The issue of evaluating strength and service life is discussed as applied to structures, the exploitation properties of which are characterized by multi-parametric nonstationary thermal mechanical effects. The main requirements to mathematical models of the related processes are formulated. In the framework of mechanics of damaged media, a mathematical model describing processes of inelastic deformation and damage accumulation due to creep is developed. The mechanics of damaged media model consists of three interconnected parts: relations defining inelastic behavior of the material accounting for its dependence on the failure process, equations describing kinetics of damage accumulation, and a strength criterion of the damaged material. The results of numerically simulating the carrying capacity of a nuclear power plant reactor vessel in the event of a hypothetical emergency are presented. Emergency conditions were modeled by applying pressure modeling the effect of melt-down, the constant internal pressure and temperature varying within the part of the vessel in question. The analysis of the obtained numerical results made it possible to note a number of characteristic features accompanying the process of deformation and failure of such facilities, connected with the time and place of the forming macrocracks, the stressed-strained state history and the damage degree in the failure zone, etc. The results of comparing the numerical and experimental data make it possible to conclude that the proposed defining relations of mechanics of damaged media adequately describe degradation of the initial strength properties of the material for the long-term strength mechanism and can be effectively used in evaluating strength and service life of structures under thermal mechanical loading.