The Euler-Bernoulli equation for a homogeneous beam is

$$ EI w^{(4)}(x) = q(x),$$

where $w$ is beam height and $q$ is load density.

Inspired by the deflection in a multi-support cantilever bridge near my home, I wondered: what if we have an infinitely long beam with a constant load and periodic supports? Then, up to change of variables, the bridge's deflection has the equation

$$ w^{(4)}(x) = q_0 $$

Since the bridge is periodic, I interpret $w : S^1 \to \mathbb{R}$ as a function on the circle.

As for boundary conditions, the supports translate to periodic conditions $w(0) = w(2\pi) = w_0$. It also seems correct that $w$ should be an even function, so $w'(0) = w'''(0) = 0$, etc.

But no matter what, I keep running into the conclusion that $w$ is a constant function. Here is one line of reasoning. Express both sides of the E-B equation as Fourier series. Then the $n$th coefficient obeys

$$n^4 \hat x[n] = \hat q[n] = w_0 \delta[n] $$ which implies that $\hat x[n] = 0$ when $n \neq 0$.

This can't be right though. I definitely saw a catenary-like deflection between those periodic bridge supports. Am I missing something in my reasoning?

  • $\begingroup$ Since you are totally ignoring the applied loads of the supports, $w$ will indeed be constant as the constant load of its weight will put it in free fall while remaining horizontal. $\endgroup$
    – Bill Watts
    Mar 22, 2022 at 2:50

1 Answer 1


What it means, "periodic supports"?

Note that x is the coordinate along the beam varying x=0 to x=L, where L is the beam length. Thus, x cannot be equal to 2π.

It's very strange a periodic behavior regardless of the time.

The w(x) will never be constant as it represents the vertical displacements along the beam.

  • 1
    $\begingroup$ If you have a new question, please ask it by clicking the Ask Question button. Include a link to this question if it helps provide context. - From Review $\endgroup$
    – Miyase
    Jun 6, 2022 at 21:42
  • 1
    $\begingroup$ Just to clarify the standard comment above: this is Q&A site, not a discussion forum, so please don't use the answer section for something that isn't an answer to the question. That's what the comment section is for. $\endgroup$
    – Miyase
    Jun 6, 2022 at 21:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.