The fatigue analysis of structural components is a relevant research topic in both scientific and industrial communities. Despite major advances in understanding, fatigue damage remains a significant issue for both metallic and non-metallic components, sometimes leading to unexpected failures of in-service parts. Among the different assessment methodologies, critical plane methods have gained significance as they enable identification of a component’s critical location and direction of early crack propagation. However, the standard plane scanning method for calculating critical plane factors is computationally intensive and, for that, it is only applied when the component critical regions are already known. When critical areas are not easily identifiable due to complex geometries, loads or constraints, a more efficient method for evaluating critical plane factors would be required. This work presents a closed form solution for efficiently evaluating the Fatemi-Socie critical plane factor, in case of linear-elastic material behaviour and proportional loading conditions, based on tensor invariants and coordinates transformation laws. The proposed algorithm was tested on different test cases (i.e. hourglass, notched and welded joint geometries) under different loading conditions (i.e. tensile, bending and torsion) and showed a significant reduction in computation time compared to the standard plane scanning method.