Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

CHAPTER 3 Torsion of Solid Sections...............................................................

Theelasticitysolutionofthetorsionproblemforbarsofarbitrarybutuniformcrosssectionisaccom-
plishedbythesemi-inversemethod(Section2.3)inwhichassumptionsaremaderegardingeitherstress
ordisplacementcomponents.TheformermethodowesitsderivationtoPrandtl,thelattertoSt.Venant.
Bothmethodsarepresentedinthischaptertogetherwiththeusefulmembraneanalogyintroducedby
Prandtl.

3.1 PrandtlStressFunctionSolution...................................................................

ConsiderthestraightbarofuniformcrosssectionshowninFig.3.1.Itissubjectedtoequalbutopposite
torquesTateachend,bothofwhichareassumedtobefreefromrestraintsothatwarpingdisplacements
w—thatis,displacementsofcrosssectionsnormaltoandoutoftheiroriginalplanes—areunrestrained.
Further,wemakethereasonableassumptionsthatsincenodirectloadsareappliedtothebar

σx=σy=σz= 0

andthatthetorqueisresistedsolelybyshearstressesintheplaneofthecrosssection,giving

τxy= 0

Toverifytheseassumptions,wemustshowthattheremainingstressessatisfytheconditionsofequilib-
riumandcompatibilityatallpointsthroughoutthebarand,inaddition,fulfilltheequilibriumboundary
conditionsatallpointsonthesurfaceofthebar.
Ifweignorebodyforces,theequationsofequilibrium(1.5)reduceasaresultofourassumptions,to
∂τxz
∂z

= 0

∂τyz

∂z

= 0

∂τzx

∂x

+

∂τyz

∂y

= 0 (3.1)

ThefirsttwoequationsofEqs.(3.1)showthattheshearstressesτxzandτyzarefunctionsofxandy
only.Therefore,theyareconstantatallpointsalongthelengthofthebar,whichhavethesamexand
ycoordinates.Atthisstage,weturntothestressfunctiontosimplifytheprocessofsolution.Prandtl
introducedastressfunctionφdefinedby

∂φ

∂x

=−τzy

∂φ

∂y

=τzx (3.2)

Copyright©2010,T.H.G.Megson. PublishedbyElsevierLtd. Allrightsreserved.
DOI:10.1016/B978-1-85617-932-4.00003-8 65