Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

484 CHAPTER 16 Shear of Beams

UsingtherelationshipsofEqs.(15.23)and(15.24)—thatis,∂My/∂z=Sx,andsoon—thisexpression
becomes

∂σz

∂z

=

(SxIxx−SyIxy)

IxxIyy−Ixy^2

x+

(SyIyy−SxIxy)

IxxIyy−Ixy^2

y

Substitutingfor∂σz/∂zinEq.(16.2)gives

∂q

∂s

=−

(SxIxx−SyIxy)

IxxIyy−Ixy^2

tx−

(SyIyy−SxIxy)

IxxIyy−Ixy^2

ty (16.12)

IntegratingEq.(16.12)withrespecttosfromsomeoriginforstoanypointaroundthecrosssection,
weobtain

∫s

0

∂q

∂s

ds=−

(

SxIxx−SyIxy

IxxIyy−Ixy^2

)∫s

0

txds−

(

SyIyy−SxIxy

IxxIyy−Ixy^2

)∫s

0

tyds (16.13)

Iftheoriginforsistakenattheopenedgeofthecrosssection,thenq=0whens=0,andEq.(16.13)
becomes

qs=−

(

SxIxx−SyIxy

IxxIyy−Ixy^2

)∫s

0

txds−

(

SyIyy−SxIxy

IxxIyy−I^2 xy

)∫s

0

tyds (16.14)

ForasectionhavingeitherCxorCyasanaxisofsymmetryIxy=0andEq.(16.14)reducesto

qs=−

Sx

Iyy

∫s

0

txds−

Sy

Ixx

∫s

0

tyds

Example 16.1
Determine the shear flow distribution in the thin-walled Z-section shown in Fig. 16.6 due to a shear
loadSyappliedthroughtheshearcenterofthesection.

Theoriginforoursystemofreferenceaxescoincideswiththecentroidofthesectionatthemidpoint
oftheweb.Fromantisymmetry,wealsodeducebyinspectionthattheshearcenteroccupiesthesame
position.SinceSyisappliedthroughtheshearcenter,notorsionexistsandtheshearflowdistribution
isgivenbyEq.(16.14)inwhichSx=0;thatis,

qs=

SyIxy

IxxIyy−Ixy^2

∫s

0

txds−

SyIyy

IxxIyy−I^2 xy

∫s

0

tyds

or

qs=

Sy

IxxIyy−Ixy^2

⎛

⎝Ixy

∫s

0

txds−Iyy

∫s

0

tyds

⎞

⎠ (i)