本研究針對鍍鋅鋼方形管在不同彎曲方向下之力學行為進行探討，重點分析其在對稱控制曲率循環彎曲負載作用下的力學響應與破壞特性。試驗中，方形管外邊長分別設定為20、30、40與50 mm，控制曲率條件為±0.5、±0.55、±0.6、±0.65、±0.7及±0.75 m⁻¹，彎曲方向則分別為0°、22.5°與45°，所有方形管壁厚均為1 mm。
實驗結果顯示，彎矩–曲率關係表明所有鍍鋅鋼方形管在循環彎曲過程中最終均形成穩定的彈塑性迴圈。在固定彎曲方向的條件下，外邊長越大，其對應的最大彎矩值亦隨之上升；相對地，當外邊長固定時，彎曲方向角度越大，所需的彎矩值亦越高。由外邊長變化–曲率關係可觀察到，無論彎曲方向或外邊長尺寸為何，隨著循環次數增加，此關係皆呈現對稱且具棘齒狀增長的特徵，而外邊長或彎曲方向越大，外邊長變化幅度亦更為明顯。
在控制曲率與循環至斷裂圈數的關係方面，結果顯示於相同控制曲率條件下，外邊長較大的方形管其損壞所需循環圈數較少。若將此結果繪製於雙對數座標圖後，可見不同外邊長之方形管各自形成四條直線。最後，本研究建立一套理論模型，以描述不同外邊長的鍍鋅鋼方形管在各彎曲方向下，於對稱控制曲率循環彎曲負載條件中的控制曲率與循環至斷裂圈數之關聯，理論分析結果與實驗數據高度一致，驗證本模型能有效描述鍍鋅鋼方形管之彎曲破壞行為。

This study investigates the mechanical behavior of galvanized square steel tubes subjected to different bending directions. The focus is on analyzing their mechanical responses and failure characteristics under symmetric curvature-controlled cyclic bending loads. In the experiments, the outer side lengths of the square tubes were set to 20, 30, 40, and 50 mm. The controlled curvature levels were ±0.5, ±0.55, ±0.6, ±0.65, ±0.7, and ±0.75 m⁻¹, while the bending directions were 0°, 22.5°, and 45°. All square tubes had a uniform wall thickness of 1 mm.
The experimental results indicate that the moment–curvature relationships show that all galvanized square steel tubes eventually develop stable elastoplastic hysteresis loops during cyclic bending. Under a fixed bending direction, an increase in the outer side length leads to a corresponding increase in the maximum bending moment. Conversely, for a fixed outer side length, a larger bending direction angle requires a higher bending moment. From the outer side length variation–curvature relationships, it can be observed that regardless of the bending direction or tube size, these relationships exhibit symmetric ratcheting-type growth with increasing loading cycles. Moreover, larger outer side lengths or greater bending direction angles result in more pronounced variations in the outer side length.
Regarding the relationship between the controlled curvature and the number of cycles to fracture, the results show that under the same controlled curvature condition, square tubes with larger outer side lengths require fewer loading cycles to failure. When these results are plotted on a double logarithmic coordinate system, four straight lines are observed for square tubes with different outer side lengths. Finally, a theoretical model is established to describe the relationship between the controlled curvature and the number of cycles to fracture for galvanized square steel tubes with various outer side lengths under symmetric curvature-controlled cyclic bending loads and different bending directions. The theoretical predictions show excellent agreement with the experimental data, confirming that the proposed model can effectively characterize the bending behavior of galvanized square steel tubes.

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. W. F. Pan and C. H. Fan, “An experimental study on the effect of curvature-rate at preloading stage on subsequent creep or relaxation of thin-walled tubes under pure bending”, JSME International Journal, Series A, Vol. 41, No. 4, pp. 525-531 (1998).
10. 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).
11. 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).
12. K. H. Chang, C. M. Hsu, S. R. Sheu and W. F. Pan, “Viscoplastic response and collapse of 316L stainless steel under cyclic bending”, Steel and Composite Structures, Vol. 5, No. 5, pp. 359-374 (2005).
13. 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).
14. 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).
15. 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).
16. 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).
17. 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).
18. 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).
19. 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).
20. 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).
21. 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, 1891, (2023).