Functionally graded materials (FGMs) are revolutionizing various industries with their customizable properties, a key advantage over traditional composites. The rise of voxel-based 3D printing has furthered the development of FGMs with complex microstructures. Despite these advances, current design methods for FGMs often use abstract mathematical functions with limited relevance to actual performance. Furthermore, conventional micromechanics models for the analysis of FGMs tend to oversimplify, leading to inaccuracies in effective property predictions. To address these fundamental deficiencies, this paper introduces new gradation functions for functionally graded beams (FGBs) based on bending strain energy density, coupled with a voxel-based design and analysis approach. For the first time, these new gradation functions directly relate to structural performance and have proven to be more effective than conventional ones in improving beam performance, particularly under complex bending moments influenced by various loading and boundary conditions. This study reveals the significant role of primary and secondary gradation indices in material composition and distribution, both along the beam axis and across sections. It identifies optimal combinations of these indices for enhanced FGB performance. This research not only fills gaps in FGB design and analysis but also opens possibilities for applying these concepts to other strain energy density types, like shearing and torsion, and to different structural components such as plates and shells.