本研究旨在探究 S45C 直角扇形管在循環彎曲負載下的力學行為，實驗設計涵蓋了 20、30、40 與 50 mm 四種不同外半徑的試件，並透過彎管實驗機施加範圍介於±0.5至 ±0.75 m-1共六組（±0.5、±0.55、±0.6、±0.65、±0.7和±0.75 m-1）的對稱控制曲率，藉以觀測管件在循環負載下的變形過程；研究成果透過「彎矩曲率」與「外半徑變化曲率」的關聯曲線來釐清其結構響應，並進一步分析控制曲率循環至損壞圈數的對應關係，藉此定義該材質管件的失效行為。
經由實驗數據分析發現，試件之彎矩–曲率響應在往復載荷下均建構出閉合的遲滯迴圈軌跡。當外半徑較小時，內部應力分布能迅速達到平衡並展現出穩定的力學迴圈；反之，當外半徑增大時，迴圈在收斂至穩定週期前，會先誘發顯著的循環軟化特質與截面收縮現象。而外半徑變化與曲率的雙軸軌跡，不論在彈性或塑性區間均表現出顯著的非線性響應，並呈現出對稱、累積且遞增的演變趨勢。橫向對比顯示，管件外半徑尺寸越大時，其外半徑變化的累積速率亦隨之顯著加劇。
當維持相同的控制曲率時，具備較大外半徑之直角扇形管試件，其結構的循環至損壞圈數呈現顯著的縮減。若將此控制曲率與循環至損壞圈數之對應關係轉換至雙對數座標系中，四類不同外半徑規格之離散數據點經線性回歸擬合後，均各自對應出高度規律且互不交疊的四條軌跡。最終，本研究提出修正之理論解析模型，並將數值估算數據與實驗量測軌跡進行比對，結果確證兩者在循環至損壞圈數的判定上展現出高度的一致性與重合度，充分證實了本研究所建構之幾何修正經驗公式具備理論合理性。

This study investigates the mechanical behavior of S45C right-angled fan -shaped tubes under cyclic bending loads. The experiments involved specimens with four different outer radius (20, 30, 40 and 50 mm). A tube-bending machine applied symmetric controlled curvatures in six groups from ±0.5 to ±0.75 m-1(±0.5, ±0.55, ±0.6, ±0.65, ±0.7, and ±0.75 m-1) to observe the cyclic deformation process. The structural responses were clarified using "bending moment-curvature" and "outer radius change-curvature" curves. Additionally, the relationship between controlled curvature and cycles to failure was analyzed to define the failure behavior of the tubes.
The experimental results show that the bending moment-curvature responses form closed hysteresis loops under cyclic loading. For smaller outer radii, the internal stress distribution quickly reaches equilibrium and shows stable loops. In contrast, for larger outer radius, the loops exhibit significant cyclic softening and cross-sectional contraction before stabilizing. The biaxial paths of outer radius change versus curvature display strong non-linear responses in both elastic and plastic regions, showing a symmetric, cumulative, and increasing trend. Comparisons show that a larger outer radius accelerates the accumulation rate of the Outside radius variation change.
At the same controlled curvature, specimens with larger outer radius show a major decrease in the number of cycles to failure. When this relationship is plotted in a double logarithmic coordinate system, linear regression of the discrete data points from the four sizes produces four highly regular, non-overlapping lines. Finally, this study proposes a modified analytical model. The comparison between the theoretical predictions and experimental data shows high consistency and overlap in predicting the number of cycles to failure, which proves the theoretical rationality of the geometrically modified empirical formula.

1. L. G. Brazier, “On the flexure of thin cylindrical shells and other thin sections”, Proceedings of the Royal Society, Series A, Vol. 116, No. 773, pp. 104-114 (1927).
2. R. M. Korol, “Critical buckling strains of round tubes in flexure”, International Journal of Mechanics and Science, Vol. 21, No. 12, pp. 719-730 (1979).
3. P. K. Shaw and S. Kyriakides, “Inelastic analysis of thin-walled tubes under cyclic bending”, International Journal of Solids and Structures, Vol. 21, No. 11, pp. 1073-1100 (1985).
4. S. Kyriakides and P. K. Shaw, “Inelastic buckling of tubes under cyclic loads”, Journal of Pressure Vessel Technology, Vol. 109, No. 2, pp. 169-178 (1987).
5. E. Corona and S. Kyriakides, “On the collapse of inelastic tubes under combined bending and pressure”, International Journal of Solids and Structures, Vol. 24, No. 5, pp. 505-535 (1988).
6. E. Corona and S. Kyriakides, “An experimental investigation of the degradation and buckling of circular tubes under cyclic bending and external pressure”, Thin-Walled Structures, Vol. 12, No. 3, pp. 229-263 (1991).
7. W. F. Pan, T. R. Wang and C. M. Hsu, “A curvature-ovalization measurement apparatus for circular tubes under cyclic bending”, Experimental Mechanics, Vol. 38, No. 2, pp. 99-102 (1998).
8. W. F. Pan and Y. S. Her, “Viscoplastic collapse of thin-walled tubes under cyclic bending”, ASME Journal of Engineering Materials and Technology, Vol. 120, No. 4, pp. 287-290 (1998).
9. K. L. Lee, W. F. Pan and J. N. Kuo, “The influence of the diameter-to-thickness ratio on the stability of circular tubes under cyclic bending”, International Journal of Solids and Structures, Vol. 38, No. 14, pp. 2401-2413 (2001).
10. W. F. Pan and K. L. Lee, “The effect of mean curvature on the response and collapse of thin-walled tubes under cyclic bending”, JSME International Journal, Series A, Vol. 45, No. 2, pp. 309-318 (2002).
11. K. H. Chang and W. F. Pan, “Buckling life estimation of circular tubes under cyclic bending”, International Journal of Solids and Structures, Vol. 46, No. 2, pp. 254-270 (2009).
12. K. L. Lee, C. Y. Hung, H. Y. Chang and W. F. Pan, “Buckling life estimation of circular tubes of different materials under cyclic bending”, Journal of Chinese Institute Engineers, Vol. 33, No. 2, pp. 177-189 (2010).
13. K. L. Lee, C. Y. Hung and W. F. Pan, Variation of ovalization for sharp-notched circular tubes under cyclic bending, Journal of Mechanics, Vol. 26, No. 3, pp. 403- 411 (2010).
14. K. L. Lee, C. M. Hsu and W. F. Pan, “The influence of diameter-to-thickness ratios on the response and collapse of sharp-notched circular tubes under cyclic bending”, Journal of Mechanics, Vol. 28, No. 3, pp. 461- 468 (2012).
15. K. L. Lee, C. C. Chung and W. F. Pan, “Growing and critical ovalization for sharp-notched 6061-T6 aluminum alloy tubes under cyclic bending”, Journal of Chinese Institute of Engineers, Vol. 39, No. 8, pp. 926935 (2016).
16. K. L. Lee, K. H. Chang and W. F. Pan, “Effect of notch depth and direction on stability of local sharp-notched circular tubes subjected to cyclic bending”, International Journal of Structural Stability and Dynamics, Vol. 18, No. 7, 1850090 [23 pages] (2018).
17. K. L. Lee, M. L. Weng and W. F. Pan, “On the failure of round-hole tubes under cyclic bending”, Journal of Chinese Society of Mechanical Engineering, Vol. 40, No. 6, pp. 663 - 673 (2019).
18. K. L. Lee, Y. C. Tsai and W. F. Pan, “Mean curvature effect on the response and failure of round-hole tubes submitted to cyclic bending”, Advances in Mechanical Engineering, Vol. 13, No. 11, pp. 1-14 (2021).
19. M. C. Yu and W. F. Pan, “Failure of elliptical tubes with different long-short axis ratios under cyclic bending in different directions”, Metals, Vol. 13, No. 11, (2023).
20. W. F. Pan and Y. A. Chen, “Response and fracture of EMT carbon steel round-hole tubes with different hole orientations and different hole diameters under cyclic bending”, Applied Sciences, Vol. 14, No. 13, (2024).
21. C. J. Su and W. F. Pan, “Minor axis variation and critical minor axis variation of elliptical tubes under cyclic bending”, Informatica Journal, Vol. 36, No. 2, pp. 1-20 (2025).