Is Roark's Formula Suitable for Shaft Hertzian Contact Stress Calculation?

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Discussion Overview

The discussion centers around the calculation of contact stress for a shaft supported by a cylindrical hole in a diaphragm. Participants explore the suitability of Roark's Formula for this calculation, particularly in the context of determining whether the diaphragm can withstand the radial load from the shaft. The conversation includes considerations of different formulas and assumptions regarding the geometry of the shaft and hole.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks a more accurate formula for contact stress than the basic bearing stress formula, specifically for a shaft in a cylindrical hole.
  • Another participant questions the relevance of stress on the inner diameter if the shaft and hole diameters are equal, suggesting a potential misunderstanding of the scenario.
  • A clarification is provided that the shaft is under radial load and supported by the hole, indicating that the plate will experience contact stress that could lead to failure if too thin.
  • A participant suggests that the scenario could be modeled as a cantilevered beam under load, given the described conditions.
  • One participant argues that the original poster may be overcomplicating the problem and emphasizes the importance of bearing stress rather than focusing on the zero denominator issue in Roark's formula.
  • A later reply proposes using Roark's formula with an assumption that the shaft diameter is approximately 99% of the hole diameter, considering this a conservative approximation to avoid an infinite Kd.

Areas of Agreement / Disagreement

Participants express differing views on the complexity of the problem and the appropriateness of using Roark's formula. There is no consensus on the best approach or formula to use, and the discussion remains unresolved regarding the suitability of the proposed assumptions.

Contextual Notes

Participants have not reached a consensus on the definitions or assumptions regarding the diameters of the shaft and hole, nor on the implications of using Roark's formula in this context. The discussion includes unresolved mathematical considerations related to the formulas being discussed.

servaa
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Hello

I am trying to calculate contact stress for a shaft on a hole. The purpose of this calculation is to see if the diaphram on which the hole exists is thick enough to handle the shaft radial load. Easiest approximation would be to use a simple formula for bearing stress (stress = force / (t*d)), but more accurate approximation is required.

A formula for cylinder on a cylinder seems to be available from many different sources, but I can't find a formula for cylinder in a cylindrical hole. The closest thing I've come up with is from Roark's Formulas 7th Edition Ch.14.4 Table 14.1 Case 2.c "cylinder in a cylindrical socket", but this equation results in a zero denominator if the cylinder and hole diameters are equal.

Can anyone help me with this..?

Thanks
 
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I might not quite get what you're saying, but if the cylinder and the hole have the same diameter why would the hole have any stress on it's inner diameter? Are you talking about a press fit of some sort?
 
I guess I should've explained my question more clearly: a shaft is under radial load on one end, and is supported radially on the other end by a hole in a plate. I believe the plate will be under some contact stress, and it will fail if it is too thin..? The fit between the shaft and the hole is a close running fit.
 
A picture would really help a lot.
 
servaa said:
Hello

I am trying to calculate contact stress for a shaft on a hole. The purpose of this calculation is to see if the diaphram on which the hole exists is thick enough to handle the shaft radial load. Easiest approximation would be to use a simple formula for bearing stress (stress = force / (t*d)), but more accurate approximation is required.

A formula for cylinder on a cylinder seems to be available from many different sources, but I can't find a formula for cylinder in a cylindrical hole. The closest thing I've come up with is from Roark's Formulas 7th Edition Ch.14.4 Table 14.1 Case 2.c "cylinder in a cylindrical socket", but this equation results in a zero denominator if the cylinder and hole diameters are equal.

Can anyone help me with this..?

Thanks

It sounds like from what you are describing, a cylindrical hole with a rod in it with a tight fit (such that the diameters are equal) and an end load, would behave as a cantilevered beam.

CS
 
Personally I think you are over complicating this. I would think that the bearing stress/tear out stress is what you really need.

You shouldn't run into a 0 in the denominator for Kd because the two diameters are not equal. Kd may be very large, but it is definable.
 
Here's a picture of the problem. The shaft is supported at two points by diaphragm 1 & 2, and a downward force is applied at the left end of the shaft. The point of interest is whether the diaphragm is strong enough to handle contact stress between itself and the shaft, not how the shaft would behave.

I have decided to use the formula in Roark's with an assumption that shaft diameter is ~99% of the hole diameter. I guess this would be a conservative approximation, and Kd will not be infinite. Would this be a reasonable assumption?
 

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