Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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