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.

为了分析具有阶梯和锥形柱的非均匀起重机结构的稳定性，提出了从传递矩阵法 (TMM) 和有限元法 (FEM) 混合导出的等效单元刚度。应用二阶理论建立任意截面阶梯柱的微分方程，并通过TMM推导出等效单元的精确斜挠度方程。此外，在 FEM 中，斜率-挠度方程被公式化为精确对称的切线刚度矩阵。基于有限元屈曲准则，可以精确地获得伯努利-欧拉意义上的阶梯柱的临界力。给出了在起重机结构中具有三个广泛使用的约束的阶梯柱的屈曲准则方程作为建议刚度的应用。同时，将提出的阶梯柱刚度矩阵扩展到分析锥形柱的屈曲，这可以通过具有选择性等效惯性矩的四节阶梯柱进行模拟。最后，验证实例表明所提出的方法可用于准确有效地评估非均匀起重机结构的临界载荷。