Determining max shear stress

Question

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?

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

Answer 1

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.