How should you change the diameter/radius (while keeping constant overall volume) of the cross-section along the length of a round cantilever beam so that you minimise deflection with concentrated load at the free end?
- In other words: Looking for a shape of solid of revolution that would deflect the least with an end point load.
Thought that a simple cylindrical shape is not the best for this kind of situation. Let's leave aside the other cross-sectional shapes (I-beams etc.) and other types of loads (for now..)
Assuming these are constant:
- Load - P
- Length of the beam - l
- beam type - cantilever with concentrated load at the free end
- beam material
- cross-sectional shape - round
- amount of material used (volume)
- anything else important that I haven't thought of (you get the deal)
I think there has to be an exact solution (shape/formula) to this problem, but getting to it is way out of reach for a CS student who just got interested in engineering.
In order to make my point clear here's a quick draft of what I have in my mind - radius (/diameter) decreasing (in purple) from the support along the length of the beam to (almost) zero at the free end. Rotate the curve around the x axis and you get the new shape (green line represents 'regular' round beam for comparison).
If there is indeed a solution to this, how can it be generalized to include other types of loads and cross-sectional shapes of beams? (for example I-beam type - assuming instead of the radius we just change the scale of the cross-section along the beam length)