Rational topology of steel i-beams with various gradients of changing wall height and shelf width at specified sections along the length of the beam

A methodological approach has been developed for determining the rational topology of steel I-beams with variable stiffness under uniformly distributed loading along the beam length. It has been shown that for such beams, with varying web height and flange width, the maximum stress does not occur in the section where the maximum bending moment acts. The problem of finding the optimal cross-sectional height is solved using the Lagrange multipliers method in conjunction with the Kuhn–Tucker conditions. For steel I-beams with variable web height and flange width, the sufficient condition for structural optimality is confirmed: the area of the web is equal to the total area of the two flanges. However, under linear reduction of web height and flange width in the direction of decreasing bending moments, new critical cross-sections arise along the beam length in which the normal stresses in the flanges exceed those in the section with the maximum bending moment. This indicates that beams with variable stiffness may have multiple governing sections. An improved physical–mathematical model of the stress–strain state of I-beams in bending is proposed. A steel I-beam with the proposed new topology has the ability to adapt to its stress–strain state by introducing reverse variation of flange width: in selected sections, the beam height decreases or remains constant, while the flange width and accordingly the flange cross-sectional area increases relative to the section where the maximum bending moment acts. This improved design approach allows for achieving stress levels in all current cross-sections that do not exceed the yield strength of steel along the entire length of the I-beam. The numerical studies conducted demonstrate the possibility of finding new rational design solutions for variable cross-section steel I-beams. The existence of an admissible set of rational solutions based on the obtained results has also been confirmed. Thus, the problem of determining the rational topology of steel I-beams with linearly varying flange width and web height represents a design task with appropriately formulated and adequate design condition.