Asymptotic Analysis 52 (2007) 259–297 259 IOS Press On slender shells and related problems suggested by Torroja’s structures J.I. Díaz a and E. Sanchez-Palencia b a Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias, 3, 28040 Madrid, Spain E-mail: ildefonso.diaz@mat.ucm.es b Laboratoire de Modélisation en Mécanique, Université Pierre et Marie Curie, 4, place Jussieu, Paris, France E-mail: sanchez@lmm.jussieu.fr Abstract. We study the rigidification phenomenon for several thin slender bodies or shells, with a small curvature in the transversal direction to the main length, for which the propagation of singularities through the characteristics is of parabolic type. The asymptotic behavior is obtained starting with the two-dimensional Love–Kirchoff theory of plates. We consider, in a progressive study, a starting basic geometry, we pass then to consider the “V-shaped” structure formed by two slender plates pasted together along two long edges forming a small angle between their planes and, finally, we analyze the periodic extension to a infinite slab. We introduce a scalar potential ϕ and prove that the equation and constrains satisfied by the limit displacements are equivalent to a parabolic higher-order equation for ϕ. We get some global informations on ϕ, some on them easely associated to the different momenta and others of a different nature. Finally, we study the associate obstacle problem and obtain a global comparison result between the third component of the displacements with and without obstacle. Keywords: thin shells, V-shaped structures, asymptotic behavior, scalar potential, parabolic higher-order equations, one-side problems 1. Introduction The experience shows that when considering a slender or shell a small curvature in the transversal direction to the main length supply an extra rigidification with respect to the planar case. Think, for instance, about the familiar flexible steel retractable meter tape measure, which enjoys rigidity from its special transversal curvature. Here we shall carry out the study of the asymptotic modeling of such kind of shell structures (see Fig. 1 concerning the “basic problem” considered in Section 2). We also will consider more sophisticated structures formed by coupling two of such basic shells by means of an edge with slight folding (see Fig. 2 in Section 3), as well as the case of an infinity set of shells obtained by the periodic repetition of the basic structure (see Fig. 3 in Section 4). The consideration of this type of periodic structures is motivated by some of the structures designed by the outstanding engineer Eduardo Torroja (Madrid, 1899–1961). For instance, the shell roofs of the Madrid Racecourse (1935) are a brilliant result of the forms of the reinforced concrete consisting of a system of portal frames, spread at 5 m intervals and connected longitudinally by small reinforced concrete double curvature vaults. The cantilever roof, with a minimum thickness of 5 cm, overhangs to a distance of 12,8 m. Although Torroja also produced some theoretical works (see, for instance, [39]), the mentioned structure was calculated 0921-7134/07/$17.00  2007 – IOS Press and the authors. All rights reserved