Thermoelastic damping is one of the main mechanisms of energy loss in nano- and microresonators. Thermoelastic damping assessment is traditionally explored for thermal boundary conditions of thermal insulation. In this work a classical Lifshitz-Roukes’ model is extended to the case of a generalized Newton’s law of cooling. Based on Bernoulli-Euler beam model, refined analytical expressions for complex eigenfrequencies and quality factor damping coefficient are derived. Taking into account heat flow in the longitudinal direction of the resonator. The results reveal that there are three qualitatively different cases, corresponding to thermal boundary conditions of thermal insulation, constant temperature and a mixture of both. Additionally, it is shown that the profile of each case significantly relies on geometric properties of the model.