Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

22.4 Shear 595

determinatestructure.Thesystemofsimultaneousequationsfromwhichthefinalshearflowsarefound
willthenbe“wellconditioned”andwillproducereliableresults.Thesolutionofan“ill-conditioned”
systemofequationswouldprobablyinvolvethesubtractionoflargenumbersofasimilarsizewhich
wouldthereforeneedtobeexpressedtoalargenumberofsignificantfiguresforreasonableaccuracy.
Although this reasoning does not apply to a completely idealized wing section, since the calculated
values of shear flow are constant between the booms, it is again advantageous to “cut” either top or
bottomskinpanelsfor,inthespecialcaseofawingsectionhavingahorizontalaxisofsymmetry,a
“cut” in, say, the top skin panels will result in the “open section” shear flows (qb)beingzerointhe
bottomskinpanels.Thisdecreasesthearithmeticallaborandsimplifiesthederivationofthemoment
equation,aswillbecomeobviousinExample22.4.
The“opensection”shearflowqbinthewingsectionofFig.22.8isgivenbyEq.(19.6),thatis,

qb=−

(

SxIxx−SyIxy

IxxIyy−Ixy^2

)⎛

⎝

∫s

0

tDxds+

∑n

r= 1

Brxr

⎞

⎠

−

(

SyIyy−SxIxy

IxxIyy−Ixy^2

)⎛

⎝

∫s

0

tDyds+

∑n

r= 1

Bryr

⎞

⎠

Weareleftwithanunknownvalueofshearflowateachofthe“cuts,”thatis,qs,0,I,qs,0,II,...,qs,0,N,
plustheunknownrateoftwistdθ/dz,which,fromtheassumptionofanundistortedcrosssection,isthe
sameforeachcell.Therefore,asinthetorsioncase,thereareN+1unknownsrequiringN+1equations
forasolution.
ConsidertheRthcellshowninFig.22.9.Thecompletedistributionofshearflowaroundthecellis
givenbythesummationofthe“opensection”shearflowqbandthevalueofshearflowatthe“cut,”
qs,0,R.Wemaythereforeregardqs,0,Rasaconstantshearflowactingaroundthecell.Therateoftwist
isagaingivenbyEq.(16.22);thus,

dθ

dz

=

1

2 ARG

∮

R

q

ds

t

=

1

2 ARG

∮

R

(qb+qs,0,R)

ds

t

Fig.22.9

Redundant shear flow in theRthcellofanN-cell wing section subjected to shear.