Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

488 CHAPTER 16 Shear of Beams

SubstitutingforIxxinEq.(i),wehave

qs=

− 12 Sy

h^3 ( 1 + 6 b/h)

∫s

0

yds (ii)

Theamountofcomputationinvolvedmaybereducedbygivingsomethoughttotherequirementsof
the problem. In this case, we are asked to find the position of the shear center only, not a complete
shearflowdistribution.Fromsymmetry,itisclearthatthemomentsoftheresultantshearsonthetop
andbottomflangesaboutthemidpointofthewebarenumericallyequalandactinthesamerotational
sense. Furthermore, the moment of the web shear about the same point is zero. We deduce that it is
onlynecessarytoobtaintheshearflowdistributiononeitherthetoporbottomflangeforasolution.
Alternatively,choosingaweb/flangejunctionasamomentcenterleadstothesameconclusion.
Onthebottomflange,y=−h/2sothatfromEq.(ii)wehave

q 12 =

6 Sy

h^2 ( 1 + 6 b/h)

s 1 (iii)

Equatingtheclockwisemomentsoftheinternalshearsaboutthemidpointofthewebtotheclockwise
momentoftheappliedshearloadaboutthesamepointgives

Syξs= 2

∫b

0

q 12

h

2

ds 1

or,bysubstitutionfromEq.(iii)

Syξs= 2

∫b

0

6 Sy

h^2 ( 1 + 6 b/h)

h

2

s 1 ds 1

fromwhich

ξs=

3 b^2

h( 1 + 6 b/h)

(iv)

Inthecaseofanunsymmetricalsection,thecoordinates(ξS,ηS)oftheshearcenterreferredtosome
convenientpointinthecrosssectionwouldbeobtainedbyfirstdeterminingξSinasimilarmannerto
thatofExample16.2andthenfindingηSbyapplyingashearloadSxthroughtheshearcenter.Inboth
cases,thechoiceofaweb/flangejunctionasamomentcenterreducestheamountofcomputation.

16.3 SHEAR OF CLOSED SECTION BEAMS

The solution for a shear-loaded closed section beam follows a similar pattern to that described in
Section16.2foranopensectionbeambutwithtwoimportantdifferences.First,theshearloadsmaybe
appliedthroughpointsinthecrosssectionotherthantheshearcentersothattorsionalaswellasshear
effectsareincluded.Thisispossible,since,asweshallsee,shearstressesproducedbytorsioninclosed
sectionbeamshaveexactlythesameformasshearstressesproducedbyshear,unlikeshearstressesdue
toshearandtorsioninopensectionbeams.Secondly,itisgenerallynotpossibletochooseanoriginfor