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For example if I have a piece of metal tube or round bar that is 13 inches and the same metal tube or round bar that is the exact same thickness but the second one is 20 inches I know that the 20 inch tube is going to be more brittle than the tube that is 13 inches when supporting its own weight and bend a little (even though we could not see it at those differences).

So my question is what math and strength formulas could I use to figure out how thick the 20 inch or any inch tube has to be to be the same strength of the same tube that is not as long.

Like with using random numbers if I have a 13 inch piece of steel that is .25 inch thick how thick does a 20 inch piece of same steel have to be to have the same strength at the end of the tube and have the same amount of stiffness as the shorter one. I am just wondering I can not remember what strength that is or what formulas I can use to measure this across the board with different metals like Steel, Stainless, Aluminum, Titanium and other types of metals to figure out these types of problems at different lengths.

I hope that I explained this correctly I know what I mean I just hope I am explaining it correctly.

Thank you guys