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So I understand that shear stress can be either in the the form of transverse or torsional or both at the same time. And normal stress is due to the moment or a force normal to the surface. My question is why is it that sometimes when it asks for the max shear stress we use the $\tau_{max}= \sqrt{\left( \frac{\sigma}{2}\right)^2 + \tau^2}$ and sometimes we use $\tau_{max} = \tau_v + \tau_T$ to find the max shear stress?

Why sometimes do we consider the normal stress in calculating the max shear stress and sometimes we do not?

This is an example where it says find the max shear stress and we use <span class=$\tau_{max}= \tau_v + \tau_T$" />

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  • $\begingroup$ I am not certain the notation you are using for $\tau_{max} = \tau_v + \tau_T$. What is $\tau_v$ and what is $\tau_T$? $\endgroup$ – NMech Apr 16 at 11:43
  • $\begingroup$ Additionally, does your question regarding the $\tau_{max}= \sqrt{\left( \frac{\sigma}{2}\right)^2 + \tau^2}$ refers mainly to the Mohr's circle? Because a similar expression is also encountered at failure theories. $\endgroup$ – NMech Apr 16 at 11:47
  • $\begingroup$ @NMech tV is the transverse shear stress and tT is the torsional shear stress $\endgroup$ – user15588486 Apr 16 at 11:47
  • $\begingroup$ IMHO, I think you should provide at least one (maybe two) concrete examples, in order to answer that efficiently (because this can have a multitude of answers). E.g. in the general case $\tau_{max} = \tau_v + \tau_T$ is not correct. The more general form would be $\tau_{max} = \sqrt{\tau_v^2 + \tau_T^2}$. $\endgroup$ – NMech Apr 16 at 11:51
  • $\begingroup$ @NMech I edited my question with an example from our lecture. That is one step of the problem and we found max shear stress by tV + tT as shown in the picture but in just a few steps we considered the stress at point A and the max shear stress formula we used was the square root one $\endgroup$ – user15588486 Apr 16 at 12:12
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In the example you are presenting the shear stresses on cross-section of beam AB are as in the image below.

enter image description here

  • with $\color{green}{green}$ is the shear stress $\tau_v$ which has a single direction (parallel to the shear force)
  • with $\color{red}{red}$ is the torsional stress $\tau_T$, which has a changing direction (see the following graph)

enter image description here

if you notice at points

  • B and D the stresses are perpendicular. In that case $\tau=\sqrt{\tau_v^2 +\tau_t^2}$
  • C the stresses are cancelling each other. In that case $\tau_C=\tau_v - \tau_t$
  • A the stresses are contributing to each other. In that case $\tau_C=\tau_v + \tau_t$

Now, regarding the other question of your post (i.e. $\tau_{max}= \sqrt{\left( \frac{\sigma}{2}\right)^2 + \tau^2}$), again its best if you post an example you have in mind.

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  • $\begingroup$ I couldn't be any more appreciative of your help and your effort. Thank you very much. $\endgroup$ – user15588486 Apr 16 at 13:21

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