Theory of Tire Shape

  • Yukio NakajimaEmail author


The theory of tire shape has been well studied since the early 1900s because it has been easier to develop theory for the sidewall shape of a tire than to develop theory for other tire design elements, such as the crown shape, bead structure, belt structure, pattern and material. Three important theories have been established in the history of the development of theory of the sidewall shape: natural equilibrium theory for a bias tire, natural equilibrium theory for a radial tire and ultimate tire shape theory where the finite element method is combined with optimization technology. These theories and applications are discussed in this chapter.


  1. 1.
    H.F. Schippel, Fabric stresses in pneumatic tires. Ind. Eng. Chem. 15(11), 1121–1131 (1923)CrossRefGoogle Scholar
  2. 2.
    R.B. Day, J.R. Purdy, Goodyear research report (1928)Google Scholar
  3. 3.
    J.F. Purdy, Mathematics Underlying the Design of Pneumatic Tires (Hiney Printing Co., 1963)Google Scholar
  4. 4.
    W. Hofferberth, Zur Statik des Luftreifens. Kaustschuk Gummi Jahrg 8(5), 124–130 (1955)Google Scholar
  5. 5.
    R.S. Rivlin, Plane strain of a net formed by inextensible cords. J. Rotational Mech. Anal. 4, 951–974 (1955)MathSciNetzbMATHGoogle Scholar
  6. 6.
    W.H. Walston, W.F. Ames, Design and analysis of inflated membranes reinforced with extensible cords. Tex. Res. J. 35(12), 1078–1098 (1965)CrossRefGoogle Scholar
  7. 7.
    V.L. Biderman, Trudy Nauch. Issledovatel, Inst. Shinnio Promy, vol. 3 (Gpskhimimsdat, Moscow, 1957), pp. 57–64Google Scholar
  8. 8.
    H.G. Lauterbach, W.F. Ames, Cord stresses in inflated tires. Tex. Res. 29, 890–900 (1959)CrossRefGoogle Scholar
  9. 9.
    R.B. Day, S.D. Gehman, Theory for meridian section of inflate cord tires. Rubber Chem. Tech. 36(1), 11–27 (1963)CrossRefGoogle Scholar
  10. 10.
    E. Robecchi, L. Amici, Mechanics of pneumatic tire, part I, the tire under inflation alone. Tire Sci. Technol. 1(3), 290–345 (1973)CrossRefGoogle Scholar
  11. 11.
    E. Robecchi, Mechanics of pneumatic tire, part II, the laminar model under inflation and in rotation. Tire Sci. Technol. 1(4), 382–438 (1973)CrossRefGoogle Scholar
  12. 12.
    S.K. Clark et al., Tire shape calculations by the energy method. Kaustschuk Gummi Jahrg 25(12), 587–596 (1972)Google Scholar
  13. 13.
    H.K. Brewer, Tire stress and deformation from composite theory. Tire Sci. Technol. 1(4), 46–76 (1973)Google Scholar
  14. 14.
    D. Bozdog, W.W. Olson, An advanced shell theory based tire model. Tire Sci. Technol. 33(4), 227–238 (2005)CrossRefGoogle Scholar
  15. 15.
    F. Böhm, Zur Statik und Dynamik des Gurtreifens, Automobeltechnische Zeitschrift. Jahrg 69, 255–261 (1967)Google Scholar
  16. 16.
    F. Frank, Therie und Berechung des statischen Krafte und des Querschnittsform von Kreuglagen und Gurtelreifen, in Rubber Conference of the DKG (Berlin, 1968)Google Scholar
  17. 17.
    T. Akasaka, Y. Sakai, On the standing waves in radial tire. Fukugo Zairyo 1(1), 26–34 (1972)Google Scholar
  18. 18.
    F. Koutny, A method for computing the radial deformation characteristics of belted tires. Tire Sci. Technol. 4(3), 190–212 (1976)CrossRefGoogle Scholar
  19. 19.
    F. Koutny, Geometry and Mechanics of Pneumatic Tires (Zlín, CZE, 2007)Google Scholar
  20. 20.
    K. Yamagishi et al., Study on contour of radial tire: rolling optimization theory—RCOT. Tire Sci. Technol. 15(1), 3–29 (1987)CrossRefGoogle Scholar
  21. 21.
    H. Ogawa et al., A study on the truck and bus radial tire—tension control optimization theory (TCOT). Tire Sci. Technol. 18(4), 236–261 (1990)CrossRefGoogle Scholar
  22. 22.
    Y. Nakajima et al., Theory of optimum tire contour and its application. Tire Sci. Technol. 24(3), 184–203 (1996)CrossRefGoogle Scholar
  23. 23.
    H. Sakai, Tire Engineering (Guranpuri-Shuppan, 1987) (in Japanese)Google Scholar
  24. 24.
    K. Kabe, Study on structural mechanics of tire deformation characteristics. Ph.D. thesis (Chuo University, 1980) (in Japanese)Google Scholar
  25. 25.
    J.D. Walter, Centrifugal effects in inflated rotating bias ply tires. Tex. Res. J. 40, 1–7 (1970)CrossRefGoogle Scholar
  26. 26.
    H. Nakayama, K. Furukawa, Satisficing trade-off method with an application to multiobjective structural design. Large Scale Syst. 8, 47–57 (1985)zbMATHGoogle Scholar
  27. 27.
    Y. Nakajima, New tire design procedure based on optimization technique. SAE Paper, No. 960997 (1996)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mechanical Science and Engineering, School of Advanced EngineeringKogakuin UniversityHachiojiJapan

Personalised recommendations