本研究以S45C中碳鋼直角扇形管為研究對象，探討不同外半徑尺寸於循環彎曲負載下之變形行為及失效特性，並建立失效預測模型，選用外半徑20、30、40及50 毫米之S45C中碳鋼直角扇形管，利用循環彎曲實驗機施加六種控制曲率進行循環彎曲實驗。實驗過程中量測試件外半徑變化，並記錄循環圈數，以分析外半徑變化量與控制曲率間之關係，並配合循環圈數探討不同尺寸試件之失效壽命。
研究結果顯示，直角扇形管在循環彎曲負載作用下，外半徑變化量會隨循環圈數持續累積增加，循環曲線逐漸向外擴張，顯示材料產生不可回復之變形，並可將循環變形歷程分為初始變形、穩定變形與裂紋萌生，以及裂紋快速擴展與最終失效三個階段，其中穩定變形與裂紋萌生階段占整體循環圈數之主要部分，是影響試件失效之關鍵階段。比較不同尺寸試件可發現，當試件外半徑增加，變形累積速率提高，循環圈數則明顯降低；此外，在相同控制曲率下，臨界外半徑變化量會隨外半徑增加而提高，而同一尺寸試件之臨界外半徑變化量也隨控制曲率增加而提高，顯示幾何尺寸及控制曲率皆為影響循環彎曲失效行為之重要因素。
為建立不同尺寸直角扇形管之外半徑變化預測模式，本研究建立適用於S45C中碳鋼直角扇形管之經驗式。經擬合求得材料參數值，進而完成不同尺寸試件之外半徑變化預測經驗式。此外，針對臨界外半徑變化量與控制曲率之非線性關係，透過雙對數分析建立預測關係式。討論之理論與實驗結果具有良好一致性，驗證關係式可有效描述不同尺寸之S45C中碳鋼直角扇形管在循環彎曲負載下之外半徑變化行為及臨界外半徑變化特性，可提供相關試件失效評估及工程設計之參考依據。

This study investigates the deformation behavior of S45C medium-carbon steel right-angle fan-shaped tubes with different outer radii under cyclic bending loading and develops predictive models for outer radius variation. Specimens with outer radii of 20, 30, 40, and 50 mm were tested under six levels of controlled curvature using a cyclic bending testing machine. During the experiments, the variation in the outer radius was continuously measured, and the number of cycles was recorded to investigate the relationship between outer radius variation and controlled curvature, as well as to compare the deformation behavior of specimens with different geometric dimensions.
The experimental results show that the outer radius variation increases progressively with the number of cycles, while the cyclic deformation curves gradually expand outward, indicating the accumulation of irreversible plastic deformation. The cyclic deformation process can be divided into three stages: the initial deformation stage, the stable deformation and crack initiation stage, and the rapid crack propagation stage. Among these, the stable deformation and crack initiation stage occupies the largest proportion of the cyclic deformation process and plays a dominant role in the overall deformation behavior. Comparisons among specimens of different sizes reveal that increasing the outer radius results in a higher deformation accumulation rate and a corresponding decrease in the number of cycles. Furthermore, under the same controlled curvature, the critical outer radius variation increases with increasing outer radius. For specimens with the same outer radius, the critical outer radius variation also increases as the controlled curvature increases. These results demonstrate that both geometric dimensions and controlled curvature are important factors influencing the cyclic bending behavior of right-angle fan-shaped tubes.
To predict the outer radius variation of right-angle fan-shaped tubes with different dimensions, an empirical equation applicable to S45C medium-carbon steel right-angle fan-shaped tubes was established. The material parameters were determined through curve fitting, and empirical prediction equations for specimens with different outer radii were subsequently developed. Furthermore, a double-logarithmic analysis was employed to establish a predictive relationship describing the nonlinear correlation between the critical outer radius variation and the controlled curvature. The theoretical predictions are in good agreement with the experimental results, demonstrating that the proposed equations can accurately describe the outer radius variation behavior and the critical outer radius variation characteristics of S45C medium-carbon steel right-angle fan-shaped tubes under cyclic bending loading. The proposed equations provide a useful reference for deformation evaluation and engineering design of related structural components.

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