title: ÐÐ½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¸ ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ Ð½ÐµÐ»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ ÑÑÑÐ¾Ð¹ÑÐ¸Ð²Ð¾ÑÑÐ¸ Ð±Ð°Ð»ÐºÐ¸ Ð¿ÑÐ¸ ÑÐ¸ÑÑÐ¾Ð¼ Ð¸Ð·Ð³Ð¸Ð±Ðµ
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: Analytical and numerical solution of the nonlinear problem of beam stability in pure bending
Creators: name: Ð£ÑÐ¸Ð¼Ð¸ÑÐµÐ²Ð°, Ð¯Ð½Ð°
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2021
Subjects: ÐÐ°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ°Ñ ÑÐ¸Ð·Ð¸ÐºÐ°
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description: This work is devoted to solving the problem of stability in the exact formulation. The subject of research is the stress-strain state and stability of a geometrically nonlinear elastic beam under static loading. The tasks of the work: 1. Obtain and compare the exact solution of the stability problem and the asymptotic solution of the linearized stability problem for a cantilever beam loaded with a potential following moment; 2. Obtain a numerical solution to the stability problem using the finite element method (FEM) and compare it with the exact analytical solution. The work uses a geometrically exact beam theory and a variational approach. A spatial model of a beam is considered, which takes into account all six stiffnesses. The results are: 1. The exact solution to the stated stability problem and the asymptotic solution to the linearized stability problem are obtained; 2. The error of the standard formula is about 10%; 3. Using the FEM, an exact stiffness matrix was obtained, which nonlinearly depends on the load parameter; 4. Using Newton's method, an algorithm for solving a nonlinear eigenvalue problem is developed. The results of the work can be applied to solve any sustainability problem. The use of the exact formula makes it possible to make a more economical solution in comparison with the classical approach (the critical force is less than ~ 10%).