Free out of plane vibration characteristics of curved beams with variable cross-sections on elastic foundations are studied by TMM (Transfer Matrix Method) since the mathematical model of the present system based on the coupled differential eigenvalue problem with variable coefficients which can not be solved easily by exact methods. Vibrations of beams on different elastic foundations are reviewed. Out of plane vibration of curved beams on different elastic foundations are investigated. TMM is detailed with its applications to vibration problems. To solve the vibration problems, TMM is examined with several computer programs developed in Mathematica. The accuracy of the TMM results obtained from the developed program is evaluated by comparing with FEM results found from model created in ANSYS. Finally, the effects of the variation of cross-section of the curved beams and elastic foundation parameters on natural frequencies are investigated.