Anyone have a convenient equation to find the centroid of a section of a non-elliptical conic thin shell and its moment of inertia (Ix)?

If the curve is elliptical, like the one above, I can find Ix of the fuselage skin by subtracting the Ix of the inside of the skin from the Ix of the outside of the skin. I can also work out its centroid, which I will need to find the neutral axis of the whole fuselage section buildup at any station (upper skin + lower skin + longerons).

However, I've used non-elliptical sections, so I've made life more difficult because I don't believe the y-bar and Ix equations apply any longer. I suppose I could integrate these curves graphically but it would be tedious. Any suggestions?

Edit: It occurs to me that I will need the torsion constant,

If the curve is elliptical, like the one above, I can find Ix of the fuselage skin by subtracting the Ix of the inside of the skin from the Ix of the outside of the skin. I can also work out its centroid, which I will need to find the neutral axis of the whole fuselage section buildup at any station (upper skin + lower skin + longerons).

However, I've used non-elliptical sections, so I've made life more difficult because I don't believe the y-bar and Ix equations apply any longer. I suppose I could integrate these curves graphically but it would be tedious. Any suggestions?

Edit: It occurs to me that I will need the torsion constant,

*J*, in addition to*I*.
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