# Analysis of Interference-Fit Joints
## 1 Introduction  
Interference-fit joints are commonly used in engineering constructions for connecting the shaft and the hub [1]. They are particularly common in the design of electric motors. An interference-fit joint is obtained by selecting appropriate dimensions of the joined elements [2]. The initial difference in dimensions of these elements is an important factor influencing the load capacity of the joint. Depending on the assembly method, a distinction is made between press-fit joints obtained by pressing one element into another and shrink-fit joints obtained by heating or cooling one component before assembly and allowing it to return to the ambient temperature after assembly, employing the phenomenon of thermal expansion. Press-fit and shrink joints ensure good coaxiality. They are also characterized by the ability to carry variable and impact loads. The lack of additional connecting elements makes these joints simple and quick to make, and therefore relatively cheap. To determine the surface pressure in the joint zone and the load capacity of an interference-fit joint, formulas derived from analogy to the circular-symmetric Lame's task are used [3]. According to this analogy, the condition of load transfer by the interference-fit joint is the initial difference between the hub inside diameter and the shaft diameter. After the joint has been assembled, surface pressure \(\mathrm{p_{s}}\) equal to the radial pressure occurs on the contact surface of the shaft and the hub (Figure 1). This pressure can be calculated from the following Formula (1) [4; 5]:  
\[\mathrm{p_{s}}=\mathrm{E}\cdot\Delta\cdot(\mathrm{d_{w}}^{2}-\mathrm{d_{ww}}^{ 2})(\mathrm{d_{zo}}^{2}-\mathrm{d_{w}}^{2})/\mathrm{d_{w}}^{3}\cdot(\mathrm{d_{ zo}}^{2}-\mathrm{d_{ww}}^{2}) \tag{1}\]  
where:E--is Young's modulus,  
\(\Delta=\Delta\mathrm{d_{o}}-\Delta\mathrm{d_{w}}\)  
d\({}_{\mathrm{w}}\)--outer diameter of the shaft,  
d\({}_{\mathrm{ww}}\)--inner diameter of the shaft,  
d\({}_{\mathrm{zo}}\)--outer diameter of the hub. The value of axial force breaking the interference-fit joint can be calculated from the following formula:  
\[\mathrm{F=\pi\cdot\mu\cdot p_{s}\cdot d\cdot l} \tag{2}\]  
where:  
\(\mu\)--coefficient of friction of the joined parts contact surfaces,  
d--nominal shaft diameter,  
l--length of the press-fit joint. Calculating the load capacity of a press-fit joint from the formulas using the assumptions of the linear theory of elasticity is possible assuming that the value of the surface pressure is lower than the yield point. In the literature on design issues of interference-fit joints, one can find a modification of Formula (1) which takes into account surface roughness of joined elements [6]. The modification assumes reducing the size of parameter \(\Delta\) by the smoothing \(\mathrm{G}\) dimension factor according to the following Formula (3):  
\[\mathrm{G=0.8(Rz_{w}+Rz_{o})} \tag{3}\]  
where:  
Rz\({}_{\mathrm{w}}\)--roughness coefficient Rz of the shaft,  
Rz\({}_{\mathrm{0}}\)--roughness factor Rz of the hub. Formula (1) takes the form:  
\[\mathrm{p_{s}=E\cdot(\Delta-G)\cdot(d_{w}{}^{2}-d_{ww}{}^{2})(d_{\mathrm{zo}}{ }^{2}-d_{w}{}^{2})/d_{w}{}^{3}\cdot(d_{\mathrm{zo}}{}^{2}-d_{\mathrm{ww}}{}^{2})} \tag{4}\]  
This formula is recommended for calculating the surface pressure in press-fit joints. Calculating the load capacity of press-fit joints using formulas based on the assumptions of the linear theory of elasticity is possible only on the condition that the surface pressure value is lower than the yield point. However, depending on the type of fit, it is difficult to assume in advance that the pressing process will occur in the elastic range. Figure 1: The idea of the interference-fit joint. Therefore, when determining the value of the load breaking the joint, it is problematic to use Formula (2) based on Formula (1) or (4). The literature on the design of load-bearing capacity of press-fit joints is not very extensive.