.

Ans. 89.7N/mm^2.