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Distributions of load stresses and residual stresses at notches


The fatigue strengh reduction factor K, can be mitigated or eliminated by suitable surface treatments. Analysis of these affects requires the knowledge of the distributions of load stresses and of residual stresses below the surface of notches. This paper describes a simple, approximate formula to determine load stress distributions and residual stress distributions at notches. The load stress distributions by the present approach were compared with finite element analysis under tension, bending and torsion loading. Residual stress distributions by the simple formula were compared with measured date by shot peening. An example of optimization in surface treatments by such analysis is shown.

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a, b, c :

Constants in the load stress distribution

d :

Total depth to which the compressive residual stress extends (mm)

D :

Diameter, Large diameter of filleted shaft (mm)

K f :

Fatigue strength reduction factor, fatigue notch factor

K t :

Stress concentration factor

n :

Exponent in the load stress distribution

N f :

Number of cycles to failure

r :

Radius of fillet or semi-circular notch (mm)

R :

Radius of shaft (mm)

x :

Distance from notch root (mm)

w :

Half width of fillet or notch in their narrow section (mm)


Distance from surface to the point where stress gradient at the surface meets the x-axis (mm)

σa :

Alternating stress (MPa)

σeq :

Equivalent stress (MPa)

σf :

Fatigue strength coefficient (MPa)

σF :

Fatigue strength (MPa)

σL :

Load stress at point x (MPa)

σm :

Mean stress (MPa)

σN :

Net stress (MPa)

σo :

Nominal stress (MPa)

σR :

Residual stress (MPa)

σR,P :

Maximum compressives residual stress or peak compressive stress (MPa)


Half width of the notch in plates or the radius at the narrow section of shafts (mm)


  1. Al-Hassani, S. T. S., 1981, Mechanical Aspects of Residual Stress Development in Shot Peening, ICSP CETIM Pergamon Press, pp. 583–602.

  2. Almen, J. O. and Black, P. H., 1963, Residual Stresses and Fatigue in Metals, McGraw-Hill Co., New York.

  3. Berkovitas, A., 1986, “Variation of Fatigue Notch Factor With Lifetime, Stress Ratio and Temperature,” J. of Engineering Materials and Technology, Trans. ASME, Vol. 108, pp. 179–185.

  4. Braski, D. N. and Royster, D. M., 1966, “X-Ray Measurement of Residual stress in Titanium Alloy Sheet,” in J. B. Newkirk and G. R. Mallett, ed., Advances in X-Ray Anaysis. Vol. 10, pp. 295–310.

  5. Brodrick, R. F., 1955, “Protective Shot Peening of Propellers,” WADC Tech. Report 55-56, U. S. Dept. of Defense, pp. 306–309.

  6. Dietrich, G. and Potter, J., 1977, “Stress Measurements on Cold Worked Fastener Holes,” Advances in X-Ray Analysis, Vol. 20, pp. 321–328.

  7. Fuchs, H. O., 1972, “Regional Tensile Stress as a Measure of the Fatigue Strength of Notched Parts,” Mechanical Behavior of Materials, Vol. 2, Society of Materials Science, Kyoto, Japan, pp. 478–488.

  8. Fuchs, H. O., 1981, “The Strength of Shot Peened Parts, Design Calculations and Specifications,” Niku-Lari, A. ed., Shot Peening, Proc. 1st Intnl. Conf., Pergamon Press, pp. 323–332.

  9. Fuchs, H. O., 1982, “The Effect of Self Stresses on High Cycle Fatigue,” J of Testing and Evaluation, JTEVA, Vol. 10, No. 4, pp. 168–173.

  10. Fuchs, H. O., ed., 1984, Shot Peening. Proc. 2nd Intnl. Conf. SAE Warrendale PA.

  11. Fuchs, H. O., ed., 1987, Shot Peening Stress Profiles Metal Improvement Co. Paramus. NJ.

  12. Fuchs, H. O., 1988, “Approximate Analysis for Optimizing Prestress Treatments,” Analytical and Experimental Methods for Residual Stress Effects in Fatigue, ASTM STP 1004, R.L. Champoux, J.H. Underwood, and J.A. Kapp, Eds., American Society for Testing and Materials, Philadelphia, pp. 13–20.

  13. Gerber, T. L. and Fuchs, H. O., 1970, “Improvement of the Fatigue Strength of Notched Bars by Compressive Self-Stresses,” Achievement of High Fatigue Resistance in Metals and Alloys, ASTM STP 467, pp. 276–295.

  14. Heller, R. A., Seki, M. and Freudenthal, A. M., 1964, “The Effects of Residual Stress on Random Fatigue Life,” Proceedings ASTM, Vol. 64, pp. 516–535.

  15. Horger, O. J., 1950, “Residual stresses,” ch. 11 in Hetenyi, M. ed. Handbook of Exp. Stress Analysis, John Wiley & Sons, New York, pp. 459–578.

  16. Klesnil, M. and Lukas, P., 1980, Fatigue of Metallic Materials Science Monographs, Vol. 7, Elsevier Scientific Publishing Co., pp. 187–188.

  17. Lauchner, E., 1974, “Peening of High Strength Steel Using Hard Shot,” Presentation at WESTEC.

  18. Leverent, G. R., Langer, B. S., Yuen, A. and Hopkins, S. W., 1979, “Surface Residual Stresses, Surface Topography, and the Fatigue Behavior of Ti-6A1-4V,” Metallurgical Trans, Vol. 10 A. pp 251–257.

  19. Mattson, R. L. and Robinson, G. H., 1965, “Case Carburizing,” in Horger, O. J. ed. Metal Engineering-Design, McGraw-Hill, New York, pp. 284–290.

  20. MIL-HDBK-5C, 1976, Metallic Materials and Elements f. Acrospace Structures, U. S. Dept. of Defense, pp. 3–284.

  21. Morrow, J., Ross, A. A., and Sinclair, G. M., 1960, “Relaxation of Residual Stresses Due to Fatigue Loading,” SAE Trans. Vol. 68, p. 40.

  22. Neuber, H., 1946, Theory of Notch Stresses, J. W. Edwards, Ann Arbor MI.

  23. Niku-Lari, A. ed., 1981, Shot Peening, Proc. 1st Intnl. Conf., Pergamon Press.

  24. Niku-Lari, A., 1984, “Contraintes Residuelles et Fatigue d' aluminium des Alliages Grenailles,” in Proceedings, 2nd Intnl. Conf. on Shot Peening, ICSP-2, H. O. Fuchs ed., pp. 102–114.

  25. Peterson, R. E., 1974, Stress Concentration Factors, John Wiley, New York.

  26. Peterson, R. E., 1961, “The Role of Stress Distribution in Fatigue.” Experimental Mechanics, Vol. 1, No. 4, pp. 105–115.

  27. Roark, R. J. and Young, W. C., 1975, Formulas for Stress and Strain, 5th ed., McGraw-Hill Co., pp. 590–606.

  28. SAE Handbook, 1983, “Technical Report on Fatigue Properties,” SAE J 1099, Society of Automotive Engineers, pp. 3.57–3.65.

  29. Schijve, J., 1980, “Stress Gradients Around Notches,” Fatigue of Engineering Materials and Structures, Vol. 3, No. 4, pp. 325–338.

  30. Sharma, V. K., Walter, G. H. and Breen, D. H., 1979, “Predicting Case Depth for Gears,” Product Engineering, p. 49.

  31. Shot Peening Applications, 1980, 6th ed., Metal Improvement Co., Inc. Paramus, NJ.

  32. Siebel, E. and Stieler, M., 1955, “Non-uniform Stress Distribution in Fatigue,” (in German) VDI-Zeitchrift, Vol. 97, No. 5. pp. 121–126.

  33. Smith, K. N. Watson, P. and Topper, T. H., 1970, “A Stressstrain Function for the Fatigue of Metals,” J. of Materials, Vol. 5. No. 4. pp. 767–778.

  34. Starker, P, Wohlfahrt, H. and Macherauch, E., 1979, “Subsurface Crack Initiation During Fatigue as a Result of Residual Stresses,” Fatig. of Eng. Matl. and Structures, Vol. 1, pp. 319–327.

  35. Todd, R. H., 1971, Self Stress Concentrations. Dissertation, Stanford University. (1970) Also Experimental Mechanics, Vol. 11, No. 12, pp. 548–553.

  36. Underwood, J. H. and Kendall, D. P., 1984, “Fracture Analysis of Thick-wall Cylinder Pressure Vessel,” Theor. & Applied Fracture Mechanics, Vol. 2, pp. 47–58.

  37. Vasilakis, J. D., 1986, Thermal and Transformation Stresses in., Tubes, First Army Conference on Applied Math., George Washington University.

  38. Waisman, J. L., 1952, Report from Technical Service Co. to Metal Improvement Co.

  39. Walker, K., 1970, “The Effect of Stress Ratio During ... Fatigue,” ASTM STP 462, Am. Soc.for Testing and Materials, Philadelphia PA, pp. 1–14.

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Correspondence to Soon-Bok Lee.

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Lee, S. Distributions of load stresses and residual stresses at notches. KSME Journal 6, 132–139 (1992).

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Key Words

  • Load Stress
  • Residual Stress
  • Fatigue Notch Factor
  • Shot Peening
  • Notch