1. Stress on bolt

I am trying to determine the max torque before failure on a 5/16” dia bolt (pinned connection). It looks to me the bolt is in single shear at two points. See the attached image. I made an attempt at it. See the other attached image.

2. I suggest you do a quick goggle for "bolt torques". It will give you a number of references and tables for bolt torque limits.

3. First, this appears to be homework. Is it?

It appears from your question that you are talking about the maximum tightening torque on the screw before failure. Most engineers on this forum will assume that's what you are talking about. But your worksheet appears to be a calculation of the maximum torque that can be applied to the 1.25 dia shaft before the 5/16 bolt fails in shear. Your diagram could be read either way. Which is it?

If in fact you are trying to calculate the maximum torque on the shaft, I have some comments.
- How do you know the bolt will fail before the shaft or the support bracket do? It might not.
- Are you sure you used the right information to calculate the cross sectional area?
- In my opinion you made two other errors in your calculations based on your understanding of the nature of the dynamics of free bodies. But their effect was to cancel each other out.

4. This is not homework. It is just some extracurricular work relating to something at work. Yes I am trying to calculate the maximum torque that can be applied to the 1.25 dia shaft before the 5/16 bolt fails in shear. I don't know what will fail first, but my first thought was that the 5/16" bolt would. After that I thought it would be a good idea to try and do the calculation (I like to keep my mind fresh b/c my job doesn't require calcs like this). To begin any problem the first step is the free body diagram. Would you say mine is correct?

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