Looking for a way of calculating the maximum weight (W) to the rod with the given length (L) where the rod did not break and that only bend for (b) mm.
Need only approximative solution (read: simple as possible :), nothing with finite-elements or integrals or so.
The practical background: What rod i should to use for suspension of 100kg with L=1.5m and max bending approx. b=1cm? For what constants should I looking for in the published "properties" for the different materials and different profiles? E.g. steel rod vs bamboo? 🙂
Some URL's to helpful practical solutions (read: easy math) is very welcomed – my googling failed.
Best Answer
If you are looking to limit deflections in the beam then there are three main properties you need to look out for:
The elastic deflection of a simple beam with a point load at mid-span is given by:
W * L^3 / (48 E I)
You need to make sure that the system of units are addressed when you do this calculation because the properties are usually defined using slightly different units.
You will also need to check that the strength of the beam is not exceeded by the load you have applied or the deformation will increase very quickly potentially resulting in failure of the beam.
Do this initially by calculating the mid-span bending moment using:
M = W * L / 4
The stress in the beam can then be calculated from the bending moment using:
stress = M * y / I
Where y is normally half the depth of the beam although does require some more careful calculation where the beam is not symetrical.
You need to check that your calculated stress is less than the yield stress of the material.
This all gives the basics of the calculation, although there are a lot of subtleties that I have not addressed here (shear deflection, shear failure, bearing effects etc.).