# What's the best shape (solid of revolution) for a cantilever beam to carry a point load at the free end?

## 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)

• As large a cross section as possible. Which means there is no optimal size unless there is clearly a constraint that limits the size. Mar 21 at 7:52

## 2 Answers

$$\Delta = \dfrac{PL^3}{3EI}$$

For P, L, E being held constant, the only thing you may change is $$I$$, and as it is in the denominator, so the larger the $$I$$ the smaller the deflection.

However, how to increase $$I$$ yet maintain the volume constant? Since the hint under the assignment points to a solid of revolution, depending on the definition of "solid of revolution", there are two shapes with possible solutions:

1. Hollow circular shape.

2. Uniform tappering cone.

There is no need to discuss the method to find the size of a hollow shape that satisfies the stated constraints; for the uniform tapering cone, you can find the equivalent volume and the base diameter/radius of the conical shape, and use FEM to find the deflections corresponding to the various ratios of $$D_{cone}/D_{rod}$$, using the relationship $$V_{cone} = V_{rod}$$ and the constrain $$\Delta_{cone} \le \Delta_{rod}$$.

Note, this paper presents the equation for a truncated cone with the diameter/radius at the free end defined.

• Maybe op is looking for formulation with self weight? Mar 21 at 8:01
• @joojaa Obviously not. Thanks for catching my mistake.
– r13
Mar 21 at 17:10

I-beam, without a doubt. Why do you want a revolved solid?

• The bottom section of the first page is difficult to read. Instead of having one photo of both pages, would it be possible to have larger photos of each page separately?
– Fred
Apr 22 at 0:36
• They're just simple stress max equations ... stressmax=Mmax*y/I ... You'll have to forgive me, I didn't solve for I with the I-beam shape since college. But I can confidently tell you, I beam is the strongest shape for loaded cantilevers. At this point in my career I would solve this in Solidworks Simulation. May 3 at 16:40