Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

15.4 Calculation of Section Properties 461

entiresectionoftheproductsecondmomentofareaofallsuchpairsofelementsresultsinazerovalue
forIxy.
Wehaveshownpreviouslythattheparallelaxestheoremmaybeusedtocalculatesecondmoments
ofareaofbeamsectionscomprisinggeometricallysimplecomponents.Thetheoremcanbeextendedto
thecalculationofproductsecondmomentsofarea.Letussupposethatwewishtocalculatetheproduct
secondmomentofarea,Ixy,ofthesectionshowninFig.15.30(c)aboutaxesxywhenIXYaboutitsown,
say,centroidal,axessystemCXYisknown.FromEq.(15.42),

Ixy=

∫

A

xydA

or

Ixy=

∫

A

(X−a)(Y−b)dA

which,onexpanding,gives

Ixy=

∫

A

XYdA−b

∫

A

XdA−a

∫

A

YdA+ab

∫

A

dA

IfXandYarecentroidalaxes,then

∫

AXdA=

∫

AYdA=0.Hence,

Ixy=IXY+abA (15.43)

ItcanbeseenfromEq.(15.43)thatifeitherCXorCYisanaxisofsymmetry;thatis,IXY=0,then

Ixy=abA (15.44)

Therefore, for a section component having an axis of symmetry that is parallel to either of the
section reference axes, the product second moment of area is the product of the coordinates of its
centroidmultipliedbyitsarea.

15.4.5 Approximations for Thin-Walled Sections

We may exploit the thin-walled nature of aircraft structures to make simplifying assumptions in the
determination of stresses and deflections produced by bending. Thus, the thicknesstof thin-walled
sections is assumed to be small compared with their cross-sectional dimensions so that stresses may
beregardedasbeingconstantacrossthethickness.Furthermore,weneglectsquaresandhigherpowers
oftin the computation of sectional properties and take the section to be represented by the midline
ofitswall.Asanillustrationoftheprocedure,weshallconsiderthechannelsectionofFig.15.31(a).
ThesectionissinglysymmetricaboutthexaxissothatIxy=0.ThesecondmomentofareaIxxisthen
givenby

Ixx= 2

[

(b+t/ 2 )t^3

12

+

(

b+

t

2

)

th^2

]

+t

[2(h−t/ 2 )]^3

12