# Integrated Design of Hydraulic Coupling Bolts for Large Shaft Systems
## 2 Calculation of Tightening Load
### First Tightening Load  
The thick-walled cylinder theory is applicable for analyzing structures with a cylindrical shape, where the thickness is greater than one-tenth of the diameter [21]. Based  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Yield Stress [MPa]** & **Tensile Strength [MPa]** & **Elongation [\%]** \\ \hline SNCM439 & 700 & 1100 & 12 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mechanical properties of SNCM439.  
Figure 4: Schematic illustration of the designed bolt shape.  
on the assumption that the sleeve is equivalent to a thick-walled cylinder, the upper and lower limits for the first tightening load were determined. First, the lower limit of the first tightening load (\(F_{1,1}\)) was calculated, considering the gap (\(\delta\)) between the coupling and the sleeve that exists before tightening, as depicted in Figure 5. To ensure that the sleeve sufficiently creates contact pressure with the coupling during the first tightening process, it is necessary to apply a load that expands the sleeve at least as much as the existing gap. The internal pressure required to expand the sleeve by the gap can be calculated with respect to the maximum thickness of the sleeve using Equation (6) and then converted into an axial force through Equation (7).  
\[p_{i}=\frac{\delta E\big{(}D^{2}-D_{2}^{2}\big{)}}{D\cdot D_{2}^{2}} \tag{6}\]  
\[F_{1,1}=\mu\cdot\pi\cdot D_{3}\cdot L\cdot p_{i} \tag{7}\]  
\[D_{3}=\frac{D_{1}+D_{2}}{2} \tag{8}\]  
where \(D_{1}\) and \(D_{2}\) are the maximum and minimum inner diameters of the sleeve inclination part, respectively, \(D\) is the inner diameter of the coupling, \(E\) is the elastic modulus, \(p_{i}\) is the internal pressure required to expand the sleeve by the gap, \(F_{1,1}\) is the lower limit of the axial force required to form the internal pressure \(p_{i}\), \(\mu\) is the friction coefficient, \(L\) is the sleeve length, and \(D_{3}\) is the average inner diameter of the sleeve inclination part, defined by Equation (8).  
The upper limit of the first tightening load (\(F_{1,u}\)) was determined based on the structural integrity of the sleeve. When the bolt is press-fitted to expand the sleeve, resulting in an internal pressure (\(p_{i}\)), the maximum shear stress (\(\tau_{\max}\)) generated on the inner surface of the sleeve can be calculated as follows.  
\[\tau_{\max}=\frac{p_{i}D^{2}}{D^{2}-D_{2}^{2}} \tag{9}\]  
Equation (9) can be reformulated in terms of the maximum internal pressure (\(p_{y}\)) at which yielding occurs on the inner surface of the sleeve, leading to Equation (10). Consequently, the upper limit of the axial force (\(F_{1,u}\)) required to form the maximum internal pressure (\(p_{y}\)) can be calculated using Equation (11).  
\[p_{y}=\frac{D^{2}-D_{2}^{2}}{2D^{2}}\sigma_{y} \tag{10}\]  
\[F_{1,u}=\mu\cdot\pi\cdot D_{3}\cdot L\cdot p_{y} \tag{11}\]  
The lower and upper limits of the calculated first tightening load were 348 kN and 880 kN, respectively. When the second tightening force remains constant at 825 kN, as determined in the previous section, the contact pressure and stress distribution at the interface between the sleeve and the coupling are primarily influenced by the first tightening  
Figure 5: Description of parameters for calculating the lower limit of the 1 st tightening load.  
load. In the following section, we report on the FE analysis conducted within the range between the lower and upper limit loads determined above to obtain the first tightening load that does not exceed the allowable stress while maintaining the required contact pressure between the sleeve and coupling.