The　curved　pipe　made　of　functionally　graded　material　conveying　fluid　is　considered　and　the　in-plane　free　vibration　frequency　of　the　resulting　composite　pipe　is　investigated.　The　material　properties　are　assumed　to　distribute　continuously　along　the　pipe　wall　thickness　according　to　a　power　law　and　the　effective　mass,　flexural　rigidity,　and　mass　ratio　are　used　in　the　governing　equations.　The　natural　frequencies　are　derived　numerically　by　applying　the　modified　inextensible　theory.　The　lowest　four　natural　frequencies　are　studied　via　the　complex　mode　method,　the　validity　of　which　is　demonstrated　by　comparing　the　results　with　those　in　available　literatures.　A　parametric　sensitivity　study　is　conducted　by　numerical　examples　and　the　results　obtained　reveal　the　significant　effects　of　material　distribution　gradient　index,　flow　velocity,　fluid　density,　and　opening　angle　on　the　natural　frequencies　of　the　FGM　curved　pipes　conveying　fluid.