Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

514 CHAPTER 17 Torsion of Beams

DifferentiatingEq.(17.6)withrespecttosgives

1

δGt

=

pR
2 A
or

pRGt=

2 A

δ

=constant (17.7)

AclosedsectionbeamforwhichpRGt=constantdoesnotwarpandisknownasaNeuber beam.For
closedsectionbeamshavingaconstantshearmodulus,theconditionbecomes

pRt=constant (17.8)

Examplesofsuchbeamsareacircularsectionbeamofconstantthickness;arectangularsectionbeam
forwhichatb=bta(seeExample17.2);andatriangularsectionbeamofconstantthickness.Inthelast
casetheshearcenter,andhencethecenteroftwist,maybeshowntocoincidewiththecenterofthe
inscribedcirclesothatpRforeachsideistheradiusoftheinscribedcircle.

17.2 TorsionofOpenSectionBeams....................................................................

Anapproximatesolutionforthetorsionofathin-walledopensectionbeammaybefoundbyapplying
theresultsobtainedinSection3.4forthetorsionofathinrectangularstrip.Ifsuchastripisbentto
formanopensectionbeam,asshowninFig.17.10(a),andifthedistancesmeasuredaroundthecross
section is large compared with its thicknesst, then the contours of the membrane—that is, the lines
of shear stress—are still approximately parallel to the inner and outer boundaries. It follows that the
shearlinesinanelementδsoftheopensectionmustbenearlythesameasthoseinanelementδyofa
rectangularstripasdemonstratedinFig.17.10(b).Equations(3.27),(3.28),and(3.29)maytherefore
be applied to the open beam but with reduced accuracy. Referring to Fig. 17.10(b), we observe that
Eq.(3.27)becomes

τzs= 2 Gn

dθ

dz

, τzn= 0 (17.9)

Equation(3.28)becomes

τzs,max=±Gt

dθ

dz

(17.10)

andEq.(3.29)is

J=

∑st^3

3

or J=

1

3

∫

sect

t^3 ds (17.11)

InEq.(17.11),thesecondexpressionforthetorsionconstantisusedifthecrosssectionhasavariable
wall thickness. Finally, the rate of twist is expressed in terms of the applied torque by Eq. (3.12)—
thatis,

T=GJ

dθ

dz

(17.12)