# Strength of a chain link

FEAnalyst
TL;DR Summary
How to calculate the stresses in a chain link?
Hi,

I am interested in the topic of hand calculations of chain link's strength. I am talking about a regular industrial chain with hanging weight. From what I've read, there are 3 potentially possible approaches:
- Lame's problem (circular cross-section has to be replaced with equivalent rectangular one) - I've even found a formula based on this approach in a Polish book but it gives me very low stresses when compared with FEA
- curved beam theory - not sure how to apply it to this particular case
- Hertz contact stress - as above plus I don't know if it can be used in this case

Which of these methods would you use ? Or maybe I have to combine two of them to get a full picture of the stress distribution (keep in mind that I'm mainly interested in the maximum stress).

Dr.D
I'd go straight to Seely & Smith, Advanced Mechanics of Materials, 2nd ed, p. 185 ff and again p. 578 ff. The discuss chain links in detail.

jim mcnamara, berkeman and FEAnalyst
FEAnalyst
I'd go straight to Seely & Smith, Advanced Mechanics of Materials, 2nd ed, p. 185 ff and again p. 578 ff. The discuss chain links in detail.
Thank you for the recommendation. I've found this book and it indeed contains a short chapter on chain link strength. The formulas there are very simple:

However, the stresses that I get from the numerical analysis at equivalent locations are orders of magnitude higher than those predicted by the equations from the book. I suppose that the approach discussed there simply doesn't take into account the contact stresses that are predominant in FEA results. Thus, some kind of combined approach mentioned in my original post (or just the Hertz approach) could be the right choice for such a comparison. What do you think about that ? Can you provide some further suggestions ?

Mentor
Exactly where are the FEA stresses too high? The stresses shown in the copy of Fig. 111 from Seely & Smith in the above post do not include Hertzian contact stress, while FEA might or might not, depending on how the model was built.

Note that Seely & Smith page 185 discusses the fully elastic case, while on page 578 the fully plastic case is discussed. The real world case will be in between, as discussed in the last paragraph on page 187. The discussion in Seely & Smith implies that, while elastic FEA analysis of chain links might be useful as an academic exercise, it is not very useful for real world calculations.

FEAnalyst
Exactly where are the FEA stresses too high? The stresses shown in the copy of Fig. 111 from Seely & Smith in the above post do not include Hertzian contact stress, while FEA might or might not, depending on how the model was built.
Thanks for the reply. The FEA model utilizes symmetry in 3 planes (so only 1/8 of each link in pair is modeled) and includes frictionless contact. Interestingly, the stress at point B (section h) seems to agree with the analytical solution in one link but not in the other (where it’s 4,5 times higher in FEA).

Would it be possible to account for Hertzian stresses in the analytical solution ? I’m not sure if any of the standard contact cases discussed in literature fits here due to the curvature of the bodies in contact.

Mentor
An analytical solution that includes Hertzian contact stress would apply only to a perfectly brittle chain link, such as one made of glass. Real chains are ductile, so while elastic FEA analysis of chain links might be useful as an academic exercise, it is not very useful for real world calculations.

If you are determined to play with academic exercises, keep in mind that Hertzian contact stress calculations assume a linear elastic system. In a linear system, superposition holds. You calculate the bending case separately from the Hertzian contact stress case, then superimpose the two solutions.

FEAnalyst
An analytical solution that includes Hertzian contact stress would apply only to a perfectly brittle chain link, such as one made of glass. Real chains are ductile, so while elastic FEA analysis of chain links might be useful as an academic exercise, it is not very useful for real world calculations.

If you are determined to play with academic exercises, keep in mind that Hertzian contact stress calculations assume a linear elastic system. In a linear system, superposition holds. You calculate the bending case separately from the Hertzian contact stress case, then superimpose the two solutions.
Thanks again, that’s exactly what I would like to do. I’m not going to design any actual chain, I only want to do this study for educational purposes. So I can assume linear elastic behavior. I just wasn’t entirely sure if Hertzian stress can be directly added to normal stress obtained from equations given in the book. But it should be possible if it’s done in the same spot.

Now there’s only one doubt left - how to solve this particular Hertzian contact problem ? In various books there are formulas for cases like 2 spheres, 2 cylinders, sphere on a plane, cylinder on a plane and so on. But none of these fits curved bars like chain links in contact. Maybe the case with two cylinders with perpendicular axes would be the most appropriate, at least up to points where curvature starts. Unfortunately, book chapters about the strength of chain links (like the one in Seely & Smith book) focus only on bending stresses, contact is not discussed there. I haven’t found any details regarding chain link calculations with contact included so it seems that only general cases mentioned above can be used.

jrmichler
FEAnalyst
I used the Hertz contact case involving two cylinders with perpendicular axes and the results are in very good agreement with FEA. I suppose that the curved beam theory would also be helpful but for sections away from the contact region.