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This paper presents an efficient method for the analysis of printed periodic structures, consisting of a multilayered array of metal patches in a stratified dielectric medium, illuminated by a uniform plane-wave. These structures are widely used as frequency selective surfaces and mirrors. The analysis is based on the solution of an integral equation by the method of moments (MoM) with entire-domain basis functions. The basis functions are calculated numerically by the boundary integral-resonant mode expansion (BI-RME) method. The patches may have an arbitrary shape, and both metal conductivity and dielectric losses are considered. Some examples are reported to show the accuracy and rapidity of the proposed method.