- STRUCTURAL MECHANICS
- Calculation of Rational Gradation of Sections for Statistically Determinable Bending Systems as a Non-Linear Programming Problem
- UDC 624.014.046

**Nikolai N. DEMIDOV**, å-mail: melirina08@mail.ru

Moscow State University of Civil Engineering (National Research University), Yaroslavskoe shosse, 26, Moscow 129337, Russian Federation

**Abstract**. The article is devoted to the actual issue of steel economy, by searching for optimal places for the stepwise change of moments of inertia of the I-beams. The step change in the cross-section of beams has proved itself in practice and is widely used in construction. This is one of the most frequently used methods of reducing the steel consumption without reducing the reliability of bearing structures. It is shown that such a problem can be considered as an optimization problem. On a number of concrete examples it is shown that, with the appropriate formulation, the optimization problem reduces to the problem of non-linear programming. The chosen formulation of the objective function is not the only possible one, for example, thus it is possible to solve the maximization problem, but a slightly different objective function. Stepwise reduction of moments of inertia when decreasing the steel consumption leads to decrease in the rigidity of the structure. When solving the optimization problem, a beam with several large deflections is obtained, so the second limiting state must be taken into account in the actual design. All the problems posed in this paper are of practical interest. The formulas obtained can be used in design practice.

**Key words**: objective function, non-linear problem, constraints, inequalities, partial derivatives, Hesse matrix, Sylvester criterion, moment of inertia, moment diagram, step change of cross sections, minimum steel consumption, statically defined systems, deflections. - REFERENCES

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