Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

15.4 Calculation of Section Properties 463

ahorizontalaxisthroughitscentroidisgivenby

Ixx= 2

∫a/^2

0

ty^2 ds= 2

∫a/^2

0

t(ssinβ)^2 ds

fromwhich

Ixx=

a^3 tsin^2 β

12

Similarly,

Iyy=

a^3 tcos^2 β

12

Theproductsecondmomentofareais

Ixy= 2

∫a/^2

0

txyds

= 2

∫a/^2

0

t(scosβ)(ssinβ)ds

whichgives

Ixy=

a^3 tsin2β

24

We note here that these expressions are approximate in that their derivation neglects powers oft^2
and upward by ignoring the second moments of area of the elementδsabout axes through its own
centroid.
Propertiesofthin-walledcurvedsectionsarefoundinasimilarmanner.Thus,Ixxforthesemicircular
sectionofFig.15.33is

Ixx=

∫πr

0

ty^2 ds

Expressingyandsintermsofasinglevariableθsimplifiestheintegration,so

Ixx=

∫π

0

t(rcosθ)^2 rdθ