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Think about it conceptually instead of leaping straight into the maths - the maths is just a formal expression of the underlying concept.
Imagine you have a solid bar 10 mm in diameter and you bend it into a circle with a diameter of 1000 mm. The outside circumference of the bar must be 31.4 mm longer than the inside (10 mm x pi). The metal on the outside must stretch and the material on the inside must compress to account for a 1% difference.
Now imagine that you take that same bar and use it to make a tube with a diameter of 100 mm and then try to bend that into a circle of the same 1000 mm diameter. In this case, the stretching and compression of the metal has to account for a 10% difference, so it will take much more force to bend it because it must be stretched much more. This is why larger diameter tubes are much stiffer than solid bars of the same mass as long as they don't crumple. As soon as they crumple, they lose most of their stiffness.
Have a search for something called section modulus, that should answer your questions
That makes sense hols, thanks. I really couldn't get my head around the "tube of infinite radius with infinity thin walls" being 2 or 4 times stiffer (can't recall which now) than the solid bar of same mass. Of course an infinitly thin wall would crumple under any sort of point load...
I had always conceptually looked at round bar as rounded off squares. You can imagine the square inside it and it is easy to think of a load acting on the 'slices' as if they were beams. Even though the middle third of the beam is basically along for the ride, it prevents the top third from deflecting into space etc.
If you hollow out a square bar, it is easy to visualize a load placed on the top. The load will travel laterally to the sides, which will carry the load via compression and tension in each side. Redistribution of the material to make a taller vertical member accounts for the stiffness. As long as the load is centred and the middle of the vertical members don't bulge outward and the top is strong enough to transfer the stress to the sides it is all really easy to imagine
But how is the load distributed in a cylinder? Is the top 1/4 the equivalent of a peaked roof carrying the stress to the 1/4 on each side which act as vertical 'beams' and if so how does the curve affect stiffness of the vertical members. Or is a cyclinder a top semi circle which wants to flatten laterally under stress and cannot because the lower semi circle acts as a tension member to hold the bottom edges of the top semi circle in place, therefore acts a bit like a beam shaped like an upside down u.
That's the part I cannot grasp about hollow tubes of the non square variety.
I was working on one of my old bikes today when I realised just how relevant this topic is in a bike forum.
On my bike there were 11 threaded parts with holes through them.
The obvious ones were the canti mounts = 4, chainring bolts = 5.
If we count the ISIS crank bolts, that's another 2, although they are capped.
If you have QR axles with cup and cone bearings, add 2 more.
All of those get cranked up pretty tight.
And then there's the cable adjusters which even have a slot in them - I didn't count those because they're used finger tight. 🙂
I also noticed last night that Shimano SPD-SL cleat bolts are hollow.