Abstract: The model of the main roof was treated as the thin plate model，according to the principle of the minimum potential energy of the elasticity，the displacement of the roof based on the thin plate model laid on the Winkler foundation was obtained.The immediate roof and coal which support the main roof were regarded as viscoelastic foundation，the model of the foundation was the Generalized Kelvin-Voigt model，on the basis of the analytic results which based on the plate on the elastic foundation，according to elastic-viscoelastic theory，then the analytic solutions of the displacement and the stress of the roof based on the plate on viscoelastic foundation were got by Laplace transform method，provided theoretical basis for analyzing the stability of the roof and forecasting the subsiden ce of the old mine goaf area.

Key words: Elastic foundation, Viscoelastic foundation, Thin plate model, Roof, Mechanical analysis