## 4 Conclusions  
This study primarily investigates the response and fracture of EMT carbon steel RHTs under cyclic bending with different \(\phi\) of \(0^{\circ}\), \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\) and different \(d\) of 2, 4, 6, 8, and 10 mm. The behavior of 6061 aluminum alloy RHTs under cyclic bending with different \(\phi\) and \(d\) tested by [PERSON] et al. [27] is compared. Based on tested, analyzed, and compared data, the following conclusions can be drawn:  
1. From the experimental \(M\)-\(\kappa\) relationships, it is observed that as the \(\kappa\) increases, the \(M\) also increases. Under cyclic bending loads, the \(M\)-\(\kappa\) relationship exhibits a stable loop until eventual buckling failure. Results also indicate that for different \(\phi\) and \(d\), the \(M\)-\(\kappa\) relationships show similar loops. This phenomenon is consistent with the experimental results of [PERSON] et al. [27]. 2. In the experimental \(\Delta D/D_{\rm o}\)-\(\kappa\) relationships, higher \(\kappa\) leads to increased \(\Delta D/D_{\rm o}\). Larger \(d\) or smaller \(\phi\) values correspond to increased \(\Delta D/D_{\rm o}\). This phenomenon is  
Figure 12: (**a**) Relationships between log\(\alpha\) and \(d/t\) for different \(\phi\), (**b**) relationship between \(\gamma\) and \(\phi\), and (**c**) relationship between \(\alpha_{\rm o}\) and \(\phi\). Figure 13: Experimental and theoretical \(\kappa_{\rm c}/\kappa_{\rm o}\)–\(N_{\rm f}\) relationships for EMT carbon steel RHTs under cyclic bending on double logarithmic coordinates with \(\phi=\) (**a**) \(0^{\circ}\), (**b**) \(30^{\circ}\), (**c**) \(60^{\circ}\), (**d**) \(90^{\circ}\) and different \(d\). consistent with the experimental results of [PERSON] et al. [27]. Additionally, under cyclic bending loads, the \(\Delta D/D_{\rm o}\)-\(\kappa\) relationships exhibit a ratcheting, unsymmetrical, bowtie, and growing trend. The larger \(d\) or smaller \(\phi\) is, the more serious the asymmetry is. This phenomenon is consistent with the experimental results of [PERSON] et al. [27]. 3. The experimental \(\kappa_{\rm c}/\kappa_{\rm o}\)-\(N_{\rm f}\) relationships reveal that, when \(\phi\) and \(d\) are fixed, larger \(\kappa_{\rm c}/\kappa_{\rm o}\) leads to fewer \(N_{\rm f}\). Fixing \(\phi\) and \(\kappa_{\rm c}/\kappa_{\rm o}\), larger \(d\) results in fewer \(N_{\rm f}\). Conversely, when fixing \(d\) and \(\kappa_{\rm c}/\kappa_{\rm o}\), larger \(\phi\) leads to more \(N_{\rm f}\). When \(\phi\) is fixed, the relationships of \(\kappa_{\rm c}/\kappa_{\rm o}\)-\(N_{\rm f}\) for five different \(d\) on double logarithmic coordinates exhibit five straight lines. This phenomenon is consistent with the experimental results of [PERSON] et al. [27]. However, for a certain \(\phi\), they show almost-parallel lines, while we found non-parallel lines. 4. This study adopts the formula (Equation (1)) proposed by [PERSON] and [PERSON] [1] to describe the \(\kappa_{\rm c}/\kappa_{\rm o}\)-\(N_{\rm f}\) relationships. The formulation of the material parameter \(C\) in Equation (2), proposed by [PERSON] et al. [27], was used in this study. The material parameters \(\beta\) and \(C_{\rm o}\) were derived from Equations (3) and (4), respectively. Additionally, the material parameter \(\alpha\) (Equation (5)) was newly proposed based on the experimental data. The material parameters \(\gamma\) and \(\alpha_{\rm o}\) were derived from Equations (6) and (7), respectively. The theoretical equations (Equations (1)-(4) and (5)-(7)) were employed to describe the \(\kappa_{\rm c}/\kappa_{\rm o}\)-\(N_{\rm f}\) relationships for EMT carbon steel RHTs under cyclic bending with \(\phi\) = 0\({}^{\circ}\), 30\({}^{\circ}\), 60\({}^{\circ}\), and 90\({}^{\circ}\) and various \(d\) (dashed lines in Figure 13a-d), and the results of experimental and theoretical analyses are reasonably consistent. The research was conducted with the generous support of the National Science and Technology Council, through grant NSTC 112-2221-E-006-174.