ANALYSIS OF SHELLS REINFORCED BY MASSIVE STIFFENING RIBS Y. H. Savula 1* , K. Jarmai 2 , and I. S. Mukha 1** The paper presents results of applying a heterogeneous mathematical model “elastic body–Timoshenko shell” to design shells with massive ribs. Numerical results are obtained for a cylindrical shell with ribs. They are compared with results obtained using the theory of elasticity and the theory of Timoshenko shells with piecewise-constant thickness Keywords: heterogeneous mathematical model elastic body–Timoshenko shell, shells with massive ribs, cylindrical shell with ribs 1. Introduction. An approach for the mathematical modeling of junctions in elastic multistructures, i.e., H-shaped beams, shells with stiffeners, shells clamped in three-dimensional foundations, etc. on the basis of heterogeneous models was suggested in [1, 2, 5–7, 14–16]. Within the framework of this approach, the heterogeneous model “elastic body–Timoshenko shell” is considered in [8–11]. The theoretical background of the heterogeneous model is given in [12]. FEM analysis of structures on the basis of the “elastic body–Timoshenko shell” model is performed in [8, 10, 11]. The problem of coupling the BEM and the FEM for the analysis of junctions in elastic multistructures on the basis of the heterogeneous mathematical model was considered in [9]. The present paper applies the heterogeneous mathematical model “elastic body–Timoshenko shell” to design shells with massive ring ribs. Within the framework of the heterogeneous model, a ring of a stiffened shell is modeled by the 3D theory of elasticity and a thin-walled part of stiffened structures is modeled by 2D Timoshenko’s shell theory. The boundary and variational formulation of the heterogeneous mathematical model are given. To illustrate the approach, the results of numerical analysis of cylindrical shells with stiffeners performed on the basis of the heterogeneous model (model 1) are presented. These results are compared with those obtained on the basis of the 3D theory of elasticity (model 2) and the 2D Timoshenko’s shell theory (model 3). 2. Heterogeneous Mathematical Model. Let an elastic body occupy a bounded, connected domain W W 1 2 È * (Fig. 1), where W 1 and W 2 * are three-dimensional domains with Lipshitz boundaries GG 1 2 , * [2]. Let the three-dimensional domain W 1 be referred to a Cartesian coordinate system x 1 , x 2 , x 3 . We denote three orthogonal unit vectors on the boundary G 1 by r n 1 , r n 2 , r n 3 (Fig. 2) ( r n 1 is normal outward vector to G 1 ). Let also the three-dimensional domain W 2 * be thin, i.e., one of its dimensions— thickness h—is considerably smaller then the other two. We refer the domain W 2 * to an orthogonal curvilinear coordinate system z 1 , z 2 , z 3 { } W W 2 1 2 3 1 2 2 3 2 2 * , , : , , / / = Î - ££ zzzzz z h h on the middle surface S Ì = ¡ 3 3 0 ( ) z , which is a mapping of the set W 2 2 Ì ¡ (the middle surface is referred to the lines of principal curvature) (Fig. 3): ( ) x i i = Î = jz z zz 1 2 1 2 2 123 , , , , ,, W . (1) Let us denote the three orthogonal unit vectors of the curve ¶S by r n 1 , r n 2 , r n 3 (Fig. 2), where r n 1 is unit normal vector, which lays in the tangent plane of the middle surface S; r n 2 is the unit tangent vector to the curve G 2 (Fig. 4); r n 3 is the unit normal vector to the middle surface S. International Applied Mechanics, Vol. 44, No. 11, 2008 1063-7095/08/4411-1309 ©2008 Springer Science+Business Media, Inc. 1309 1 Department of Applied Mathematics, National University of Lviv, 1 University St., Lviv, Ukraine, e-mail: * savula@franko.lviv.ua, ** imukha@franko.lviv.ua; 2 Department of Materials Handling and Logistics, University of Miskolc, Miskolc-Egyetemvaros, Hungary, e-mail: altjar@gold.uni-miskolc.hu. Published in Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 132–142, November 2008. Original article submitted March 20, 2007.