If I have a uniform rod --

And then I push on each end toward the center >--<

So it bends >^<

What shape does it make? Is it a parabola, a catenary, or something else?

Assume constant stiffness along the rod. I'm using the rod to draw an aerodynamic shape, and I'd like to know what shape I'm drawing. I've read about parabolas and catenaries, but I don't know the answer to this question.



1 Answer 1


For a slender column, pinned at both ends, the typical shape would be:

$$y(x)=\delta_{max}\sin\left(\frac{\pi x}{L}\right)$$

where $\delta_{max}$ is the maximum deflection.

see https://www.continuummechanics.org/columnbuckling.html for more details.

  • $\begingroup$ Is that when perfectly aligned? $\endgroup$
    – Solar Mike
    Feb 19, 2020 at 13:13
  • $\begingroup$ That's great. Thanks. What does "pinned" mean in this context? I thought the ends would be moving in as the column bends. $\endgroup$ Feb 19, 2020 at 13:56
  • $\begingroup$ So I guess instead of parabolic or catenary, would you describe it as sinusoidal? $\endgroup$ Feb 19, 2020 at 14:02
  • 1
    $\begingroup$ Yep - sinusoidal is correct. Pinned means free to rotate, there is no moment applied to the ends. As opposed to fixed. $\endgroup$ Feb 19, 2020 at 14:20
  • $\begingroup$ The site you linked to is total crap. "The first step is to assume a deformed shape." The shape of a deformed rod in compression, where the length is fixed and the force of compression is what ever it has to be, does not have a closed form. It involves a set of differential equations, at least one is elliptical, and it has to be solved numerically after specifying the boundary conditions of the case. In general, the neutral fiber varies and variable stiffness may need accounting for. The bending moment at s is proportional to the actual deflection at s everywhere. $\endgroup$
    – Phil Sweet
    Feb 20, 2020 at 0:22

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