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A thick-walled cylindrical tube made of a Functionally Graded Material (FGM) and undergoing thermomechanical loads has been investigated. The tube material properties have been considered to be isotropic and vary as a power function of the radius. A yield variable parameter has been defined according to von Mises criterion, has been introduced into the stress governing equation. The value of the radius where yielding may appear has been assessed in relation to internal pressure, temperature difference and heat generation. An analytical solution has been derived, allowing the prediction of radial position of yeld onset as related to the elastic limit domain configuration according to FGM parameter settings and thermomechanical loads. Moreover, elastic limit region changes have been investigated for two different designs of FGM materials. Stress and yield parameter variations along the thickness have been obtained. The effects of FGM parameters and of the shape factor of the cylinder cross-section on initial yielding due to thermomechanical loads have been studied.