Wednesday 11 April 2007

The powers that be

When trying to make parts of my machine stiffer I got to wondering about the relationship between a material's thickness and its resistance to bending. It is obviously not a linear relationship because as a sheet gets thicker not only is there more material to resist bending, but the outer layers have more leverage than the inner, so it must be at least a square law. I tried googling this for some time but failed to find a formula. I did find a comment on CNCzone by somebody that thought he recollected it being a fourth power law. I can believe this because we recently had two versions of a metal box made at work, one in 0.5 mm steel and the other 0.8 mm. While the thin one was quite flimsy the 60% thicker one was very solid. Any mechanical engineers out there?

I came across another fourth power law recently on the website of the company that made my XY table. If you have a servo system moving something from A to B as fast as it can go, then going twice as fast requires 16 times the power. Some video lectures, and a lot of other useful info about servo systems, are here

The highest power law I have ever come across is that incandescent bulb lifetime is inversely proportional to the 12th power of voltage, see Can anybody beat that?


  1. The senior mech eng at work says that when he was doing finite element analysis (admittedly some time ago), the stiffness depended on the square of the thickness. Googling for stiffness and thickness probably wouldn't help much ;->

  2. Thanks sOlstice. I have lots of offcuts of MDF with different thicknesses so I might have a go at measuring it some time and see if it fits a square law.

  3. I have just seen a spring formula where the force was proportional to the fourth power of the wire diameter. This would imply that the stiffness of a rod is a forth power of its diameter so a sheet would be a square law on its thickness as sOlstice suggested.

  4. The stiffness of a section is proportional to it's second moment of area.

    For a sheet, the stiffness will be proportional to the Width of the sheet multiplied by the cube of the thickness.

    For references, google "section properties".