The Southwell and the Dunkerley Theorems

  • T. Tarnai
Part of the International Centre for Mechanical Sciences book series (CISM, volume 354)


Summation formulae are used in the theory of elastic stability so that approximate estimates of the critical load factors of a complex problem are obtained by combining the load factors of subproblems in different ways. If the critical load factors are directly added, then the formula is called a Southwell type formula. If the reciprocals of the critical load factors are added, then the formula is called a Dunkerley type formula. The practical advantage of the summation formulae is that, for the subproblems, there are usually solutions available, or it is easy to determine them while, for the original problem the solution would be difficult to obtain. In this paper we will present the main theorems and formulae (Southwell theorem, Dunkerley theorem, Föppl-Papkovich theorem, Kollár conjecture, Melan theorem), and show, where possible, the conditions under which the results are on the safe side. The proofs are given in mathematical way, based on the relationships for eigenvalues of linear operators in Hilbert space. The theoretical results are illustrated by examples.


Eigenvalue Problem Critical Load Elastic Foundation Stability Domain Rayleigh Quotient 


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  1. 1.
    Timoshenko, S.P. and J.M. Gere: Theory of Elastic Stability, McGraw-Hill, New York, 1961.Google Scholar
  2. 2.
    Weinberger, H.: Variational Methods for Eigenvalue Approximation. SIAM Philadelphia, Pa. 1974.CrossRefMATHGoogle Scholar
  3. 3.
    Weinberger, H.: Some mathematical aspects of buckling. Chapter 1 in this volume.Google Scholar
  4. 4.
    Kollár L.(Editor): Special problems of engineering theory of stability (in Hungarian), Akadémiai Kiadó, Budapest, 1991.Google Scholar
  5. 5.
    Strigl, G.: Das nicht lineare Überlagerungsgesetz für die Lösungen von zusammengesetzten Stabilitätsproblemen mit Verzweigungspunkt, Der Stahlbau, 24 (1955), 33–39, 51–61.Google Scholar
  6. 6.
    Plantema, F.J.: Theory and Experiments on the Elastic Overall Instability of Flat Sandwich Plates, Doctoral Thesis, Delft, 1952.Google Scholar
  7. 7.
    Kollár L.: Recent results in the theory of stability through the eyes of a designer (in Hungarian), Magyar Építöipar, 20 (1971), 333–337.Google Scholar
  8. 8.
    Bleich, F.: Buckling Strength of Metal Structures, McGraw-Hill, New York 1952.Google Scholar
  9. 9.
    Kármán, T. and M.A. Biot: Mathematical Methods in Engineering, McGraw-Hill, New York, 1940.Google Scholar
  10. 10.
    Melan, H.: Kritische Drehzahlen von Wellen mit Langsbelastung . Zeitschrift der Österr. Ingenieur- und Architekten-Vereines, 69 (1917), 610–612, 619–621.Google Scholar
  11. 11.
    Horne, M.R.: The Rankine-Merchant load and its application. Chapter 3 in this volume.Google Scholar

Copyright information

© Springer-Verlag Wien 1995

Authors and Affiliations

  • T. Tarnai
    • 1
  1. 1.Technical University of BudapestBudapestHungary

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