A dynamic problem for a cylindrical shell on an elastic foundation is formulated. A numerical algorithm for solving this problem is outlined. The results obtained are analyzed Keywords: nonlinear theory of shells, dynamic problems, elastic foundation, numerical methods Introduction. The dynamic interaction of cylindrical shells with elastic media is a problem of current importance because cylindrical shells are widely used in modern engineering structures such as tunnels, tanks, pipelines, and drill and casing pipes. Mathematically, such problems are quite complicated, in both formulation and solution (use of the theory of elasticity and the theory of shells, formulation of medium-shell interface conditions, development of a numerical algorithm, etc.) [1,5,6]. There are more simple approaches to analyze the interaction between a structure and an ambient medium [2,4,9]. An example is the Winkler model [3,9]. The dynamic interaction of spherical and cylindrical shells with bilateral and unilateral Winkler foundations was analyzed in [4] in the axisymmetric case. Forced vibrations of an infinite cylindrical shell in an elastic medium under nonaxisymmetric loading were studied in [3] considering Winkler and Pasternak foundations. A number of dynamic problems for one-and three-layer beams and rods on an elastic foundation were addressed in [13,14]. Static stress-strain problems for plates and shells on an elastic foundation are solved in [15,16].The present paper addresses nonstationary problems of dynamic deformation of a cylindrical shell in an elastic medium. We will examine the cases where the medium is described by the equations of continuum and is modeled by a Winkler foundation. The results obtained will be compared.1. Cylindrical Shell on Elastic Foundation. The dynamic behavior of a cylindrical shell is described by the vibration equations of the Timoshenko theory of shells [6] including terms modeling the elastic foundation. The vibration equations are derived using a geometrically nonlinear version of the theory (Timoshenko theory of shells of the second order). These equations have the form

The dynamic behavior of reinforced shells of revolution in an elastic medium is modeled. Pasternak's model is used. A problem of vibration of discretely reinforced shells of revolution is formulated and a numerical algorithm is developed to solve it. Results from an analysis of the dynamic behavior of a reinforced spherical shell on an elastic foundation are presented as an example Keywords: reinforced shells, elastic foundation, nonstationary vibration, numerical method Introduction. The static interaction of shells and rods with an elastic medium is well understood [2,7]. Of interest is the dynamic behavior of shells and elastic elements with media. The properties of an elastic medium with a deformable system inside are best described by the equations of elasticity. However, this problem is very complicated mathematically [3]. The problem of describing the elastic medium can be given a relatively simple mathematical formulation, which would allow us to determine the reaction of the elastic medium to the structure with adequate accuracy. Noteworthy are studies of the deformation of single-layer and multilayer beams and cylindrical shells on an elastic foundation under nonstationary loads [4,5,9,11], where the Winkler model was mainly used. There are very few studies on shells of other shapes (except for cylindrical) interacting with elastic media under dynamic loads. The dynamic behavior of reinforced shells on an elastic foundation is of interest. Dynamic problems for reinforced cylindrical shells on an elastic foundation were solved in [10]. Nonstationary vibrations of reinforced cylindrical and ellipsoidal shells are studied in [12,13]. Free vibration of reinforced open cylindrical shells is addressed in [11].The objective of the present paper is to study the dynamic behavior of reinforced shells of revolution in an elastic medium under a nonstationary distributed load. Use will be made of Pasternak's model [7].1. Problem Formulation. Consider a reinforced isotropic shell of revolution in an elastic medium under an internal impulsive load. The elastic medium is modeled by a Pasternak foundation [7]. This model is characterized by two moduli of subgrade reaction C 1 and C 2 to tension/compression and shear, respectively.The inhomogeneous shell structure consists of a smooth shell (casing) of revolution and ring ribs rigidly attached to it. To describe the forced vibration of this structure, we will use a hyperbolic system of nonlinear differential equations of the Timoshenko theory of shells and curvilinear rods. We assume that the displacements vary as follows throughout the thickness of the casing (in coordinates s z , ):