A kind of viscoelastic spectral stochastic finite element method (VSSFEM) was developed and stochastic analysis of solid rocket motor grains was presented. Considering the randomness of solid propellant Poisson's ratio, viscoelastic spectral stochastic finite element formulations were derived from the incompressible or nearly incompressible viscoelastic incremental finite element method and orthogonal expansion theory. Finally, stochastic analysis of a hollow cylindrical grain with elastic restraint was presented. Analysis of examples indicates that this method is efficient and effective for stochastic analysis of viscoelastic body.