A genetic algorithm of optimal design of beam at restricted sagging

Abstract

A genetic algorithm for solving the problem of optimal beam material distribution along length at a given restriction on maximum sagging value is suggested. A review of literature sources is conducted and it was shown that existing solutions cover partial cases only in which the position of the point with maximum sagging was defined previously. In the paper presented I-section beam with constant proportions is considered, i.e., beam width, caps, and web thickness are proportional to beam height in the current cross-section. A statically determined beam is being considered. The load applied to a beam can be arbitrary, including cases of non-symmetrical loads and differently oriented ones. The position of point(s) at which beam sagging is maximum are unknown at the beginning of optimization and are found in the process solution. The problem is solved in the linear definition. Beam mass was assumed to be an optimization criterion. The method of finite differences is used for beam sagging finding, i.e., for the solution of the differential equation of the bending beam with a variable cross-section. Discretization allows transforming the problem of design into the problem of beam height determination at a system of reference points. At this stage, found values of beam height must satisfy restrictions on reference point displacements. The suggested technique allows controlling beam displacement quite flexibly because restrictions on point displacement are considered separately and do not depend on each other. The suggested objective function is the linear superposition of beam mass and the possible penalty in case of beam maximum sagging over exceeding predefined values. The application of a genetic algorithm allows getting sets of beam thicknesses those, which guaranty reaching the minimum of the objective function. The model problem is solved. It is shown that the suggested algorithm allows effectively solves problems of optimal design of beams with restrictions on the maximum sagging value. The suggested approach can be developed for strength restrictions, statically undetermined structures, etc.

Application of Bezier curves for description of structure shape in optimization of adhesive joints

“…Одним із можливих способів розв'язання цієї задачі є дискретизація шуканої функції. У цьому випадку задача зводиться до знаходження товщини елементів конструкції в системі точок [11]. Перехід від неперервних функцій до дискретних дозволяє оптимізувати з'єднання з покроковою зміною товщини [12].…”

“…Для цього беремо за шукані змінні довжину з'єднання L і товщину накладки у вузлових точках   1 i  і знаходимо такі їх оптимальні значення, які забезпечують мінімум маси накладки (5) під час виконання обмежень за міцністю (6) і (7). Однак, на відміну від задачі про знаходження оптимального розподілу матеріалу вздовж балки [11], якщо значення товщини   . Якщо за критерій оптимальності взяти масу конструкції (5) і накласти обмеження на максимальні перші головні напруження у клейовому шарі (6) і мінімальну товщину накладки (8), то фітнес-функцію можна подати у вигляді…”

Topological optimization of a symmetrical adhesive joint. Island model of genetic algorithm

“…One of the possible way to solve this problem is the discretization of the desired function. In this case, the problem is to find the thickness of structural elements in a system of points [13]. The transition from continuous to discrete functions makes it possible to optimize the joint with a stepped thickness change [14,15] (this solution is often used in joints of layers composites).…”

“…In addition, the one-dimensional setting of the problem makes it possible to use a fairly simple and efficient finite difference method to calculate the stress state of the adhesive joint. This method is successfully used both for solving one-dimensional problems [7,13] and for solving two-dimensional problems of joint mechanics [28].…”

“…It means to determine the optimal values, which provide, for example, a minimum mass of the overlay (8) when the strength restrictions (10) are fulfilled. However, in contrast to the problem of finding the optimal distribution of the material along the beam [13], if the thickness values  …”