本研究旨在探討不同外邊長的鍍鋅鋼方形管在不同曲率比控制循環彎曲負載下的力學行為與疲勞失效。實驗採用四點純彎曲實驗機，對外邊長分別為20、30、40與50 mm，壁厚皆為1 mm的鍍鋅鋼方形管進行三種曲率比為-1、 -0.5與0的循環載荷試驗。試驗利用荷重元、雷射測距儀及橢圓化量測裝置即時記錄彎矩與曲率的關係、外邊長變化與曲率的關係以及曲率與循環至破壞圈數的關係。
由彎矩–曲率關係可知，在相同曲率比條件下，隨著外邊長增加，其最大彎矩明顯提升，顯示外邊長對彎曲承載能力具有顯著影響。循環迴圈初期雖出現輕微鬆弛現象，但隨循環次數增加，最終皆可發展為穩定之彈塑性迴圈。由外邊長變化量–曲率關係可觀察到，方形管於循環彎曲載荷下之外邊長變化呈現隨曲率正負交替之對稱張縮行為，反映截面於受拉與受壓階段中之交替變形特性。當外邊長增加時，外邊長變化量之最大變化量顯著提升；此外，當曲率比= −1時曲線呈現近似對稱，而曲率比= −0.5 及0 則顯示不對稱循環行為，並偏向平均曲率方向。由曲率範圍–循環至破壞圈數關係可知，在雙對數座標下曲率範圍與疲勞壽命呈現近似線性關係；在固定外邊長條件下，疲勞壽命隨曲率範圍增加而降低，而在固定曲率範圍條件下，曲率比越接近−1，循環至破壞圈數越大。進一步根據壽命關係式，並結合實驗結果建立一套考慮外邊長與厚度比與曲率比影響之循環彎曲疲勞壽命預測模型，其理論分析結果與實驗數據具有良好一致性，顯示本模型可合理描述鍍鋅鋼方形管於不同幾何尺寸與曲率比下之循環疲勞失效行為。

This study investigates the mechanical behavior and fatigue failure of galvanized steel square tubes with various side lengths under cyclic bending loads controlled by different curvature ratios. Using a four-point pure bending test machine, cyclic loading tests were conducted on square tubes with outer side lengths of 20, 30, 40, and 50 mm (all with a wall thickness of 1 mm) at three curvature ratios: -1, -0.5, and 0. Load cells, laser displacement sensors, and ovalization measurement devices were employed to record real-time data, including moment-curvature relationships, side length deformation versus curvature, and curvature range versus cycles to failure.
Experimental results indicate that under the same curvature ratio, the maximum moment increases significantly with larger side lengths, demonstrating the substantial impact of geometry on bending capacity. Although slight relaxation occurs during the initial cycles, the moment-curvature response eventually stabilizes into steady elastoplastic loops. Regarding cross-sectional deformation, the variation in side length exhibits a symmetrical expansion and contraction behavior that alternates with the sign of the curvature, reflecting the alternating deformation characteristics during tension and compression stages. The maximum deformation of the side length increases significantly as the tube size increases. Furthermore, the deformation curves are nearly symmetrical at a curvature ratio of -1, while they exhibit asymmetric behavior biased toward the mean curvature at ratios of -0.5 and 0.
Analysis of the curvature range versus cycles to failure reveals a near-linear relationship in a log-log scale. At a fixed side length, fatigue life decreases as the curvature range increases; conversely, at a fixed curvature range, fatigue life increases as the curvature ratio approaches -1. Based on these findings, a fatigue life prediction model was developed, incorporating the effects of the width-to-thickness ratio and curvature ratio. The theoretical predictions show good agreement with experimental data, indicating that the proposed model can reasonably describe the cyclic fatigue failure of galvanized steel square tubes across different geometries and loading conditions.

[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] 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).
[3] 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).
[4] 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).
[5] 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).
[6] S. Kyriakides and G. T. Ju, “Bifurcation and localization instabilities in cylindrical shells under bending – I. Experiments”, International Journal of Solids and Structures, Vol. 29, No. 9, pp. 1117-1142 (1992).
[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
[10] pure bending”, JSME International Journal, Series A, Vol. 41, No. 4, pp. 525-531 (1998).
[11] 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).
[12] 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).
[13] 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).
[14] K. L. Lee, C. M. Hsu, W. F. Pan, “Endochronic Simulation for the response of 1020 carbon steel tubes under symmetric and unsymmetric cyclic bending with or without external pressure”, Steel and Composite Structures, Vol. 8, No. 2, pp. 99-114 (2008).
[15] 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).
[16] 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).
[17] 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).
[18] 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).
[19] 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).
[20] K. L. Lee, C. M. Hsu and W. F. Pan, The influence of mean curvatures on the collapse of sharp-notched circular tubes under cyclic bending, Journal of Chinese Society of Mechanical Engineering, Vol. 34, No. 5, pp. 461-468 (2013).
[21] 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).
[22] 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).
[23] 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).
[24] K. L. Lee, Q. Y. Wen and W. F. Pan, “Response of round-hole tubes with different hole sizes and positions under pure bending relaxation”, Informatica Journal, Vol. 32, No. 8, pp. 48-65 (2021).
[25] 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).
[26] 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).