# Abstract

Relevance. Single-connected and multi-connected plate systems are widely used in construction, aircraft, shipbuilding, mechanical engineering, instrument making. As a result, the study of the stability of geometrically nonlinear plate systems is an urgent topic. But, despite significant achievements in this area, there are still many unsolved problems. Thus, the requests of the above-mentioned areas of application of thin-walled spatial systems require further study of the issue of static and dynamic stability. The aim of the work - development of a method of the dynamic stability analysis of geometrically nonlinear plate systems such as prismatic shells under the action of dynamic compression loads. Methods. A plate system, which is subject to dynamic compression loads in the longitudinal direction, is considered. Kirchhoff - Love hypotheses are taken into account. The material stress-deformation diagram is linear. The displacement of points in the normal direction to the median plane of the plates is determined in the form of the Vlasov expansion. To derive the basic differential equations of stability, the energy method and the variational Vlasov method are used. The extreme value of the total energy is determined using the Euler - Lagrange equation. As a result, a set of basic nonlinear differential equations for studying the buckling of the plate system under the action of dynamic compression loads is obtained. Results. The developed method is used to stability analysis of a geometrically nonlinear prismatic shell with a closed contour of the cross section, under central compression under the action of dynamic loading. The edges of the shell rest on the diaphragm. The buckling of the prismatic shell in the longitudinal direction along one and two half-waves of a sinusoid is studied. The numerical integration of nonlinear differential equations is performed by the Runge - Kutta method. Based on the calculation results, graphs of the dependence of the relative deflection on the dynamic coefficient are constructed. The influence of the rate of change of compression stress, the initial imperfection of the system, and other parameters on the criteria for the dynamic stability of the plate system is investigated.

### Sergey P. Ivanov

Volga State University of Technology; Mari State University

Author for correspondence.
Email: sp-ivanov@mail.ru
3 Lenin Sq, Yoshkar-Ola, 424000, Russian Federatio; 1 Lenin Sq, Yoshkar-Ola, 424000, Russian Federation

Doctor of Science, Professor, Head of the Department of Strength of Materials and Applied Mechanics of VSUT; Professor of the Department of Electromechanics of MarSU

### Anastasia S. Ivanova

Volga State University of Technology

Email: sp-ivanov@mail.ru
3 Lenin Sq, Yoshkar-Ola, 424000, Russian Federatio

senior lecturer, Department of Strength of Materials and Applied Mechanics

### Oleg G. Ivanov

Volga State University of Technology

Email: sp-ivanov@mail.ru
3 Lenin Sq, Yoshkar-Ola, 424000, Russian Federatio

Cand. Sc., Associate Professor, Department of Strength of Materials and Applied Mechanics

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