An edge‐based smoothed finite element method (ES‐FEM) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the elastic solid mechanics problems. In this paper the ES‐FEM is further extended to a more general case, n‐sided polygonal edge‐based smoothed finite element method (nES‐FEM), in which the problem domain can be discretized by a set of polygons, each with an arbitrary number of sides. The simple averaging point interpolation method is suggested to construct nES‐FEM shape functions. In addition, a novel domain‐based selective scheme of a combined nES/NS‐FEM model is also proposed to avoid volumetric locking. Several numerical examples are investigated and the results of the nES‐FEM are found to agree well with exact solutions and are much better than those of others existing methods. Copyright © 2010 John Wiley & Sons, Ltd.