Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

22.3 Torsion 591

Fig.22.5

Shear flow distribution in theRth cell of anN-cell wing section.

TheshearflowinEq.(22.5)isconstantalongeachwallofthecellandhasthevaluesshowninFig.22.5.
Writing

∫

ds/tforeachwallasδ,Eq.(22.5)becomes

dθ

dz

=

1

2 ARG

[qRδ 12 +(qR−qR− 1 )δ 23 +qRδ 34 +(qR−qR+ 1 )δ 41 ]

or,rearrangingthetermsinsquarebrackets,

dθ

dz

=

1

2 ARG

[−qR− 1 δ 23 +qR(δ 12 +δ 23 +δ 34 +δ 41 )−qR+ 1 δ 41 ]

Ingeneralterms,thisequationmayberewrittenintheform

dθ

dz

=

1

2 ARG

(−qR− 1 δR−1,R+qRδR−qR+ 1 δR+1,R) (22.6)

inwhichδR−1,Ris

∫

ds/tforthewallcommontotheRthand(R−1)thcells,δRis

∫

ds/tforallthewalls
enclosingtheRthcell,andδR+1,Ris

∫

ds/tforthewallcommontotheRthand(R+1)thcells.
ThegeneralformofEq.(22.6)isapplicabletomulticellsectionsinwhichthecellsareconnected
consecutively—thatis,cellIisconnectedtocellII,cellIItocellsIandIII,andsoon.Insomecases,
cellImaybeconnectedtocellsIIandIII,andsoon(seeproblemP.22.4)sothatEq.(22.6)cannotbe
usedinitsgeneralform.Forthistypeofsection,theterm

∮

∫ q(ds/t)shouldbecomputedbyconsidering
q(ds/t)foreachwallofaparticularcellinturn.
There areNequations of the type (22.6) which, with Eq. (22.4), comprise theN+1 equations
requiredtosolvefortheNunknownvaluesofshearflowandtheoneunknownvalueofdθ/dz.
Frequently, in practice, the skin panels and spar webs are fabricated from materials possessing
different properties such that the shear modulusGis not constant. The analysis of such sections is
simplifiediftheactualthicknesstofawallisconvertedtoamodulus-weightedthicknesst∗asfollows.