Mathematical modeling of the calculation of survivability of deformable solids

The article deals with the problem of solving the problems of dynamics and statics which is based not only on the fundamental methods of mechanics, but also using common hypotheses and assumptions of the resistance of materials. This makes it possible to study the state of the rods and plates in case of sudden structural changes, to analyze their sensitivity in longitudinal stratification. Modeling of various constructions for use as first approximations and tests in numerical methods is the basis for solving applied problems of dynamics and statics, due to the integration of differential equations with variable coefficients, the use of power seriesin approximate calculations. Fourth-order differential equations with variable coefficients allow us to create a solution algorithm that takes into account all initial conditions. The construction of a mathematical model for inhomogeneous rods in statics and dynamics allows us to calculate the impact on structures that are subjected to sudden disconnections of bonds and changes in the structure. This approach allows to solve optimization problems for the choice of the of distribution law of cross-sectional areas, providing for a given amount of material minimum tension in the dangerous section with a sudden decrease of the of static uncertainty degree.