74 CHAPTER 3 Torsion of Solid Sections
Fig.3.6
Torsion of a bar of elliptical cross section.
Thesemimajorandsemiminoraxesareaandb,respectively,sothattheequationofitsboundaryis
x^2
a^2
+
y^2
b^2
= 1
Ifwechooseastressfunctionoftheform
φ=C
(
x^2
a^2
+
y^2
b^2
− 1
)
,(i)
thentheboundaryconditionφ=0issatisfiedateverypointontheboundaryandtheconstantCmay
bechosentofulfilltheremainingrequirementofcompatibility.Thus,fromEqs.(3.11)and(i)
2 C
(
1
a^2
+
1
b^2
)
=− 2 G
dθ
dz
or
C=−G
dθ
dz
a^2 b^2
(a^2 +b^2 )
(ii)
giving
φ=−G
dθ
dz
a^2 b^2
(a^2 +b^2 )
(
x^2
a^2
+
y^2
b^2
− 1
)
(iii)
SubstitutingthisexpressionforφinEq.(3.8)establishestherelationshipbetweenthetorqueTandthe
rateoftwist
T=− 2 G
dθ
dz
a^2 b^2
(a^2 +b^2 )
(
1
a^2
∫∫
x^2 dxdy+
1
b^2
∫∫
y^2 dxdy−
∫∫
dxdy
)
The first and second integrals in this equation are the second moments of areaIyy=πa^3 b/4and
Ixx=πab^3 /4,whereasthethirdintegralistheareaofthecrosssectionA=πab.Replacingtheintegrals