The aim of this paper is to develop an adequate mathematical model, methods and algorithms for solving three-dimensional problems for axisymmetric spatial inhomogeneous viscoelastic systems (shells, foundations and bases) and to assess the dynamics of protective shell (containment) of a nuclear power plant (NPP) under resonant modes of vibration. The problem is solved using the semi-analytical finite element method. Firstly, the eigenmodes of vibration of the system are determined in an elastic three-dimensional statement, secondly, the solution to the problem of forced vibrations of viscoelastic systems is constructed using the expansion of these eigenmodes of vibration. Viscoelastic properties of the material are described using the hereditary Boltzmann-Volterra theory. The principle of virtual displacements is used to simulate dynamic processes in inhomogeneous viscoelastic systems. The convergence and accuracy of the solutions obtained are investigated by test problems. The frequency response characteristics (FRC) in various points of the NPP containment are estimated at various viscosity parameters of the material. It was revealed that the highest amplitude of vibrations in resonance modes occurs at close values of the frequency of external effect to the first eigen frequencies of the system; in the presence of dense spectra of eigen frequencies of the system, the highest amplitudes can occur at higher frequencies of external effect.

The paper is devoted to improving the theory of bending and vibrations of three-layer plates with transverse compressible filler and thin outer bearing layers. For the outer layers, the Kirchhoff-Love hypothesis is accepted and the motion of their points is described by the equations of the theory of thin plates relative to forces and moments. Unlike bearing layers, a filler is considered as a three-dimensional body that does not obey any simplifying hypotheses. The equations of the bimoment theory of thick plates with respect to forces, moments and bimoments, created in the framework of the three-dimensional theory of elasticity, taking into account the nonlinearity of the distribution law of displacements and stresses over the thickness, are taken as the equations of motion of the filler. Expressions of forces, moments, and bimoments in the layers, as well as boundary conditions at the edges of a three-layer plate with respect to force factors, are given. In the conjugate zones of the layers, the complete contact conditions for the continuity of displacements and stresses are set. An example is considered and numerical results are obtained.

A mathematical model, methods and algorithm for estimating the natural vibrations of inhomogeneous axisymmetric systems in a three-dimensional statement are presented in the paper. The problem is solved by the semi-analytical finite element method (FEM). The reliability of the developed methods and algorithms was validated by comparing the results obtained with the exact solution of a number of test problems for three-dimensional bodies, as well as by comparing the results with the results of field experiments. Spatial natural vibrations of inhomogeneous spatial axisymmetric systems, i.e. the models of nuclear power plants (NPPs) shielding shell in a three-dimensional statement were investigated. The results were analyzed to detect mechanical effects and revealed that the presence of multiple (close) eigenfrequencies in real spatial axisymmetric systems is not an exception, but a rule.

Nonstationary deformation of a cylindrical shell located in an infinite elastic medium is studied under plane pressure wave with a front parallel to the shell axis. A combined solution method is used in the paper. Separating the angular coordinate by the Fourier method, the plane problem is reduced to a series of one-dimensional problems for each harmonic. Applying the Laplace transform over time to these systems of equations, the exact solution in the images is obtained. Asymptotic solutions of the stress-strain state at large values of time t → ∞ are obtained. In parallel with this, a truncated system is numerically solved using an explicit finite-difference scheme and a method for minimizing numerical dispersion, which gives an accurate description of front discontinuities. A comparison of both solutions made it possible to determine the applicability limits of the asymptotic forms and to obtain an assessment of dynamic state of shell and medium during the entire time of interaction.