Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

506 CHAPTER 17 Torsion of Beams

Itfollowsthatθ=Az+B,u=Cz+D,v=Ez+F,whereA,B,C,D,E,andFareunknownconstants.
Thus,θ,u,andvarealllinearfunctionsofz.
Equation(16.22),relatingtherateoftwisttothevariableshearflowqsdevelopedinashearloaded
closedsectionbeam,isalsovalidforthecaseqs=q=constant.Hence,

dθ

dz

=

q

2 A

∮

ds

Gt

whichbecomes,onsubstitutingforqfromEq.(17.1)

dθ

dz

=

T

4 A^2

∮

ds

Gt

(17.4)

Thewarpingdistributionproducedbyavaryingshearflow,asdefinedbyEq.(16.25)foraxeshaving
theiroriginatthecenteroftwist,isalsoapplicabletothecaseofaconstantshearflow.Thus,

ws−w 0 =q

∫s

0

ds

Gt

−

AOs

A

q

∮

ds

Gt

ReplacingqfromEq.(17.1),wehave

ws−w 0 =

Tδ

2 A

(

δOs

δ

−

AOs

A

)

(17.5)

where

δ=

∮

ds

Gt

and δOs=

∫s

0

ds

Gt

ThesignofthewarpingdisplacementinEq.(17.5)isgovernedbythesignoftheappliedtorqueTand
thesignsoftheparametersδOsandAOs.Havingspecifiedinitiallythatapositivetorqueisanticlockwise,
thesignsofδOsandAOsarefixedinthatδOsispositivewhensispositive;thatis,sistakenaspositive
inananticlockwisesense,andAOsispositivewhen,asbefore,p(seeFig.17.3)ispositive.
Wehavenotedthatthelongitudinalstrainεziszeroinaclosedsectionbeamsubjectedtoapure
torque.Thismeansthatallsectionsofthebeammustpossessidenticalwarpingdistributions.Inother
words,longitudinalgeneratorsofthebeamsurfaceremainunchangedinlengthalthoughsubjectedto
axialdisplacement.

Example 17.1
Athin-walledcircularsectionbeamhasadiameterof200mmandis2mlong;itisfirmlyrestrained
against rotation at each end. A concentrated torque of 30kNm is applied to the beam at its midspan
point. If the maximum shear stress in the beam is limited to 200N/mm^2 and themaximumangleof
twistto2◦,calculatetheminimumthicknessofthebeamwalls.TakeG=25000N/mm^2.