在大多數的平面對稱結構物中，各振態多為在單一個水平方向位移或是扭轉變形，而在平面不對稱的結構物中，各個振態同時存在水平位移及扭轉變形。在地震力作用於平面不對稱的結構物時，由於扭轉變形顯著且與水平位移同時發生，這樣的問題在靜力分析中難以準確模擬結構之受震行為，而須採用動力分析以求得更精準的結果。然而因傳統數值模型自由度數高，每筆歷時分析所花費的時間很長，且在設計時會考慮各種外力加載以及不同的地震歷時。若是採用簡化模型，因其自由度數量較少，能大大的降低分析時長，對於結構初步設計有極大幫助。而以往的簡化模型都能很有效重現原始模型受震時的平移向位移歷時，關於扭轉向變形問題多是透過旋轉彈簧解決。本研究參考日本笠井和彥教授針對二維高樓結構所提出的簡化方法，將建築結構視為一懸臂柱，在結構頂部施加彎矩並取得各層樓的位移及曲率，依其曲率進而求取結構各層的勁度。本研究將其從二維模型推展至三維模型，並延伸其概念，透過在結構頂層施加扭矩並取得各樓層的轉角，進而計算結構各層的扭轉參數。本研究開發兩種包含扭轉及撓曲變形均有良好準確度的簡化方法，分別為Bifurcated Column Model (BC)與Single Column Model (SC)，讓簡化模型也能重現包含平移與扭轉方向的模態振形與結構受震反應。為了討論本研究所提出的簡化模型之適用範圍，分別設計了兩大不同的模型。第一種為平面對稱模型，並以質心點與剛心點之偏移量為變數，討論當質心與剛心不重合所產生的扭轉效應，同時結構斷面設計成兩種，分別為隨樓高固定斷面與變換斷面兩種情況。第二種為平面不對稱模型，以模型質量參與比在各方向的分量總和為變數，探討簡化模型在單一模態中包含兩向平移與扭轉模態皆參與時的適用性。將各個數值模型轉為本研究所提出的簡化模型後，分別對其進行模態分析與一系列的動力分析，輸入的地震歷時共選擇了3筆地震，分別為日本建築研究所BCJL2人造地震、台灣921地震 TCU052資料與1940年墨西哥El Centro地震，並正規化至SLE地震等級，最後依據分析的結果，提出本研究的簡化方法適用範圍。本研究的簡化模型會探討與目標模型的模態振形、結構週期與動力受震反應等項目，在本研究所設計的各目標模型中，週期誤差由0%分布至10%，而模態振形採方均根誤差，各模型誤差由0.1分布至0.55，在各方向受震反應的效果評估則是以位移最大值為主。

For structures with symmetrical plans, each mode mainly contains only one translational or one torsional deformation. However, for structures with asymmetrical plans, each mode mainly contains both translational and torsional responses. Under the earthquakes, the torsional deformation can be enhanced. Detailed and time-consuming response history analysis (RHA) is needed to investigate the torsional responses. This study aims to propose a stick model which can simulate both translational and rotational responses together by material parameters. By applying a torque at the top of the structure, the torsional stiffness of each story can be obtained. Then, the stick model is assigned with the flexural, shear, and torsional stiffness together for each story. Beside the structure stiffness, the separation of center of mass (CM) and center of rigidity (CR) must consider. There are two types of simplified model proposed in this study. There are Bifurcated Column Model (BC) and Single Column Model (SC). To verify the effectiveness of the proposed stick model, two types of benchmark MBM models are designed. The first type is plane-symmetric models, and the torsional response is triggered by adjusting the locations of the centers of mass and rigidity. The second type is the plane-asymmetric models, which include different amount of model mass participation ratio in each direction. This study concludes with a procedure for constructing the proposed stick model and assumes all the responses remain in elastic state. Nonlinear simplified model is still in developing.

[1] K.Kasai, K.Watai, S.Maeda, D.Sato, andY.Suzuki, “Equivalent mass-spring modeling method for super-tall buildings of increasing height (part 1) overview of past and proposed methods for bending-shear model,” J. Struct. Constr. Eng., vol. 85, no. 722, pp. 791–801, Jun.2020, doi: 10.3130/aijs.85.791.
[2] K.Watai, S.Maeda, K.Kasai, D.Sato, andY.Suzuki, “Equivalent mass-spring modeling method for super-tall buildings of increasing height (Part 2): Shear and bending components separating method for the building responses using a new bending-shear model,” J. Struct. Constr. Eng., vol. 86, no. 779, pp. 21–31, Jan.2021, doi: 10.3130/aijs.86.21.
[3] Y.Suryanto, “Seismic Design and Assessment for Mid-rise Building Equipped with Damped-outrigger System,” no. January, 2022.
[4] E.Miranda, “Approximate Seismic Lateral Deformation Demands in Multistory Buildings,” J. Struct. Eng., vol. 125, no. 4, pp. 417–425, Apr.1999, doi: 10.1061/(ASCE)0733-9445(1999)125:4(417).
[5] R.Soleimani andH.Hamidi, “General Substitute Frame Model (GSF) for efficient estimation of seismic demands of steel and RC moment frames,” Eng. Struct., vol. 246, Nov.2021, doi: 10.1016/j.engstruct.2021.113031.
[6] A. R.Khaloo andH.Khosravi, “Modified fish-bone model: A simplified MDOF model for simulation of seismic responses of moment resisting frames,” Soil Dyn. Earthq. Eng., vol. 55, 2013, doi: 10.1016/j.soildyn.2013.09.013.
[7] A. K.Chopra andR. K.Goel, “A modal pushover analysis procedure to estimate seismic demands for unsymmetric-plan buildings,” Earthq. Eng. Struct. Dyn., vol. 33, no. 8, pp. 903–927, 2004, doi: 10.1002/eqe.380.
[8] A. K.Chopra andR. K.Goel, “A modal pushover analysis procedure for estimating seismic demands for buildings,” Earthq. Eng. Struct. Dyn., vol. 31, no. 3, pp. 561–582, 2002, doi: 10.1002/eqe.144.
[9] J. P.Moehle andL. F.Alarcon, “Seismic Analysis Methods for Irregular Buildings,” J. Struct. Eng., vol. 112, no. 1, pp. 35–52, 1986, doi: 10.1061/(asce)0733-9445(1986)112:1(35).
[10] Jui-Liang Lin andKeh-Chyuan Tsai, “Simplified seismic analysis of asymmetric building systems,” Earthq. Eng. Struct. Dyn., vol. 36, no. 4, pp. 459–479, 2007, doi: 10.1002/eqe.635.
[11] Y. T.Weng, J. L.Lin, C. Y.Tsai, andK. C.Tsai, “Analytical assessment of a 2-story BRBF for full-scale 3D sub-structural pseudo-dynamic testing,” Int. Conf. Adv. Exp. Struct. Eng., vol. 2005-July, pp. 347–354, 2005.
[12] J. L.Lin andK. C.Tsai, “Simplified seismic analysis of one-way asymmetric elastic systems with supplemental damping,” Earthq. Eng. Struct. Dyn., 2007, doi: 10.1002/eqe.653.
[13] J. L.Lin andK. C.Tsai, “Seismic analysis of two-way asymmetric building systems under bi-directional seismic ground motions,” Earthq. Eng. Struct. Dyn., vol. 37, no. 2, pp. 305–328, 2008, doi: 10.1002/eqe.759.
[14] 曹智嘉 (Chih-Chia Tsaur), 林瑞良 (Jui-Liang Lin), and蔡克銓 (Keh-Chyuan Tsai), “立面不規則建築受震反應簡化分析方法,” 結構工程, vol. 32, no. 4, pp. 88–109, 2017, doi: 10.6849/SE.201712_32(4).0004.
[15] B. Z.Lin, M. C.Chuang, andK. C.Tsai, “Object-oriented development and application of a nonlinear structural analysis framework,” Adv. Eng. Softw., 2009, doi: 10.1016/j.advengsoft.2008.03.012.
[16] F. T.McKenna, “Object-oriented finite element programming: Frameworks for analysis, algorithms and parallel computing,” ProQuest Diss. Theses, 1997.
[17] 林保均, 蔡克銓, 吳安傑, and莊明介, “挫屈束制支撐與接合設計 雲端運算流程解說,” 2020.
[18] “建築技術規則,” 2021.
[19] “耐震設計規範,” 2011.
[20] A. K.Chopra, Dynamics of Strucutures 4 Edition. 2012.