<p>This study proposes Chebyshev polynomials-based different shear deformation theories to analyse the buckling, bending, and free vibration behaviours of axially loaded functionally graded (FG) curved beams for the first time. The Chebyshev-third-order shear deformation theory satisfies the condition of eliminating shear stress at the bottom and top surfaces of the FG curved beam without requiring a shear correction factor. Furthermore, this theory can be simplified to the first-order shear deformation and classical beam theories. The material characteristics of the FG curved beam exhibit variability through its thickness by a power law distribution. The governing equations are derived from Lagrange’s equations. The Ritz method, utilising polynomial shape functions based on the Fibonacci sequence, has been developed to solve the problem. FG curved beams with four boundary conditions are examined, including simply-supported, clamped-free, clamped-simply supported, and clamped-clamped. Numerical examples are carried out to assess the accuracy and efficacy of the present theory. Additionally, the study elucidates the behavioural patterns of FG curved beams regarding various parameters such as boundary condition, slenderness ratio, curvature, and power-law index. The findings of this study serve as a benchmark for future research endeavours. Moreover, they have the potential to enhance the design and optimisation of FG curved beams across a diverse array of engineering applications.</p>