Abstract: An explicit expression of element compliance matrix of the finite element method for arbitrary meshes is derived using the base forces. The governing equations expressed by the base forces are obtained using Lagrangian multiplier method, and the displacements formulation of nodes is given. A program of the finite element method is developed. The method is used to solve several problems of arbitrary polygonal meshes and aberrant meshes, and the numerical results are compared with those of the theoretical solutions and traditional finite element method solutions. The results show that the method proposed is efficient to employ the arbitrary meshes and is free from mesh aberrant.