Axially　functionally　graded　(AFG)　beams,　with　variable　coefficients　in　the　governing　equation,　are　a　novel　class　of　composites　structures　that　have　continuous　variations　in　material　properties　from　one　component　to　another.　In　this　paper,　the　asymptotic　development　method　(ADM)　is　utilized　to　investigate　the　free　vibration　of　uniform　AFG　beams　with　different　boundary　conditions.　By　decomposing　the　variable　flexural　stiffness　and　mass　per　unit　length　into　reference　invariant　parts　and　variant　parts,　perturbation　theory　is　introduced　to　obtain　an　approximate　formula　of　the　natural　frequencies　of　the　uniform　AFG　beams.　The　numerical　results　of　the　proposed　method　are　confirmed　by　comparing　the　obtained　results　with　those　obtained　via　finite　element　analysis　and　the　published　literature　results,　the　comparison　reveals　the　proposed　method　yields　an　accurate　estimate　of　the　first　three　order　natural　frequencies　of　the　AFG　beam.　Moreover,　the　influences　of　the　gradient　parameter　and　support　conditions　on　the　first　three　natural　frequencies　are　discussed.　The　proposed　analytical　method　is　simple　and　efficient　and　can　be　used　to　conveniently　analyze　uniform　AFG　beams　with　arbitrary　changes　in　the　material　properties　along　the　axial　direction.