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For a rod in compression, the max shear stress at a point is $\pm\sigma_x/2$, i.e. half the compressive stress applied at either end.

This value is obtained using the equation for $\sigma_\theta$, the longitudinal stress at any point in a material if it is subjected to a set of stresses (equivalently represented by normal and shear stresses), then rotated through an angle $\theta$, while these same stresses remain in place.

So, does this max shear stress in a rod in compression refer to when the rod is rotated by 45 degrees such that it experiences max shear stress under the same compressive load? Or is this a value that the rod experiences at all times as a result of weaknesses in the material along different axes? Because at an angle of 0 degrees, there wouldn't be any shear stress in a perfectly homogeneous rod under compression, would there?

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What you are referring to is the shear stress in Mohr circle at a plane rotated counterclockwise 45 degrees from the horizontal $ \sigma_n $ axis.

It means if we consider a differential small cube of the rod under just plane compression, normal stress, at a plane with an angle of 45 the average $ \sigma_n = \frac{ (o+\sigma_{n\ max})}{2 }= \tau_{xy\ max} $.

It does not refer to rotating the actual rod.

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  • $\begingroup$ Oh I see, so it's the components of the compressive / tensile force acting in that inclined plane which gives rise to shear stress. I always thought forces acted only in their specified direction, not in a 180 degree arc around them; and components were only used to simplify a problem! $\endgroup$
    – Candidium
    Commented Feb 16, 2020 at 23:41
  • $\begingroup$ A force vector acts only in 1 direction but stresses are made of many force vectors. When you say components of a force you're usually resolving/finding the effect of the force in 2 directions. It's impossible for a force to act in a direction perpendicular to itself. When we say stress however its made of vectors acting in all directions so saying longitudinal stress means summing them up in that direction. Along a plane at 45 degrees you'd be summing up all stress vectors with a component in that direction $\endgroup$
    – Skawang
    Commented Jul 15, 2020 at 6:56

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