Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

72 CHAPTER 3 Torsion of Solid Sections

whichmaybewritten,fromFig.3.5,as

τzx=−

∂φ

∂x

dx

dn

−

∂φ

∂y

dy

dn

=−

∂φ

∂n

(3.16)

where, in this case, the direction cosineslandmare defined in terms of an elemental normal of
lengthδn.
Therefore,wehaveshownthattheresultantshearstressatanypointistangentialtothelineofshear
stressthroughthepointandhasavalueequaltominusthederivativeofφinadirectionnormaltotheline.

Example 3.1
Determine the rate of twist and the stress distribution in a circular section bar of radiusRwhich is
subjectedtoequalandoppositetorquesTateachofitsfreeends.

Ifweassumeanoriginofaxesatthecenterofthebar,theequationofitssurfaceisgivenby

x^2 +y^2 =R^2

Ifwenowchooseastressfunctionoftheform

φ=C(x^2 +y^2 −R^2 ) (i)

theboundaryconditionφ=0issatisfiedateverypointontheboundaryofthebarandtheconstantC
maybechosentofulfilltheremainingrequirementofcompatibility.Therefore,fromEqs.(3.11)and(i)

4 C=− 2 G

dθ

dz

sothat

C=−

G

2

dθ

dz

and

φ=−G

dθ

dz

(x^2 +y^2 −R^2 )|2(ii)

SubstitutingforφinEq.(3.8)

T=−G

dθ

dz

(∫∫

x^2 dxdy+

∫∫

y^2 dxdy−R^2

∫∫

dxdy

)

ThefirstandsecondintegralsinthisequationbothhavethevalueπR^4 /4, whereasthethirdintegralis
equaltoπR^2 ,theareaofcrosssectionofthebar.Then,

T=−G

dθ

dz

(

πR^4

4

+

πR^4

4

−πR^4

)