eli5: Can any engineers explain second moment of area intuitively?
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The second moment of area, or moment of inertia, can be understood as a measure of how an object’s mass is distributed around its axis of rotation. It tells us how resistant the object is to changes in its rotational motion. A larger second moment of area means it’s harder to change the object’s rotation, while a smaller one makes it easier to rotate. In engineering, this concept is essential for designing structures and machines, ensuring they can withstand loads and stresses.
Think about a 2 by 4 being used as a bridge over a little ditch. If you lay it flat with the 4” side facing up, when you walk across it it’s gonna bend a bit once you get to the middle.
Now imagine the same beam but you twist it 90 degrees so the 2” side is facing up. It’s gonna be harder to walk across with the smaller face but once you get to the middle intuition says it’s going to bend way less ( think also of bending a ruler with the flat but facing up is easy but it can’t be bent with the sharp edge facing up)
But why?
The simple explanation is that the mass of the beam is further away from the bending axis. Both beams will bend halfway between the top and the bottom but flipping the beam 90 degrees means the top gets further away from the bending axis (i.e it’s 1” away in the first example and 2” in the second)
This is second moment of area. It’s a measure of how concentrated the mass is around the bending axis. Less concentrated means the beam is more resistant to bending (this is why I beams are shaped like that to move all the mass away from the bending axis in the middle.) Take note that as in the first example the same beam can have many different second moment of areas as it depends where the force on the beam is being applied.
I’m not going to explain exactly how to find it but there’s general solutions to find the second moment of area for common shapes (squares, triangles etc) and a complicated method for any shape. Feel free to ask questions if you don’t understand and I’ll try get back to you.
Imagine putting a simple beam across a gap and walking out into the middle of it so as it bends. You’ll see that the top surface compresses and the bottom stretches. The middle stays neutral. Now imagine you look at very thin length ways sections of that beam. The section right at the bottom stretches the most and slowly, as you look at each section in turn, they stretch a little less until the middle section which neither stretches nor compresses, from there they compress a bit more all the way to the top which compresses the most.
In other words how much they have to stretch or compress depends on how far they are from the middle section. Now we know that compressing or stretching any substance takes force. And we know that relationship between how much you strech or compress most substances is linear in the elastic range. Finally we know that torque is a product of force and the distance from the rotational centre.
So if you consider each of those sections earlier as a spring with it’s (k) spring constant dependent on the total area, how much it stretches dependent on how far it is from the middle section, you can see that the torque, or moments, it applies to resist bending will be it’s area times the distance from that middle section squared. If you make these sections infinitesimally small and add them all together you get an accurate figure in m^(4) showing how resistant to bending around the middle axis (called the neutral axis) the whole shape is. In other words you integrate the area with respect to the x axis squared
Sometimes the beam won’t be bending around the neutral axis, in this case you can just integrate to the new axis or use a formula that allows conversion.
If you have a 2×4 beam and try to bend it so the 2 is the depth it is easier to bend than if the 4 is the depth. The reason being is because the bending moment you are applying is distributed across 4 inches rather than 2, it’s like if you have a pole on a rectangular base that is 2 foot by 4 foot it’s easier to push it over if you push it in the two foot direction rather than the four foot direction.
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