Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

3.3 The Membrane Analogy 77

simplifiesto
(
∂ψ
∂y

+x

)

m+

(

∂ψ

∂x

−y

)

l= 0 (3.21)

Itmaybeshown,butnotaseasilyasinthestressfunctionsolution,thattheshearstressesdefinedin
termsofthewarpingfunctioninEqs.(3.19)producezeroresultantshearforceovereachendofthebar
[Ref.1].ThetorqueisfoundinasimilarmannertothatinSection3.1where,byreferencetoFig.3.3,
wehave

T=

∫∫

(τzyx−τzxy)dxdy

or

T=G

dθ

dz

∫∫[(

∂ψ

∂y

+x

)

x−

(

∂ψ

∂x

−y

)

y

]

dxdy (3.22)

BycomparisonwithEq.(3.12)thetorsionconstantJisnow,intermsofψ

J=

∫∫[(

∂ψ

∂y

+x

)

x−

(

∂ψ

∂x

−y

)

y

]

dxdy (3.23)

Thewarpingfunctionsolutiontothetorsionproblemreducestothedeterminationofthewarping
functionψwhichsatisfiesEqs.(3.20)and(3.21).Thetorsionconstantandtherateoftwistfollowfrom
Eqs. (3.23) and (3.22); the stresses and strains from Eqs. (3.19) and (3.18); and, finally, the warping
distributionfromEq.(3.17).

3.3 TheMembraneAnalogy............................................................................

Prandtlsuggestedanextremelyusefulanalogyrelatingthetorsionofanarbitrarilyshapedbartothe
deflectedshapeofamembrane.Thelatterisathinsheetofmaterialwhichreliesforitsresistanceto
transverseloadsoninternalin-planeormembraneforces.
Supposethatamembranehasthesameexternalshapeasthecrosssectionofatorsionbar(Fig.3.7(a)).
Itsupportsatransverseuniformpressureqandisrestrainedalongitsedgesbyauniformtensileforce
N/unit length as shown in Fig. 3.7(a) and (b). It is assumed that the transverse displacements of the
membranearesmallsothatNremainsunchangedasthemembranedeflects.Considertheequilibrium
ofanelementδxδofthemembrane.ReferringtoFig.3.8andsummingforcesinthezdirection,we
have

−Nδy

∂w

∂x

−Nδy

(

−

∂w

∂x

−

∂^2 w

∂x^2

δx

)

−Nδx

∂w

∂y

−Nδx

(

−

∂w

∂y

−

∂^2 w

∂y^2

δx

)

+qδxδy= 0

or

∂^2 w

∂x^2

+

∂^2 w

∂y^2

=∇^2 w=−

q

N

(3.24)