Dislocation pile-ups at grain boundaries determine the back-stress opposing plastic deformation while promoting yielding in neighbouring grains. A regular array of parallel pile-ups of edge dislocations was analysed numerically to determine the equilibrium positions of all dislocations and compared with the result of various simplifications. A model based on infinite low angle boundaries could not reproduce the long-range stress field of the numerical calculations; an analytical correction for this simplification is presented. Infinite parallel dislocations overestimated the long-range stresses compared with finite segments. The effect of randomness in dislocation distributions was studied and average stress fields calculated, which were used to estimate the stress fields of more complex pile-up arrangements. Results for multiple pile-ups do not support classical arguments for the Hall–Petch relationship. Distributions of excess dislocations produce considerable long-range stresses which are not effectively screened by pile-ups of the opposite sign.