Using　the　concept　of　base　forces　given　by　Gao　[12]　as　fundamental　variables　to　describe　the　stress　state　of　an　elastic　system,　this　paper　presents　an　explicit　expression　of　two-dimensional　element　compliance　matrix　on　the　complementary　energy　principle　with　concave　polygonal　meshes.　The　detailed　2-D　formulations　of　governing　equations　for　the　new　finite　element　method　the　base　force　element　method　(BFEM)　are　written　in　terms　of　the　base　forces　concept　using　the　Lagrange　multiplier　method.　The　explicit　displacements　expression　of　nodes　is　given.　To　verify　the　model　proposed　in　this　paper,　a　program　on　the　2-D　BFEM　using　MATLAB　language　is　made　and　a　number　of　examples　on　concave　polygonal　meshes　and　aberrant　meshes　are　provided　to　illustrate　the　BFEM.　Good　agreement　has　been　obtained　comparing　numerical　results　using　the　proposed　BFEM　to　available　theoretical　results.　The　concave　polygonal　element　model　can　be　used　efficiently　for　the　elasticity　analysis　with　increasing　element　aspect　ratios　and　distortion　meshes　compared　with　the　traditional　finite　element　method　(FEM).　(C)　2013　Elsevier　Ltd.　All　rights　reserved.