A　hybrid　＇FE-Meshfree＇　four-node　quadrilateral　element　with　continuous　nodal　stress　using　radial-polynomial　basis　functions　(Quad4-RPIMcns),　was　recently　proposed　for　static　analysis.　The　Quad4-RPIMcns　element　can　be　considered　as　a　development　of　the　previous　partition-of-unity　(PU)　based　＇FE-Meshfree＇　QUAD4　element　(Quad4-RPIM)　which　uses　FE　shape　functions　to　construct　the　PU　and　radial-polynomial　basis　functions　to　construct　the　local　approximation　(LA),　so　as　to　synergize　the　individual　strengths　of　finite　element　and　meshfree　methods.　As　a　result,　high　order　global　approximations　in　Quad4-RPIMcns　element　could　be　easily　constructed　without　adding　extra　nodes　and　DOFs,　thereby　achieving　high　accuracy　and　convergence　rate.　In　this　paper,　the　element　is　further　applied　to　conduct　free　vibration,　forced　vibration　and　geometric　nonlinear　analyses　of　two-dimensional　solids.　Several　numerical　test　problems　are　solved　and　the　performance　of　the　element　is　compared　with　that　of　the　three-node　triangular　element　(Trig3)　and　four-node　isoparametric　quadrilateral　element　(Quad4).　Numerical　results　show　that　Quad4-RPIMcns　element　has　higher　tolerance　to　mesh　distortion　and　gives　more　accurate　solution　as　compared　to　Trig3　and　Quad4　elements.