Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

16.3 Shear of Closed Section Beams 489

Fig.16.10

Shear of closed section beams.

satwhichthevalueofshearflowisknown.Considertheclosedsectionbeamofarbitrarysectionshown
inFig.16.10.TheshearloadsSxandSyareappliedthroughanypointinthecrosssectionand,ingeneral,
causedirectbendingstressesandshearflowswhicharerelatedbytheequilibriumequation(16.2).We
assumethathoopstressesandbodyforcesareabsent.Therefore,

∂q

∂s

+t

∂σz

∂z

= 0

Fromthispoint,theanalysisisidenticaltothatforashearloadedopensectionbeamuntilwereachthe
stageofintegratingEq.(16.13),namely,

∫s

0

∂q

∂s

ds=−

(

SxIxx−SyIxy

IxxIyy−Ixy^2

)∫s

0

txds−

(

SyIyy−SxIxy

IxxIyy−Ixy^2

)∫s

0

tyds

Let us suppose that we choose an origin forswhere the shear flow has the unknown valueqs,0.
IntegrationofEq.(16.13)thengives

qs−qs,0=−

(

SxIxx−SyIxy

IxxIyy−Ixy^2

)∫s

0

txds−

(

SyIyy−SxIxy

IxxIyy−Ixy^2

)∫s

0

tyds

or

qs=−

(

SxIxx−SyIxy

IxxIyy−Ixy^2

)∫s

0

txds−

(

SyIyy−SxIxy

IxxIyy−Ixy^2

)∫s

0

tyds+qs,0 (16.15)

WeobservebycomparisonofEqs.(16.15)and(16.14)thatthefirsttwotermsontheright-handside
of Eq. 16.15 represent the shear flow distribution in an open section beam loaded through its shear
center.Thisfactindicatesamethodofsolutionforashearloadedclosedsectionbeam.Representing
this“open”sectionor“basic”shearflowbyqb,wemaywriteEq.(16.15)intheform

qs=qb+qs,0 (16.16)