Solution of the dynamic problem of the effect of a mobile massive load on a structure by direct methods

Objective. The article presents the modeling of dynamic processes in buildings and structures. The general formulation of the dynamic problem of the motion of a massive load along a massive structure is considered. Method. The equation of motion is obtained in the form of the finite element method. The equations are solved using direct methods of integrating dynamics problems. Absolutely stable direct integration schemes are constructed, in which the system of resolving equations is trivial - the matrix of the system is diagonal. The complexity at the time step is as low as in explicit schemes. The proposed methods can be considered as explicit absolutely stable direct integration schemes for a dynamic problem with a time-varying mass. Recommendations are given for assessing the accuracy of the numerical solution. The problem of moving a massive load along a massive structure is considered. Discretization of the spatial domain is performed by the finite element method. Discretization of the time domain is based on a step-bystep single-layer process. An approach similar to Wilson's theta method, applied on the basis of Gartin's variational principle, is used to construct the main equations. Result. Differential equations of the problem of the motion of a moving massive load on an arbitrary massive structure are constructed in the form of the finite element method, and numerical methods for solving the equations of motion using absolutely stable direct integration schemes are presented. The system of resolving equations can have a diagonal structure, which allows classifying the obtained schemes as explicit. Conclusion. The proposed approach can be applied to the analysis of the stress-strain state and bearing capacity of arbitrary spans when a massive load moves along them. The effect of the load velocity on the stress-strain state of the structure is taken into account. There are no restrictions on boundary conditions and types of loading.

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