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If we look at a cross section at any point, stress will be the same $\sigma=F/A$ and the hooke law says that $\sigma=E*\epsilon$

It means that the deformation will be constant throughout the beam, which obviously can't be correct. So what am I missinterpreting here?

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  • $\begingroup$ Why do think the deformation won't be uniform (constant is no change wrt time, uniform is no change wrt space)? If we neglect the beam's weight (lets say it's really small compared to the applied compressive force), the stress and strain will be uniform over the cross section. $\endgroup$
    – Phil Sweet
    Jan 26 '20 at 13:24
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You seem to be mixing up strain and deformation. The strain, $\epsilon$ or change in length per length, is clearly uniform throughout the beam, and therefore the deformation is not. You get the deformation by integrating the strain, so the deformation varies linearly with zero at one end and maximum at the other.

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  • $\begingroup$ Yes that's probably the case. So what is the formula for deformation? $\endgroup$ Jan 26 '20 at 21:41
  • $\begingroup$ The formula for deformation is quite simply the strain integrated over the length. The integration will leave you with one unknown variable — an integration constant — that you'll have to figure out based on the diagram of the statical system, specifically that the deformation will always be zero at A. $\endgroup$
    – ingenørd
    Jan 27 '20 at 6:23
  • $\begingroup$ Or more formally, ϵ=du/dl, with u being the deformation. $\endgroup$
    – ingenørd
    Jan 27 '20 at 6:37
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Strain, $\epsilon = \Delta L/L$, is the ratio of deformation, not the deformation. Deformation along the beam even with the same $\epsilon$ is proportional to the length of the section we consider, like deformation for one foot of the beam is half of the deformation of two feet. It means deformation is not constant. The stress $\ \sigma $ is constant no matter which cross-section we consider.

We can think of the beam like a spring under compression. It compresses uniformly along its length but if we consider only 1/10 of its length the compression is 1/10 of total compression of the spring. And deformation for that section is 1/10th of the total deformation.

EDIT

I had confused the strain with deformation. I corrected my answer to clear this mix-up.

It is the deformation that varies with section length, not the strain, apologies. And thanks to @JohnHoltz for commenting on that.

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It will be the same at each point while in the elastic stress range ( about 0.5 % strain for typical steels). When it reaches yield it can become non-uniform. For a research project on torsional strain; Strain was uniform ( indicated by ink lines on the polished specimen) until yield , then all plastic strain occurred in a small zone.

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