MATHEMATICAL MODELING OF STRESS-STRAIN STATE OF LOADED RODS WITH ACCOUNT OF TRANSVERSE BENDING

Abstract

Mathematical simulation of static and dynamic processes of strain taking into account of transverse bending under loading is presented in the paper in linear and geometrically nonlinear statements. An extensive analysis of research work carried out in universities and science centers all over the world is given, the relevance of the problem and the areas of solution application are emphasized. The mathematical correctness of the problem statement is shown. Variations of kinetic, potential energy, volume and surface forces are determined for mathematical models of static and dynamic processes of strain with taking into account of transverse bending of loaded rods in linear and geometrically nonlinear statements. Based on the theory of elastic strains and the refined theory of Vlasov-Dzhanelidze-Kabulov, and using the Ostrogradsky-Hamilton variation principle, a mathematical model of the statics and dynamics of the process of rod points displacement is developed for transverse bending in linear and geometrically nonlinear statements. Equations of a mathematical model with natural initial and boundary conditions in a vector form are given. A computational algorithm is developed for calculating the statics and dynamics of rods under loading in linear and geometrically non-linear statements using the central finite differences of the second order of accuracy. The strain processes when the rod is rigidly fixed at two edges are considered in linear and geometrically nonlinear statements. The calculation results obtained are given in the form of graphs. The propagation of longitudinal, transverse vibrations and the angle of inclination along the length of the rod was studied at different points of times. Linear and geometrically non-linear vibration results are analyzed and compared.