Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

82 CHAPTER 3 Torsion of Solid Sections

Weshouldnotclosethischapterwithoutmentioningalternativemethodsofsolutionofthetorsion
problem.Theseinfactprovideapproximatesolutionsforthewiderangeofproblemsforwhichexact
solutionsarenotknown.Examplesofthisapproacharethenumericalfinitedifferencemethodandthe
Rayleigh–Ritzmethodbasedonenergyprinciples[Ref.5].

References

[1] Wang,C.T.,AppliedElasticity,McGraw-Hill,1953.
[2] Hetényi,M.,HandbookofExperimentalStressAnalysis,JohnWiley,1950.
[3] Roark,R.J.,FormulasforStressandStrain,4thedition,McGraw-Hill,1965.
[4]HandbookofAeronautics,No.1,StructuralPrinciplesandData,4thedition.Publishedundertheauthorityof
theRoyalAeronauticalSociety,TheNewEraPublishingCo.Ltd.,1952.
[5] Timoshenko,S.,andGoodier,J.N.,TheoryofElasticity,2ndedition,McGraw-Hill,1951.

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

P.3.1 Showthatthestressfunctionφ=k(r^2 −a^2 )isapplicabletothesolutionofasolidcircularsectionbarof
radiusa.Determinethestressdistributioninthebarintermsoftheappliedtorque,therateoftwist,andthewarping
ofthecrosssection.
Isitpossibletousethisstressfunctioninthesolutionforacircularbarofhollowsection?
Ans. τ=Tr/Ip, whereIp=πa^4 /2,
dθ/dz= 2 T/Gπa^4 , w=0everywhere.

P.3.2 DeduceasuitablewarpingfunctionforthecircularsectionbarofP.3.1andhencederivetheexpressions
forstressdistributionandrateoftwist.

Ans. ψ=0, τzx=−

Ty

Ip

, τzy=

Tx

Ip

, τzs=

Tr

Ip

,

dθ

dz

=

T

GIP

P.3.3 Show that the warping functionψ=kxy,inwhichkis an unknown constant, may be used to solve the
torsionproblemfortheellipticalsectionofExample3.2.

P.3.4 Showthatthestressfunction

φ=−G

dθ

dz

[

1

2

(x^2 +y^2 )−

1

2 a

(x^3 − 3 xy^2 )−

2

27

a^2

]

isthecorrectsolutionforabarhavingacrosssectionintheformoftheequilateraltriangleshowninFig.P.3.4.
Determinetheshearstressdistribution,therateoftwist,andthewarpingofthecrosssection.Findthepositionand
magnitudeofthemaximumshearstress.

Ans.

τzy=G

dθ

dz

(

x−

3 x^2

2 a

+

3 y^2

2 a

)

τzx=−G

dθ

dz

(

y+

3 xy

a

)