Exact stiffness matrix of multi-step columns and its application in non-uniform crane structure stability analysis

To analyze the stability of non-uniform crane structures with stepped and tapered columns, equivalent element stiffness derived from hybrid of the transfer matrix method (TMM) and the finite element method (FEM) is proposed. The second-order theory is applied to establish the differential equations of the stepped column with any number of sections, and the exact slope-deflection equations of the equivalent element are derived by TMM. Furthermore, the slope-deflection equations are formulated into the exact symmetric tangent stiffness matrix in FEM. Based on the FEM buckling criterion, the critical forces of stepped columns can be obtained precisely in Bernoulli-Euler sense. Buckling criterion equations of the stepped column with three widely used constraints in crane structures are given as the application of the proposed stiffness. Meanwhile, the proposed stiffness matrix of the stepped column is extended to analyze the buckling of the tapered column, which can be simulated by the four-section stepped column with the selective equivalent inertia moment. Finally, verification examples demonstrate that the proposed method can be used to accurately and efficiently evaluate the critical load of the non-uniform crane structures.