Comparative analysis of approaches for developing a generalised formula for buckling load of n-stage hydraulic cylinders

Accurate prediction of buckling in telescopic hydraulic cylinders remains a major challenge as with the increase in the number of stages, variations in stiffness become significant. Traditional analytical approaches often fail to capture these variations, leading to erroneous predictions. Although in some studies recursive formulas have been proposed for the buckling load, however, they typically assume equal-slopes at the stage interfaces. This assumption neglects the actual variation in bending stiffness across stages. To address this limitation in the present study, a generalised formula has been developed for the buckling load of n-stage hydraulic cylinders by incorporating variations in bending stiffness and slope discontinuities into the moment boundary conditions. Bending stiffness is evaluated through two distinct approaches. In the first approach, stiffness is determined at each connection by assuming that the rod within the cylinder behaves as a supported beam. While this method simplifies the formulation and supports a generalised solution, it assumes constant stiffness and may not fully capture the deformed behaviour at the interface. To improve accuracy, the second approach evaluates bending stiffness at the actual buckling point, where the deformed segment is modelled as fixed-free with a bending moment acting at the free end. This method redefines boundary conditions at the buckling location using the Successive Approximation Method (SAM), thus better representing the physical response. Both approaches lead to a generalised stiffness matrix applicable to any n-stage hydraulic cylinder configuration. To validate the proposed methods, single-stage, two-stage, and three-stage hydraulic cylinders of varying lengths are analysed. Finite Element Analysis (FEA) is conducted to compare predicted buckling loads and buckling points. While both methods provide consistent predictions, the second approach is better suited for complex, real-world applications.