Lateral–Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads

Cantilever steel beams are an essential structural element in civil engineering fields such as bridges and buildings. However, there is very little research on the critical moment (Mcr) of cantilever beams subjected to a concentrated load (CL) or a combination of concentrated load and uniformly distributed load (CUDL) when the concentrated load is not limited to the free end. Therefore, the focus of the current paper is the calculation of Mcr for cantilever steel beams under CL and CUDL. This paper proposes a program and a simple closed-form solution for Mcr that are applicable to the elastic buckling analysis of cantilever I-beams under CL and CUDL. Based on the Rayleigh–Ritz method, a matrix equation and the corresponding procedure about Mcr under CL and CUDL are derived by using infinite trigonometric series for the buckling deformation functions. The value of Mcr and the corresponding mode of buckling can be obtained efficiently by considering the symmetry of the section, the ratio of two load values and the load action position. Experimental results and finite element calculations validate the numerical solutions of the procedure. A closed-form solution for Mcr is derived according to the assumption of a small torsion angle and the specific values of each coefficient in the closed-form solution of Mcr are calculated by the proposed procedure. The results show that the procedure and closed-form solution for Mcr presented in this paper have a high degree of accuracy in calculating the Mcr of the cantilever beam under CL and CUDL. The deviations between the results calculated by the proposed procedure and data from existing literature are less than 8%. These conclusions are capable of solving the calculation problem of Mcr for cantilever beams under CL or CUDL, which are both significant load cases in engineering. The study provides a reference for the design of cantilever steel beams.