# Roymech's Fatigue Calculator

### Description

Fatigue considerations are important because the consequent failure is generally sudden and at a stress level much lower than the ultimate stress. Fatigue properties of materials are generally determined by producing Wohler /S-N Plots. These are simply plots with stress as the vertical axis and log (number of complete stress reversals) as the horizontal axis. A number of material specimens are tested and the points at which they break are plotted on the S-N curve.

The fatigue strength is the maximum completely reversed stress under which a material will fail after it has experienced the stress for a specified number of cycles. (The strength is accompanied by the number of cycles).Fatigue Strength (fixed number of cycles) = Sn. It is a useful property of steel (and titanium) that when the stress level fall below a certain value the specimen is effectively never likely to fail. Generally other materials do not exhibit this effect. The Fatigue limit is the maximum completely reversed stress for which it is assumed that the material will never fail regardless of the number of cycles. Fatigue Limit = S'n

Experiments have shown little direct relationship between the fatigue limit and the yield strength, ductility etc. However some relationship between the fatigue limit and the tensile strength has been established for un-notched polished specimens tested using the rotating beam method. This method loads the specimens by reversed bending.

Purpose of calculation:

- Produce Wohler /S-N Plots and Goodman Diagram Plot
- Determine cut off stress.
- Determine the fatigue strength of a material.
- Determine the fatigue limit.
- Determine the relationship between the fatigue limit and the tensile strength.

**Calculation Reference**

Many references have been considered in producing this calculation. I have named the calculation after the Roymech website in recognition of his great website.

- https://roymech.org/Useful_Tables/Fatigue/Fatigue_index.html
- http://materials.open.ac.uk/mem/mem_mf.htm
- http://www.fatiguecalculator.com/
- Petersons Stress Concentration Factors

**Calculation Procedure**

*1) Input Applied Stress*

*2) Define Material*

- Material ultimate tensile strength.
- Material yield
- Type of loading
- Mean low cycle fatigue limit factor ('mean' represents a 50% survival limit).
- Number of cycles related to mean low cycle fatigue limit
- Mean high cycle fatigue limit factor ('mean' represents a 50% survival limit).
- Number of cycles related to mean high cycle fatigue limit
- Factor of safety

*3) Calculate Low Cycle Fatigue Limit*

- Fatigue limit at zero mean stress
- Allowable alternating stress - Soderberg (conservative)
- Allowable alternating stress - Goodman (steel, aluminium & titanium)
- Allowable alternating stress - Gerber (less conservative)
- Allowable alternating stress - Smith (cast iron & magnesium)

*4) Determine Modifying Factors for High Cycle Fatigue Limit*

Size Factor - The endurance limits of specimens have been observed to vary with their size. This is possibly related to the probability of a high stress interacting with a critical flaw within a certain volume, i.e., when the volume is large there is a higher probability of failure. Hence, when the size increases, the endurance decreases. Alternatively, since there appears to be a more pronounced size effect in reversed bending and/or torsion than in the reversed axial loading situation this suggests that the stress gradient at the surface is partially responsible for the size effect.

Surface finish factor.

- As Forged
- Hot Rolled
- Rough turned (peak to valley height 30μm)
- Turned/rough ground (peak to valley height 12μm)
- Fine turned (peak to valley height 6.5μm)
- Fine ground (peak to valley height 2.5μm)
- Lapped/rough polished (peak to valley height 1μm)
- Mirror polished

Probability of survival factor - The basic values are mean values implying a 50% survival rate. To enable determination of design strength values with a higher survival rate i.e. 90% upwards then the indicated strength values must be reduced.

- 50.0000% 1,000
- 90.0000% 0.897
- 95.0000% 0.868
- 99.0000% 0.814
- 99.9000% 0.753
- 99.9900% 0.702
- 99.9990% 0.659
- 99.9999% 0.620

Miscellaneous Factor - This factor is a general factor to allow for any other factors:

- corrosion
- electrolytic plating
- metal spraying
- cyclic frequency
- fretting corrosion
- radiation effects on materials.

Stress concentration factor - The theoretical stress concentration factor Kt of a section at subject to an internal stress resulting from a change of shape and/or geometry :

Kt = Highest value of stress at a discontinuity / Nominal stress at local minimum section

This value does not allow for the sensitivity of the material to stress concentrations.

Useful references for calculation of Kt :

- /repository/strength/stress/
- Machinery's Handbook 18th ed.
- Mechanical Engineers Data Book (J.Carvill)
- Machine Design-Theory & Practice A.D Deutschman, W.J Michels, C.E. Wilson
- The calculators in the links below (ETB and Stacie Glass)

Notch sensitivity factor - The material notch sensitivity value "q" is used to quantify the sensitivity of a material to local high stresses. The notch sensitivity of a material is a measure of how sensitive a material is to notches or geometric discontinuities.

- High notch sensitivity for Brittle/Hard Materials q=1.0
- Very perfect material is significantly damaged by addition of a notch
- Low Notch sensitivity for Soft Ductile Materials
- Material with a lot of flaws not damaged much by one more.
- Petersons estimate for steel
- Fatigue notch factor

*5) Calculate High Cycle Fatigue Limit*

Fatigue limit at zero mean stress

- Allowable alternating stress - Soderberg (conservative)
- Allowable alternating stress - Goodman (steel, aluminium & titanium)
- Allowable alternating stress - Gerber (less conservative)
- Allowable alternating stress - Smith (cast iron & magnesium)

*6) Construct SN Diagram*

- Use Excel curve fit functions for SN curve
- Calculate allowable alternating stress for known number of cycles
- Calculate allowable number of cycles for known alternating stress

*7) Construct Goodman diagram*

**Calculation Reference**

Petersons Stress Concentration Factors

Machinery's Handbook 18th ed.

Mechanical Engineers Data Book (J.Carvill)

Machine Design-Theory & Practice A.D Deutschman, W.J Michels, C.E. Wilson

The calculators in the links below (ETB and Stacie Glass)

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