Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

76 CHAPTER 3 Torsion of Solid Sections

theiroriginalunloadedshape,althoughtheymaysufferwarpingdisplacementsnormaltotheirplane.
Thefirstoftheseassumptionsleadstotheconclusionthatcrosssectionsrotateasrigidbodiesabout
a center of rotation or twist. This fact was also found to derive from the stress function approach of
Section3.1sothat,referringtoFig.3.4andEq.(3.9),thecomponentsofdisplacementinthexandy
directionsofapointPinthecrosssectionare

u=−θyv=θx

Itisalsoreasonabletoassumethatthewarpingdisplacementwisproportionaltotherateoftwistand
isthereforeconstantalongthelengthofthebar.Hence,wemaydefinewbytheequation

w=

dθ

dz

ψ(x,y), (3.17)

whereψ(x,y)isthewarpingfunction.
Theassumedformofthedisplacementsu,v,andwmustsatisfytheequilibriumandforceboundary
conditionsofthebar.Wenoteherethatitisunnecessarytoinvestigatecompatibility,asweareconcerned
with displacement forms which are single-valued functions and therefore automatically satisfy the
compatibilityrequirement.
ThecomponentsofstraincorrespondingtotheassumeddisplacementsareobtainedfromEqs.(1.18)
and(1.20)andare

εx=εy=εz=γxy= 0

γzx=

∂w

∂x

+

∂u

∂z

=

dθ

dz

(

∂ψ

∂x

−y

)

γzy=

∂w

∂y

+

∂ν

∂z

=

dθ

dz

(

∂ψ

∂y

+x

)

⎫

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎪

⎭

(3.18)

Thecorrespondingcomponentsofstressare,fromEqs.(1.42)and(1.46)

σx=σy=σz=τxy= 0

τzx=G

dθ

dz

(

∂ψ

∂x

−y

)

τzy=G

dθ

dz

(

∂ψ

∂y

+x

)

⎫

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎪

⎭

(3.19)

Ignoring body forces, we see that these equations identically satisfy the first two of the equilibrium
equations(1.5)andalsothatthethirdisfulfilledifthewarpingfunctionsatisfiestheequation

∂^2 ψ

∂x^2

+

∂^2 ψ

∂y^2

=∇^2 ψ= 0 (3.20)

The direction cosinenis zero on the cylindrical surface of the bar, and so the first two of the
boundaryconditions(Eqs.(1.7))areidenticallysatisfiedbythestressesofEqs.(3.19).Thethirdequation