C. R. Acad. Sci. Paris, t. 326, SCrie I, p. 1249-1254, 1998 Probkmes mathematiques de la mkaniquelMathemat/ Problems in Mechanics (Analyse mathCmatiquelMafhematica/ Analysis) Spectre des rhseaux de poutres Bertrand DEKONINCK, Serge NICAISE LIMAV. ISTV, Tlnivrrsitk tirl Valmcknws, R.P. 31 1, 59304 Valmciannrs cedrx, Franc-e (Rryo le 3 avril 1998, awrpti. Ir 27 avril 1008) RCsum& Nous analysons le spectre d’un modCle de r&eaux de poutres d’Euler-Bernoulli. Nous donnons tout d’abord I’Cquationcaract6ristique du spectre. Dam certainscas particuliers, nous montrons que le spectre dCpend seulement de la structure du graphe. Finalement, nous donnonsle comportement asymptotique des valeurs propres en Ctablissant la formule de Weyl. 0 Acadkmie desSciences/Elsevier, Paris The eigenvalue problem for networks of beams Abstract. We consider thr .spectral unalJsis of a mc~drl of networks (f Euler-Bernoulli beams. We jirst give the charucteristic equation ,fbr the spectrum. Secondly, in some particular situations, we show that the spectrum depends ml,: on the structure qf the graph. Third/y, we invrstigate thcl a.synptotic hehaviour of the eigenvulues by proving the so-called Weyl’s ,$wmu/~. 0 AcadCmie desSciences/Elsevier, Paris A bridged English Version Various models of multiple-link flexible structures consisting of finitely many interconnected flexible elements. like strings, beams, plates, shells have been recently given (see for instance [9], [7], [6] and the references cited there). The spectral analysis of such structures has, in addition to its own mathematical interest, some applications to control or stabilization problems (see [9]). For interconnected strings (corresponding to second order operator on each string), a lot of results were obtained: let us quote the asymptotic behaviour of the eigenvalues (see [ 11, [2], [5], [12]); the relationship between the eigenvalues and algebraic graph theory (see [3], [S], [9]); qualitative properties of solutions (SW IS], [ 131) and finally studies of the Green function (.w [ 141). In this paper, we investigate the first two types of results for networks of Euler-Bernoulli beams. Namely, on a finite network made of edges k:,) (identified to a real interval of length lj, see Section 1). .j = 1.. . , N, we consider the eigenvalue problem: (0.1) Note prbenthe par Gilles L~sne~u. 0764-4442/98/03261249 0 Acadt?mie des SciencesElsevier. Paris 1249