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..)

cantilever beam

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).

quick draft

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)

  • 1
    $\begingroup$ As large a cross section as possible. Which means there is no optimal size unless there is clearly a constraint that limits the size. $\endgroup$
    – joojaa
    Commented Mar 21, 2022 at 7:52

3 Answers 3


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

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.

  • $\begingroup$ Maybe op is looking for formulation with self weight? $\endgroup$
    – joojaa
    Commented Mar 21, 2022 at 8:01
  • $\begingroup$ @joojaa Obviously not. Thanks for catching my mistake. $\endgroup$
    – r13
    Commented Mar 21, 2022 at 17:10

enter image description here

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

  • $\begingroup$ 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? $\endgroup$
    – Fred
    Commented Apr 22, 2022 at 0:36
  • $\begingroup$ 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. $\endgroup$
    – lsk18271
    Commented May 3, 2022 at 16:40

Classical beam theory allows for section properties that vary along the length. Although the solution become a bit more difficult. Like r13 mentioned, you have a variable Moment of Inertia... I(x).

You can get a generalized solution for the beam deflection and stresses by applying solving the 2nd order differential equation (double integration method) $\dfrac{d2v}{dx2}$ = $\dfrac{M(x)}{EI(x)}$ note that your M and your I are both functions of x.

If you optimize this shape to minimize the deflection, you will see that the variation in the radius (for a circular cone say) closely follows the shape of the moment diagram.

You can see this in bridge designs where the deepest section is located over the support and it follows a nonlinear profile towards a smaller section at the midspan.

You also might be interested to search tapered steel moment frame, as the process you mentioned (varying the depth of the Wideflange members) is very common practice in the building industry.



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