A　partition-of-unity　(PU)　based　＂FE-Meshfree＂　three-node　triangular　element　(Trig3-RPIM)　was　recently　developed　for　linear　elastic　problems.　This　Trig3-RPIM　element　employs　hybrid　shape　functions　that　combine　the　shape　functions　of　three-node　triangular　element　(Trig3)　and　radial-polynomial　basis　functions　for　the　purpose　of　synergizing　the　merits　of　both　finite　element　method　and　meshfree　method.　Although　Trig3-RPIM　element　is　capable　of　obtaining　higher　accuracy　and　convergence　rate　than　the　Trig3　element　and　four-node　iso-parametric　quadrilateral　element　without　adding　extra　nodes　or　degrees　of　freedom　(DOFs),　the　nodal　stress　field　through　Trig3-RPIM　element　is　not　continuous　and　extra　stress　smooth　operations　are　still　needed　in　the　post　processing　stage.　To　further　improve　the　property　of　Trig3-RPIM　element,　a　new　PU-based　triangular　element　with　continuous　nodal　stress,　called　Trig3-RPIMcns,　is　developed.　Numerical　examples　including　several　linear,　free　vibration　and　forced　vibration　test　problems,　have　confirmed　the　correctness　and　feasibility　of　the　proposed　Trig3-RPIMcns　element.