## 3 FE Analysis for the Flange Segment  
### FE Modeling  
As stated in the GL2010 standard, \"Calculations with the aid of the finite element method that do not consider flange gaps, as well as other calculation methods leading to comparable results, are not permissible [28]. Thus, it is worthwhile to explore the effect of flange gaps on bolt fatigue quantitatively. However, the authors are aware that the reported literature is rather limited. This section will consider the flange gaps mainly originating from transportation, installation, and service. The flange gaps are artificially classified into three categories. Flange gap type I appears at the inner edge of the flange, with an opening height of \(h_{\rm I}\). Assuming that only the upper flange gap opens, it extends along the flange gap surface for a length of \(l_{\rm I}\) from the edge. Flange gap type II is opposite in direction to flange gap type I, with an opening height of \(h_{\rm II}\) and an extension length of \(l_{\rm II}\), as shown specifically in Figure 10b. When the upper flange rotates along the circumferential direction with the intersection of the upper and lower flanges on one side as the axis of rotation, it can form a type III gap, as shown in Figure 10c, with an opening height of \(h_{\rm III}\).  
Because the effect of the flange gaps on fatigue damage cannot be quantitatively evaluated in the Schmidt-Neuper algorithm, the FE analysis using the commercial software ANSYS 19.2 TM is performed. For one segment of the flange system, all components, including the flange, washer, and nut head, are discretized using hexahedral elements as plotted in Figure 11. Two nonlinear contact pairs are established between flange and washers, while one contact pair exists between the flange surfaces. The bolts and the pretension are simulated with beam elements and pretension elements. A group of umbrella-like rigid elements is adopted to imitate the connection of a bolt and nut head. In this study, geometric nonlinearity is ignored for the sake of simplification, which is extremely beneficial in terms of computing costs.  
Figure 10: Three types of the flange gap: (**a**) type I; (**b**) type II; (**c**) type III.  
Firstly, the preferred flange with no gaps is analyzed on the FE model and Schmidt-Neuper algorithm, as drawn in Figure 12, aiming to compare the similarities and differences between the two approaches. The displacement contours corresponding to the FE analysis are illustrated in Figure 13.  
Figure 11: Finite element of the flange system with an open gap.  
Figure 12: Nonlinear relationship between wall tension \(Z\) and bolt force \(F_{\text{VS}}\).  
Figure 13: Flange displacement contours under different wall tensions \(Z\): (**a**) \(Z=-600\) kN; (**b**) \(Z=-400\) kN; (**c**) \(Z=-200\) kN; (**d**) \(Z=0\) kN; (**e**) \(Z=200\) kN; (**f**) \(Z=400\) kN; (**g**) \(Z=600\) kN.  
Compared to the Schmidt-Neuper algorithm, the FE-derived bolt internal force curve is asymmetrical, as clearly demonstrated in Figure 12. Once the tower wall is pressed, the bolt force fluctuation can be negligible. Under the same external load, the bolt force is relatively minimal in the FE results. The universally adopted Schmidt-Neuper algorithm embodies the engineering relevance of conservative design, as can be concluded.  
It is noticeable that the maximum flange displacement is consistently greater under positive tension compared to negative tension in Figure 13. At \(Z\) values of 400 kN and 600 kN, a fault line appears between the upper and lower flanges, indicating a clear separation between them.