PSI - Issue 81

Artem Bilyk et al. / Procedia Structural Integrity 81 (2026) 177–183

178

problem in implementing optimization results into real design is taking into account production and technological limitations associated with the assortment and features of manufacturing structures (Vasileios et al. (2014); Pidgurskyi et al. (2024); Shevchenko (2015)). Artificial approximation of discrete assortments, which is performed to simplify the application of mathematical methods of linear and nonlinear programming, leads to the fact that the obtained value of the local optimum requires inverse mapping to a discrete scale. If in the case of one-dimensional problems this leads only to an increase in material consumption compared to the theoretical ones (the closest higher value is found), then in the case of multidimensional problems (which, in particular, are problems of optimal design of beams composed from steel sheets) the inverse mapping process requires the development of special strategies for discretization of cross-sectional elements, which affects the final result (Permyakov et al. (2008)). After all, a theoretically selected cross-section based on artificially continuous functions may turn out to be “impossible” from the point of view of production or the available assortment.

Nomenclature S

feasible region of solutions (FRS)

optimal design solution

x opt

some alternative from a discrete set of solutions

x

discrete set of solutions

S = { x 1 , x 2 ,…x N } C j = { C 1 , C 2 ,… C M }

set of criteria of the design at the different levels of analysis

list of parametric constraints

h k (x)

independent variables of model constraints (system boundaries) discrete set of solutions based on model constraints

p m

S v = { x

m1 , x m2 , …, x mN }

reduced final discrete set of solutions

S red =S v \{ H

1 U H 2 U … U H K }

This article is devoted to solving the problem of finding a global optimal solution for steel structures using the example of symmetrical equal-flanges I-beams made of sheets. The problem of rational arrangement of a three-sheet steel I-beam in terms of local optimization is a classic one, but differs greatly in approaches. Basically, the problem was considered as the optimum height of the beam with a given moment of resistance from the condition of minimum area of its cross section, at a given ratio of wall height to its thickness, etc. (Horev et al. (1997); Nilov et al. (2010)). Along with the optimal one, the so-called minimum (rational) I-beam height was usually considered, based on the condition of the maximum permissible deflection of the beam. Later, the problem of an I-beam of minimal mass was posed as a mathematical programming problem with structural constraints inequalities: for strength in bending and shear and for deflection, with solutions for individual cases (Banichuk and Kobelev (1983); Belskiy and Tamarchenko (1990)). In modern times, solving problems of optimizing steel I-beams was performed both analytically and numerically, with an increase in variable parameters and the application of a larger number of constraints (Aykut (2019); Lobanov et al. (2003); Gordeev (2009); Bilyk (2008)). However, the task of finding a globally optimal solution for equal flange monosteel I-beams, taking into account design, production, and technological constraints, is posed here for the first time. 2. Numerical research The exhaustive search method remains guaranteed to achieve the global optimum, although it is the most laborious in terms of the number of operations. For continuous functions, a certain satisfactory step of enumerating values is chosen, which, however, as a rule, does not correspond to the aggregation of real objects. Modern computer capabilities allow us to largely remove the limitations on machine time, which many optimization methods were oriented towards. At the same time, for high-dimensional problems, a method is needed that would be more efficient than exhaustive search, but would allow us to achieve the global optimum with certainty. In general, the optimal choice problem is formulated in the space of states: given the main quantitative and qualitative input data in the Feasible region of solutions (FRS), choose the best design solution according to the selected quality criterion. Let x - some alternative from a discrete set S - FRS, containing all possible alternatives: ∈ S= { 1 , 2 ,…, }. Criteria ∈ { 1 , 2 ,… } are indicators (superconvolutions) of the design at different levels of analysis. Then the choice of the optimal design should be carried out according to the decision rule : =argmin ∈ ( )|ℎ ( ) (1) where ℎ ( ) – list of parametric constraints. In exhaustive search, to find the global optimum, according to the rule all alternatives must be analyzed. This paper proposes a new method, called the directed selection method of design solutions, which is a modified exhaustive search method. In terms of the sequence of stages, it conceptually consists of the following: 1. Consideration of model constraints (system boundaries) and selection of independent variables . 2. Generation of a set of discrete object solutions based on model constraints (system boundaries): =∈ S|( ) , which is forming a discrete FRS: S = { 1 , 2 ,…, }.