# How to calculate stress in a cylindrical-uniaxial-loaded bar with sudden changes in its diameter

How can I calculate the von Mises stress in the cross-section of a bar like this that is axially loaded?

• If it was a uniform bar, how would you approach it? Then, consider each section... Nov 29, 2018 at 15:41
• I would just divide the load [F], over the area [A]. But I dont think it is right to apply that approach to the sections where the area where I'm applying the force is smaller than the diameter of the cylinder. Nov 29, 2018 at 15:46
• Note that I'm particularly interested on how stress behaves around the steps of the bar, not so much on the overall bar itself. Nov 29, 2018 at 15:49
• How sharp are the corners inside those steps? Not as critical in tension as bending, but stress concentration may be an issue if they are very sharp... Nov 29, 2018 at 15:54
• @JonathanRSwift They are not rounded at all, they are as sharp as the CNC tool that made the piece allowed to. But those are the kind of problems I'm concerned about. Nov 29, 2018 at 15:57

Stress Concentration factors on the internal corner of a Stepped Shaft are well understood, and have some standardised calculations/equations: I'd recommend this excellent online calculator, which shows all the background equations further down the page. This allows you to calculate the stress concentration factor in various loading conditions.

https://www.amesweb.info/StressConcentrationFactor/SteppedShaftWithShoulderFillet.aspx

You will see for yourself how reducing the fillet radius towards zero drastically raises the stress concentration factor, and you should try to remove the requirement for a 'sharp' corner if at all possible.

Here's an FEA rendering of the shaft illustrated in your question, showing how the root of the smallest diameter is the critical area (N.B. this does have a radius!)

You get two effects near the area of sudden changes in cross section.

1 - Stress concentration and contour lines of stress distribution.

2 - Stress wave propogation.

Sharp corners are stress concentration traps. Anticipating the path of stress and desgning the part's geometry according to that helps.

Both of these items one and two affect the performance of the member significantly, specially under repetitive or shock loading.

There are computer simulation sofware that can analyze these up to a high level of precision.