Grain boundaries (GBs) serve as barriers dividing the bulk material into distinct regions with varying orientations and discontinuous tensorial properties. When subjected to external loading, the misorientation of grains and inherent crystal anisotropy create stress disparities near the GBs, resulting in stress states significantly deviating from the average. The stress conditions at grain boundaries play a crucial role in influencing microscopic phenomena such as diffusion and segregation, as well as initiating failure mechanisms such as fatigue [1], creep, and corrosion. In this study, an analytical bicrystal model [2] is employed to systematically investigate the stress state at grain boundaries. For bicrystals with grain boundaries perpendicular to uniaxial external stress, the magnitude of incompatibility stress is linked to the difference in Young’s modulus on either side of the grain boundary. The maximum incompatibility stress level is expressed as a function of the elastic components of the cubic crystal. In the case of bicrystals with inclined grain boundaries, we introduce a simple criterion that categorizes cubic materials into three groups. The extreme value of incompatibility stress and the corresponding configurations are semi-analytically determined. Alkali metals and high-temperature phases such as Fe (FCC), Ti (BCC), and NiTi (B2) exhibit the highest stress concentration at grain boundaries inclined at approximately 47 degrees, while common metals exhibit the highest incompatibility at grain boundaries perpendicular to the external stress.