This paper describes the analysis procedure and FEM model considering the parallel effects of stress and thermal problem coupled with the electromagnetic field problem of a 10 MJ Superconducting Magnetic Energy Storage (SMES) magnet. Since the coupled processes in most analyses are very simplified or even not considered, the general aim of this paper was to create, solve, and validate a numerical model of the coupled stress and thermal phenomena that occur within an SMES magnet. A finite element model has been developed to solve the coupling of the magnetic field, stress, and thermal problems. Temperature and stress distribution of the SMES magnet in two operating conditions are simulated, taking into account the eddy current, ac loss, and heat conduction. This provides useful insight for optimized coil design and condition assessment of SMES.