Rational topology of cantilever steel beams with variable flange width and web height under deflection and strength constraints

The article solves the problem of selecting the optimal topology of a cantilever steel I-beam with variable web height and flange width, subject to deflection constraints and assuming an optimal distribution of steel in each cross-section based on strength conditions. The problem is solved using the method of Lagrange multipliers. The optimal design criterion is taken to be the objective function minimizing steel consumption for the structure. The condition of optimal distribution of steel between the flanges and the web based on strength criteria is adopted for each cross-section. Possible deviations from the optimal ratio between the flange area and the web area are accounted for by an additional coefficient. The problem belongs to nonlinear programming. The strength condition of the web is considered inactive and is ensured by structural measures (web stiffeners). Out-of-plane stability of the beam (against lateral bending) is provided by an appropriate system of horizontal bracing along the flanges. An analytical function describing the cross-sectional variation along the length of a cantilever tapered I-beam is obtained under the optimization conditions for a uniformly distributed load and a given relative design deflection. The derived analytical function for the relative optimal height of the I-beam along the length of the structure is a power-law function and depends on the load, the deflection constraints, and the optimal or rational distribution of steel in each cross-section. The optimal height of the I-beam for the support section (where the maximum bending moment occurs) is determined under the conditions of the problem. The identified patterns of variation in the optimal beam height allow one to select the optimal topology of the structure and to account for the possibility of higher stresses arising in sections of smaller height. It is confirmed that the optimal structural solution depends on the load distribution law. The obtained results allow determining the degree of variability of the cross-section height for the optimal topology.  The analytical formulas derived for the optimal height of a beam with variable flange width and variable web height enable, at the first stage of variant design, an evaluation of the efficiency of the design solution.