An explicit solution for stresses and strains in pyrolytic-carbon coatings on spherical fuel particles is presented. The resulting model is a modification of the Prados-Scott model which accounts for stresses arising due to anisotropic, radiation-induced dimensional change and the buildup of internal fission gas pressure, and for stress relaxation due to radiation-induced creep. Finite displacements are shown to amplify the effects of anisotropic dimensional changes. The use of the explicit solution allows a reduction in computation time by several orders of magnitude. The reduction in computation time makes feasible the use of Monte Carlo analyses (which have previously been precluded by high computation costs) to establish the effects of random variations in coated-particle parameters.