Rational steel canopy structures over stadium stands using welded I-beams with variable flange width and web height

An improved methodological approach has been developed for determining the optimal structural form of a cantilever welded steel I-beam with variable flange width and web height under combined loading: uniformly distributed along the length and an applied concentrated bending moment at the free end. These steel structures demonstrate a high efficiency of steel use, serving as primary load-bearing elements of canopies over stadium stands. The structural efficiency is achieved through the redistribution of steel along the length of the element and over the section height in such a way as to attain an optimal structural form. The selection of the optimal configuration is based on numerical studies focused on determining the rational gradient of the I-beam’s height variation and selecting the gradient value of flange width, while satisfying the strength conditions for each cross-section. It is assumed that the rational distribution of steel over the section height corresponds to the optimal ratio between the flange and web areas according to the criterion of minimizing steel consumption for each elementary segment of the beam. The optimization problem of a conical welded I-beam under the adopted loading conditions with a linear variation of flange width and web height is formulated in the traditional manner as a single-criterion constrained optimization problem, where the criterion is the minimization of steel consumption, and strength constraints in each cross-section (which depend on the gradients of variation of flange width and web height under combined loading) are imposed. The problem is solved with the aim of complying with the strength constraints and determining the limiting values of the gradients of change of web height and flange width of the cross section of the steel I-beam for a generalized loading scheme. Numerical investigations were carried out to find the gradients of variation of web height and flange width based on the criterion of minimizing steel consumption and satisfying strength constraints in each cross-section along the length of the structure. The results obtained show that the optimal gradient of variation of section height is γh≤0,6 for P1l/Mx,0≤2,0, and γh≤0,5  for  P1l/Mx,0≤1,0.. It has been established that for P1l/Mx,0≤0,25 and γh≤0,9, the strength conditions are satisfied for all cross-sections of the rational structural form of the welded steel I-beam. A methodological approach has been developed for solving problems of this type in a generalized form.