МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ОПТИМІЗАЦІЇ ПОКРИТТЯ КРУГЛОЇ У ПЛАНІ БУДІВЛІ З РАДІАЛЬНИМ РОЗТАШУВАННЯМ БАЛОК

Abstract

The article presents the results of optimization of the angle between radial beams in the floor of a circular building in the plan. On the one hand, they rest on the central post, and on the other, on vertical supporting structures along the circle. Steel decking is laid on the beams. The angle between the beams is determined so that the mass of the beam and the deck is minimal. This angle is considered optimal. To solve the problem, the target function of the cost of flooring and radial beams per unit floor area is used. This function depends on the angle between the beams. Using mathematical methods of differentiation, the minimum of the objective function and the corresponding value of the optimal angle were found. The thickness of the flooring was determined on the basis of ensuring its rigidity. It is assumed that composite welded radial beams have I-beams with two axes of symmetry. The height of the beam corresponds to the equality of the areas of the shelves and the wall. The problem of determining the optimal angle between the beams was solved on the basis of ensuring the strength of the beams under normal stresses. In the design diagram of the beam, a triangular distributed load is adopted. The dimensions of the cross-section of the beam were determined based on the equality of the required and actual moments of resistance, and were included in the target cost function. The study took into account that the deflection of the beam at the optimal angle between them can exceed the limiting standard value. Based on the solution of the system of equations of strength and stiffness, a formula is obtained for the minimum angle between the beams from the stiffness condition. The carried out mathematical studies have shown that at the optimal angle between the beams, it is possible to ensure its rigidity. This is possible when the flexibility of the beam wall exceeds a certain minimum value. Analysis of the formula for the minimum value of the wall flexibility showed that it is proportional to the design steel resistance to the sixth power. Therefore, to ensure that the deflection of the beam does not exceed the limiting value at the optimum angle, it is necessary to use low strength steel. To confirm the practical feasibility of using the proposed method, the problem was solved with certain numerical data. The results obtained have confirmed that the problem has a practical meaning at a relatively low steel strength. In addition, it turned out that the optimal angle between the beams does not depend on its span.