ELASTIC CURVE-BASED ANALYSIS FOR BENT BEAM PROFILE OPTIMIZATION

Abstract

This paper investigates the selection of cross-sectional shapes for beams based on the theory of elastic bending and the shapes of elastic curves under various forms, intensities, and distributions of loads on beams with different boundary conditions. Choosing the cross-section profile shape of a bent beam is crucial for optimizing its performance under various loading conditions. Different shapes offer varying levels of efficiency in terms of strength, stiffness, and material usage. The linear elastic theory of bending is utilized to derive the differential equation of the elastic curve for bent beams. By solving this second-order differential equation using the direct integration method and the Clebsch procedure, the elastic curves are obtained. Various loading positions and scenarios are analyzed for overhanging and cantilever elastic beams, demonstrating the correlation between beam bending stiffness, shape factor, and degree of utilization of the cross-sectional shape. We apply the criterion of ultimate bending strength (flexural strength) of steel for dimensioning beams. Thus, the characteristic dimension of the profile is determined according to the maximum value of the bending moment for each type of beam, considering the values and distribution of the load. For the same bent beam and loading values, different cross-section profiles are suggested, and the maximal deflection in each case is obtained. Based on these findings, the best choice for profile selection is considered. The study also examines the relationship between shape factors and the degree of utilization of the cross-section, and the concept of the ideal shape of beams with variable stiffness along their length. The results provide insights into the optimal selection of cross-sectional shapes for beams subjected to bending, helping engineers and designers develop a sense for the most efficient selection of cross-sectional shapes.