Moment of Inertia (Composite Shape) without using a Neutral Axis

StrEng007 · Mar 15, 2022

For composite shapes, I've typically calculated Ix by means of a parallel axis (located at the neutral axis for the overall shape). For the image below, Ix would be:

Ix = [Is + As(d1)²] + [Ip + Ap(d2)²]

Where:
Is = moment of inertia for the stiffener (angle)
As= area of stiffener (angle)
d1 = distance from centroid of the stiffener to the centroid of the composite shape

Ip = moment of inertia for the panel (plate)
Ap= area of the panel (plate)
d2 = distance from centroid of the panel to the centroid of the composite shape

To use this approach, you much first calculate the location of the Neutral Axis.

However, I came across this formula that calculates the same Ix.

The first two values are the moment of inertia of the stiffener and plate, respectively. The third values provides a short cut to using the parallel axis (ie, you don't need to calculate the Neutral Axis of the cross section). Is there a particular name for this method? It seems like a much quicker way to calculate Ix of a composite shape. I'm trying to find any text that has a more thorough discussion on this.

Note: This method comes from Blodgett's welded structures, section 6.4.

ProgrammingPE · Mar 15, 2022

It looks like you understand how the second equation works. It is the same as the first equation but it just has the calculation for determining the neutral axis of the composite shape built into it.

You're correct that the first equation includes Is + Ip which is the same as Is + Ap*t²/12 in the second equation since Ip = b*t³/12 = (b*t)*t²/12 = Ap*t²/12.

This just leaves As*(d1)² + Ap(d2)² from the first equation and As*Ap*d²/(As+Ap) from the second equation. I'll show you that these terms are also identical once you include the equation for determining the composite shape centroid.

d1 = Ap*d/(As+Ap)
d2 = d - d1 = d - Ap*d/(As+Ap)

Plug these into what remains in the first equation and you get:
As*(Ap*d/(As+Ap))² + Ap*(d - Ap*d/(As+Ap))²

If you go through all of the messy algebra to simplify this, you see that it reduces to the last term of the second equation:
As*Ap*d²/(As+Ap)

(I've also included the output from Wolfram Alpha if you prefer to avoid the algebra which I've also done myself if you really want it. Equation uses A,B,C for As,Ap,d respectively since their program sometimes gets confused by other variable names.)

Composite_Member_Moment_of_Inertia_Simplification_kpldeo.png

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StrEng007 · Mar 15, 2022

This is great. I had not run through the process of making those substitutions and simplifying that long equation. It’s nice how many variables get eliminated, it makes for such a clean equation to use.

I see why using the N.A. is more commonly used as it requires less work.

Thanks for the recommended resource.

rb1957 · Mar 16, 2022

the algebra wasn't That messy ...
comparing the 1st and 2nd equations we can see Is and Ip in both
the 1st has As(d1)^2 + Ap(d2)^2 ... (3)
now d = d1+d2, so d2 = d-d1
subs into (3) ... As(d1)^2+Ap(d-d1)^2 ... (4)
now 1st moment of area about the stiffener NA ...
Ap*d = (As+Ap)*d1, so d1 = d*(Ap/(As+Ap))
subs into (4) ... As*d^2*(Ap/(Ap+As))^2 + Ap*d^2*(1-(Ap/(As+Ap))^2
separate Ap*d^2 ... Ap*d^2*(As*Ap)/(As+Ap)^2 + Ap*d^2*(As/(As+Ap))^2
collect ... Ap*d^2*(As*Ap+As^2)/(As+Ap)^2
or ... Ap*d^2*As(As+Ap)/(As+Ap)^2
or ... Ap*d^2*As/(As+Ap)
= (As*Ap)/(As+Ap)*d^2

another day in paradise, or is paradise one day closer ?

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Moment of Inertia (Composite Shape) without using a Neutral Axis

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