# Confusion between TWO shear stress formulas

There are two formulas that I have encountered so far for shear stress in my Engineering class.

1. Let S be shear stress, F be force, and A be cross sectional area:

S = F/A

1. Note: This formula is for circular cross sections. Let S be shear stress, T be torque, C be max radius (for maximum shear stress), and J be polar moment of inertia

S = TC/J

When do I use which formula? So far 1. has been used when dealing with forces perpendicular to longitudinal axis and 2. has been used when a twisting force is applied (torque). However, I am doing a question and am stumped because they using 1. to calculate max torque of bolts in a flange. As shown below:

Why don't they use S = TC/J like every other problem in this chapter???

• The first equation you presented is basically the definition of shear stress, so it holds in all situations. The second equation you presented is the "Torsional Shear Stress Equation". It describes the maximum shear stress in a rotating shaft. Look it up in your textbook and try to clearly understand when it does and when it does not apply to a problem. – Samuel Weir Oct 4 '15 at 19:20

In this case the bolts aren't being twisted, they are being cut or sheared. In this case the entire surface is being sheared with equal magnitude, so the first formula is used.

When a bolt is in torsion there is more shear on the outside surface than the center so only outside surface gets up to the shear limit before the bolt fails. That manes you have to take into account exactly how the shape will deform in order to calculate hoe much torque it can handle.

However, you could still use the second formula with the bolts:

$$C \approx \frac{120\,\mathrm{mm}}2=60\,\mathrm{mm}$$

$$J\approx 4 \left(\frac{120\,\mathrm{mm}}2\right)^2\,\pi\,\left(\frac{14\,\mathrm{mm}}2\right)^2\approx 2.2 e6 \,\mathrm{mm}^4$$

$$T=\frac{S\,J}{C}=\frac{90 \mathrm{MPa}\,\ 2.2 e6 \,\mathrm{mm}^4}{60 \mathrm{mm}}\approx 3325 \, \mathrm{N\,m}$$

This matches the answer given in the solution.