# The Southwell and the Dunkerley Theorems

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

## Abstract

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.

## Keywords

Eigenvalue Problem Critical Load Elastic Foundation Stability Domain Rayleigh Quotient

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