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The stresses in the coupling superconducting solenoid coil assembly, which is applied in the Muon Ionization Cooling Experiment (MICE), are critical for the structure design and mechanical stability because of a large diameter and relative high magnetic field. This paper presents an analytical stress solution for the MICE coupling coil assembly. The stress due to winding tension is calculated by assuming the coil package as a set of combined cylinders.The thermal and electromechanical stresses are obtained by solving the partial differential equations of displacement based on the power series expansion method. The analytical stress solution is proved to be feasible by calculating stresses in a tested superconducting solenoid with 2.58 m bore at room temperature. The analytical result of the MICE coupling coil is in good agreement with that ofthe finite element which shows that the transverse shear stress induced by Lorentz force is principally dominant to magnet instability.