Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

15.4 Calculation of Section Properties 459

Fig.15.29

Second moments of area of a circular section.

TheequivalenceofthesetwoexpressionsforIxxismosteasilydemonstratedbyanumericalexample.
Also,fromEq.(15.36),

Iyy= 2

tfb^3

12

+

dwt^3 w

12

It is also useful to determine the second moment of area, about a diameter, of a circular section. In
Fig.15.29,wherethexandyaxespassthroughthecentroidofthesection,

Ixx=

∫

A

y^2 dA=

∫d/^2

−d/ 2

2

(

d

2

cosθ

)

y^2 dy (15.38)

IntegrationofEq.(15.38)issimplifiedifanangularvariable,θ,isused.Thus,

Ixx=

∫π/^2

−π/ 2

dcosθ

(

d

2

sinθ

) 2

d

2

cosθdθ

thatis,

Ixx=

d^4

8

∫π/^2

−π/ 2

cos^2 θsin^2 θdθ