Solution of axisymmetric dynamic problems for cylindrical shells on an elastic foundation

Abstract

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

Nonaxisymmetric vibrations of ellipsoidal shells under nonstationary distributed loads

“…The free vibrations of complex shell elements are analyzed numerically and analytically in [2] using different theories. The forced vibrations of smooth and reinforced canonic shells under nonstationary loads are numerically studied in [7,8,14].…”

“…The free vibrations of complex shell elements are analyzed numerically and analytically in [2] using different theories. The forced vibrations of smooth and reinforced canonic shells under nonstationary loads are numerically studied in [7,8,14].This paper considers the forced nonaxisymmetric vibrations of an ellipsoidal shell under nonstationary distributed load. We will use a geometrically nonlinear version of the Timoshenko shell theory of the second order.…”

Solving axisymmetric dynamic problems for reinforced shells of revolution on an elastic foundation

“…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.…”

“…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].…”

Analysis of the nonaxisymmetric vibrations of flexible ellipsoidal shells discretely reinforced with transverse ribs under nonstationary loads

“…According to reviews and monographs on the subject, the axisymmetric and nonaxisymmetric harmonic vibrations of reinforced shells of simple geometry (cylindrical, conical, and spherical) were mainly studied [1-3, 6, 18]. Results on the forced vibrations of reinforced shells under impulsive loads are presented in [7][8][9][10][11][15][16][17]. Studies on the dynamic behavior of reinforced shells of more complex geometry are very few.…”