The nonstationary behavior of reinforced compound structures is analyzed within the framework of the geometrically nonlinear Timoshenko theory of shells and rods. The numerical method used to solve the problem is based on the finite-difference approximation of the original differential equations. The dynamic behavior of a reinforced compound structure under nonstationary loading is demonstrated by way of a numerical example. The numerical results are compared with experimental data Keywords: reinforced discrete shell, Timoshenko shell theory, nonstationary loading, numerical method Introduction. Reinforced compound shells are complex, spatially inhomogeneous, elastic structures that include discrete inclusions and areas where the geometrical and material parameters change. These factors complicate problem formulations. In particular, the partial differential equations describing the stress-strain state of the original structures include discontinuities of the first kind of the strain and stress components. This is why special numerical algorithms have been developed to adequately simulate wave processes in shells with singularities.The handbook [4] presents algorithms and programs for design of compound shells under stationary and dynamic loads. The dynamic behavior of compound shells under nonstationary loads is analyzed in the monograph [5]. The nonstationary behavior of reinforced shells of revolution with discrete ribs is studied in [2]. The axisymmetric nonlinear vibrations of discretely reinforced conical shells are examined in [9]. Dynamic problems for reinforced ellipsoidal shells are solved in [13]. Initial deflections are accounted for in [11] in solving dynamic problems for discretely reinforced cylindrical shells under nonstationary loads. The nonaxisymmetric forced vibrations of sandwich cylindrical shells with ribbed core are studied in [12].Here we formulate dynamic problems for compound shells and outline a numerical algorithm for solving them. The algorithm is further used to solve model problems.1. Problem Formulation. Governing Equations. Consider a compound inhomogeneous structure consisting of a compound shell and ring ribs rigidly fixed to it along contact lines (the ribs are supposed to be placed along the coordinate line α 2 ) [1]. Forced vibrations of the structure are modeled by a hyperbolic system of nonlinear differential equations of the Timoshenko theory of shells and curvilinear rods [2]. The variation of the displacements of the shell components across the thickness is described in the coordinate frame ( , ) s z by the formulas
Dynamic problems for cylindrical shells reinforced with discrete ribs are examined. A numerical algorithm based on Richardson extrapolation is developed. Specific problems are solved, and the results are analyzed Keywords: reinforced cylindrical shell, Timoshenko model, discrete ribs, forced vibration, numerical algorithm, Richardson extrapolationThere are two basic approaches to the stress-strain analysis of reinforced shells: use of a structurally orthotropic model [1] and use of a model that allows for the discreteness of ribs [2]. During forced vibration of reinforced thin-walled shells, the presence of inhomogeneities and the wave nature of the process strongly affect the redistribution of the fields of material parameters. This situation suggests incorporating discrete ribs into the shell model, which would in turn complicate problem statements. The complexity of problem statements (boundary conditions, geometrical nonlinearity or geometrical and physical nonlinearities, the inhomogeneity of shells across the thickness, etc.) in the theory of discretely reinforced shells [4-6, 8-13, 17-21] calls for the use of numerical methods. If finite-difference methods are used, then the following two approaches are possible: shock capturing and isolation of spatial discontinuities [3]. The former approach helped to solve several problems for reinforced shells of revolution (cylindrical, spherical, and conical ones) under forced vibration. A numerical analysis shows that with such an approach the site of a rib smears over several spacings. Satisfactory solutions can be obtained from fairly fine meshes for reinforced shells under distributed loads [5,16,17,21]. In the case of boundary loads, the approach produces significant errors, which makes it impossible to accurately locate spatial discontinuities [3,8,16]. Finite-difference schemes with isolation of spatial discontinuities allow increasing the accuracy of solutions for shells reinforced with discrete ribs. Such an approach was used to solve dynamic problems for reinforced shells under distributed and boundary loads [4,6,[10][11][12][13][18][19][20]. However, even this approach provides in some cases poor convergence because of spatial discontinuities.The present paper gives a geometrically nonlinear formulation to problems for shells reinforced with discrete ribs and outlines an algorithm of numerical solution based on the Timoshenko theory of shells and rods. It is proposed to use finite-difference schemes with Richardson extrapolation [7]. We conducted computations for reinforced cylindrical shells. The numerical results allow us to judge the efficiency of our method.
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