# Euler-Bernoulli equation for a periodically supported static beam

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?

• 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. Mar 22, 2022 at 2:50