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
)