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Calculating Torque Limit on a hollow shaft type component

  1. Sep 30, 2015 #1
    I am designing a custom socket for an application at work and I am concerned about the wall thickness in the area marked by the red box (see attached drawing). The socket will be used to torque a nut to 500Nm.

    I have a degree in mechanical engineering and I can do stress calcs on straight forward shapes but I struggle when things get more complex or are composed of multiple shaft sizes and features.
    I normally either design big, ignore or pass the work on to a stressman but I would like to understand how to do it myself.

    Could someone explain to me the steps you would take to analyse the component to ensure that it will be able to withstand the applied torque?

    Thank you very much for your help.

    https://slack-files.com/files-tmb/T093Q5WNN-F0BJ303MW-77e14a6610/component_1024.jpg [Broken]
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Sep 30, 2015 #2


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    Hi Steve,

    Ordinarily my analysis "order of operations" will go something like this:
    1. First order- Is the part simple enough to analyze via traditional means using analytical equations (e.g. Roark's Formulas for stress & Strain)? If so, I do this and stop (rarely happens).
    2. Second order- Does the part have relatively simple confounding factors such as fillets, notches, or other features that can be analyzed through analytical equations in combination with derived stress concentration factor (e.g. Roark's Formulas for Stress & Strain + Peterson's Stress Concentration Factors)? If so, I do this, and then usually follow up with FEA in case the simplified stress concentration factor missed something.
    3. Third order- If the two above yield no obvious analysis path, best bet is to run an FEA analysis.
    In most cases (maybe because I'm spoiled with the software availability) I will likely run an FEA analysis on nearly everything if I'm curious about deformation/stress/temperature, but honestly I'm rarely designing something that can be easily analyzed via Roark's any more. Even in these cases though, it's always a good idea to try and make a simplified analysis through an independent means (like analytical analysis or experimental testing) to see if your FEA model is making sense.

    In your case I'd say the design shown probably falls somewhere between 2 and 3, but you might be able to make some assumptions to make an educated guess at where it might fail. Roark's unfortunately only has equations for axially symmetric torsional anslysis, so by itself we can't use it for what you seek. On the other hand it might be possible to use a set of equations from Peterson's Stress Concentration Factors, I looked through my copy of the book and found a table on page 169 (Google Books Link Here, picture attached) which is titled "Chart 3.15a, Stress concentration factors of a torsion tube with a shoulder fillet (Rushton 1964; ESDU 1981)" The geometry pictured isn't too far off from what you're doing, so it might be this could give you your second-order analysis.

    Even with the second-order analysis, I'm thinking you'll need to follow-up with an FEA analysis anyway. Usually stress concentration factors for torsion or tension assume a long shaft, so the analysis area is far away from the ends; your design probably won't meet this criteria. Also, you might be incorrectly assuming where the design will fail. Looking at the design, it might fail at the thin wall near your hex wrench flats. An FEA analysis with the right boundary conditions could help you find the weak spots.
  4. Sep 30, 2015 #3


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    For a total understanding of the torque loading stress analysis you should Goggle " shaft torque stress analysis" and " hollow shaft torque stress analysis" where you will find a number of references describing the theory and method of stress analysis to be utilized. For a quick analysis of your shaft refer to the below site that I located under the second category.

    For a quick analysis see: http://www.amesweb.info/Torsion/TorsionalStressCalculator.aspx

    At the same time, in order to give you some guidance, I am offering you the below of what appear in my opinion, to be the critical areas of analysis for your tool; however, these should not be considered to be a complete analysis as required by your organization. Regardless of these recommendations, any analysis you perform should be submitted to your organization's "stressmen" for approval.

    As for your area of concern, as long as the analysis shows the shearing stress along the shaft is acceptable and there are no bending stresses on the tool shaft then the transition section should be safe as well because it has a smooth radius transition which at any point has a greater effective wall thickness than the rest of the shaft.

    At the same time, you should also analyze the driving hex end of the tool. You could use the average of the flat 36 diameter and the points' 42 diameter for that analysis; but, I strongly recommend that you use only the 36 OD for that analysis to be safe. As a general recommendation, unless you need the tool's 32 bore hole through the entire length of the tool in order to insert the tool around a shaft, I would recommend that you consider only drilling a blind end hole from the socket end that stops at a point 40 or more from the hex end to provide a solid center under the hex drive area.

    The next area that you should be concerned about is the socket. Unless you have a prior tool design that specifies the socket design that you are using there are two areas of concern.

    First, the thickness of the sockets back wall should be analyzed for the applied tool torque load shearing stress at the ID of the of the socket. The applied torque Ss is:
    Ss = the socket ID /2 x torque / t wall x ID circumference.
    Note: If you had a sharp corner at the shaft connection to this wall instead of the large translation radius from the shaft to the wall; then, the critical point of analysis would be at the shaft OD. In this case, that does not appear to be an issue but for a thorough analysis you might want to include that analysis point as well.
    I also recommend that you revise your drawing's longitudinal length dimensions to specify the actual thickness of the back wall of your socket since this is more critical than the overall length of the tool.

    Second, for the the socket's cylindrical wall to have an Ss equal to that of the shaft, the socket wall thickness should be at least:
    t socket = t shaft x shaft OD / socket OD

    Third, analyze the shearing stress on the socket splines due to the torque loading. The formula for that is:
    Ss = (torque x socket ID/2) / (no. of splines x one spline base area)
    ie, Ss = (500Nm x 63 / 2) / ( 9 x 5.9 x 25)
    If the shearing stress beyond the materials allowable, then a potential solution is to increase the depth of the socket and therefore increase the base area of the splines; however, if this length is limited by the length of the application shaft splines, then an alternative is consider a hiigher strength material for your tool.

    I hope this helps; and, I realize that you may well be aware of some or most of what I have included but I wanted to be as complete as possible.

    PS: The previous post was added while I was putting all of this together and with regard to the use of computer stress anlaysis, since the majority of my career was spent before it availability, my analysis have primarily been using classic theory methods. In the more recent years I the finite element method for analyzing complex segments of a component; but even then, because I was there when finite element analysis was first introduced, and errors in errors in selecting the correct meshing often resulted in wrong results, I still utilize classic analysis as much as possible to verify its results. Just as a bit of advice in that respect, while generally there is no way to compare stress values in complex shapes, deflections determined by each method are always a solid basis for accurate comparison for verification.
    Last edited: Sep 30, 2015
  5. Oct 1, 2015 #4
    Wow, thank you both for replying with such detail.
    Those answers were exactly what I was expecting/hoped for!

    JBA - I really want to get to grips with the classical methods for analysis of basic and semi-complex components, as you said this will help get an initial understanding and will help to validate any further FEA modelling.

    Mech_Engineer - With the FEA methods, the one sticking point I have is load placement and boundary conditions. Do you have any advice on how to constrain parts like the example above or how to determine where the loads will be applied (without experimental work). For example, I could apply a torque but should this be applied to all faces of the socket flats or at the end..? Also what would be your approach on constraining the opposite end?

    Thanks again!
  6. Oct 1, 2015 #5


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    Regarding FEA boundary conditions, if I were doing the analysis I would probably include the parts that this drive socket is interfacing with as rigid (zero deformation, infinite stiffness) solids. Make sure those solids have similar gaps in the interface to what might be expected in the real parts. From there I would apply a torque to the "drive socket" which is driving your hex wrench flats, and apply a fixed condition to the splined part being driven, and maybe a cylindrical support on the part itself to enhance convergence.

    Without the mating parts being modeled, your next best bet is applying a 500 N-m moment to the 6 wrench flat faces, and either fixed condition or maybe non-linear "compression only" conditions to one side of the drive splines in your socket. Additionally you'll need some kind of axial constraint (maybe a cylindrical axial-only constraint) to constrain rigid-body motion in the axial direction.

    As JBA and I pointed out, I think your failure point will be at the sharp corner at the base of the hex wrench flats where the wall is thinnest. You should consider putting a fillet there. Also, make sure you know what material you're making this part out of, what strength certification is required, and what your acceptable safety factor is. If this part is going to be driven by a high-impact device like an impact wrench, you will want a high safety factor.
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