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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?

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  • $\begingroup$ 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

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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.

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    $\begingroup$ 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
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    $\begingroup$ 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

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