No abstract

The reliability of the results obtained in the work is determined by the rigor and correctness of the statements of the initial problems; theoretical substantiation of the finite-difference schemes used; controlled accuracy of numerical calculations; conducting test calculations; compliance of the established regularities with the general properties of oscillations of thin-walled structural elements. The correctness of the formulation of the problems is achieved by using the well-known equations of the theory of shells and rods of the Tymoshenko type, which are approximations of the original equations of the three-dimensional theory of elasticity. When deriving the equations, the equations of oscillations of the multilayer shell in the smooth region and the equations of oscillations of reinforcing ribbed elements (transverse ribs) were obtained. It is not difficult to show that the indicated equations by the classification of equations in partial derivatives are equations of the hyperbolic type, which are an approximation of the oscillating equations of three-dimensional elastic bodies and sufficiently correctly reproduce wave processes in non-homogeneous shell structures, taking into account spatial gaps. Numerical algorithms for approximate solutions of the original equations are based on the use of the integro-interpolation method of constructing difference schemes. When constructing difference schemes, kinematic quantities refer to difference points with integer indices, and the values of deformations and moment forces refer to difference points with half-integer indices. This approximation of the initial kinematic and static values allows the fulfillment of the law of conservation of the total mechanical energy of the elastic structure at the difference level. The numerical algorithm is based on the use of separate finite-difference relations in the smooth domain and on the lines of spatial discontinuities with the second order of accuracy in spatial and temporal coordinates.

The purpose of this scientific-essay is constructionation numerical dynamical task solving algorithm of multilayered discretely substantiated shells, that based on using Richardson finite-difference approximations types. Multilayered discretely substantiated shells refer to complex nonuniformity by thickness elasticity-structures. In one reason, nonuniformity existed because of shell-flakiness structure, in other case-because of existing discretely substantiated edges. Including the discretely count of substantiated elements brings to exist new bursting coefficients on spatial coordinates in output equations. Numerical method usage (finite-dіfference method, finite-elements method etc.) for solving dynamic-progressions tasks in listed structures observing convergence of the worsening numerical results. For the constructing more effective numerical algorithms used the method, which based on finding approximation solutions by Richardson.

A problem of non – linear deformation of multiplayer conical shells with allowance for discrete ribs under non – stationary loading is considered. The system of non – linear differential equations is based on the Timoshenko type theory of rods and shells. The Reissner’s variational principle is used for deductions of the motion equations. An efficient numerical method with using Richardson type finite difference approximation for solution of problems on nonstationary behaviour of multiplayer shells of revolution with allowance distcrete ribs which permit to realize solution of the investigated wave problems with the use of personal computers. As a numerical example, the problem of dynamic deformation of a five-layer conical shell with rigidly clamped ends under the action of an internal distributed load was considered.