Is it possible to solve this moment of Inertia by integration?

Aug 2014
507
1
India
Find the moment of Inertia of the cross-sectional area of an I section about its centroidal axis:



Is it possible to solve this moment of Inertia by integration?
 

topsquark

Math Team
May 2013
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The Astral plane
Yes. The formula is \(\displaystyle I = \int m^2 dr\). The usual way to do this is to write m as a function of r (using any density information you have. Usually at this level the density is constant.)

However you have three basic objects. All of them are essentially cylinders (The disks at top and bottom are simply solid cylinders which aren't very tall) so you can just look up the formula. The length of the cylinders don't matter.

Here's a list of standard moments of inertia. It's good practice to calculate them on your own but then you can freely use the tables.

Moments of inertia are additive along the same axis.

-Dan
 
Aug 2014
507
1
India
Sol: Here A1, A2 and A3 are the areas:





Centriod ; \(\displaystyle X_{C} = \large \frac {A_{1}X_{1} + A_{2}X_{2}+A_{3}X_{3}}{A_{1} + A{2}+ A_{3}}= 15 cm\)
Centriod ; \(\displaystyle Y_{C} = \large \frac {A_{1}Y_{1} + A_{2}Y_{2}+A_{3}Y_{3}}{A_{1} + A{2}+ A_{3}}= 10.96 cm
\)
Moment of Inertia w.r.t Centroid X-X:

\(\displaystyle I_{XX} = I_{XX1} + I_{XX2} + I_{XX3}\)

\(\displaystyle I_{XX1} = I_{G1.X} + A_{1}Y^{2}\)

\(\displaystyle = I_{G1.X} + A_{1}.(Y_{1} - \overline {Y})^{2}\)

= \(\displaystyle \large \frac {30*5^{3}}{12}+ 150(25 - 10.96^{2}\)) (Here \(\displaystyle I_{G1.X} = \large \frac {b.d^3}{12}\))

= \(\displaystyle 11048.24 cm^{4}\)

\(\displaystyle I_{XX2} = I_{G2.X} + A_{2}Y^{2}\)

\(\displaystyle = I_{G1.X} + A_{2}.(Y_{2} - \overline {Y})^{2}\)

\(\displaystyle = \large \frac {5*15^{3}}{12} + 75(12.5 - 10.96^{2})\)

\(\displaystyle =1584.12 cm^{4}\)

\(\displaystyle I_{XX3} = I_{G3.X} + A_{3}.Y^{2}\)

\(\displaystyle = I_{G3.X} + A_{3}.(Y_{3} - \overline {Y})^{2}\)

=\(\displaystyle \large \frac {20*5^{3}}{12} + 100(22.5 - 10.96^{2})\)

\(\displaystyle = 13525.25 cm^{4}\)

\(\displaystyle I_{xx} = 11048.28 + 1584.12 + 13525.5 = 26137.86 cm^{4}\)

Moment of Inertia w.r.t Centroid Y-Y:

\(\displaystyle I_{YY} = I_{YY1} + I_{YY2} + I_{YY3}\)

\(\displaystyle I_{YY1} = I_{G1 }+ A_{1}.X^{2}\)

= \(\displaystyle I_{G1.Y} + A_{1}. (X_{1} - \overline X)^{2}\) (Here\(\displaystyle I_{G1.X} = \large \frac{d.b^{3}}{12}\))

=\(\displaystyle \large \frac {5*30^{3}}{12} + (150)(15 - 15)^{2}\)

= \(\displaystyle 11250 cm^{4}\)

\(\displaystyle I_{YY2} = I_{G2} + A_{2}.X^{2}\)

= \(\displaystyle I_{G2.Y} + A_{2}. (x^{2} - \overline X)^{2}\)

= \(\displaystyle \large \frac {15*5^{3}}{12} + (75)(15-15)^{2}\)

= \(\displaystyle 156.25 cm^{4}\)

\(\displaystyle I_{YY3} = I_{G3} + A_{3}.X^{2}\)

=\(\displaystyle I_{G3.Y} + A_{3}. (x^{2} - \overline X)^{2}\)

= \(\displaystyle \large \frac {5*20^{3}}{12} + (100)(15-15)^{2}\)

= \(\displaystyle 3333.33 cm^{4}\)

\(\displaystyle I_{yy} = 11250 + 156.25 + 3333.33 = 14739.58 cm^{4}\)

I following this lengthy method to proceed, is there any easy method to solve it?
 
Last edited:

topsquark

Math Team
May 2013
2,533
1,052
The Astral plane
Yes. The formula is \(\displaystyle I = \int m^2 dr\). The usual way to do this is to write m as a function of r (using any density information you have. Usually at this level the density is constant.)
This is what I get for not double checking the equation. It should be \(\displaystyle I = \int r^2 dm\).

Sorry about that.

-Dan
 
Last edited:

topsquark

Math Team
May 2013
2,533
1,052
The Astral plane
I following this lengthy method to proceed, is there any easy method to solve it?
Yes. The moment of inertia of a solid cylinder is \(\displaystyle I = \dfrac{1}{2} mr^2\) where r is the radius of the cylinder.

The new problem is that I didn't notice before that you aren't given any masses. The best you can do is assume that the density of each piece is the same. So you would have
\(\displaystyle I_{tot} = \dfrac{1}{2} \rho (0.150)^1 + \dfrac{1}{2} \rho (0.025)^2 + \dfrac{1}{2} \rho (0.100)^2\)
where \(\displaystyle \rho\) is the density of the material.

Factoring out the density to simplify a bit:
\(\displaystyle I_{tot} = \rho \left ( \dfrac{1}{2} (0.150)^2 + \dfrac{1}{2}(0.025)^2 + \dfrac{1}{2} (0.100)^2 \right )\)

-Dan