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On Wikipedia derivation on Timoshenko beam equation we arrive at this:

$$ \delta U = \int_L \left[-M_{xx}\frac{\partial (\delta\varphi)}{\partial x} + Q_{x}\left(-\delta\varphi + \frac{\partial (\delta w)}{\partial x}\right)\right]~\mathrm{d}L $$

then:

Integration by parts, and noting that because of the boundary conditions the variations are zero at the ends of the beam, leads to

$$ \delta U = \int_L \left[\left(\frac{\partial M_{xx}}{\partial x} - Q_x\right)~\delta\varphi - \frac{\partial Q_{x}}{\partial x}~\delta w\right]~\mathrm{d}L $$

But I don't understand what are the boundary conditions. Here we are just deriving the general differential equation, I don't see what boundary conditions we have imposed yet. They are not mentioned earlier in the article either. So what am I missing here?

S. Rotos
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  • Isn't it assumed that the displacements at the ends of the beams are zero (e.g. they are supported/clamped)? – Sohom Paul Jul 05 '19 at 13:23
  • @Sohom Paul No, I don't think it was assumed. We are deriving the equation in general, and it should describe the beam under various conditions. If we had a cantilever beam supported at one end, we cannot assume displacement is zero at the other end. And even in the case of both ends supported, the angle at the ends would not be zero. – S. Rotos Jul 05 '19 at 13:53

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