Practical Stability of the “Cross” Scheme in the Numerical Integration of Dynamic Equations for Flexible Thin-Walled Structural Elements Obeying the Hypotheses of the Timoshenko Theory

Abstract

In the von Kármán approximation, we formulate the initial boundary-value problem of the dynamics of flexible isotropic and composite elastic beams-walls within the framework of two versions of the Timoshenko theory. We perform a qualitative analysis of the resolving system of equations of motion. It is shown that, in the geometrically linear statement, the dynamics of elastic beams is described by a hyperbolic system. At the same time, in the case of deformation of flexible beams, the system of resolving motion equations may change its type degenerating from a hyperbolic system into a system of mixed (composite) type. We develop finite-difference and variation-difference versions of the explicit (in time) “cross” scheme for the numerical integration of the posed initial boundary-value problems. On the basis of these numerical methods, we perform the numerical analyses of the dynamic flexural deformation of flexible metallic and composite beams under explosive-type loads. The result of these calculations demonstrate that, in almost cases, one can indicate the levels of loading of flexible beams under which the “cross” scheme becomes unstable, although the condition of stability obtained in the linear approximation is satisfied with a considerable margin. Thus, it is shown that, in the case of dynamic analysis of flexible beams, one can speak only about the practical stability of the “cross” scheme but not about its conditional stability.