Analytical Solution of Optimization Problem of Stability of Frame Systems

International Multi-Conference on Industrial Engineering and Modern technologies

Shein, A.I.^1 ; Zemtsova, O.G.^1

Department of Mechanics, Penza State University of Architecture and Construction, Titova str., 28, Penza

440028, Russia^1

关键词: Nodal displacement;    Nonlinear mathematical programming;    Optimization problems;    Optimized parameter;    Stability condition;    Stationary conditions;    System of equations;    Transcendental functions;   

Others  :  https://iopscience.iop.org/article/10.1088/1757-899X/463/4/042065/pdf
DOI  :  10.1088/1757-899X/463/4/042065

来源: IOP

【 摘 要 】

The article deals with the problem of optimizing multi-story frame systems from the condition of stability. When solving the problem of optimizing the cross-sections of the elements of frame systems, the bearing capacity of which is determined by their stability, the known parameters are configuration, support fastenings, material and nodal loads. As an optimized parameter, the constant bending stiffness within each rod is assumed. The optimization problem is solved by the criterion of minimum volume of the frame material. The mathematical model compiled by the authors in a general form is a nonlinear mathematical programming task. To solve it, the method of Lagrange multipliers is used. The problem is solved in two stages. At the first stage of the stationary conditions, the authors obtained the relationships between the nodal displacements for free and non-free frames. In the second stage, at the found values of these relations, the determinant condition of the critical state of the frame structure can be represented in the form of a system of equations. In this case, the system of equations obtained by the method of Lagrange multipliers becomes analytically solvable with respect to unknown linear stiffnesses, since the equations entering into it do not contain reciprocal products of transcendental functions from unknowns. In this article, in closed form, equations are obtained that allow determining the optimal linear stiffnesses of columns and crossbars of free and non-free frames. The analytical solution obtained has been checked. To select the optimal stiffnesses from the stability condition, it is necessary to define the critical parameters using the constructive flexibility of the elements, using the functions presented in the paper, and then determine the linear stiffnesses of the posts and crossbars.