纜索構件為各類索支承橋梁之重要承重或輔助支撐構件，其力學行為受到自重、垂度及初始索力影響，並呈現張力相依之有效勁度。傳統線性弦理論雖便於工程分析，惟未能完整反映纜索自重與構形變化；當支承擾動幅值增大或擾動頻率接近自然頻率時，纜索可能產生明顯之位移、索力變化及頻率偏移，使線性模型之適用性降低。
本研究以彈性懸鏈線理論為基礎，採拉格朗日座標描述大變形行為，建立三維非線性索元素並推導其切線勁度矩陣。動力分析採絕對座標法，結合 Newmark-β 積分與 Newton–Raphson 迭代程序，求解非同步支承擾動下之線性與非線性動力歷時反應。透過不同初始索力、擾動幅值及擾動頻率之參數分析，並以動態索力增量、極值誤差、正規化均方根誤差及相關係數等指標，評估線性分析之適用性。另結合 Irvine 參數λ 與無因次擾動幅值D/L，分別表徵纜索之幾何非線性特性及外部擾動程度，建立分析前可供初步判斷之示警門檻與非線性分析建議門檻。
工程案例以成功吊橋主索為對象，將吊索傳遞之橋體重量等效為離散集中力，建立主索靜力平衡模型，比較實際線形與設計線形之索力狀態及模態頻率，並探討支承擾動下之動力反應與線性分析適用性。此外，針對斜張橋縮尺模型之 C10 斜索，結合振動台試驗與數值分析，評估端部彈簧、抗彎連桿及支承拘束所形成之複合邊界條件對振動頻率、位移與索力反應之影響。結果顯示，提高初始索力可降低位移反應與動態索力增量，並延後明顯非線性行為之發生；然而，當擾動頻率接近自然頻率時，即使輸入幅值較小，仍可能因共振放大而造成線性與非線性結果之顯著差異。索力損失則會增加垂度，並降低幾何勁度與自然頻率。因此，纜索非線性動力分析之啟用，應綜合考量初始索力、擾動幅值與頻率、邊界條件及頻率偏移等因素。

Cables are important load-carrying components in cable-supported bridges, and their mechanical behavior is strongly influenced by self-weight, sag, and initial tension. This study develops a three-dimensional nonlinear cable element based on elastic catenary theory using a Lagrangian formulation and derives the corresponding tangent stiffness matrix. Dynamic responses under asynchronous support excitation are solved using the Newmark-β method combined with Newton–Raphson iteration. Parametric analyses are conducted for different initial cable tensions, excitation amplitudes, excitation frequencies, and cable tension-loss conditions. The Irvine parameter λ and the dimensionless excitation amplitude D/L are further used to establish warning and nonlinear-analysis thresholds.
The Chenggong Suspension Bridge and Cable C10 of a scaled cable-stayed bridge model are examined as engineering and experimental cases, respectively. Results show that higher initial tension generally reduces displacement and cable-force variations and delays significant nonlinear behavior. Resonance substantially narrows the applicable range of linear analysis, while cable tension loss increases sag and reduces geometric stiffness and natural frequency. The Cable C10 results also indicate that measured frequencies under complex boundary conditions should not be interpreted directly using ideal string theory. Overall, the proposed framework provides a practical reference for selecting between linear and nonlinear dynamic analysis.

[1] Michalos, J., & Birnstiel, C. (1960). Movements of a cable due to changes in loading. Journal of the Structural Division, 86(12), 23-38.
[2] Ernst, H. (1965). Der E-Modul von Seilen unter Berücksichtigung des Durchhanges. Der Bauingenieur, 40(2), 52–55.
[3] Irvine, H. M., & Caughey, T. K. (1974). The linear theory of free vibrations of a suspended cable. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 341(1626), 299-315.
[4] Jayaraman, H. B., & Knudson, W. C. (1981). A curved element for the analysis of cable structures. Computers & Structures, 14(3-4), 325-333.
[5] Ahmadi-Kashani, K., & Bell, A. J. (1988). The analysis of cables subject to uniformly distributed loads. Engineering Structures, 10(3), 174-184.
[6] Thai, H. T., & Kim, S. E. (2011). Nonlinear static and dynamic analysis of cable structures. Finite elements in analysis and design, 47(3), 237-246.
[7] Li, J., Spencer Jr, B. F., Elnashai, A. S., & Phillips, B. M. (2012). Substructure hybrid simulation with multiple-support excitation. Journal of engineering mechanics, 138(7), 867-876.
[8] Guo, J., Gu, Y., Wu, W., Chu, S., & Dang, X. (2022). Seismic fragility assessment of cable-stayed bridges crossing fault rupture zones. Buildings, 12(7), 1045.
[9] 游宗熹 (2007)。三維纜索非線性動態放大因子之探討(碩士論文)。國立臺灣科技大學營建工程系。
[10] 袁傳智 (2008)。非線性纜索動態放大效應之研究(碩士論文)。國立臺灣科技大學營建工程系。
[11] 徐毓宏 (2014)。大垂度索彎曲行為與小垂度索振動分析(博士論文)。國立臺灣科技大學營建工程系。
[12] Gong, J., Zhi, X., Fan, F., & Shen, S. (2020). Static and dynamic stiffness in the modeling of inclined suspended cables. Journal of Constructional Steel Research, 172, 106210.
[13] 葉治頡 (2025)。人行橋縮尺風洞模型結合地盤反應模擬之振動台試驗研究(碩士論文)。國立成功大學土木工程學系。
[14] Liao, Qi-Yang (2026). Comprehensive Identification and Multi-Scale Experimental Study of Bridge Structures via Shaking Table, Wind Tunnel Testing, and In-Situ Monitoring. Ph.D. dissertation, Department of Civil Engineering, National Cheng Kung University.
[15] Wu, C. C., Wu, J. H., Liao, Q. Y., Chu, S. Y., Fang, C., & Liu, P. H. (2026). Design, construction, and preliminary validation of an asymmetric pedestrian suspension bridge. Journal of the Chinese Institute of Engineers, 1-19.
[16] Lan, F., Guo, T., Qiao, W., & Kang, H. (2024). Nonlinear hardening/softening dynamic analysis and its application to cables: a geometrical framework. Journal of Sound and Vibration, 583, 118433.
[17] Li, C., He, J., Zhang, Z., Liu, Y., Ke, H., Dong, C., & Li, H. (2018). An improved analytical algorithm on main cable system of suspension bridge. Applied Sciences, 8(8), 1358.
[18] Lepidi, M. (2024). Catenary configuration and geometric stiffness matrix of inextensible cables: Analytical high-order asymptotic solutions for parametric design. Applied Mathematical Modelling, 128, 1-25.
[19] Zhou, Y., & Chen, S. (2019). Iterative nonlinear cable shape and force finding technique of suspension bridges using elastic catenary configuration. Journal of Engineering Mechanics, 145(5), 04019031.
[20] Zhang, W.-M., Chang, J.-Q., & Feng, D.-D. (2022). Determination of main cable shape and hanger tensions of a suspension bridge based on the measured live-load deflection of the main beam: An analytical algorithm. Engineering Structures, 272, 115031.
[21] Zui, H., Shinke, T., & Namita, Y. (1996). Practical formulas for estimation of cable tension by vibration method. Journal of structural engineering, 122(6), 651-656.
[22] 吴庆雄, 李浏, & 陈宝春. (2010). 考虑弯曲刚度的拉索面内固有振动的理论计算公式. 工程力学, 27(11), 9-15.
[23] 巫文勝（2014）。以軸力梁理論提昇預力鋼纜傳統評估公式實用性之研究（碩士論文）。朝陽科技大學營建工程系。
[24] Chen, C. C., Wu, W. H., Chen, S. Y., & Lai, G. L. (2018). A novel tension estimation approach for elastic cables by elimination of complex boundary condition effects employing mode shape functions. Engineering Structures, 166, 152–166.
[25] 陳建州, 吳文華, 陳欣怡, & 賴國龍 (2020)。結合振形之微振索力估算法中量測點位配置對精度的影響。結構工程, 35(3), 99–117。
[26] 劉彥廷 (2021)。基於部分振形之索力多點量測法研析(碩士論文)。國立雲林科技大學。
[27] 陳建州, 吳文華, 柯翊珮, & 賴國龍 (2025)。基於振形之索力估算法於不同量測佈點模式下的準確性與適用性。中國土木水利工程學刊，37(1)，37–51。
[28] 日本道路協會(1984)。小規模吊橋指針・同解說。
[29] Benedettini, F., & Rega, G. (1989). Planar non-linear oscillations of elastic cables under superharmonic resonance conditions. Journal of Sound and Vibration, 132(3), 353-366.
[30] 目崎勇太(2026)。應用數位影像相關法於橋梁模型之多點動態位移監測(碩士論文)。國立成功大學土木工程學系。