# Determining max shear stress

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?

$\tau_{max}= \tau_v + \tau_T$" />

• I am not certain the notation you are using for $\tau_{max} = \tau_v + \tau_T$. What is $\tau_v$ and what is $\tau_T$?
– NMech
Apr 16, 2021 at 11:43
• 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.
– NMech
Apr 16, 2021 at 11:47
• @NMech tV is the transverse shear stress and tT is the torsional shear stress Apr 16, 2021 at 11:47
• 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 $\tau_{max} = \sqrt{\tau_v^2 + \tau_T^2}$.
– NMech
Apr 16, 2021 at 11:51
• @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 Apr 16, 2021 at 12:12

In the example you are presenting the shear stresses on cross-section of beam AB are as in the image below.

• 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)

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.

• I couldn't be any more appreciative of your help and your effort. Thank you very much. Apr 16, 2021 at 13:21