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# Bearing stress calculation, and why (caution: MATH ahead)

#### SShep71

##### Hot Rolled
Here I am working on some bearing stress calculations for a Strength of Materials class. The way being taught is to look at the contact from a "projected" approach, i.e. a cylindrical bearing projection will be calculated as simply rectangular length * width.

I have looked for an explanation, and asked for an explanation but all I get told/find is the projection method is industry standard because the actual method is too complex. THIS I have a problem with. Now, I know that in consideration of surface finish, bearing clearances and theoretical contact area being a simple tangent line of length L this is a simply complicated answer. I am approaching this from the perspective of contact area bearing surface under load after break in period (where the two surfaces have had time to wear together creating more than a theoretical contact). Wouldn't bearing stress being, Tau=(Shear force/area), be the shear force divided by the length multiplied by the contact area. The contact area being the length multiplied by half the circumference.

For example, a 20mm diameter shaft passing through a 15mm wide member according to what I can find the bearing surface area is calculated as 20mm x 15mm = 300mm^2. This is a 2 dimensional area calculation.

I argue that the calculation should, for the same setup, one half x (2 x π x radius of shaft) x length.
[1/2 x (2 x π x 10mm) x 15mm] = 471.238mm^2. This is a 3 dimensional calculation and more representative of the two surfaces in contact with one another under load.

I would expect that this is obviously more complicated than either method, as it would have to involve Integration to account for the round surface. I just cannot find the information, everything keeps going back to the "industry standard" method. Is the industry standard done that way because it is a "happy medium" between the extremes. I just find it obnoxious that something in engineering can be simplified to that but not explained better. Then again, considering the quality of some of the engineers I work with I can see why so many items are just garbage anymore.

I just don't want to be THAT kind of engineer, I learned all the trades and got all the experience first, now engineering school isn't turning out to be what I thought it was. Any insights here are greatly expected.

This Youtube video is the shortest and most to the point about bearing stress and it is the same reference I used for the calculation. All the other videos I have come across explain it the same way.

#### Ulrich von Ludendorff

##### Aluminum
Certain details are often overlooked not through the negligence of modern engineers, but rather because the use of more precise methods for these standards is not justified, since there is no observable difference between the different approaches in normal conditions of use. Therefore, to observe a measurable difference in practice, one must apply this very precise method to all components, which is done when necessary, mainly for practical reasons.

Take the example of bearings: according to your method, to measure a significant difference in any measurement, you would need to know the exact type of grease used, the metal alloy of the balls, the temperature, the humidity, and many other factors. This approach would quickly become exponentially complex for insignificant results.

#### technocrat

##### Hot Rolled
I totally agree with Quebec, most situations are well within known safety factors. There is no justification to go beyond simple models unless the situation is high performance, where operating as close to edge may confer some specific advantage and other risks are also considered. That gets a whole lot more complicated.
Precision oiled brass bearing operating close to dynamic lubrication limit or a high spec ball bearing in a machining spindle - which way would you go for job surface finish?

#### nt1953

##### Hot Rolled
The actual load profile is very complicated. First of all both the part with the hole and the shaft deflect with load, so belief that the loads are constant or linear is out the window. Loads along the length of the hole are more sinusoidal or parabolic with highest loads at the point of contact and lower loads as you move away from the contact. Loads are also higher at the exit from the hole and lower inside the part. So, simplifying a solution becomes a necessity. What you hope for is that someone has done enough work to compare the simplified solution with reality so that the solution is shown to be adequate. Then you apply safety factors to give yourself and reality a fighting chance.

Generally speaking, if the parts have wear, then there may be more clearance between the parts, and because of that, there may be less contact surface. That would mean higher contact loads.

Real life is complex. Figuring exact solutions using calculus and differential equations in multiple directions at the same time can become daunting. That is why you fall back to models (experience based or finite element). The trick is to know the limitations of the models, and how much faith to put into them.

#### Gard

##### Aluminum
Well said NT1953.
In my opinion, Engineering is the process of measuring something and coming up with a formula that predicts the measured results over the full range of the experiment. The formula is then used to predict future results measured with similar conditions. Attempting to understand the fundamental physics of the situation and develop the math and equations is an excellent first step. However using theory to predict results without measured experience can be dangerous because reality often does not follow your best theory. If there is an industry standard for something, it is reasonable to assume it has worked well enough for lots of people.

Finite element modeling is a great tool but it is not uncommon to use it to design a part then make the actual part and stress it to failure in the real world and find it fails in a manner not predicted by the first finite element model.

If you want to dig into this further look into the subject of Hertz stress. The math is not that hard to calculate stress and deformation when a curved surface is pressed into a flat or another curved surface.

#### jaguar36

##### Hot Rolled
This method (Fbru= P/Dt) is commonly used for fasteners in aerospace structure. Keep in mind though, that it is just a formula to predict a failure, with some reasonable level of confidence. Thats the difference between science and engineering. The goal is not to get some theoretical stress profile that is perfectly accurate, the goal is to be able to design a joint that doesn't fail at the prescribed load. The allowable bearing stress (Fbru) in this case, is not the ultimate tension or compression stress of the material, but it is its own unique value. Joints need to be tested to establish what that value is, and it typically would have different values for different edge distances.

The great thing about this method is that it goes directly back to test data. Therefore it is quite accurate, as long as your test is a reasonable approximation of your actual joint. There are factors that have been determined (by test again) to account for some common things like countersinking and so on. It also is a nice simple formula which makes errors less likely to occur, and makes it easy to check. It also has shown over ~100 years of use, to work well.

It does have some limitations however, first being that you do need the actual test generated allowables for it. You could approximate Fbru based on other allowables, but its not going to be very accurate with out test data. This method also only works for static analysis, it does not apply to fatigue failures. For fatigue failures, much more complicated analysis must be done that does include alot of the things you mentioned.

Finally Fbru is also frequently misused with in FEM analysis as folks will just pull a peak von mises stress from the model and compare it to Fbru.

#### gbent

##### Diamond
Per your example, the only loads that can be resisted are loads normal to the surface. If you do a free body of a segment of the hole near 90 degrees from the direction of the load, the surface is near parallel to to the load, so when you figure the force vector there is very area to carry the load.

The most exacting analysis becomes flawed once you add the deflection.

It all boils down to "The only difference between theory and practice is in theory, there is none".

##### Diamond
We talkin' rolling element bearings, plain lubricated bearings, limits of constraint, deflection considerations, solid Vs porous bearing materials, shock loading, - there's so many variables in real bearing applications, theoretical problems have to be limited, unrealistic, and hedged around with painstaking qualifications. So much so, experienced shop hands delight in suffocating an engineering intern's innocent question with an infinity of "what abouts" and "yabbuts."

I'm guessing from your problem statement you're referring to plain bearings self-lubricated by a hydrodynamic film. A projected load area bounded by an arc length of the journal diameter times the bearing length and other common assumptions are but rules of thumb that works so well in preliminary calculations, few analylists bother to go much further.

Further discussion, if competently developed, leads to 300 level ME course material. I regret I'm too long retired from the trade. I suggest you look up textbooks on bearings and lubrication. I'll look around and maybe suggest a few in a future post.

#### SShep71

##### Hot Rolled
All these replies are excellent and more along the lines of what I was hoping to hear. I am a firm believer that the mathematical approach to some instances is far inferior to actual experience and physical testing, I am just trying to follow the process in where math first: practical second. I would have liked to have seen the math but after digging out my old school texts from the early 70s (yes I know some of you used these books when they were brand new), I found that in "Formulas For Stress and Strain" the 5th edition, that this area of research is listed out by reference. I looked up a few references and I saw the "Hertz" approach as well as a significant study done by Whittemore and Petrenko related to deflection in spherical races. I saw there was a significant amount of differential equations and partial derivatives involved but that the results were minimal and there was significant signs of diminishing returns.

Essentially what I am saying is I was wrong in my approach in that the complications added aren't really worth the payout, so the industry standard it is for me. The
method of visual projection seems to be the most valid method.

Awesome job all that replied.

#### technocrat

##### Hot Rolled
do the calculations, use your experience, trust your intuition, communicate with other stakeholders.

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