The dynamic stability of physically nonlinear plate systems under biaxial compression

The article presents the method of dynamic stability analysis of plate systems with nonshifting ribs. A plate system under the biaxial dynamic compression loads is considered. The Kirchhoff - Love hypotheses, the nonlinear-elastic body hypothesis are considered the basis of the calculations. The material of the plate system is assumed to be physically nonlinear, stress-deformation diagram is approximated in the form of a cubic polynomial. The displacement of points in normal direction to middle plane of plates is presented in the form of Vlasov expansion. To derive the basic differential equations of stability, the strainenergy method and Vlasov's variation method are used. The extreme value of total energy of the system is defined using Euler - Lagrange equation, after solving of which the set of basic nonlinear differential equations of buckling of the plate system with non-shifting ribs under dynamic compression loads is given. As an example, the stability calculation of physically nonlinear T-shaped plate system hinge-supported along the contour is carried out. Buckling of the plate system occurs longitudinally on one half-wave of sinusoid. At the solution of a task in the first approximation, a nonlinear differential equation is derived, the numerical integration of which was carried out by the Runge - Kutta method. Based on the results of the calculations, graphs of the relative magnitude of deflection against the dynamic coefficient are plotted. The influence of the degree of physical nonlinearity of the material, the rate of change of the dynamic compressive load on the dynamic criterion of buckling of the plate system was studied.