Design/methodology/approach This study establishes an analytical model to calculate the conformal contact in spherical plain bearings and verifies the new model by finite element analysis. Findings The new model proposed in this paper overcomes the limitations of elastic half-space and small-deformation assumptions. After conducting accuracy validation, it was observed that the computational error of the new model has significantly decreased in comparison to the Johnson model. For a conformal contact with a clearance of 0, the error is nearly 0. Practical implications The analytical model can calculate the contact pressure distribution of self-lubricating spherical plain bearings bonded with a self-lubricating layer and can be extended to the conformal contact problem of spherical contact surfaces in biomechanics and other fields. Originality/value The model presented here overcomes the limitations of the elastic half-space and small deformation assumptions. It accurately calculates the contact pressure distribution of self-lubricating spherical bearings. Moreover, the complex nonlinear relationship between variables such as normal force, clearance, maximum contact pressure and contact radius was investigated using this model. The model can also be extended and applied to the conformal contact problem of spherical contact surfaces in various fields, including biomechanics. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-08-2024-0296/