When designing the axial joint pressing process, a relevant issue is the process parameters which directly influence the nature of the deformation of the housing, and, ultimately, on the formation of an all-in-one connection of a ball stud with a joint housing. These parameters are press force, punch displacement and matrix geometry. The quality of the all-in-one connection depends on a correct selection of the indicated parameters. If the axial joint insert is compressed evenly, the connection will be considered to be the most rational, that is the load will be distributed uniformly over the entire insert surface from the spherical surfaces of the ball stud and the deformed housing. In this way, the press must provide necessary force and displacement; the matrix geometry must provide the required shape of the deformed part of the housing. The all-in-one connection quality is evaluated by a number of operational characteristics of finished products, whose values must be located in intervals acceptable from the customer's point of view. Such characteristics include the following: ball stud rocking resistance torque, breakaway torque at ball stud rocking, axial spring travel and axial stiffness of the joint. In order to determine the specified parameters without many experiments on physical objects, a computer simulation model of the axial joint pressing process based on the finite element method was developed. Required press force, punch displacement and matrix geometry are determined by a computer simulation of the pressing process based on the finite element method. The accomplished experimental research led to the conclusion about acceptable repeatability of the results of the developed mathematical model and experimental data.