Mechanical couplings in engineering usually use interference fits to connect the shaft and hub. A railway wheel axle is a press fit that is connected by interference and can be subjected to bending stress. In loaded press fits, a high concentration of contact stresses can be generated in the area of the axle-fillet beam, which in most cases leads to the failure of the axle due to fatigue and fretting fatigues. Therefore, it is crucial to determine the ability of the press-fitted joints to provide sufficient frictional resistance that can withstand the loads and torques by evaluating the safety factor, especially when the mechanical or structural system is loaded. In this paper, the contact pressure and stress distribution along the radius of the wheel axle are studied using the analytical calculation of Lame’s equation, and the numerical method used is by ANSYS software. It was found that interference fits have a great influence on the connection strength of interference fits, which are directly related to the contact pressure. Increasing the interference increases the contact pressure, which allows higher torque and load capacity to be transmitted. The finite element analysis showed good agreement for the highest interference value of 230 µm with a relative error of 1.4%, while this error increased to the maximum relative error of 14.33% for a minimum interference of 100 µm.