Post-Buckling Analysis of Arch and Serpentine Structures Under End-to-End Compression

Arch and serpentine structures are two fundamental structural forms with significant applications in various fields. When subjected to compressive loading at both ends, these structures undergo flexural-torsional post-buckling, resulting in complex deformation modes that are challenging to describe using basic functions (e.g., trigonometric functions and polynomial functions), posing significant challenges in finding analytical solutions. In this study, we propose a novel approach to address this issue. By representing the lateral displacement with a trigonometric series expansion and utilizing the equilibrium equation, the angular displacement is expressed in terms of special functions known as Mathieu functions. Furthermore, the energy method is employed to obtain analytical solutions for the flexural-torsional post-buckling deformation components. The theoretical findings are validated through experiments and finite element analysis. Based on the theoretical results, explicit analytical expressions for the maximum principal strain and the bending-torsion ratio of the structures are derived, offering valuable insights for the design of arch and serpentine structures in practical applications.