On numerical solution of dynamic problems in the theory of reinforced shells

Abstract

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.

Non-linear vibrations of shells: A literature review from 2003 to 2013

“…In a series of papers, Lugovoi et al [145 147] used the same theory and extended the work of Ref. [144] by solving the problem via a new technique based on the Richardson extrapolation method in [145], studying the dynamics of compound cylindrical shells with spherical caps in [146], and investigating the dynamic behavior of cylindrical shells in an elastic medium in [147]. Moreover, the stability of vibration mode transformation between extensional and flexural modes, and two to one internal resonance in thin walled circular cylindrical shells subjected to large initial radial velocity or impulse was investigated by Shi et al [148].…”

Dynamic problems for and stress–strain state of inhomogeneous shell structures under stationary and nonstationary loads

“…Comparing numerical results and experimental data confirm the high efficiency of the method. A problem formulation for and an equation of motion of discretely reinforced cylindrical shells and a method and a numerical algorithm to solve them based on Richardson approximations are presented in [34]. This approach makes it possible to reduce the number of numerical operations by two-thirds.…”

Stress-strain state near mine workings in anisotropic rock masses under the action of discontinuous waves

“…The efficiency of the approach is demonstrated with a specific example Keywords: mine workings, discontinuous waves, ray-path method Introduction. Problems of theoretical modeling of propagating discontinuous waves are of importance for analyzing the strength and stability of the rock near and the ground support of mine workings subjected to the action of seismic waves and waves generated by blasting operations and rock bursts [4][5][6][18][19][20][21]. The dynamic theory of elasticity solves such problems by studying the propagation and evolution of discontinuity fronts of derivatives of field functions.…”