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A simple analytical model of intergranular normal stresses is proposed for a general elastic polycrystalline material with arbitrary shaped and randomly oriented grains under uniform loading. The model provides algebraic expressions for the local grain-boundary-normal stress and the corresponding uncertainties, as a function of the grain-boundary type, its inclination with respect to the direction of external loading and material-elasticity parameters. The knowledge of intergranular normal stresses is a necessary prerequisite in any local damage modeling approach, e.g., to predict the intergranular stress-corrosion cracking, grain-boundary sliding or fatigue-crack-initiation sites in structural materials. The model is derived in a perturbative manner, starting with the exact solution of a simple setup and later successively refining it to account for higher order complexities of realistic polycrystalline materials. In the simplest scenario, a bicrystal model is embedded in an isotropic elastic medium and solved for uniaxial loading conditions, assuming 1D Reuss and Voigt approximations on different length scales. In the final iteration, the grain boundary becomes a part of a 3D structure consisting of five 1D chains with arbitrary number of grains and surrounded by an anisotropic elastic medium. Constitutive equations can be solved for arbitrary uniform loading, for any grain-boundary type and choice of elastic polycrystalline material. At each iteration, the algebraic expressions for the local grain-boundary-normal stress, along with the corresponding statistical distributions, are derived and their accuracy systematically verified and validated against the finite element simulation results of different Voronoi microstructures.