Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

3.4 Torsion of a Narrow Rectangular Strip 79

and
q
N

=−F= 2 G

dθ
dz
Theanalogynowbeingestablished,wemaymakeseveralusefuldeductionsrelatingthedeflected
formofthemembranetothestateofstressinthebar.
Contourlinesorlinesofconstantwcorrespondtolinesofconstantφorlinesofshearstressinthe
bar.Theresultantshearstressatanypointistangentialtothemembranecontourlineandequalinvalue
tothenegativeofthemembraneslope,∂w/∂n,atthatpoint,thedirectionnbeingnormaltothecontour
line(seeEq.(3.16)).Thevolumebetweenthemembraneandthexyplaneis

Vol=

∫∫

wdxdy

andweseethatbycomparisonwithEq.(3.8)

T=2Vol

The analogy therefore provides an extremely useful method of analyzing torsion bars possessing
irregular cross sections for which stress function forms are not known. Hetényi [Ref. 2] describes
experimentaltechniquesforthisapproach.Inadditiontothestrictlyexperimentaluseoftheanalogy,it
isalsohelpfulinthevisualappreciationofaparticulartorsionproblem.Thecontourlinesoftenindicate
aformforthestressfunction,enablingasolutiontobeobtainedbythemethodofSection3.1.Stress
concentrationsaremadeapparentbytheclosenessofcontourlines,wheretheslopeofthemembrane
islarge.Theseareinevidenceatsharpinternalcorners,cut-outs,discontinuities,andsoon.

3.4 TorsionofaNarrowRectangularStrip............................................................

InChapter17,weshallinvestigatethetorsionofthin-walledopensectionbeams,thedevelopmentof
thetheorybeingbasedontheanalysisofanarrowrectangularstripsubjectedtotorque.Wenowconve-
nientlyapplythemembraneanalogytothetorsionofsuchastripshowninFig.3.9.Thecorresponding
membrane surface has the same cross-sectional shape at all points along its length except for small
regionsnearitsendswhereitflattensout.Ifweignoretheseregionsandassumethattheshapeofthe
membraneisindependentofy,thenEq.(3.11)simplifiesto

d^2 φ

dx^2

=− 2 G

dθ

dz

Integratingtwice

φ=−G

dθ

dz

x^2 +Bx+C

Substitutingtheboundaryconditionsφ=0atx=±t/2,wehave

φ=−G

dθ

dz

[

x^2 −

(

t

2

) 2 ]

(3.26)