# Integrated Design of Hydraulic Coupling Bolts for Large Shaft Systems
## 2 Calculation of Tightening Load
### FE Analysis for Tightening Process  
During the tightening process, the hydraulic pressure applied to the tensioner causes displacement in the adapter, resulting in the tensioning of the bolt. We simplify the shape of the adapter for analysis, excluding the threaded part. The detailed adapter shape will be considered in the following section. The shape of a coupling bolt with a spiral oil groove is not perfectly symmetric. However, the spiral angle is small, and the effect of the oil groove on the tightening process is insignificant. Therefore, a quarter three-dimensional (3D) model was used for FE analysis, as shown in Figure 6a, based on the results of a previous study [18]. The FE analysis was conducted using the ANSYS Mechanical APDL commercial software package. The overall mesh sizes were determined by referencing the preceding study [18], ensuring continuity with established methodologies and maintaining consistency in the modeling approach. To improve the accuracy of the analysis, finer elements were created on the areas where contact occurs, such as between couplings and sleeves and between bolts and sleeves. A mesh independent test was performed on the stress concentration zone of the bolt shown in Figure 4, due to its significance for structural integrity. By refining the element size from 0.0015 to 0.0012 mm, the stress concentration factor converged to a constant value of 1.72 at an element size of 0.0012 mm. Consequently, a mesh consisting of approximately 340,000 elements and 670,000 nodes was generated. The FE model included four distinct types of 3D elements: tetrahedral 4-node elements, hexahedral 8-node elements, wedge 6-node elements, and pyramid 5-node elements. Each of these elements featured three translational degrees of freedom per node [22]. The selection of elements was based on their unique advantages: tetrahedral elements for modeling complex geometries, hexahedral elements for superior computational efficiency and accuracy in regular structures, and pyramid and wedge elements for compatibility with a combination of tetrahedral and hexahedral elements [23; 24; 25]. A full integration scheme was consistently applied to all elements, demanding approximately one hour of computational time for each simulation. The tightening load was applied in three distinct steps, as shown in Figure 6b. Step 1 involves the first tightening process in which the bolt is inserted into the coupling hole, causing radial compressive stress. Step 2 is the unloading process of the first tightening load, and Step 3 is the second tightening process of applying axial tensile force by extending the bolt. The boundary conditions of the analysis are depicted in Figure 6c. In Step 1, the axial displacement of the thickest part of the sleeve was constrained, and in Step 2, this constraint was removed. Furthermore, in Step 3, the end of the bolt opposite to the one connected to the adapter was constrained in the axial direction to simulate the effect of being tightened with a nut. Symmetry conditions were consistently applied to the longitudinal cross-section of the bolt throughout the entire process to restrict displacement in the direction perpendicular to the plane. For the coupling, constraints were applied in directions other than the radial direction. The friction coefficient at the contact surfaces was set to 0.18 with reference to [13; 19], and nonlinear analysis was conducted to account for the contact nonlinearity between the coupling and the sleeve and between the sleeve and the bolt. The FE analysis results for the first and second tightening processes are shown in Figure 7. First, Figure 7a represents the von-Mises stress distribution on the bolt when arbitrary first and second tightening loads were applied to a quarter FE model, set at 87 kN and 250 kN, respectively. The analysis revealed a maximum stress of 425 MPa in the fillet part of the minimum cross-section of the bolt after the second tightening process.