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— We are interested in a buckling model for a non shallow arch, in which the critical buc kling load appears as the s mallest generalized eigenvalue of a line ar operator in a Hilbert space. The mechanical orïgin of the model is recalled. Then we give a mathematical proof of existence of a smallest eigenvalue. Then we give a mathematical proof of differentiability of the smallest eigenvalue with respect to the shape of the midsurface. Ifit is simple, it is Fréchet differentiable, ifnot, it has a directional derivative which is regular enough to gei necessary optimality conditions. Finalty we give an analytical formula for this derivative, and we explain how to compute it numerically despite the heaviness of the computations. Resumé. — Nous nous intéressons à un modèle de flambement pour une arche profonde, dans lequel la valeur critique de flambement apparaît comme la plus petite valeur propre d'un opérateur linéaire dans un espace de Hilbert. On rappelle tout d'abord l'origine mécanique du modèle. Ensuite, une démonstration mathématique de l'existence d'une plus petite valeur propre est donnée. Puis on démontre mathématiquement un résultat de différentiabilité de cette plus petite valeur propre par rapport à la forme de la surface moyenne : si elle est simple, elle est Fréchetdifférentiable, sinon, elle possède une dérivée directionnelle qui est suffisamment régulière pour donner des conditions nécessaires d'optimalité. Enfin, on donne une formule explicite pour cette dérivée, et on explique comment en faire un calcul numérique, malgré la lourdeur des calculs. INTRODUCTION The strength of an arch dépends on its shape, that is the shape of its mideurve and its cross section. As a matter of f act, the dependence on the (*) Rcceivcd in July 1988. Q) Département de Mathématiques, Faculté des Sciences, Parc Valrose F-06034 Nice Cedex. MAN Modélisation mathématique et Analyse numérique 0764-583X/90/03/307/35/$ 5.50 Mathematical Modelling and Numerical Analysis © AFCET Gauthier-Villars 308 D. CHENAIS, B. ROUSSELET midcurve is much more complicate than the cross section. This is what we are interested in here. The cross section could be added easily. In référence [1], we have studied the differentiability dependence of the static response of an arch with respect to the shape of its midcurve. In this paper, we study the dependence of the buckling load of a nonshallow arch with respect to the shape of the midcurve. The model we consider is the simple Eulerian buckling one : the prebuckled state comes from the linear model, the buckling équation is an eigenvalue équation. It is now well known that in optimization problems involving eigenvalues, these may be repeated at the optimum [2]. A genera! account of perturbations theory of eigen-elements may be found in [3] ; these gênerai results may be used to obtain semi-differentiability of repeated eigenvalues in shape optimization [4] ; related results may be found in [5] and [6], In référence [7], we have proved a gênerai resuit of dependence of eigenvalues and eigenvectors of a linear operator with respect to a functional parameter. We have used this result to study the dependence of the buckling load of a beam with respect to its thickness. In this paper, we use this gênerai result for the arch problem. The eigenvalue problem is self-adjoint. It has a compactness property, like in the beam case. But the dependence on the midcurve is much more complicated that the dependence on the thickness in the beam problem. Moreover, the buckling équation dépends on the prebuckled state, which dépends on the midcurve shape. In paragraph I, we briefly recall the origin of the buckling model we work on. In paragraph II, we describe the set of eigenvalues of the problem, and prove that there is a smaller one in modulus. In paragraph III, we study the differentiability. Paragraph III. 1 recalls the results of référence [7] we use. Then, in paragraph III.2 we prove the directional differentiability of the eigenvalue if it is multiple, its Fréchet differentiability if it is simple. Then we show how to lead a numerical computation of this derivative.

Dependence of the buckling load of a nonshallow arch with respect to the shape of its midcurve