Statistical structural health monitoring methodologies and applications

STATISTICAL STRUCTURAL HEALTH MONITORING: METHODOLOGIES AND APPLICATIONS WANG ZENGRONG NATIONAL UNIVERSITY OF SINGAPORE 2008 STATISTICAL STRUCTURAL HEALTH MONITORING: METHODOLOGIES AND APPLICATIONS WANG ZENGRONG (B.Eng., XJTU) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2008 To my parents ACKNOWLEDGEMENTS I would like to express my cordial gratitude to my supervisor, Associate Professor Ong Khim Chye Gary. Throughout my graduate study, I have been greatly benefiting from his informative guidance and invaluable suggestions, without which this research would not have been possible. Compelling justifications for my gratitude also come from his constant encouragement and unscheduled help, with which this research goes towards a successful completion. It is his thoroughness, professionalism and commitment that make my graduate study a rewarding experience. I would like to appreciate Associate Professor Mohamed Maalej for his edifying instructions during the early days of my graduate study. I would also like to appreciate Professor Quek Ser Tong for his constructive comments and for giving his insights in Hilbert-Huang transform. Many thanks are also due to my fellow graduate students, both inside and outside the Department of Civil Engineering, for the good time spent together. Finally, the financial support of the Research Scholarship and the state-of-the-art research facilities provided by the University are acknowledged. iii TABLE OF CONTENTS Title Page i Dedication ii Acknowledgements iii Table of Contents iv Summary vii List of Tables ix List of Figures xi List of Abbreviations xiv List of Symbols xvi CHAPTER ONE INTRODUCTION 1.1 1.2 1.3 1.4 Background Literature Review 1.2.1 Vibration Characteristics Based SHM 1.2.2 Statistical Techniques for Vibration Characteristics Based SHM Objective and Scope of Work Organization of Thesis CHAPTER TWO MULTIVARIATE EXTENSION OF THE STRUCTURAL HEALTH MONITORING SCHEME USING AUTOREGRESSIVE COEFFICIENTS BASED STATISTICAL PROCESS CONTROL TECHNIQUES 2.1 2.2 Introduction Vibration Response Data Representation and Characteristics Monitoring 2.2.1 Representation of Vibration Response Data Based on Time Series Analysis 2.2.2 Monitoring of Vibration Response Data Characteristics Based on Multivariate Statistical Process Control 2.2.3 Effects of the Autocorrelation in the Characteristics Data 1 2 6 8 9 11 15 iv 2.3 2.4 Case Study 2.3.1 Structural Health Monitoring Using Autoregressive Coefficients Based Hotelling’s T2 Control Chart: without Addressing the Autocorrelation 2.3.2 Structural Health Monitoring Using Autoregressive Coefficients Based Hotelling’s T2 Control Chart: with Addressing the Autocorrelation Summary CHAPTER THREE IMPROVING STRUCTURAL DAMAGE DETECTION SENSITIVITY USING AUTOREGRESSIVE-MODEL-INCORPORATING MULTIVARIATE EXPONENTIALLY WEIGHTED MOVING AVERAGE CONTROL CHART 3.1 3.2 3.3 3.4 Introduction Autoregressive-Model-Incorporating Multivariate Exponentially Weighted Moving Average Control Chart for Structural Damage Detection 3.2.1 Procedures Based on the Reference State 3.2.2 Procedures Based on the Current State Case Study 3.3.1 Damage Detection 3.3.2 Parametric Study Summary CHAPTER FOUR A MULTIVARIATE STATISTICAL APPROACH TO STRUCTURAL DAMAGE DETECTION 4.1 4.2 4.3 4.4 Introduction Representation and Monitoring of Vibration Response Data: A Multivariate Statistical Approach Case Studies 4.3.1 Numerical Example of a Five-Story Shear Frame 4.3.2 Numerical Example of a Shear Wall 4.3.3 Experimental Example of the I-40 Bridge Benchmark Summary CHAPTER FIVE A NONPARAMETRIC STATISTICAL FRAMEWORK FOR STRUCTURAL HEALTH MONITORING 5.1 5.2 Introduction Formulation of the Nonparametric Statistical Structural Health Monitoring Framework 15 17 19 24 45 45 46 46 52 55 56 59 60 83 83 84 92 92 95 97 98 116 116 117 v 5.3 5.4 Case Studies 5.3.1 A 20-Degree-of-Freedom System 5.3.2 A Hyperbolic Paraboloid Roof Shell Summary 124 124 126 129 CHAPTER SIX CONCLUSIONS 150 6.1 6.2 150 151 Conclusions Future Work REFERENCES 153 APPENDIX A DATA OF THE DAMAGE INDICATOR PROFILES FOR THE HYPERBOLIC PARABOLOID ROOF SHELL CONSIDERED IN SECTION 5.3.2 161 APPENDIX B PUBLICATIONS RELEVANT TO THIS RESEARCH 166 vi SUMMARY This research is concerned with statistical structural health monitoring (SHM). First, an innovative SHM scheme based on time series analysis and multivariate statistical process control (MSPC) techniques is presented. The scheme consists of two major procedures, viz vibration response data representation and characteristics monitoring. First, a series of autoregressive (AR) models is fitted to the response time histories of a structure to be monitored. Representing the health condition of the structure, the coefficients of these AR models are extracted to form a set of multivariate data known as vibration response data characteristics. Hotelling’s T2 control chart is then applied to monitor these characteristics obtained. As an MSPC tool, Hotelling’s T2 control chart has the capacity of simultaneously monitoring the multivariate characteristics data without having to neglect the inherent relation between the components of the data. The efficacy of the proposed SHM scheme is demonstrated by numerically simulated acceleration time histories based on a progressively damaged reinforced concrete (RC) frame, either with or without addressing the autocorrelation in the characteristics data. The results are compared with those obtained by using univariate Shewhart X control chart to show the advantages of the proposed scheme in terms of the sensitivity of the defined damage indicator with respect to damage severity. A parametric study is also included to investigate the effects of the number of data points used for AR model fitting, the order of AR models and the number and locations of sensors on the proposed scheme, as well as to further illustrate its potential as a promising SHM approach. To further improve structural damage detection sensitivity, a scheme using autoregressive-model-incorporating multivariate exponentially weighted moving average (MEWMA) control chart is presented. This scheme comprises the procedures based on the undamaged or reference state of the structure being monitored and those based on its damaged or current state. In the procedures based on the reference state, sets of multivariate data are formulated by a series of AR model fitting, and these data are then subjected to MEWMA control chart analysis to establish a benchmark damage indicator. The damage indicator obtained in the procedures based on the current state is compared with the benchmark for the purpose of structural damage detection. The autocorrelation in the multivariate data is addressed, and special procedures to allow vii for the uncertainty involved in process parameter estimation as well as those for control limit determination are proposed for structural damage detection application. A numerically simulated case study is used to verify the efficacy of the proposed scheme and to show its advantages. A parametric study is also included to study the effects of some parameters and to demonstrate the robustness of the scheme against parameter selection. The issue of structural damage detection is then addressed through an innovative multivariate statistical approach. By invoking principal component analysis (PCA), the vibration responses acquired from the structure being monitored are represented by the multivariate data of the sample principal component coefficients (PCCs). A damage indicator is then defined based on a MEWMA control chart analysis formulation, involving special procedures to allow for the effects of the estimated parameters and to determine the upper control limit (UCL) in the control chart analysis for structural damage detection applications. Also, a data shuffling procedure is proposed to address the autocorrelation probably present in the obtained sample PCCs. This multivariate statistical structural damage detection scheme can be applied to either the time domain responses or the frequency domain responses. The efficacy and advantages of the scheme is demonstrated by the numerical examples of a five-story shear frame and a shear wall as well as the experimental example of the I-40 Bridge benchmark. Finally, a nonparametric statistical framework for SHM is presented. Vibration response data are first represented by the coefficients of a series of fitted AR models in the time domain or by the averages of binned power spectral density (PSD) estimates in the frequency domain. Three types of statistical hypotheses are then formulated and tested by nonparametric techniques to monitor these characteristics. Specifically, twosample Kolmogorov-Smirnov test, Mann-Whitney test and Mood test are used in this study. For each type of hypothesis formulation, a function of the resulting P-values is used to define a damage indicator profile (DIP) whereby damage locations are identified. The highlight of this framework is that, due to its nonparametric nature, it does not require a particular functional form for the underlying population of an extracted vibration response data characteristic. Two numerically simulated case studies, i.e., a 20-degree-of-freedom system and a hyperbolic paraboloid roof shell, demonstrate the efficacy of the proposed nonparametric SHM framework. Multiple damage locations are also considered in the case studies. viii LIST OF TABLES Table 2.1 Total number of outliers and average sensitivity based on multivariate Hotelling’s T2 control chart and those based on univariate Shewhart X control chart (without addressing the autocorrelation) 26 Total number of outliers and average sensitivity based on multivariate Hotelling’s T2 control chart and those based on univariate Shewhart X control chart (with addressing the autocorrelation) 27 Total number of outliers and average sensitivity based on Hotelling’s T2 control chart for different numbers of data points used for AR model fitting (with addressing the autocorrelation) 28 Total number of outliers and average sensitivity based on Hotelling’s T2 control chart for different orders of AR models (with addressing the autocorrelation) 29 Total number of outliers and average sensitivity based on Hotelling’s T2 control chart for different numbers of sensors (with addressing the autocorrelation) 30 Total number of outliers and average sensitivity based on Hotelling’s T2 control chart for different locations of sensors (with addressing the autocorrelation) 31 Table 3.1 Values of the damage indicator in the false alarm test 62 Table 3.2 Values of the damage indicator based on MEWMA control chart and those of the damage indicators based on other control charts 63 Values of the damage indicator based on MEWMA control chart for different AR model orders 64 Values of the damage indicator based on MEWMA control chart for different numbers of data points used for AR model fitting 65 Values of the damage indicator based on MEWMA control chart for different smoothing parameters 66 Table 2.2 Table 2.3 Table 2.4 Table 2.5 Table 2.6 Table 3.3 Table 3.4 Table 3.5 Table 4.1 Results for the numerical example of a five-story shear frame 100 Table 4.2 Results for the numerical example of a shear wall 101 ix CHAPTER SIX (d) The autocorrelation present in the characteristics of the vibration response data may in some case cause increase in false alarms. To address the autocorrelation, sampling less frequently as in Chapter Two and Three or data shuffling as in Chapter Four are used. (e) Along with MEWMA control chart, sample PCCs based on either time-domain data and frequency-domain data are calculated to formulate a multivariate SHM approach. The case studies of a shear frame, a shear wall and the I-40 Bridge benchmark demonstrate its effectiveness. (f) An open-ended nonparametric statistical framework for SHM is formulated. In particular, the efficacy of the DIPs based on either fitted AR coefficients or PSD estimates is illustrated by the case studies of a 20-DOF system and a hyperbolic paraboloid roof shell. 6.2 Future Work Based on the results of this research, some possible future work is suggested below: (a) Potential of incorporating multivariate time series models, e.g., VAR model, vector autoregressive moving average (VARMA) model, for vibration response data representing worth further research efforts. 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(2006). “A multivariate change-point model for statistical process control.” Technometrics, 48(4), 539-549. 160 APPENDIX A DATA OF THE DAMAGE INDICATOR PROFILES FOR THE HYPERBOLIC PARABOLOID ROOF SHELL CONSIDERED IN SECTION 5.3.2 The DIP values obtained in Section 5.3.2 are presented in this appendix, wherein Table A.1 lists the values based on the fitted AR coefficients, and Table A.2 gives those based on the PSD estimates. 161 Location IDa 162 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Formulation I 1.6241E+01 1.7747E+01 1.4442E+01 4.3389E+01 6.3085E+01 7.7733E+01 1.5446E+02 8.4672E+01 2.7820E+02 3.4549E+01 9.8615E+06 5.2854E+01 1.6817E+01 5.1758E+01 9.2815E+02 9.8699E+02 2.8351E+01 2.1871E+01 3.3422E+01 3.6735E+01 1.8380E+04 1.8225E+01 1.6249E+01 1.2575E+02 1.5045E+01 1.4075E+01 1.8048E+05 Table A.1 DIPs based on the fitted AR coefficients for the hyperbolic paraboloid roof shell HPRS1 HPRS2 HPRS3 Formulation II Formulation III Formulation I Formulation II Formulation III Formulation I Formulation II Formulation III 1.9236E+01 1.0271E+01 3.2966E+01 2.3837E+01 1.7328E+01 8.4810E+03 4.2043E+03 1.5199E+01 1.2280E+01 6.1667E+01 1.0691E+01 1.1469E+01 1.3372E+01 5.4148E+01 2.2201E+02 1.6684E+01 1.4012E+01 2.1031E+01 1.9572E+02 4.7029E+02 1.8155E+01 2.2549E+01 1.3816E+01 2.1726E+02 6.0471E+01 2.3163E+01 3.0515E+02 1.9003E+03 2.0548E+01 3.8904E+02 4.8428E+02 1.7461E+01 1.0212E+03 1.5813E+01 1.1325E+01 3.7869E+01 1.6891E+01 5.3222E+01 9.7324E+01 1.2595E+01 6.1776E+01 3.6859E+01 1.8812E+01 2.6567E+01 1.0171E+01 2.3084E+02 8.9318E+01 3.7295E+01 1.0337E+02 2.2728E+01 2.4167E+01 3.6796E+01 1.1449E+01 7.5133E+05 9.9686E+06 2.8506E+01 1.7084E+02 4.3397E+01 1.7108E+02 2.5640E+02 1.3414E+01 2.4975E+02 4.8356E+02 4.0845E+01 4.5289E+03 1.5123E+01 5.1406E+05 1.9472E+07 2.5916E+01 4.9338E+03 7.3790E+05 2.2576E+02 4.5915E+01 3.5929E+01 1.6126E+04 5.5824E+05 2.0691E+01 4.1015E+02 5.0799E+03 1.6404E+01 9.9743E+06 5.6854E+01 1.0030E+07 1.0180E+07 1.3618E+03 1.0001E+07 1.0011E+07 3.2047E+01 7.8470E+01 1.2842E+02 3.8370E+02 1.5583E+03 2.1779E+01 1.9816E+06 9.8899E+06 1.1208E+02 1.7508E+01 1.3904E+01 2.5191E+02 1.6483E+02 1.6183E+01 4.9803E+06 9.5524E+06 2.8443E+01 6.9758E+01 2.2788E+01 6.5765E+02 3.9423E+02 1.4762E+01 7.2154E+06 9.9629E+06 2.0753E+01 8.3873E+03 1.1019E+01 1.8581E+02 3.9808E+02 3.7040E+01 9.1612E+01 1.0249E+02 3.6435E+01 1.1353E+03 3.3268E+01 3.1168E+01 8.8707E+01 1.3689E+01 4.1888E+03 1.1608E+05 2.4534E+01 2.0345E+01 3.4858E+01 1.5855E+02 2.2698E+02 2.7286E+01 1.8018E+01 2.8549E+01 9.2118E+00 3.3645E+01 7.9767E+00 1.2358E+01 1.6168E+01 1.8063E+01 4.9613E+03 4.4653E+04 8.7039E+00 8.2194E+01 2.2545E+01 3.9587E+01 5.9131E+01 1.6453E+01 6.1513E+06 1.0026E+07 2.0592E+01 1.0769E+02 3.6332E+01 5.3011E+01 3.3489E+01 1.9788E+02 9.9999E+06 1.0000E+07 2.1348E+01 4.9869E+06 2.1297E+01 2.4014E+06 9.9445E+06 1.0366E+02 8.9986E+06 1.0050E+07 2.0224E+01 2.0751E+01 4.7069E+01 7.9338E+01 3.5895E+02 2.0862E+01 8.3876E+03 2.4401E+04 1.8649E+01 2.3283E+01 1.4435E+01 4.2522E+01 8.1369E+01 1.5440E+01 1.0554E+02 5.4883E+02 2.2388E+01 1.6235E+01 5.2011E+01 1.7050E+01 2.3710E+01 1.4306E+01 2.7098E+05 7.7444E+06 9.9525E+00 1.2037E+01 5.9778E+01 1.5395E+01 1.2527E+01 3.1273E+01 4.6175E+03 8.7750E+04 2.2078E+01 1.6601E+01 2.6286E+01 9.4885E+00 1.3347E+01 1.9054E+01 1.3967E+01 2.4213E+01 1.2086E+01 4.2572E+05 3.8130E+01 8.7448E+02 3.5942E+02 2.1291E+01 2.1203E+02 1.6793E+02 5.7392E+01 28 1.6059E+01 2.7424E+01 1.8213E+01 29 4.8557E+01 4.5911E+01 1.9580E+01 30 5.2718E+01 8.3871E+01 9.7399E+00 31 4.2190E+01 1.0636E+02 1.9007E+01 32 3.0532E+01 7.0776E+01 1.1475E+01 33 1.7305E+01 2.1262E+01 1.1035E+01 34 2.9314E+03 8.3486E+03 1.5174E+01 35 2.2022E+02 8.4816E+02 1.2139E+01 36 4.1305E+01 3.8325E+01 1.5355E+01 37 4.4563E+01 4.8901E+01 1.2993E+01 38 3.7758E+01 5.4147E+01 5.8875E+01 39 1.4434E+01 1.9955E+01 1.0687E+01 40 1.9259E+01 2.4408E+01 1.6865E+01 41 9.4834E+00 1.0390E+01 1.3683E+01 42 1.3414E+01 1.5557E+01 1.3825E+01 43 2.9997E+03 6.4712E+03 4.2736E+01 44 1.8062E+03 1.0918E+04 3.3715E+01 45 2.8498E+01 2.8150E+01 1.7981E+01 46 7.4578E+01 5.2308E+02 1.8007E+01 47 1.5481E+01 1.1258E+01 1.4549E+01 48 1.0338E+01 1.2170E+01 1.3414E+01 49 5.2231E+02 6.9634E+03 2.1570E+02 50 9.3803E+02 3.8205E+02 1.4755E+02 51 9.7525E+01 9.9814E+01 1.0545E+01 52 8.2456E+02 6.3362E+02 3.3267E+02 53 3.2596E+01 5.1956E+01 1.4435E+01 54 6.1863E+01 1.8841E+01 2.1489E+01 a “Location ID”=6×(“Location ID 1”-1)+“Location ID 2” 2.1056E+02 9.6874E+00 2.0588E+01 4.1125E+02 2.4741E+02 1.9208E+05 4.3537E+06 5.3053E+02 2.1085E+03 3.8655E+03 3.3247E+01 5.6031E+02 4.0230E+02 3.5601E+01 9.7684E+01 1.0432E+04 3.1527E+02 1.5478E+01 2.0273E+01 5.4161E+02 2.8356E+02 7.0554E+02 1.5989E+01 1.2857E+01 1.9386E+01 2.3806E+01 2.4757E+01 2.1476E+02 8.8075E+00 2.0149E+01 7.5272E+03 2.3181E+02 4.6499E+06 1.7721E+07 3.3298E+04 4.2068E+03 4.1424E+03 6.6598E+01 8.0653E+03 8.2292E+02 3.3735E+02 3.5728E+02 1.9214E+04 1.0445E+03 2.3793E+01 3.8771E+01 2.0161E+02 6.6058E+02 2.5373E+02 1.7342E+01 1.8350E+01 1.5268E+01 2.3510E+01 6.8667E+01 1.2440E+01 2.3740E+01 1.4390E+01 1.7591E+01 2.2384E+01 2.2809E+01 1.3828E+01 4.1915E+01 3.1628E+01 4.5897E+01 2.9874E+01 1.5319E+01 2.1464E+01 8.3712E+00 4.1869E+01 1.1070E+01 1.0494E+01 1.2020E+01 2.0935E+01 1.5607E+01 1.4061E+01 1.9802E+01 1.7445E+01 3.2314E+01 1.9355E+01 2.9309E+01 1.0766E+01 2.0169E+01 3.9769E+01 1.5745E+02 1.7906E+02 2.0908E+03 8.0879E+06 3.0649E+03 5.0253E+01 2.5463E+01 6.8131E+02 3.2655E+01 1.3414E+02 3.2328E+02 1.6932E+01 4.7265E+01 5.8222E+02 6.1995E+01 8.4117E+01 2.4963E+01 2.7664E+01 7.6318E+01 9.9965E+01 1.7302E+01 3.5892E+01 2.8496E+01 7.5150E+01 1.4348E+03 2.4865E+01 1.0627E+02 7.8164E+02 5.3058E+02 5.3095E+04 9.9748E+06 5.9988E+04 1.1094E+02 1.0702E+02 3.0846E+03 9.4856E+01 1.0682E+02 6.2170E+02 2.1816E+01 1.8758E+02 5.2475E+02 1.3394E+02 1.0481E+02 3.9940E+01 7.0203E+01 1.9778E+02 4.2983E+01 2.1887E+01 2.1726E+01 4.5495E+01 1.4802E+03 2.9192E+03 1.3328E+01 3.6483E+01 1.3985E+01 2.2425E+01 1.2317E+01 3.6279E+01 1.8046E+01 1.2068E+01 1.0206E+01 6.1889E+01 1.2586E+01 9.4609E+00 1.2301E+01 7.5233E+00 2.7992E+01 1.4429E+01 9.6576E+00 1.7823E+01 8.5927E+01 1.8027E+01 2.4187E+01 1.6020E+03 4.1285E+01 4.7292E+01 3.8753E+01 1.2042E+02 1.9211E+01 163 Location IDa 164 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Formulation I 8.1213E+01 1.8393E+02 6.7872E+01 2.0622E+02 2.4471E+02 4.7055E+07 2.2650E+02 1.4479E+02 3.1626E+06 1.7712E+02 9.8135E+06 6.8467E+04 1.4980E+03 1.7778E+02 1.2608E+04 1.5274E+02 1.3070E+02 9.0873E+01 7.0220E+02 1.0738E+03 1.7573E+06 1.1380E+02 7.7705E+01 2.3088E+02 2.4299E+02 1.3209E+02 7.3791E+02 Table A.2 DIPs based on the PSD estimates for the hyperbolic paraboloid roof shell HPRS1 HPRS2 Formulation II Formulation III Formulation I Formulation II Formulation III Formulation I 1.0558E+02 4.4916E+02 4.3960E+02 1.1358E+03 2.7118E+02 9.9953E+06 2.7992E+02 8.6347E+01 2.2177E+02 3.2828E+02 2.2297E+02 8.6625E+06 7.8225E+01 1.0481E+02 5.6486E+02 3.5191E+03 1.7088E+02 2.2153E+03 2.9817E+02 7.4847E+01 1.9009E+05 9.2119E+06 3.2175E+02 8.5532E+05 3.4861E+02 1.1280E+02 1.0776E+04 6.5840E+04 1.7930E+02 2.3482E+06 1.2316E+08 2.2695E+02 7.3549E+07 1.3827E+08 3.5529E+02 4.2140E+07 2.6104E+02 2.2559E+02 4.1472E+02 1.5174E+03 2.6348E+02 2.2475E+06 1.8308E+02 7.3941E+01 3.1190E+02 3.4792E+02 1.0650E+02 1.9242E+05 1.3798E+07 1.5586E+02 7.7448E+04 5.7777E+06 6.5912E+02 8.5433E+06 6.4333E+02 1.0303E+02 8.9981E+04 1.7876E+06 3.7801E+02 4.7641E+03 1.8585E+07 1.8649E+02 4.6535E+06 2.4129E+07 3.5301E+02 1.1310E+07 2.9506E+05 9.3723E+01 9.9388E+06 1.6851E+07 1.2063E+02 2.4443E+07 1.8610E+03 1.3077E+02 7.4800E+02 1.5345E+03 2.3650E+02 6.7254E+07 2.0985E+02 6.7692E+02 1.0634E+03 3.3997E+03 1.9487E+02 2.2060E+05 4.2919E+05 2.6813E+03 5.4565E+03 3.9349E+03 2.9421E+02 3.1826E+04 2.3411E+02 1.6708E+02 1.1215E+04 3.1669E+05 1.9726E+02 1.3241E+07 2.9572E+02 2.3238E+02 4.1836E+02 1.5751E+03 2.5991E+02 2.1277E+04 1.0774E+02 1.2554E+02 5.1570E+02 3.5959E+02 2.3391E+02 2.6107E+05 1.4111E+04 1.0705E+02 5.8565E+03 5.0800E+04 1.7573E+02 1.1311E+08 7.1471E+02 1.4128E+02 1.2192E+03 2.7268E+03 7.2426E+01 1.4813E+07 6.5218E+06 1.4902E+02 6.7972E+06 2.0568E+07 2.9832E+02 1.0426E+07 4.8019E+02 1.7584E+02 1.4085E+03 4.1735E+02 3.9448E+02 3.9997E+07 7.9204E+01 1.2341E+02 5.8711E+03 2.2282E+05 9.0578E+01 3.2690E+06 2.8822E+02 1.2962E+02 3.6623E+03 1.6753E+04 1.5266E+02 2.0078E+06 2.7050E+02 1.4577E+03 8.8199E+02 3.8953E+03 2.3956E+02 1.1501E+07 1.9314E+02 1.3208E+02 1.4035E+05 3.7133E+06 4.1463E+02 9.8293E+06 5.8579E+03 7.5565E+02 1.9197E+07 1.9943E+07 5.5313E+02 2.1903E+07 HPRS3 Formulation II Formulation III 1.8072E+07 1.8971E+02 9.9680E+06 9.1516E+01 2.4434E+04 4.0102E+03 9.9070E+06 2.0614E+02 9.8444E+06 1.6070E+02 6.7999E+07 9.2805E+02 2.4862E+07 3.0373E+02 4.0307E+05 8.1703E+01 2.6831E+07 3.4316E+02 2.5798E+04 4.9372E+02 2.7972E+07 7.6360E+02 4.8791E+07 4.7145E+02 8.8603E+07 8.1095E+02 1.4018E+06 1.8635E+02 1.8911E+04 2.7503E+02 2.2240E+07 1.6183E+03 9.0112E+05 2.4997E+02 1.1657E+06 2.1307E+02 1.3710E+08 1.4157E+02 5.0073E+07 1.3905E+02 2.2457E+07 3.3744E+02 4.0004E+07 3.3840E+02 1.3426E+07 1.5659E+02 2.1573E+07 3.7746E+02 1.6702E+07 1.3950E+05 1.0039E+07 8.1777E+04 3.5874E+07 1.8210E+03 28 2.5325E+02 1.1963E+03 1.1495E+02 29 3.7780E+02 1.4944E+03 2.6383E+02 30 2.4620E+03 1.5514E+03 1.3304E+02 31 1.2881E+05 6.8554E+05 4.3549E+02 32 8.1205E+03 7.5899E+03 1.3851E+02 33 5.3230E+02 3.0359E+03 3.5554E+02 34 1.4267E+03 4.0867E+04 1.8265E+02 35 5.0654E+02 5.6514E+02 9.8328E+01 36 1.0861E+02 1.9079E+02 1.3012E+02 37 1.6542E+02 1.9328E+02 3.7525E+02 38 1.3287E+02 1.6161E+02 2.7908E+02 39 1.9314E+02 1.7150E+02 2.3023E+02 40 4.2735E+02 5.9590E+02 2.3099E+02 41 1.0635E+02 1.0086E+02 8.1805E+01 42 8.6180E+01 1.3964E+02 1.0927E+02 43 2.8078E+02 6.6227E+02 1.1656E+02 44 2.2250E+03 8.5314E+02 2.5913E+02 45 1.4500E+02 3.2127E+02 1.3770E+02 46 2.9179E+02 3.6295E+02 1.4055E+02 47 1.1737E+02 1.3425E+02 1.0361E+02 48 3.6644E+02 7.1033E+02 1.2573E+02 49 1.2044E+04 7.7260E+04 1.0332E+02 50 1.3255E+03 1.7868E+03 6.2190E+02 51 1.7017E+02 2.1727E+02 4.2349E+02 52 9.0141E+01 9.3934E+01 1.0888E+02 53 2.4966E+02 4.2662E+02 1.1203E+02 54 3.5957E+02 6.4261E+02 1.0478E+02 a “Location ID”=6×(“Location ID 1”-1)+“Location ID 2” 1.0142E+07 1.4044E+05 3.3227E+04 2.2779E+05 1.1662E+07 1.0401E+07 1.9670E+07 3.4017E+06 1.0113E+06 5.1965E+04 3.1952E+06 3.1822E+06 4.2174E+06 6.1479E+04 8.7291E+03 5.3562E+03 1.1456E+04 3.6873E+03 2.0216E+03 9.4819E+02 2.3526E+04 1.0146E+07 3.7408E+02 6.6538E+02 7.5826E+02 5.0231E+02 1.1314E+04 2.0507E+07 7.3945E+06 2.2028E+05 1.0916E+07 2.3096E+07 1.4995E+07 3.7605E+07 1.8506E+07 1.2447E+07 2.8760E+06 9.9992E+06 9.3335E+06 9.3164E+06 6.6924E+04 2.5985E+06 2.6164E+05 2.0219E+05 5.5948E+03 2.7704E+03 2.1810E+03 1.6823E+05 2.0133E+07 2.3018E+03 2.9959E+03 2.6017E+03 6.6590E+02 3.5073E+05 3.7502E+02 4.0909E+02 2.7467E+02 2.2461E+02 3.0259E+02 3.3835E+02 5.9336E+03 5.0841E+02 1.6158E+02 9.5868E+01 1.1469E+02 1.6099E+02 1.6362E+02 3.3665E+02 1.7463E+02 1.3111E+02 1.2925E+03 2.3373E+02 1.8294E+02 1.1689E+02 1.4168E+02 5.7502E+02 1.8837E+02 1.0598E+02 1.7181E+02 2.3567E+02 2.1173E+02 2.6871E+07 1.0009E+07 1.0388E+07 7.0562E+06 2.6682E+07 1.8710E+07 1.9208E+07 1.0018E+07 2.0008E+07 1.3041E+07 1.7026E+07 4.1679E+06 8.6956E+06 1.5847E+04 1.8710E+05 2.4379E+06 5.0976E+04 6.1740E+04 2.9846E+04 5.8074E+05 1.7241E+07 1.2766E+07 2.7597E+05 3.1737E+06 2.8764E+03 6.0197E+05 2.9423E+07 3.6929E+07 1.0118E+07 2.0030E+07 1.4946E+07 3.2048E+07 2.4087E+07 2.0535E+07 1.1389E+07 2.0071E+07 1.5990E+07 1.8478E+07 1.0034E+07 1.0874E+07 1.1265E+05 1.0981E+06 3.5066E+05 1.1216E+05 7.4387E+05 1.1886E+05 9.2264E+06 2.0000E+07 2.5035E+07 9.1959E+06 8.1878E+06 3.9515E+03 8.6194E+06 3.0237E+07 2.9993E+03 3.4635E+02 7.6200E+02 7.7644E+02 1.6158E+03 5.8041E+02 5.4766E+02 2.4816E+02 1.7259E+02 7.1684E+02 4.5253E+02 1.3720E+02 5.2080E+02 1.9061E+02 1.9144E+02 3.9198E+02 1.1340E+03 1.4901E+02 3.9496E+02 1.0910E+02 6.4428E+02 1.4918E+03 6.5132E+02 1.7265E+02 3.2179E+02 1.3598E+02 1.8998E+02 165 APPENDIX B PUBLICATIONS RELEVANT TO THIS RESEARCH Journal Papers Ong, K. C. G., Wang, Z., and Maalej, M. (2008). “Adaptive magnitude spectrum algorithm for Hilbert-Huang transform based frequency identification.” Eng. Struct., 30(1), 33-41. Wang, Z., and Ong, K. C. G. (2008). “Autoregressive coefficients based Hotelling’s T2 control chart for structural health monitoring.” Comput. Struct., 86(19-20), 1918-1935. Wang, Z., and Ong, K. C. G. “Structural damage detection using autoregressivemodel-incorporating multivariate exponentially weighted moving average control chart.” (Under review.) Wang, Z., and Ong, K. C. G. “A multivariate statistical approach to structural damage detection.” (Under review.) Wang, Z., and Ong, K. C. G. “A nonparametric statistical framework for structural health monitoring.” (Under review.) Conference Papers Ong, K. C. G., Maalej, M., and Wang, Z. (2006). “Sensitivity analysis of adaptive magnitude spectrum algorithm identified modal frequencies of reinforced concrete frame structures.” Proc., 31st Conf. on Our World in Concrete & Structures, Singapore, 287-296. Wang, Z., and Ong, K. C. G. (2007). “Structural health monitoring of reinforced concrete frames for progressive damage using Hotelling’s T2 control chart.” Proc., 6th Int. Workshop on Structural Health Monitoring, Stanford University, Stanford, Calif., 322-331. 166 [...]... and Ching and Beck (2004) proposed a Bayesian probabilistic framework for structural health monitoring based on the probabilistic model updating scheme developed by Beck and Katafygiotis (1998), and Katafygiotis and Beck (1998) Worden (1997) and Worden et al (2000) proposed a pattern recognition scheme based on transmissibility functions and squared Mahalanobis distance Nair et al (2006) proposed statistical. .. for TDS2 based on (c) AR coefficients and Formulation II, and (d) PSD estimates and Formulation III; and those for TDS3 based on (e) AR coefficients and Formulation I, and (f) PSD estimates and Formulation III 135 A hyperbolic paraboloid roof shell: (a) Three-dimensional view; and (b) plan view 138 Examples of the vibration response data characteristics at Point C and Point I: (a) 1st fitted AR coefficients... Five presents the nonparametric statistical framework for SHM; and (f) Chapter Six summarizes major conclusions, and suggests future work 7 CHAPTER TWO MULTIVARIATE EXTENSION OF THE STRUCTURAL HEALTH MONITORING SCHEME USING AUTOREGRESSIVE COEFFICIENTS BASED STATISTICAL PROCESS CONTROL TECHNIQUES 2.1 Introduction Sohn et al (1999; 2000b) and Fugate et al (2001) addressed the structural damage detection... incorporate the principal component analysis (PCA), a dimension reduction technique, into the control chart based structural damage detection formulation (Wang and Ong 2007b) This is to arrive at a multivariate statistical approach to the damage detection and health monitoring of structures, and to broaden the applicability of the formerly developed formulation Flexibility of domain choice (i.e., time... Formulation II using (c) AR coefficients and (d) PSD estimates; and those based on Formulation III using (e) AR coefficients and (f) PSD estimates 144 For HPRS3, DIPs based on Formulation I using (a) AR coefficients and (b) PSD estimates; those based on Formulation II using (c) AR coefficients and (d) PSD estimates; and those based on Formulation III using (e) AR coefficients and (f) PSD estimates 147 Fig 5.3... (HPRS2); and (d) averages of the PSD estimates in the 10th frequency bin (HPRS2) 139 For HPRS1, DIPs based on Formulation I using (a) AR coefficients and (b) PSD estimates; those based on Formulation II using (c) AR coefficients and (d) PSD estimates; and those based on Formulation III using (e) AR coefficients and (f) PSD estimates 141 For HPRS2, DIPs based on Formulation I using (a) AR coefficients and. .. average; MSPC = multivariate statistical process control; PCA = principal component analysis; PCC = principal component coefficient; PSD = power spectral density; xiv RC = reinforced concrete; RMS = root mean square; SNR = signal-to-noise ratio; SHM = structural health monitoring; SPC = statistical process control; UCL = upper control limit; VAR = vector autoregressive; and VARMA = vector autoregressive... coefficients of the fitted AR models 1.2.2 Statistical Techniques for Vibration Characteristics Based SHM Recently, with the development in sophisticated and readily available hardware for data acquisition and analysis, and, more importantly, with the capability of addressing 3 CHAPTER ONE the essentially involved uncertainty quantitatively, SHM using probabilistic and statistical techniques has been drawing... vibration response data and, in turn, the structural health condition, these sets of the multivariate data of the AR coefficients il will then be monitored by Hotelling’s T2 control charts 2.2.2 Monitoring of Vibration Response Data Characteristics Based on Multivariate Statistical Process Control Once the vibration response data are represented by sets of the AR coefficients i 1, 2, , p; and l 0, 1, il ,... state, and that in the reference state xxiv CHAPTER ONE INTRODUCTION 1.1 Background The challenges of improving structural safety have been stimulating continuous interest among civil engineering researchers to further the state of the art in the related design, construction, inspection and retrofitting techniques Particularly, the last several years have seen rapid research development in structural health . STATISTICAL STRUCTURAL HEALTH MONITORING: METHODOLOGIES AND APPLICATIONS WANG ZENGRONG NATIONAL UNIVERSITY OF SINGAPORE 2008 STATISTICAL STRUCTURAL HEALTH MONITORING: . FIVE A NONPARAMETRIC STATISTICAL FRAMEWORK FOR STRUCTURAL HEALTH MONITORING 116 5.1 Introduction 116 5.2 Formulation of the Nonparametric Statistical Structural Health Monitoring Framework. coefficients and Formulation III, and (b) PSD estimates and Formulation III; those for TDS2 based on (c) AR coefficients and Formulation II, and (d) PSD estimates and Formulation III; and those