Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

3.1Prandtl Stress Function Solution 73

whichgives

T=

πR^4

2

G

dθ

dz

thatis,

T=GJ

dθ

dz

(iii)

inwhichJ=πR^4 / 2 =πD^4 /32(Disthediameter),thepolarsecondmomentofareaofthebar’scross
section.
SubstitutingforG(dθ/dz)inEq.(ii)from(iii)

φ=−

T

2 J

(x^2 +y^2 −R^2 )

andfromEqs.(3.2)

τzy=−

∂φ

∂x

=

Tx

J

τzx=

∂φ

∂y

=−

T

J

y

Theresultantshearstressatanypointonthesurfaceofthebaristhengivenby

τ=

√

τzy^2 +τzx^2

thatis,

τ=

T

J

√

x^2 +y^2

thatis,

τ=

TR

J

(iv)

Theprecedingargumentmaybeappliedtoanyannulusofradiusrwithinthecrosssectionofthe
barsothatthestressdistributionisgivenby

τ=

Tr

J

andthereforeincreaseslinearlyfromzeroatthecenterofthebartoamaximumTR/Jatthesurface.

Example 3.2
A uniform bar has the elliptical cross section that is shown in Fig. 3.6 and is subjected to equal and
oppositetorquesTateachofitsfreeends.Deriveexpressionsfortherateoftwistinthebar,theshear
stressdistribution,andthewarpingdisplacementofitscrosssection.