A model of gear contact has been established based on Hertz theory. Local coordinates are constructed on different meshing points because curvature of gear profile is not constant and then distributions of residual stresses and strains are given. Upper and lower bounds for the shakedown limits have been obtained. An approximate algorithm based on a stress invariant hypothesis and a stress/strain relaxation procedure is developed to solve residual stresses and strains accumulation in elastic-plastic stress analysis of gear contact. The steady-state residual stresses of gear under repeat meshing are calculated. The results are useful information for the strength design of gear.