混凝土常因不同的材料界面、養護不佳或疲勞應變增大而導致裂縫發生。混凝土裂縫主要可分為表面裂縫、貫穿裂縫和深層裂縫三種。其中，貫穿裂縫對混凝土的整體受力及防滲效果的影響與淺層表面裂縫比較起來，無疑是大得多。表面裂縫也可能成為深層裂縫的誘發因素，對混凝土的抗風化能力和耐久性有一定影響，且在有害物入侵後將加速混凝土劣化速度，終至損壞結構，造成無法挽回的後果，甚至危害人身安全。若能事先了解裂縫的寬度、深度及走向，可及時採取適當的補救方式，以達到較佳的補強效果。
在多變數圖形識別的領域中，馬氏距離及灰關聯度的分析已被廣為應用，本文則運用敲擊試體(標準抗彎試體)所得到的時間域訊號，以此二法對七種不同的混凝土結構裂縫型式加以識別做探討。在研究小樣本空間問題時，經常會遇到共變異矩陣發生奇異現象，本文採用聯合共變異矩陣取代共變異矩陣，以避免此類情形發生，而導致無法求得正確的馬氏距離。試驗結果顯示，儘管本試驗可供訓練的資料只有20筆，即在小樣本空間的情況下，馬氏距離及灰關聯度能對時間域訊號加以區別，故可對不同混凝土裂縫型式做有效的分類。另外，學習向量量化的方法也加入比較，結果發現在小樣本空間時，無法藉由有效的訓練獲取特徵值，故其分類效果不佳。
本文也提出用以評估破壞程度的破壞指標，並以數值模擬得出不同垂直裂縫的混凝土結構的破壞指標值，而根據此破壞指標將破壞程度區分為四類，然後以試驗加以驗證。此外，將數值模擬的結果做迴歸分析後可得一拋物線方程式，進而利用此方程式做為評估未知裂縫的深度的參考依據。

Different material interface, poor conservation and fatigue strain are the main causes of cracking in concrete structures. Cracks are divided into three categories: surface crack, through crack and deep crack. Compared to surface crack, through crack could have much influence on reducing the strenghth and anti-seepage effect of the structure. Surface crack may also be a predisposing factor of deep crack, and lower the weathering resistance and durability. Invasion of harmful substances will accelerate the speed of deterioration of concrete, which finally led to the damage to the structure, resulting in irreversible consequences, even danger to life.
Concrete cracks are almost impossible to avoid, so appropriate repair can reduce the rate of strength reduction, furthermore, extend the life of the structures. With prior knowledge of the crack width, depth and direction, better reinforcement effect could be achieved.
Mahalanobis Distance (MD) and grey relational analysis (GRG) are useful methods for analyzing patterns in multivariate cases. Developed in this paper is the application of Mahalanobis Distance and grey relational analysis for crack pattern recogition in concrete structure. In case of small data sizes, the sample group covariance matrices uesd in Mahalanobis Distance analysis are singular. This paper uses the pooled covariance matrix as an alternative estimate for the sample group covariance matrix to overcome this kind problem. The results show that Mahalanobis Distance and grey relational analysis are capable of classifying the distinction among the data sets in time domain and thus identify the type of crack developed in concrete structure. In small training sample sizes situation, LVQ ( learning vector quantization ) artificial neural network has poorer classification accuracy because small size of sample sets are not weighted appropriated to reflect the complexity of each patten.
This paper also proposes a damage index. Numerical simulation is conducted in order to acquire the damage index value to relevant category. Numerical results are verified by experimental study.

參考文獻
1.沈永年、王和源、林仁益、郭文田，混凝土技術，全華書局，台北，(2011)。
2.黃兆龍，混凝土性質與行為，詹氏書局，台北市，(2007)。
3.ACI Committee 224, “Causes, Evaluation, and Repair of Cracksin Concrete Structures,” American Concrete Institute (ACI 224.1R-07), (2007)
4.Thanoon, W.A., Jaafar, M.S., Kadir, M.R.A., Noorzaei, J., “Repair and Structural Performance of Initially Cracked Reinforced Concrete Slabs,” Construction and Building Materials, Vol. 19, No. 8, pp. 595-603 (2005)
5.Price, W.H., “Control of Cracking of Concrete During Construction,” Concrete International, Vol. 4, No. 1, pp. 40-43 (1982)
6.王國榮，「環氧樹脂修補破壞混凝土之成效研究」，碩士論文，國立成功大學，台南市（2007）。
7.Issa, C.A., Debs, P., “Experimental Study of Epoxy Repairing of Cracks in Concrete,” Construction and Building Materials, Vol. 21, No. 1, pp. 157-163 (2007)
8.Kelly, J.W., “Cracks in Concrete - 1,” Concrete Construction - World of Concrete, Vol. 26, No. 9, pp. 725, 727, 729, 731-725, 727, 729, 731 (1981)
9.Kelly, J.W., “Cracks in Concrete - 2,” Concrete Construction - World of Concrete, Vol. 26, No. 9, pp. 733-734 (1981)
10.Felicetti, R., De Domenico, V.H.,Cracked Concrete Repair with Epoxy-Resin Infiltration, 2nd International Conference on Concrete Repair, Rehabilitation and Retrofitting, ICCRRR 2008, November 24, 2008 - November 26, 2008, Cape Town, South africa, pp. 293-294 (2009)
11.Cai, H., Li, Y., Li, R.,Experimental Research on New and Old Crack Comprehensive Repair to Concrete Beams,2011 International Conference on Consumer Electronics, Communications and Networks, CECNet 2011, April 16, 2011 - April 18, 2011, XianNing, China, pp. 1264-1267 (2011)
12.Shash, A.A., “Repair of Concrete Beams - a Case Study,” Construction and Building Materials, Vol. 19, No. 1, pp. 75-79 (2005)
13.Broms, B.B.,Crack Width and Spacing in Reinforced Concrete Members,ACI Journal Proceedings, pp. 1237-1256 (1965)
14.Ble, A., “Analysis of Crack Patterns in Reinforced Concrete Frames under Seismic Excitation,” master, Massachusetts Institute of Technology, Cambridge(2006)
15.D'Orazio, T., Leo, M., Distante, A., Guaragnella, C., Pianese, V., Cavaccini, G., “Automatic Ultrasonic Inspection for Internal Defect Detection in Composite Materials,” NDT and E International, Vol. 41, No. 2, pp. 145-154 (2008)
16.Kabir, S., “Imaging-Based Detection of Aar Induced Map-Crack Damage in Concrete Structure,” NDT and E International, Vol. 43, No. 6, pp. 461-469 (2010)
17.Sinha, S.K., Fieguth, P.W., “Neuro-Fuzzy Network for the Classification of Buried Pipe Defects,” Automation in Construction, Vol. 15, No. 1, pp. 73-83 (2006)
18.Drai, R., Khelil, M., Benchaala, A., “Time Frequency and Wavelet Transform Applied to Selected Problems in Ultrasonics Nde,” NDT and E International, Vol. 35, No. 8, pp. 567-572 (2002)
19.留鼎傑，「以小波轉換分析混凝土版內鋼筋及管線之敲擊回音反應」，碩士論文，朝陽科技大學，台中（2003）。
20.Mary Sansalone, W.B.S., Impact-Echo: Non-Destructive Evaluation of Concrete and Masonry Bullbrier Press, (1997)
21.Carino, N.J.,The Impact-Echo Method :An Overview Structures Congress & Exposition,, pp. 18 (2001)
22.王金圳，「以敲擊回音法檢測細長與標準圓柱及混凝土板之材料性質」，博士論文，國立壹灣科技大學，台北（2012）。
23.黃洸駿，「以敲擊回音法測漸變斷面混凝土厚度之可行性」，碩士論文，國立交通大學，新竹（2003）。
24.Xiang, Y., Tso, S.K., “Detection and Classification of Flaws in Concrete Structure Using Bispectra and Neural Networks,” NDT and E International, Vol. 35, No. 1, pp. 19-27 (2002)
25.Hsiao, C., Cheng, C.-C., Liou, T., Juang, Y., “Detecting Flaws in Concrete Blocks Using the Impact-Echo Method,” NDT and E International, Vol. 41, No. 2, pp. 98-107 (2008)
26.Liu, P.-L., Yeh, P.-L., “Vertical Spectral Tomography of Concrete Structures Based on Impact Echo Depth Spectra,” NDT and E International, Vol. 43, No. 1, pp. 45-53 (2010)
27.陳順宇，多變量分析，華泰文化，台北，(2005)。
28.呂金河譯，多變量分析，滄海書局，台北，(2005)。
29.銘傳大學統計諮詢中心, 結構方程模式, 上網日期：2012, 1/28, from: http://www.mcu.edu.tw/department/management/stat/ch_web/ch-sev/, (2003)
30.Mahalanobis, P.,On the Generalized Distance in Statistics,Natl. Inst, pp. 49–55 (1936)
31.Chen, L., Ai, Q., He, Y., Liu, Q., Meng, W.,A Real-Time Leg Motion Recognition System by Using Mahalanobis Distance and Ls-Svm,2012 3rd IEEE/IET International Conference on Audio, Language and Image Processing, ICALIP 2012, July 16, 2012 - July 18, 2012, Shanghai, China, pp. 668-673 (2012)
32.Boardman, D., Flynn, A., “A Gamma-Ray Identification Algorithm Based on Fisher Linear Discriminant Analysis,” IEEE Transactions on Nuclear Science, Vol. 60, No. 1, pp. 270-277 (2013)
33.Wang, Y., Miao, Q., Ma, E.W.M., Tsui, K.-L., Pecht, M.G., “Online Anomaly Detection for Hard Disk Drives Based on Mahalanobis Distance,” IEEE Transactions on Reliability, Vol. 62, No. 1, pp. 136-145 (2013)
34.Pushparaj, V., Gurunathan, U., Arumugam, B., “An Effective Dental Shape Extraction Algorithm Using Contour Information and Matching by Mahalanobis Distance,” Vol., No., pp. 1-10 (2012)
35.Santiago-Paz, J., Torres-Roman, D., Velarde-Alvarado, P.,Detecting Anomalies in Network Traffic Using Entropy and Mahalanobis Distance,22nd Annual International Conference on Electronics, Communications and Computers, CONIELECOMP 2012, February 27, 2012 - February 29, 2012, Cholula, Mexico, pp. 86-91 (2012)
36.Cudney, E.A., Hong, J., Jugulum, R., Paryani, K., Ragsdell, K.M., Taguchi, G., “An Evaluation of Mahalanobis-Taguchi System and Neural Network for Multivariate Pattern Recognition,” Journal of Industrial and Systems Engineering, Vol. 1, No. 2, pp. 139-150 (2007)
37.Cudney, E.A., Corns, S.M.,A Comparison of Finite State Classifier and Mahalanobis-Taguchi System for Multivariate Pattern Recognition in Skin Cancer Detection,2011 IEEE Symposium on Computational Intelligence in Bioinformatics and Computational Biology, CIBCB 2011, April 11, 2011 - April 15, 2011, Paris, France, pp. 169-175 (2011)
38.Cudney, E.A., Paryani, K., Ragsdell, K.M.,Applying the Mahalanobis-Taguchi System to Vehicle Handling,2006 ASME International Design Engineering Technical Conferences and Computers and Information In Engineering Conference, DETC2006, September 10, 2006 - September 13, 2006, Philadelphia, PA, United states, pp. (2006)
39.Cudney, E.A., Ragsdell, K.M., Paryani, K.,Applying the Mahalanobis-Taguchi System to Vehicle Ride,2006 ASME International Mechanical Engineering Congress and Exposition, IMECE2006, November 5, 2006 - November 10, 2006, Chicago, IL, United states, pp. (2006)
40.Valarmathi, B., Palanisamy, V., “Opinion Mining of Customer Reviews Using Mahalanobis-Taguchi System,” European Journal of Scientific Research, Vol. 62, No. 1, pp. 95-100 (2011)
41.Wu, T.Y., Chung, Y.L., Liu, C.H., “Looseness Diagnosis of Rotating Machinery Via Vibration Analysis through Hilbert-Huang Transform Approach,” Journal of Vibration and Acoustics, Transactions of the ASME, Vol. 132, No. 3, pp. 0310051-0310059 (2010)
42.Ren, J., Cai, Y., Xing, X.,Application of Hilbert-Huang Transform and Mahalanobis-Taguchi System in Mechanical Fault Diagnostics Using Vibration Signals,IEEE 2011 10th International Conference on Electronic Measurement and Instruments, ICEMI 2011, August 16, 2011 - August 18, 2011, Chengdu, China, pp. 299-303 (2011)
43.Iwasaki, A., Todoroki, A.,Delamination Identification of Cfrp Structure by Discriminant Analysis Using Mahalanobis Distance,Proceedings of the 11th Asian Pacific Conference on Nondestructive Testing, November 3, 2003 - November 7, 2003, Jeju Island, Korea, Republic of, pp. 1859-1865 (2004)
44.Wang, J., Duan, C.,Structural Health Monitoring Using Mahalanobis-Taguchi System,2009 International Conference on Information Engineering and Computer Science, ICIECS 2009, December 19, 2009 - December 20, 2009, Wuhan, China, pp. Wuhan University; Huazhong Normal University; Wuhan University of Technology; National Technology University of Ukraine; Columbia University (2009)
45.K. Krishnan Nair, A.S.K., Kincho H. Law, “Time Series-Based Damage Detection and Localization Algorithm with Application to the Asce Benchmark Structure,” Journal of Sound and Vibration, Vol. 291, No., pp. 349-368 (2006)
46.T.W.Anderson, An Introduction to Multivariate Statistical Analysis (3rd), A John Wiley & Sons, California, (2003)
47.Juliane Schafer, K.S., “A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics,” Statistical Applications in Genetics and Molecular Biology, Vol. 4, No. 1, pp. (2005)
48.Fauvel, M., Chanussot, J., Benediktsson, J.A., Villa, A., “Parsimonious Mahalanobis Kernel for the Classification of High Dimensional Data,” Pattern Recognition, Vol. 46, No. 3, pp. 845-854 (2013)
49.Prekopcsak, Z., Lemire, D., “Time Series Classification by Class-Specific Mahalanobis Distance Measures,” Advances in Data Analysis and Classification, Vol. 6, No. 3, pp. 185-200 (2012)
50.Hoffbeck, J.P., Landgrebe, D.A., “Covariance Matrix Estimation and Classification with Limited Training Data,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 18, No. 7, pp. 763-767 (1996)
51.Deng, J.L., “Control Problems of Grey Systems,” Systems Control Lett., Vol. 5, No., pp. 288-294 (1982)
52.溫坤禮、張簡士琨、葉鎮愷、王建文、林慧珊，MATLAB 在灰色系統理論的應用，全華圖書公司，台北，(2007)。
53.Deng, J.L., “Introduction to Grey System Theory,” J. Grey System, Vol. 1, No., pp. 1-24 (1989)
54.陳彥琴，「應用灰色理論預測新上市之生技保健食品銷售量」，碩士論文，國立成功大學，台南（2005）。
55.鄧聚龍，灰色系統理論與應用，高立圖書，台北，(2000)。
56.吳漢雄、鄧聚龍、溫坤禮，灰色分析入門，高立出版，台北，(1996)。
57.鄧聚龍，灰色系統基本方法，華中理工大學出版社，武漢，(1985)。
58.Chen, J.-K., Chen, I.S., “A Network Hierarchical Feedback System for Taiwanese Universities Based on the Integration of Total Quality Management and Innovation,” Applied Soft Computing Journal, Vol. 12, No. 8, pp. 2394-2408 (2012)
59.Guo, M., Lan, J., Li, J., Lin, Z., Sun, X., “Traffic Flow Data Recovery Algorithm Based on Gray Residual Gm(1, N) Model,” Journal of Transportation Systems Engineering and Information Technology, Vol. 12, No. 1, pp. 42-47 (2012)
60.Kuo, C.-F.J., Syu, S.-S., Lin, C.-H., Peng, K.-C., “The Application of Principal Component Analysis and Gray Relational Method in the Optimization of the Melt Spinning Process Using the Cooling Air System,” Textile Research Journal, Vol. 83, No. 4, pp. 371-380 (2013)
61.Lin, H.-L., “Optimization of Inconel 718 Alloy Welds in an Activated Gta Welding Via Taguchi Method, Gray Relational Analysis, and a Neural Network,” Vol., No., pp. 1-12 (2012)
62.Xu, J., Chen, Y., Li, W., Zhang, L., Hong, Y., Bi, X. et al., “Statistical Analysis of Groundwater Chemistry of the Tarim River Lower Reaches, Northwest China,” Environmental Earth Sciences, Vol. 65, No. 6, pp. 1807-1820 (2012)
63.Ni, C.C.,Grey Relational Grade Analysis between Vickers Hardness and Fatigue Crack Growth Data of 2024-T351 Aluminum Alloy,3rd International Conference on Manufacturing Science and Engineering, ICMSE 2012, March 27, 2012 - March 29, 2012, Xiamen, China, pp. 2435-2439 (2012)
64.Chen, X., Zhu, H., Zhang, J., Li, L., “Damage Identification in Beam-Type Structures Using Pseudo Strain Energy Density and Grey Relation Coefficient,” Transactions of Tianjin University, Vol. 16, No. 2, pp. 96-103 (2010)
65.Cheng, Z., Hu, N., Zhang, X., “Crack Level Estimation Approach for Planetary Gearbox Based on Simulation Signal and Gra,” Journal of Sound and Vibration, Vol. 331, No. 26, pp. 5853-5863 (2012)
66.Luo, S., Qian, Z., Zong, H., “Research on Flexural Performance of Epoxy Asphalt Mixture Base on Grey Relation Degree Theory,” Wuhan Ligong Daxue Xuebao (Jiaotong Kexue Yu Gongcheng Ban)/Journal of Wuhan University of Technology (Transportation Science and Engineering), Vol. 32, No. 3, pp. 393-396 (2008)
67.石傑方, 類神經網路, 上網日期：2013, 1/28, from: http://neuron.csie.ntust.edu.tw/homework/93/nn/homework2/m9304302/12.htm, (2004)
68.林昆達，「小波理論與類神經網路在橋梁非破壞檢測之應用」，碩士論文，中原大學，桃園（2002）。
69.葉怡成，類神經網路模式應與實作，儒林圖書，台北，(2001)。
70.羅華強，類神經網路: MATLAB的應用，清蔚科技公司 ，新竹，(2001)。
71.Rosenblatt, F.x., “The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain,” Psychological Review, Vol. 65, No., pp. 386-408 (1958 )
72.Minsky M .L., P.S.A., Perceptrons, MIT Press, Cambridge, (1969)
73.Hopfield, J.J.,Collective Properties of Neural Networks,Proceedings of the 35th Annual Conference on Engineering in Medicine and Biology., Philadelphia, PA, USA, pp. 189 (1982)
74.孫嘉鴻，「灰預測與演化式類神經網路應用於台指選擇權之研究」，碩士論文，朝陽科技大學，台中（2004）。
75.Barsoum, F.F., Hill, E.V.K., Zhang, Y., Korcak, A., Suleman, J.,Procedures for Fatigue Crack Growth Monitoring and Fatigue Life Prediction Using Acoustic Emission Data and Neural Networks,Structures Congress 2011, April 14, 2011 - April 16, 2011, Las Vegas, NV, United states, pp. 2341-2352 (2011)
76.Sahoo, A.K., Zhang, Y., Zuo, M.J.,Estimating Crack Size and Location in a Steel Plate Using Ultrasonic Signals and Cfbp Neural Networks,IEEE Canadian Conference on Electrical and Computer Engineering, CCECE 2008, May 4, 2008 - May 7, 2008, Niagara Falls, ON, Canada, pp. 1751-1754 (2008)
77.Lee, K., Estivill-Castro, V.,Support Vector Machine Classification of Ultrasonic Shaft Inspection Data Using Discrete Wavelet Transform,Proceedings of the International Conference on Artificial Intelligence, IC-AI'04, June 21, 2004 - June 24, 2004, Las Vegas, NV, United states, pp. 848-854 (2004)
78.Sambath, S., Nagaraj, P., Selvakumar, N., Arunachalam, S., Page, T., “Automatic Detection of Defects in Ultrasonic Testing Sing Artificial Neural Network,” International Journal of Microstructure and Materials Properties, Vol. 5, No. 6, pp. 561-574 (2010)
79.Vasudevan, M., Bhaduri, A.K., Raj, B.,Modeling the Solidification Mode in Austenitic Stainless Steel Welds, pp. 380-387 (2005)
80.Chen, Y., Ma, H.,Application of Artificial Neural Network to Flaw Classification in Ultrasonic Testing,2011 International Conference on Mechatronics and Materials Processing, ICMMP 2011, November 18, 2011 - November 20, 2011, Guangzhou, China, pp. 1876-1880 (2011)
81.Instruments, I.-E., Instrument Components, Accessed: 2013, 1/10, Available at: www.impact-echo.com, (2013)
82.Richard O. Duda , P.E.H., David G,Stork, Pattern Classification (2nd), Wiley, (2007)
83.黃國豪，「應用希爾伯特黃變換(HHT)之邊際譜分析於旋轉機械的元件鬆脫故障診斷」，碩士論文，國立中央大學，桃園（2009）。
84.Sergios Theodoridis, K.K., Pattern Recognition (4th), Academic Press, (2008)
85.Thomaz, C.E., Gillies, D.F., Feitosa, R.Q.,Using Mixture Covariance Matrices to Improve Face and Facial Expression Recognitions,3rd International Conference on Audio- and Video- Based Biometric Person Authentication, AVBPA 2001, June 6, 2001 - June 8, 2001, Halmstad, Sweden, pp. 71-77 (2001)
86.Kohonen, T.K., Self-Organization and Associative Memory, Springer-Verlag New York, (1988)
87.Hagan, M.T., Demuth, H.B., Beale, M.H., Neural Network Design, PWS publishing, (1996)
88.張智星，MATLAB程式設計入門（3版），碁峰，台北，(2011)。
89.張智星，MATLAB 程式設計－進階篇，鈦思，台北，(2008)。
90.張德豐，MATLAB神經網路編程，化學工業，北京，(2011)。
91.史峰、王小川、郁磊、李洋，MATLAB神經網絡30個案例分析，航空航天大學出版社，北京，(2010)。