Cycle Life Calculation for Specified Fatigue Strength

Apply your equation for 2 pairs of values S1&N1 and S2&N2 you obtain:

log S1=b*logN1+c and logS2=b*logN2+c

After a subtraction you get: log(S1/S2)=b*log(N1/N2) which leads to b= log(S1/S1)/log(N1/N2). Knowing "b" you can from any equation compute "c".

But, what about the graph. Why values of graph & equation don't match.

I got this equation from Shigley's book on 'Mechanical Engineering Design', there it is also mentioned that the graph is a straight line.

Good question, (I think all engineering students and practicioners wonder the same at some point in their lives; preferably earlier rather than later). I'm glad you ask.

1) Keep in mind that the log 'law for fatigue you found in your book is derived from experimental data. It is a gross (but reasonable) simplification.

2) The 'curves' are also derived from (other) experimental data, or some fitting/ interpolation of said data; by the way, there is so much scattering of experimental fatigue data, that a really good Woehler graph will present zones (50%, 90% etc), where failure may occur with said probability. As such, do not take them literally as 'lines': in this context a simple line means nothing.

3) Data from data differs, often significantly/ overwhelmingly, as do materials, test conditions etc etc.

4) By now it should be obvious that there is no precise correspondence between the eqs and curves. This simply reflects the every-day compromises engineers have to live with. It's never as simple as straight mathematics, sorry.

The kind of expertise to help you through this ambiguity (and the insights to resolve it) will come from understanding the methods of measurement, so I suggest you study these. I.e. from a book on experimental mechanics... (browse the net?)

By the way, I generally wouldn't recommend Shigley's book for trying to build an understanding of things.

And, if the above do not really apply to your case, perhaps you have found a bug in the book. It happens.

Hope this helps.

1- It is approximated with a straight line in log -coordinates.

2- as was mentioned already the line has a position depending on the "confidence" degree accepted at the design. The lower the probability the lower the stress level

3- Wöhler curves are ONLY for constant amplitude. In real life this occurs very very seldom so that a correction considering the load life profile have to be introduced.

With respect to the equations you found in the book, their deduction was done as I indicated between the 2 pairs at 1E3 and 1E6 cycles.

The "0.8*Sult" value for olygocyclic fatigue is not always true it is valid only for some materials.

Actually I want to find the torsional fatigue life of EN47 rectangular strip (UTS 133 kgf/sq mm). Since I don't have any experimental data on it's torsional endurance limit (at 1000000 cycles) & torsional fatigue strength (at 1000 cycles) I used empirical relations with a certain safety margin (after a lot of searching through books & internet sites)

σ endurance = 0.4 X σ UTS

ζ endurance = 0.577 X σ endurance

If I use formaula already mentioned in Shigley's book. Then by assuming :- ζ fatigue = 0.8 X ζ UTS (at 1000 cycles) (Sometimes it is taken as 0.75 X ζ UTS)

I get cycle life of around 22000 cycles.

If I consider ζ fatigue = ζ UTS (at 1000 cycles), then cycle life is around 38000 cycles.

Remember, these life are calculated based on induced fatigue stress obtained by Modified GOODMAN's Diagram on variable stress.

Interestingly I found one bulletin on internet site. https://www.ideals.uiuc.edu/bitstream/handle/2142/4353/engineeringexperv00000i00316.pdf?sequence=3

Here the tests are performed on a SAE3140 steel. But still I am considering the results.

The conclusion of the tests indicate that for an un-notched steel bar subjected to cyclic loading the actual results are more close to a theoretical equation given by Moore, Jasper & McAdam (based on hypothesis of constant magnitude of range of stress). The equation is :-

ζ f = (2 X ζ endurance) / (1-r)

Where, ζ endurance = to be decide experimentaly at 1000000 cycles (I took it as 0.4 X σ endurance)

r = ζ max - ζ min

ζ f = torsional fatigue strength at specified r & ζ endurance

Now by putting this value of ζ f in Equation from Shigley book (with assumption of ζ fatigue at 1000 cycles = 0.8 X ζ UTS)

Life is coming around 54000 cycles.

At the end I am considering a safe life of 20000 cycles.

Some further details.

1-Torsional stress in a square section τ= Mt/Wt with Wt= K*h*b^2

K is function of ratio h/b and has following values:

h/b 1 2 4 8 10

K .208 .246 .282 .307 .312

2- σ-1 for traction/compression = .385 Rm+30 in MPa

3- Variation of fatigue limits values (endurance) 8% = standard deviation/average

4-

The Coeff of safety is the ratio OA/OP where "P" is the loading cycle of the part. Even if you have not a stress concentrator the surface has a roughness which will generate peaks and become origin of micro-cracks. You should de-rate values for stress with about 10 to 20% according to roughness magnitude.

5- Influence of loading

As above figure shows that the loading which leads to the Wöhler curve is a constant amplitude type, obtained values are the lowest.

For real world behaviour estimation one should consider the summing of partial damages for each load amplitude and for each number of cycles at a given amplitude. Above curves and loading profiles can be used as a guide line.

I hope this will allow you to make a better estimation of the life expectancy of the torsional spring you design.

Thanks,

I have already considered K (function of h/b) & for my case it's coming 0.305.

Since my torsion plates are ground I have taken Ka as 0.9 & also considered a size factor for rectangular section kb as 0.85 & reliability factor of 0.84 for 99% reliability.

But, I don't quite understood your point no 2 (what is Rm?)& point no 3 (how to account for variation of endurance limit? where do this 8% come from?)

And point no 5 (what is P?) Please elaborate the graph.

My loading is repeated type i.e. from 0 to maximum.

Rm= mechanical resistance = ultimate stress

All data are given for values based on average in real world there is a dispersion which is defined by variation = standard deviation/average. If you want to be on the safe side the 8% have to be deducted i.e. apply a .92 Coeff.

p is the range of variation for the load= load min./load max.

Are you sure that load is always the same? If yes then use p=1 and work with the corrected Wöhler curve.

What is the application? Have you several parts working in parallel? If yes do not forget the sliding between parts. A special attention has to be given to the ends transmitting the torque to the spring in order to avoid stress concentrations.

Yes, there are 8 torsion bars, which are clamped together & twisted. There is a hole at each end, for clamping it to the device & in between there are three mechanical clamps which hold the bars together externaly.

I didn't consider sliding as I don't know how to put it in calculation - pls guide me.

Sliding will occur even if it is in the micro range. the consequence is friction and hysteresis. It is thus required to lubricate the contact faces. If galling occurs it is a source of micro cracks and earlier fatigue. The lubricant can be grease or a solid layer which will also protect against corrosion. Is is a suspension spring for a vehicle? VW used this solution over 40 years ago.

Take care that all your computations are only for torsion valid if traction appears due to torsion then you have to consider it via the von mises stress. By the way the factor 0.577 is also derived from the von Mises stress it is 1/3^0.5.

If galling occurs it is a source of micro cracks and earlier fatigue.

My application will only cause torsional fatigue. But, is there any formula to account for this earlier fatigue due to sliding contact & galling.

What is your application?

Actually it's an old machine where torsion bars are used to assist a motor while lifting heavy load. Earlier there was a torsion angle of 55 degree. But, now we are changing it to 60 degree. So, I want to calculate the life of the torsion bar after increasing the twisting.

Send the data : loads (not only amplitude but also if it is alternative or pulsating and which is the mean value for the cycle) + geometry + steel mechanical properties and results and I shall have a look and tell you how high the risk is. Of course if you want. Use since it is not interesting for every body the private message channel.

The solid lubricant I meant can be a wax. With respect to the load you mentioned a constant load as amplitude do you always have the same load to lift? It is an assistant , how do you generate the assistance? Pre load the spring and then connect the Motor? or?

Dear pc,

You are asking a very interesting question, and I have the desire to get some informations related to fatigue strength especially in piping systems. Do you have a soft copy of the Shigley's book on 'Mechanical Engineering Design'? or at least the chapter/or pages related to the subject?

With my best regards,,,

I am afraid there is no mention of fatigue strength of piping system in the book. I am also not sure whether the same method is followed for the piping system.

But, CR4 is a great place to find someone who can help you with it.