In this study, thermo-mechanical vibration analyzes of functionally graded (FG) beams made of porous material subjected various thermal loadings are carried out by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Three types of thermal loadings, namely, uniform, linear and nonlinear temperature rises through the thickness direction are considered. Thermo-mechanical material properties of FGM beam are supposed to vary continuously along the thickness direction according to the powerlaw form, which is modified to approximate the material properties with the porosity phases. The material properties of FG porous beam are assumed to be temperature-dependent. The governing equations of motion are derived through Hamilton's principle and they are solved applying DTM. According to the numerical results, it is revealed that the proposed modeling and semi analytical approach can provide accurate frequency results of the FG beams as compared to analytical results and also some cases in the literature. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as thermal effect, porosity volume fraction, material distribution profile, mode number and boundary conditions on the natural frequencies of the temperature-dependent FG beams in detail. It is explicitly shown that the vibration behaviour of porous FGM beams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FGM beams with porosity phases.

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