Buckling of a heavy tapered rod

The study of buckling of a tapered rod leads to the nonlinear eigenvalue pr oblem: [GRAPHICS] where A is an element of C([0, 1]) is such that A(s) > 0 for all s > 0 and lim(s-->0) A(s)/s(p) = L for some constants p greater than or equal to 0 an d L is an element of (0, infinity). There is a number Lambda (A) greater th an or equal to 0 such that, for mu less than or equal to Lambda (A), mu = 0 is the only solution of the problem and it minimizes the energy in the spa ce of all admissible configurations. For mu > Lambda (A), the energy is min imized by a non-trivial solution. For p = 0, this is a well understood clas sical problem studied by D. Bernoulli and Euler. For 0 < p < 2, the problem is singular but its bifurcation diagram remains similar to the case p = 0. At p = 2, striking changes occur. (1) For 0 less than or equal to p < 2, lim(s-->0) u(s) is an element of (-p i, pi) for all non-trivial solutions whereas lim(s-->0)u(s) = +/-pi if p gr eater than or equal to 2. (2) For 0 less than or equal to p less than or equal to 2, Lambda (A) > 0 w hereas Lambda (A) = 0 for p > 2. (3) For 0 less than or equal to p < 2, bifurcation from the solution u = 0 occurs only at a discrete set of eigenvalues <mu>(i) where mu (I) = Lambda (A) and lim(i-->infinity) mu (i) = infinity. For p = 2, there is a number L ambda (e)(A) is an element of [Lambda (A), infinity) such that bifurcation occurs at every value mu is an element of [Lambda (e)(A), infinity). The properties of the linearized problem, in which (1) is replaced by (4) [A (s)u'(s)}' + mu mu (s) = 0 for all s is an element of (0, 1), also change at p = 2. For 0 less than or equal to p < 2, its spectrum {<mu> (i)} is discrete and all the eigenfunctions have only a finite number of ze ros in [0, 1]. For p = 2, Lambda (e)(A) belongs to the essential spectrum a nd there may be no eigenfunctions. Furthermore, for p = 2 and mu > Lambda ( e)(A) all solutions of (4) have infinitely many zeros in [0, 1], but soluti ons of the nonlinear problem have only a finite number of zeros. (C) 2001 E ditions scientifiques et medicales Elsevier SAS.