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FIELD INVESTIGATION OF FUNDAMENTAL FREQUENCY OF BRIDGES USING AMBIENT VIBRATION MEASUREMENTS by Arden Reisham Pradeep Heerah Department of Civil Engineering and Applied Mechanics McGill University Montréal, Canada August 2009 A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of MASTER OF ENGINEERING. © Arden Reisham Pradeep Heerah, 2009. All Rights Reserved. ABSTRACT The transient nature of forces induced in structures during earthquakes requires the use of dynamic analyses to fully characterise their behaviour. A modal analysis describes the dynamic response of the structural system through modal descriptors: natural frequencies, mode shapes and damping ratios. Efficiently estimating these modal parameters for bridges allows for better structural integrity assessments and structural health monitoring of these structures. Using ambient vibration measurements to estimate modal parameters is time-saving and efficient. This research reviews the literature on the application of ambient vibration testing to the modal characterisation of bridges. Natural frequencies from ambient vibration measurements are obtained for a typical bridge in the City of Montréal in Canada. The MATLAB computing platform is used to execute spectral analyses of the field measurements. Linear, analytical models of the bridges are constructed with the SAP2000 structural analysis programme and Eigenvalue analyses are performed. The experimental and analytical results are compared and discussed, followed by recommendations for the application of this procedure to other bridges in the Montréal region. -i- RÉSUMÉ Du fait de la nature transitoire des forces se créant au sein des structures lors des tremblements de terre, la caractérisation complète de leur comportement nécessite l’utilisation d’analyses dynamiques. Une analyse modale décrit la réponse dynamique de la structure à travers ses modes de vibration : les fréquences naturelles, la forme des modes et le facteur d’amortissement. Une estimation efficace des paramètres modaux des ponts permet alors une meilleure évaluation, et donc un meilleur suivi, de leur intégrité structurale. L’utilisation des mesures de vibrations ambiantes pour estimer les paramètres modaux est donc une méthode efficace et économique. Cette étude passe en revue la littérature sur l’application de l’analyse des mesures de vibrations ambiantes pour la caractérisation modale des ponts. Les fréquences naturelles ont été obtenues sur un pont typique de la ville de Montréal. La plate-forme informatique MATLAB a été utilisée pour effectuer des analyses spectrales de ces mesures. Des modèles analytiques linéaires ont ensuite été construits avec le programme d’analyse structurel SAP2000 pour effectuer des analyses d’Eigenvalue. Les résultats expérimentaux et analytiques sont ensuite comparés et discutés, et des recommandations énoncées, pour appliquer ce procédé à d’autres ponts de la région de Montréal. - ii - ACKNOWLEDGEMENTS I extend sincere appreciation to the following, among others: My supervisor, Professor Luc E. Chouinard, for his invariable and multifaceted support and guidance throughout our research activities at McGill University and my stay in the City of Montreal. Salman Saeed, Dr. Myriam Belvaux, Dr. Alejandro de la Puente Altez, Dr. Philippe Rosset, Damien Gilles, Kuei-hua Rebecca Huang and Wendy Itagawa; their assistance throughout my research has been consequential. The Ministère des Transports du Québec, for assisting our research efforts. Lennartz electronic and LEAS électronique, manufacturers of the equipment used in this research, for their guidance and clarification. Fédon Honoré, for translating the abstract into French. Professors Saeed Mirza, Ghyslaine McClure and Colin Rogers, for their invaluable contributions toward my development throughout graduate studies. C.E.P. Ltd, to whom I am timelessly grateful. Peter and Michelle Santlal, for their friendship and limitless generosity. Staff of the Department of Civil Engineering and Applied Mechanics: Franca Della-Rovere, Anna Tzagournis, Sandy Shewchuk-Boyd, Dr. William Cook, Jorge Sayat and Ronald Sheppard, among others; and staff of the Schulich Library of Science and Engineering. My colleagues at McGill University, for their stimulating, earnest and sometimes inebriated discussions: Adriana Parada, Ali Ghafari, Hilary Ingram, Joe Mattar, John-Edward Franquet, Lai Wai Tan, Li Li, Ling Zhang, Miguel Nunes, Nabil Elias Saliba, Nicolas Desramaut, Nisreen Balh, Reza Erfani, Tatiana Tobar Valencia, Stephane Villemain, among others. I also convey gratitude to my friends and their families for their thoughts and involvement; especially to Andre Bagoo, Anuradha Gobin, Beena John, Binta Trotter, Charline Augustine, Dwayne Dubarry, Fadil Sahajad, Fédon Honoré, Navin Seebaran, Shalu Bujun, Suzette Parillon, Suzanne Seepersad, Vijay Mohan, Vince Ramlochan, Xue Zeng. - iii - ACKNOWLEDGEMENTS My mum, Gemma, and my brother, Arundel, to whom I am grateful for their unwavering support, patience and encouragement during my research efforts and for fostering the resolve within me to strive for more than only what seems possible. ”Be the change you want to see in the world.” - from Mahatma Gandhi “The art of directing the great sources of power in Nature for the use and convenience of man.” - from Institution of Civil Engineers’ Royal Charter of 1828. “Plagiarize, Let no one else’s work evade your eyes.” - from the song Lobachevsky by Tom Lehrer, Harvard Mathematics lecturer. “If we do not maintain our infrastructure, do not upgrade it, we’ll continue to have spectacular collapses.” - from Professor Saeed Mirza, McGill University. “One shall have to undergo suffering to reach truth. That is why it is said that truth is eternally victorious.” - from Rig Veda. - iv - TABLE OF CONTENTS Section Page ABSTRACT i RÉSUMÉ ii ACKNOWLEDGEMENTS iii LIST OF FIGURES viii LIST OF TABLES xii LIST OF APPENDICES xiii CHAPTER 1 - INTRODUCTION 1 1.1 Developments 3 1.2 Objectives 4 1.3 Organisation of the thesis 5 CHAPTER 2 - LITERATURE REVIEW 7 2.1 Dynamical behaviour of structures 7 2.2 Experimental modal analysis 12 2.2.1 Frequency response function (FRF) 14 2.2.2 Impulse response function (IRF) 15 2.2.3 Vibration tests 18 2.2.4 Modal testing of civil engineering structures 21 2.2.5 Ambient vibration testing of bridges 22 Modal parameter estimation 24 2.3.1 Fourier spectral analysis (FSA) 25 2.3.2 Power Spectral Density (PSD) and Cross-power 28 2.3 Spectral Density (CSD) Analysis 2.4 2.3.3 Auto-regressive moving-average (ARMA) method 30 Signal processing issues in AVT 32 2.4.1 Sampling 32 2.4.2 Averaging 33 -v- TABLE OF CONTENTS Section Page 2.4.3 Spectral leakage 33 2.4.4 Resolution 38 CHAPTER 3 - CASE STUDY 40 3.1 Bridge structure 40 3.2 Construction materials 47 3.3 Equipment 49 3.3.1 Testing 52 3.3.2 Test locations 55 3.3.3 Sampling parameters 55 3.3.4 Tests 57 3.3.5 Synchronisation 57 Analysis of experimental results 60 3.4.1 Fourier method 60 3.4.2 Welch’s PSD method 61 3.4.3 Welch’s CSD method 61 3.4.4 Auto-regressive, Modified Covariance method 62 Analysis of analytical models 62 3.5.1 Models 63 3.4 3.5 CHAPTER 4 - DISCUSSION AND RECOMMENDATIONS 69 4.1 Notes on the experimental analysis 69 4.2 Notes on the analytical analysis 79 4.3 Fundamental frequency estimates 80 4.4 Reviewing the analysis 83 4.5 Recommendations for ambient vibration testing of bridges 84 - vi - TABLE OF CONTENTS Section Page CHAPTER 5 - CONCLUSIONS AND FUTURE 88 RECOMMENDATIONS 5.1 Conclusions 88 5.2 Reviewing the analysis 90 5.3 Recommendations for ambient vibration testing of bridges 90 5.4 Proposal for future research 91 5.5 Fulfillment of objectives 93 REFERENCES APPENDICES - vii - LIST OF FIGURES Figure 1-1 Content Page Approximate distribution of seismic risk across Canada’s 2 urban population. (Adams et al., 2002) 1-2 Map of Canada showing delimitation of the eastern and 2 western seismic regions and the stable central region. (Adams and Halchuk, 2007) 2-1 Mechanical models of a multi-story building structure. 10 (Tedesco et al., 1999) 2-2 Mechanical model for a SDOF system. (Tedesco et al., 10 1999) 2-3 Coupling theoretically and experimentally derived 13 Definition of the unit impulse forcing function. (Maia and 17 dynamic modelling. (Maia and Silva, 1997) 2-4 Silva, 1997) 2-5 Definition of an arbitrary, non-periodic forcing function. 17 (Maia and Silva, 1997) 2-6 Devices used in FVT. (Cunha and Caetano, 2006) 19 2-7 Shaker devices used to excite. (Cunha and Caetano, 19 2006) 2-8 Random signals. (Ewins, 2000) 28 2-9 Description of Welch PSD analysis. (Trauth, 2007) 30 2-10 Interpretations of multi-sample averaging. (Ewins, 2000) 35 2-11 Finite-length sample and spectral leakage. (Ewins, 2000) 35 2-12 Window functions used frequently in spectral analysis. 36 (Stoica and Moses, 1997) 2-13 The effect of the window functions from Figure 2-15 in 37 the frequency domain. (Stoica and Moses, 1997) 2-14 Demonstration of how original time histories are treated with select windows. (Ewins, 2000) - viii - 38 LIST OF FIGURES Figure Content Page 3-1 Views of the monitored bridge. 41 3-2 Sketch views of the bridge deck framing system. 43 3-3 Sketch view of the bridge deck framing system: plan 44 view. 3-4 Sketch views of the moment-resisting, reinforced 44 Sketch sections through a typical column in the moment- 45 concrete frame. (De La Puente Altez, 2005) 3-5 resisting, reinforced concrete frame. (De La Puente Altez, 2005) 3-6 Sketch views of the girders’ connection to the top of the 46 moment-resisting, reinforced concrete frame.(De La Puente Altez, 2005) 3-7 Profile view (transverse direction) sketch of the beam 46 connected to the top of the abutment. (Ministry of Transport of Quebec) 3-8 Assumed stress-strain plots for the steel and concrete at 48 the time of construction of the bridge. (De La Puente Altez, 2005) 3-9 (a) Experimental equipment: seismometer and DAS. 49 3-9 (b) Experimental equipment: RCS antenna attached to DAS 50 and GPS sensor mounted on railing. 3-10 Extract from a data file indicating record details followed 50 by five rows of data points. 3-11 Typical measurement time history describing motion in each orthogonal direction (Record 6: File 27 from DAS 28 on November 30th 2007). - ix - 53 LIST OF FIGURES Figure 3-12 Content Page Testing configurations: (a) A - along the central 54 longitudinal axis; (b) B - along a transverse line; and (c) C - along one exterior longitudinal axis. 3-13 Synchronisation of a pair of measurements. 58 3-14 AM1: (a) solid view of the model; (b) schematic of the 67 structural model for AM1, Case 1; (c) schematic of the structural model for AM1, Case 2. 3-15 AM2: (a) solid view of the model; (b) schematic of the 68 structural model for AM2, Case 1; (c) schematic of the structural model for AM2, Case 2. 4-1 Typical response spectra from the four experimental 73 approaches using the ambient vibration measurements. 4-2 Frequency estimates and repeatability rates using the 75 FFT approach. 4-3 Frequency estimates and repeatability rates using the 76 Welch’s PSD approach. 4-4 Frequency estimates and repeatability rates using the 76 Welch’s CSD approach. 4-5 Frequency estimates and repeatability rates using the 77 MCOV approach. 4-6 Frequency estimates and repeatability rates in the 77 vertical direction (Z). 4-7 Frequency estimates and repeatability rates in the 78 transverse direction (Y). 4-8 Frequency estimates and repeatability rates in the longitudinal direction (X). -x- 78 LIST OF FIGURES Figure Content 4-9 Fundamental frequency estimates (frequency at first Page mode of vibration) for all experimental and analytical modal analyses. - xi - 82 LIST OF TABLES Table 3-1 Content Page Assumed material strengths for the analytical modal 48 analysis. 3-2 Specifications of the seismometers and data acquisition 51 stations used in the ambient vibration testing. 3-3 Sampling parameters used in testing. 56 3-4 Ambient conditions during testing. 58 3-5 Measurement catalogue. 59 3-6 Contributors to the reactive mass. 64 4-1 Summary of resonant frequency estimates and 74 repeatability rates by direction and processing method. 4-2 Summary of resonant frequency estimates for all four 75 algorithms in EMA and corresponding directions of occurrence. 4-3 Summary of natural frequency estimates for AM1. 81 4-4 Summary of natural frequency estimates for AM2. 81 - xii - LIST OF APPENDICES Appendix Content Page A Output spectra from FFT estimation. A -1 B Output spectra from Welch’s PSD estimation. B-1 C Output spectra from CSD estimation. C-1 D Output spectra from AR, estimation. - xiii - Modified Covariance D-1 Chapter 1 Introduction Modern society is highly dependant on its infrastructure. Safe and efficient transportation is required for economic activities as well as for emergency response services. Urban environments are highly complex with interconnected and interdependent transportation networks where bridges play a prominent role. Bridges are some of the most critical components of transportation infrastructure systems. For these structures, failure is defined as any interruption of pedestrian or vehicular traffic across or under it due to structural distress. Direct consequences of failure can range from injury to loss of life and property in the case of collapse, and indirect consequences such as disruptions to economic activities and reduced access to emergency facilities in the event of collapse or closure. One approach towards mitigating bridge failure is to ensure that it is properly designed to satisfy performance needs associated with traffic and extreme environmental loads, as well as to meet maintenance and durability objectives. It can be appreciated that evaluating the condition of existing bridges and assessing their vulnerabilities to extreme loads are essential steps towards mitigating bridge failures. The Montreal Urban Community (MUC), with its population of 3.5 million inhabitants is ranked second (behind Vancouver) among Canadian cities for seismic risk (Figure 1-1). Figure 1-2 shows that Montreal is located in a region of moderate seismic activity, an exposure which demonstrates a need to understand the health of the city’s infrastructure for the safety and convenience of its inhabitants. -1- 1 - Introduction Figure 1-1 Approximate distribution of seismic risk across Canada’s urban population. (Adams et al., 2002) Figure 1-2 Map of Canada showing delimitation of the eastern and western seismic regions and the stable central region. (Adams and Halchuk, 2007) -2- 1 - Introduction 1.1 Developments Structural vibrations are a major hazard and design limitation for civil engineering structures, including bridges (Ewins, 2000). Historically, there have been notable bridge failures that resulted in significant losses. Engineers have learnt from these experiences and refined their approaches to design, construction, monitoring and maintenance of structures (Lawson, 2005). In 1879, the Tay Bridge collapsed in Scotland from a combination of poor quality control, bad construction practices and inadequate design recommendations. A contributing factor appears to have been a dynamic system associated with wind gusts. Comparisons have been drawn between this failure and the 1940 failure of the Tacoma-Narrows Bridge; the latter bridge failed due to nonlinear mode coupling between wind forces and the flexibility and inertia of the bridge structure (Billah and Scanlan, 1991). The Hyatt Regency Skywalk in Kansas City, U.S.A. collapsed in response to vibrations from overcrowding of the skywalk in 1981. More recently, the Millennium Bridge in London, England was closed temporarily after resonance was induced by pedestrian traffic (Lawson, 2005). These few examples of bridge failures underscore the importance of determining the dynamic response of bridges to time-varying forces induced by traffic (pedestrian and vehicular) and wind and earthquake loads. The present work is part of a research project on seismic hazard analysis for the MUC. The objectives of the project are to perform seismic hazard analysis, assist in developing mitigation plans and improve emergency preparedness (Chouinard et al., 2004). Projects have already been completed on seismic microzonation (Chouinard and Rosset, 2007; Rosset et al., 2004, 2003; Madriz, 2004), seismic deficiencies of existing bridges (De La Puente Altez, 2005), seismic vulnerability of electric -3- 1 - Introduction distribution networks (Jaigirdar, 2005), seismic screening procedures for buildings and bridges (Lui, 2002), and ambient vibration analysis of buildings (Huang, 2007). 1.2 Objectives The transient nature of seismic forces induced in structures necessitates the use of dynamic analysis to characterise their response. Modal analysis defines the dynamic response of a structure through modal descriptors: natural frequencies, mode shapes and damping ratios. Efficiently formulating these modal parameters for bridges is essential for efficient and reliable structural assessments. Past work has shown experimental modal testing to be an efficient means of estimating these parameters. In particular, ambient vibration measurements provide accurate estimates of modal parameters that are used for performing structural assessments and structural health monitoring in a timely and cost-effective manner. The objective of this project is to use ambient vibration testing to characterise a typical overpass bridge in Montreal. The MATLAB computing platform is used to conduct spectral analyses on the field measurements. Linear, analytical structural models of the bridge are built with the analysis programme SAP2000 to perform eigenvalue analysis for the dynamic response. Experimental and analytical results are evaluated, compared and discussed. The main goals of this research are: 1. To review the literature on ambient vibration testing of bridges. 2. To conduct experimental and analytical modal analysis of a typical bridge in Montreal. 3. To compare and discuss the findings of the modal analyses. -4- 1 - Introduction 4. To provide guidelines for the ambient vibration testing of other bridges in Montreal. 1.3 Organisation of the Thesis Chapter 1 of the thesis presents the background and objectives of this research. Chapter 2 introduces fundamental concepts used in modal analysis and structural dynamics theory is reviewed to define modal parameters for system identification. The chapter also reviews experimental modal analysis using vibration testing for civil engineering structures, especially ambient vibration testing of bridges. This is followed by a brief description of modal analysis techniques in the frequency and time domains for civil engineering structures and bridges in particular. Emphasis is placed on the techniques utilised in this research. The chapter concludes with a review of data processing issues with ambient vibration testing. Chapter 3 describes the bridge structure selected for ambient vibration testing. The structural characteristics of the bridge pertinent to modal analysis are presented and details of the testing procedure are discussed. Subsequently, the spectral analysis techniques used on the ambient vibration records are presented. Finally, analytical models of the bridge that were created for comparative purposes are described. Chapter 4 presents a discussion of the findings obtained from the analyses that were presented in Chapter 3. Firstly, the results for the four procedures (FFT, Welch’s PSD, Welch’s CSD and AR-MCOV) used to analyse the ambient vibration measurements are outlined. This is followed by the presentation of the results for the analytical models of the bridge for different support conditions and material properties. The results from all of the analysis techniques are discussed. Finally, recommendations for the modal testing of bridges are proposed. -5- 1 - Introduction Chapter 5 presents the conclusions from the experimental and analytical modal analysis of the bridge. A review of the analysis is discussed briefly and a summary of recommendations for modal testing of bridges is provided. As well, future research directions in experimental modal analysis of bridges using ambient vibration measurements are outlined. -6- Chapter 2 Literature Review This chapter introduces fundamental concepts used in modal analysis and structural dynamics theory is reviewed to define modal parameters for system identification. The chapter also reviews experimental modal analysis using vibration testing for civil engineering structures, especially ambient vibration testing of bridges. This is followed by a brief description of modal analysis techniques in the frequency and time domains for civil engineering structures and bridges in particular. Emphasis is placed on the techniques utilised in this research. The chapter concludes with a review of data processing issues with ambient vibration testing. 2.1 Dynamical behaviour of structures The dynamic behaviour of a structure is described by the equation of motion for a system using the fundamental theory of vibrating systems. The main structural characteristics of a vibrating system are the mass, stiffness and damping properties of the structure (Tedesco et al., 1999; Craig and Kurdila, 2006). These properties determine the inertia, stiffness and damping forces of the system within the equations of motion that describe the dynamic response {x(t)} of the system (Maia and Silva, 1997). Inertia forces {Ft} are derived from the contributing mass of the system as the product of the mass {m} and the acceleration of the mass { &x& (t)} (Equation 2.1). Some systems can vibrate indefinitely even after the external excitation has ceased. However, almost all structures exhibit energy dissipation where the magnitude of the structural oscillation -7- 2 – Literature Review decreases as a function of time. This damping of the system is characterised by a damping coefficient {c} which relates the velocity of the mass { x& (t)} of the system and the damping force {FD} (Equation 2.2). There are several models for damping but the most commonly employed are viscous (dash-pot), Coulomb (friction) and Hysteresis (material) damping models (Maia and Silva, 1999). Elasticity of the system is characterised by the spring constant {k} that relates the elastic spring force {Fs} to the displacement of the mass { x(t) } (Equation 2.3). Ft = -m&x&(t) 2.1 FD = cx& (t) 2.2 Fs = kx(t) 2.3 The characteristic equation of motion is obtained from the direct application of Newton’s second law, in combination with the D’Alembert’s principle of virtual work or Hamilton’s direct integral approach (Craig and Kurdila, 2006; Clough and Penzien, 2003). The resulting equations account for all three internal forces and the excitation force {f(t)} (Equation 2.4). [m]{&x&(t)} + [c]{x& (t)} + [k]{x(t)} = {f(t)} 2.4 Dynamic analysis is performed by discretising the structure into a number of representative degrees-of-freedom (DOF). Figure 2-1 illustrates how a typical structure is idealised by a single-degree-of-freedom (SDOF) and three-degree-of-freedom (3-DOF) systems. The SDOF system is the simplest analytical model that defines the physical structure at a single spatial coordinate and is characterised by single mass, stiffness and damping properties. The equation of motion for the system can then be constructed easily from this analytical model (Equation 2.4). Figure 2-2 -8- 2 – Literature Review illustrates the analytical model for the SDOF system. The circular frequency of the single mode of vibration is described by {ω}, in terms of radians per second, and the eigenvalue is the square of this circular frequency. The cyclic frequency is defined by {f}, in Hertz, and the period of the vibration is defined by {T}, in seconds, through Equation 2.5. The dynamic response of the SDOF system is predicted from the solution of the equation of motion (Equation 2.4). f= ω 2π 2.5 a 1 f 2.5 b T= A similar approach is adopted in estimating the dynamic behaviour of multi-degree-of-freedom (MDOF) systems. The equations of motion for the 3-DOF model shown in Figure 2-1 (d) are as follows, m1&x&1 (t) + (k1 + k 2 )x1(t) − k 2 x 2 (t) = f1 (t) 2.6 a m 2 &x& 2 (t) - k 2 x1 (t) + (k 2 + k 3 )x 2 (t) − k 3 x 3 (t) = f 2 (t) 2.6 b m 3 &x& 3 (t) − k 3 x 2 (t) + k 3 x 3 (t) = f 3 (t) 2.6 c , where {f1(t)}, {f2(t)} and {f3(t)} are external forces that act on masses {m1}, {m2} and {m3}, respectively. For MDOF systems, the lowest frequency of vibration is the natural or fundamental frequency of vibration of the system. Most real structures are systems with distributed mass and stiffness properties, with an infinite number of spatial coordinates. By discretizing the continuous structure into a countable quantity of coordinates, a solution of the constituent equations of motion for the idealised representation can be attempted. Chapra (2005) and Hoffman (2001) -9- 2 – Literature Review demonstrated that standard eigenvalue and characteristic-value problems define vibrating systems. Figure 2-1 Mechanical models of a multi-story building structure. (a) Physical representation; (b) continuous (uniform distribution) model; (c) SDOF discrete model; (d) 3-DOF discrete model. (Tedesco et al., 1999) Figure 2-2 Mechanical model for a SDOF system. (Tedesco et al., 1999) Ewins (2000) noted that when the excitation function {f(t)} is considered in the complex form, +∞ f(t) = ∫ F(ω )e iωt dω −∞ - 10 - 2.7 2 – Literature Review , the system’s response function {x(t)} assumes the form, +∞ x(t) = ∫ X(ω )e iωt dω 2.8 −∞ , and the resulting eigenvalue problem can be defined for an undamped MDOF vibrating system by, {[K] − [ω 2 ][M]}{[x(t)]} = {[f(t)]} 2.9 The solution of these equations can be obtained with any of several methods outlined by Hoffman (2001) to generate estimates of eigenvalues and their matching eigenvectors. Eigenvectors {φ} yield the relative positioning of the coordinates (or mode shapes) and eigenvalues {ω} characterise the circular frequencies of the system (in radians per second). Estimates of modal damping may be derived by applying the Half-Power Bandwidth Method to response signals (Craig and Kurdila, 2006; Feng et al., 1998) and curve-fitting methods applied to crosscorrelation functions of response measurements, among other approaches (Ewins, 2000; Maia and Silva, 1997). Craig and Kurdila (2006) demonstrated how the dynamic response of vibrating systems can be described by a finite number of vibration modes. Natural frequency, mode shape and damping value are the three parameters that define each vibration mode; collectively they are called modal parameters. The fundamental mode of vibration corresponds to the lowest estimated frequency for MDOF systems. The second lowest frequency and its modal property estimates describe the second mode of vibration, and so on. For MDOF systems, the total number of vibration modes corresponds to the number of mass degrees-of-freedom comprising the system. - 11 - 2 – Literature Review Estimates of these parameters can be extracted from vibration surveys and the entire process is known as modal testing, vibration testing or system identification or dynamic characterisation. 2.2 Experimental modal analysis (EMA) EMA is used to estimate modal parameters of vibrating systems from data that has been measured on actual structures or full-scale models. This type of analysis has been performed in the industry since the early 1930’s and multiple surveys and analysis techniques have been developed to date. Several publications document modal testing and applications to civil engineering structures (Huang, 2007; Michel and Guéguen, 2007; Cunha and Caetano, 2006; Zivanovic et al., 2006; He et al., 2005; Ren et al., 2004; Dye, 2002; Ewins, 2000; Ivanovic et al., 2000; Huang et al., 1999; Muhammad, 1999; Feng et al., 1998; Maia and Silva, 1997; Xu et al., 1997; Brincker et al., 1996; Felber et al., 1995; Asmussen, 1994; Farrar et al., 1994; Rainer and Van Selst, 1976). In certain instances, the frequency of an oscillatory excitation will coincide with the natural frequency of the structure. This produces the phenomenon of resonance where there is significant amplification of vibration of the structure that can lead to damage (Tedesco, 1999). The fundamental idea behind modal testing is that natural frequencies of the structure are identified by resonant frequencies (Inman, 2006). Vibration measurements are recorded in the time domain and the analysis is performed in either the frequency or the time domain. Classical methods for system identification from EMA are based on frequency response functions (FRFs) in the frequency domain and impulse response functions (IRFs) in the time domain (He et al., 2005). Some of the objectives of modal testing are: 1. To determine mode frequencies of the structure; - 12 - 2 – Literature Review 2. To estimate mode shapes and damping information for the structure; 3. To correlate a finite-element (FE) or other theoretical model of the structure with measurements from the actual structure. 4. To formulate a dynamic model of the structure that can be used to investigate potential modifications to the structure; 5. To prepare a dynamic model that is suitable for updating a FE model of the structure and improve numerical results. 6. To construct a model to extrapolate the structural behaviour to extreme loads. 7. To add to the body of knowledge and the performance of the structure - this forms the basis for structural health monitoring. Maia and Silva (1997) present a succinct flow diagram that describes the components of numerical and experimental modal analyses and their relationship (Figure 2-3). Figure 2-3 Coupling theoretically and experimentally derived dynamic modelling. (Maia and Silva, 1997) - 13 - 2 – Literature Review 2.2.1 Frequency Response Function (FRF) Dynamic analysis in the frequency domain produces FRFs of the vibrating system. These model the amount of energy dissipation of the structure during vibration (He and Fu, 2001) and are commonly used to identify modal parameters of monitored structures. Ewins (2000) has detailed the various forms which FRF plots can take, and the theoretical basis for modal testing has been widely outlined in the literature. Fourier analysis is the basis for estimating system response in the frequency domain (Inman, 2006). It uses the Fourier Series to describe dynamic signals by representing periodic functions as a summation of harmonic functions. System vibrations are dynamic signals that are classified as either deterministic (periodic or transient) or random (stationary or non-stationary). While non-periodic (transient) functions cannot be properly addressed by the Fourier Series, they have been viewed with an infinite period (T = ∞) to help the analysis along (Maia and Silva, 1997). In other words, the Fourier Series approach is extended to a Fourier Transform as the case of a signal with an infinitely long period. This assumption allows signals to satisfy the Dirichlet condition (Equation 2.10) (Maia and Silva, 1997), +∞ ∫ x(t) dt < ∞ 2.10 −∞ , where the Fourier Transform {F(ω)} of the forcing function {f(t)} can be computed by, +∞ 1 f(t)e −iωt dt F(ω) = 2π −∫∞ - 14 - 2.11 2 – Literature Review The Fourier Transform of the response of the system {X(ω)} can then be determined from, X(ω ) = +∞ 1 x(t)e −iωt dt = H(ω )F(ω ) 2π −∫∞ 2.12 Subsequently, the dynamic response function of the system { x(t) } is derived from the Inverse Fourier Transform of {X(ω)}, and this computation includes the transfer function of the system {H(ω)}, +∞ x(t) = ∫ X(ω )e iωt dω 2.13 a −∞ +∞ +∞ −∞ −∞ x(t) = ∫ [H(ω )F(ω )]e iωt dω = ∫ X(f)e i(2πf)t df 2.13 b , which is the same as Equation 2.8. The Laplace Transform is an alternative method of solving for the frequency response of vibrating systems subjected to any type of forcing function. Inman (2001) and Maia and Silva (1997) describe its formulation and Farrar et al. (1994) employed it in modal testing. 2.2.2 Impulse Response Function (IRF) The convolution or Duhamel’s Method is the time domain alternative to Fourier analysis and is based on the formulation of the dynamic response of vibrating systems from simple (unit) impulses (Ewins, 2000). It has been shown in the literature that system response to an arbitrary, non-periodic forcing function can be described as a superposition (assuming linearity of the system) of responses to a series - 15 - 2 – Literature Review of impulses which collectively represent the forcing function. Consider the unit impulse or Dirac δ-function, +∞ ∫ δ (t − τ )dt = 1 2.14 −∞ The unit impulse function, when applied at time (t = τ), occurs for an infinitesimal period of time and has an infinite magnitude with the integral of {f(t)δ(t)} equal to unity (Figure 2-4). The corresponding IRF is denoted by, h(t − τ ) 2.15 When an arbitrary forcing function is decomposed into a series of impulses (Figure 2-5), the response of the system, at time {t}, to any one of the incremental impulses, at time {τ}, is described by, δx(t) = h(t − τ )f(τ )dτ 2.16 , and the total system response is described by integrating all incremental responses. This is the convolution or Duhamel’s integral, +∞ x(t) = ∫ h(t - τ ))f(τ )d(τ ) where h(t − τ ) = 0; t ≤ τ 2.17 −∞ , which identifies the system’s response {x(t)} as the convolution {*} of the IRF {h(t)} and the forcing function {f(t)} in the time domain, x(t) = h(t) ∗ f(t) - 16 - 2.18 2 – Literature Review , and which is the time domain formulation of the Fourier Transform of the response of the system {X(ω)} described in Equation 2.12 This demonstrates the relationship between the IRF and FRF of the system: FRF is the frequency domain representation of the IRF in the time domain. Figure 2-4 Definition of the unit impulse forcing function. (Maia and Silva, 1997) Figure 2-5 Definition of an arbitrary, non-periodic forcing function. (Maia and Silva, 1997) - 17 - 2 – Literature Review 2.2.3 Vibration tests Two classes of tests exist for conducting vibration surveys: forced vibration testing (FVT) and ambient vibration testing (AVT). The difference between them is in the nature of the excitation. In the case of FVT, an external force is imposed upon the structure. The main advantage of FVT is that precise input excitation may be controlled and measured (Huang, 2007). Common excitation sources of forced vibrations include impact hammers, mass vibrators, dynamic shakers and other impulse excitation devices (Figures 2-6, 2-7). A frequently used exciter is the eccentric mass vibrator which enables the application of sinusoidal forces with variable frequency and amplitude. Brownjohn et al. (2003) and Abdel Wahab and De Roeck (1998) describe in more detail the various types of excitation devices that are used. In some instances monitored and unmonitored sources in the form of driven test vehicles, explosives and reciprocating equipment have been used as inputs. This approach to modal testing requires force generation apparatus which is costly and difficult to transport, install and implement. In addition, He et al. (2005) note that a major difficulty with this approach is to provide controlled excitations that generate sufficient response levels. Notably, the operation is intrusive and disturbs normal activities in the building. Some testing is based on measurements of the free vibration response of the structure. To achieve free vibration response, the structure is first subjected to a predetermined displacement and then released into free response. Undoubtedly, applying this method in fullscale dynamic testing is challenging, costly and inconvenient. However, tests of this nature have been executed on civil engineering structures such as buildings, footbridges, highway bridges and offshore platforms (Lawson, 2005; Ewins, 2000; Farrar et al., 1994, Srinivasan et al., 1984; Rainer and Van Selst, 1976). - 18 - - 19 Shaker devices used to excite: (a) bridges, vertically; (b) electrohydraulic shaker from Arsenal Research; (c) dams, laterally (EMPA). (Cunha and Caetano, 2006) Devices used in FVT: (a) impulse hammer; (b) eccentric mass vibrator; (c) electrodynamic shaker; (d) impulse excitation device for bridges (K.U. Leuven). (Cunha and Caetano, 2006) Figure 2-7 Figure 2-6 2 – Literature Review 2 – Literature Review Ambient vibrations result from excitation due to wind, wave action, human activity (pedestrian and vehicular traffic, construction, etc) and micro-tremors. The key characteristic of AVT is that the excitation is unknown and unmeasured, and certain assumptions are required in order to perform modal analysis. The forcing function is assumed to be white noise which is a function containing all frequencies within the frequency spectrum; in other words the frequency domain representation of white noise is a function with constant power (Bensen et al., 2007; Bendat and Piersol, 1993). This assumption is true for random processes that are defined as ergodic, stationary random force signals with Gaussian distributions (He and Fu, 2001); that is, the average of values across a collection of time histories at a given instant in time is identical to the average of each record over time (Maia and Silva, 1997). Unfortunately, random processes inherently do not satisfy the Dirichlet condition (Equation 2.10). As a consequence, neither the excitation nor the response can be subjected to valid Fourier Transform calculations as in Equations 2.10 to 2.17. Furthermore, Bensen et al. (2007) advise that ambient noise is not truly flat in the frequency domain. In spite of this, it has been common practice to carry along the assumption of the forcing function being white noise and of white noise exciting all frequencies equally (Maia and Silva, 1997). Additionally, correlation functions and spectral densities have been introduced to describe ambient vibrations. They will be elaborated later. Since the amplitudes of vibration during AVT are small, they describe the linear behaviour of structures (He and Fu, 2001; Ivanovic et al., 2000). In fact, random excitations linearize the amount of energy dissipation of the structure during vibration, thereby modelling many structures that behave non-linearly through linearized frequency response functions (He and Fu, 2001). - 20 - 2 – Literature Review Since strict rules must be followed during the analysis, important assumptions are required for analysing ambient vibration measurements (Huang, 2007): 1. The forcing function is white noise excitation; 2. The response of the structure is a linear combination of inputs, due to system linearity. 3. The vibration measurements are ergodic and stationary random processes. Much research has been carried out on the use of forced vibration tests and ambient vibration tests, indicating that estimated modal parameters from either test have always proven to be consistent with each other and with corresponding analytical modal analyses (Crawford and Ward, 1964; Hudson, 1970; Trifunac, 1970; Felber et al., 1995a; Farrar and James, 1997; Huang et al., 1999; He et al., 2005). 2.2.4 Modal testing of civil engineering structures Modal testing evolved from mechanical engineering resonance tests. Cunha and Caetano (2006), Ivanovic et al. (2000), Ewins (2000) and Farrar and James (1994) presented extensive literature reviews of developments in modal testing of civil engineering structures and Carder (1936) identified early work by the U.S. Coast and Geodetic Survey dated in the early 1930’s. The research of Crawford and Ward (1964) advanced the use of spectral techniques for identifying the natural periods of buildings and was extended to the dynamic testing of other structures, including bridges (McLamore et al., 1971). Hudson (1970) subsequently provided detailed reviews on the dynamic testing of full-scale structures. Jennings et al. (1972) obtained estimates of frequencies, mode shapes and damping - 21 - 2 – Literature Review values for a twenty-two storey steel structure and Ellis and Jeary (1979) compared forced vibration and ambient vibration test results for a tower. Later studies by Srinivasan et al. (1981, 1984) examined several dynamic test results from nuclear power plants, buildings, dams, bridges, towers and tall chimneys as well as offshore platforms. An increasing number of structures have been instrumented in recent years, largely due to a increased awareness of dynamic issues with structures, better accessibility to equipment and greater computing capabilities (Ewins, 2000). Recent studies have been performed by Carne et al. (1988) on a turbine; by Brownjohn et al. (1992), Ventura et al. (1994) and Ivanovic and Trifunac (1995) on full-scale buildings; by Feng et al. (1998) on a tower; by Daniel and Taylor (1999) on dams; and by Wahab and DeRoeck (1998), Brownjohn et al. (1999), He et al. (2005) and Nielson et al. (2007) on bridges. 2.2.5 Ambient vibration testing of bridges Varney and Galambos (1966) summarised a series of dynamic tests performed on highway bridges in the USA between 1948 and 1965. Farrar and James (1997) reported that one of the earliest applications of modal testing to bridges using ambient vibrations was performed by McLamore et al. (1971) using an extension of spectral techniques developed by Crawford and Ward (1964). Iwasaki et al. (1972) presented tests performed in Japan while Ganga Rao (1977) conducted field and lab tests on bridge systems. Rainer and Van Selst (1976) characterized the dynamic properties of the Lions’ Gate Suspension Bridge in Canada using ambient vibration measurements. Douglas et al. (1981) use the approach to monitor a reinforced concrete bridge and a composite girder bridge while Pardoen et al. (1981) tested a steel truss bridge in New Zealand. - 22 - 2 – Literature Review Tanaka and Davenport (1983) investigated ambient vibration measurements from wind excitation for a suspension bridge in the USA. Cantieni (1984) reported on the testing of slab and girder bridges in Switzerland between 1958 and 1981. Brownjohn et al. (1987, 1989) characterized the dynamic properties of suspension bridges from ambient vibration measurements due to traffic and wind excitation. Ventura et al. (1994) estimated dynamic properties of a medium span, composite girder bridge in Canada. In 1995, Felber et al. (1995a, 1995b) presented findings of ambient vibration studies carried out on a medium span, composite steel box girder bridge in Switzerland and on a long span, steel truss bridge in Canada. Brinker et al. (1996) presented techniques for extracting modal parameters for bridges from ambient vibrations. Huang et al. (1999) studied typical multi-span, highway bridges in Taiwan and Arup tested the Millenium Bridge in UK in 2000 (Dallard et al., 2001). Ivanonvic et al. (2000) listed a number of publications addressing ambient vibration testing of bridges since the 1970’s. Cunha et al. (2001) presented findings of modal testing of a cable-stayed bridge in Portugal. Ren et al. (2004) tested a continuous steel girder bridge while He et al. (2005) presented the findings of an ambient vibration survey of a suspension bridge in the USA. Zivanovic et al. (2006) monitored a Montenegrin footbridge using ambient vibrations and Nielson et al. (2007) presented findings for typical multi-span, concrete girder highway bridges in the USA. Much of the modal testing research employed ambient vibration measurements because they were easily obtainable and did not disrupt user access to the structure. In fact, AVT was routinely used to validate the results of FVT and numerical models. In most cases, the results of the experimental modal tests were consistent and differences with numerical models were attributed to the incompleteness of structural models (Zivanovic et al., 2007; Farrar and James, 1997; Rainer and Van Selst, 1976). - 23 - 2 – Literature Review Farrar and James (1994) reviewed modal testing of a number of bridges that were investigated using ambient vibration measurements. The more significant details of each test (such as the excitation source, analytical techniques employed to extract modal properties and the modal properties that were estimated from the test) are summarised. For 3-span bridges (with 2 to 4 lanes), spans between 10 and 95 metres, internal supports of 5 to 30 metres above foundation level, and with a deck skew of less that 30 degrees and comprising a composite deck of steel reinforced concrete fastened to steel girders, the experimentally estimated fundamental frequency can range from 0.5 and 2.5 Hertz. Similar bridges with shorter internal supports and less slender decks have a larger fundamental frequency of vibration (Maia and Silva, 1997). The bridge tested in this project is not classified as slender and has a skew of 10 degrees. 2.3 Modal parameter estimation The dynamical properties of a vibrating structure are its natural frequencies, mode shapes and damping values. Every structure vibrates according to its modes of vibration. Each mode is defined by unique modal properties which collectively describe the structure’s dynamic response. Since the response of vibrating systems can be explained in the frequency domain as well as the time domain, it is not surprising to find that techniques have been developed in both of these areas. In the frequency domain, methods are based on an extension of Fourier analysis while time domain methods are based on the theory of correlation and auto-regressive, moving average and transfer-functions for parametric models. Some of the commonly used methods include the Fourier spectral analysis (He et al, 2005), power spectral density analysis, cross-power spectral density analysis (Trauth, 2007) and the curve-fitting method (Maia - 24 - 2 – Literature Review and Silva, 1997). Other extraction methods include the auto-regressive moving-average approach (He and Fu, 2001), the Ibrahim Time Domain approach (Maia and Silva, 1997), the Random Decrement technique (Feng et al., 1998), the Eigensystem Realisation Algorithm (ERA) (He et al., 2005) and the Complex Exponential methods (Ewins, 2000). These are by no means the full extent of techniques used for modal parameter estimation. Ewins (2000) and Maia and Silva (1997) conducted reviews of techniques for modal parameter estimation from vibration measurements and further subdivided established techniques according to the direct (based on spatial models) and indirect (based on modal models) nature of the processes and also on the number of modes (number of DOFs) treatable by each process. More recent work has led to the development of subspace methods of system identification such as the Stochastic Subspace Identification (SSI) technique (Watkins, 2007). Four techniques were employed to process the ambient vibration measurements in this research. They were chosen because they have been used extensively in modal testing and are user-friendly. They are discussed here in detail. 2.3.1 Fourier Spectral Analysis (FSA) FSA is one of several SDOF approaches in which a MDOF system is analysed at resonant frequencies, one by one. The basis of the method is peak-picking from FRFs in terms of Fourier amplitude; the natural frequencies are taken at peak amplitudes (Maia and Silva, 1997), the damping ratios are calculated from several methods including the sharpness of the peaks using the half-power method (Craig and Kurdila, 2006) and the mode shapes are estimated from the ratios of peak amplitudes on FRFs at various positions across the structure to corresponding peaks on the FRF of a reference position (He et al., 2005). - 25 - 2 – Literature Review The details of FSA are covered extensively in the literature, however, its important features are summarised here (Bendat and Piersol, 2000). It has been demonstrated that any periodic function {x(t)}, with a period {T}, can always be decomposed into an infinite series of sinusoids. The Fourier Series assumes the sum of two infinite cosine and sine series, ∞ 1 x(t) = a 0 + ∑ (a n cosωn t + b n sinωn t ) 2 n =1 where ω n = 2πn T 2.19 , with the Fourier coefficients {an} and {bn} related to the function {x(t)}, and where {n} is the order of cycles, T an = 2 x(t)cosωn tdt T ∫0 bn = 2 x(t)sinωn tdt T ∫0 2.20 a T 2.20 b T 2 a 0 = ∫ x(t)dt T0 2.20 c In the instance where a non-periodic function {x(t)} satisfies the condition of Equation 2.10, the response function can be written as, +∞ x(t) = ∫ [A(ω )cosωt + B(ω )sinωt ]dω 2.21 −∞ , where the coefficients are defined by, A(ω) = 1 π +∞ ∫ x(t)cosω tdt n −∞ - 26 - 2.22 a 2 – Literature Review B(ω) = 1 π +∞ ∫ x(t)sinω tdt n 2.22 b −∞ , and alternative forms of the function are expressed as Equations 2.12 and 2.13. Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) In practice, the response function {x(t)} is recorded for a finite time duration at {N} discrete points that are evenly spaced by the sampling scheme and digitised by the analogue-to-digital conversion process. Assuming the record is periodic about the length of the sample, the Fourier Transform can be estimated as a finite series with discrete points at (t = tk) and (k = 1, N) (Ewins, 2000), 1 x(t k ) ≡ (x k ) = a0 + 2 N (N −1) or 2 2 ∑ N =1 2πnk ⎞ 2πnk ⎛ + b n sin ⎟ ⎜ a n cos N ⎠ N ⎝ 2.23 , and the coefficients are defined by, an = 1 N 2πnk x k cos ∑ N k =1 N 2.24 a bn = 1 N 2πnk x k sin ∑ N k =1 N 2.24 b 2 N ∑xk N k =1 2.24 c a0 = Cooley and Tukey (Pollock, 1999) proposed an optimisation of the DFT known as the Fast Fourier Transform (FFT) where the method requires N to be an integral power of two (2) thereby reducing the - 27 - 2 – Literature Review execution time to compute the DFT of the response time history. The modal parameters of the system are then estimated from Fourier spectra generated from these relationships. 2.3.2 Power Spectral Density (PSD) and Cross-power Spectral Density (CSD) Analysis When the dynamic response function {x(t)} is now taken as a random vibration, the statistical expectation or average value of the product {x(t).x(t+τ)} yields the autocorrelation function {Rxx(τ)}, R xx = E[x(t).x(t + τ )] 2.25 , that describes how the function {x(t)} at an arbitrary time {(t+τ)} is dependent upon the function at a previous time {t}. Here, the original time history is transformed into a new function of time. Since this function obeys the Dirichlet condition (Equation 2.10) a frequency description of the original signal {f(t)} is estimated from the Fourier Transformation of {Rxx(τ)} as the power spectral density (PSD) or auto-spectral density (ASD) {Sxx(ω)}, S xx (ω ) = +∞ 1 R xx (τ )e −iωt dτ ∫ 2π −∞ 2.26 Figure 2-8 illustrates the time history, auto-correlation function and PSD of a random signal. The same logic, when applied to a pair of simultaneous functions y(t) and x(t) will generate cross-correlation {Rxy(τ)} and cross-spectral density (CSD) {Sxy(ω)} functions. The cross-correlation function is defined as, - 28 - 2 – Literature Review R xy = E[x(t).y(t + τ )] 2.27 , and the CSD is defined by its Fourier Transform as, S xy (ω ) = +∞ 1 R xy (τ )e −iωt dτ ∫ 2π −∞ 2.28 Developed in the 1960’s, the Welch’s approach to PSD analysis divides the time history into overlapping segments, computes the power spectrum for each segment and generates an average power spectrum (Trauth, 2007). Figure 2-9 demonstrates the principle of the Welch PSD analysis which has proven useful by increasing the signal-to-noise ratio of power spectra despite reducing frequency resolution. Rxx(τ) Sxx(ω) Figure 2-8 Random signals: (a) time history x(t); (b) auto-correlation function Rxx(τ); and (c) PSD Sxx(ω). (Ewins, 2000) - 29 - 2 – Literature Review Figure 2-9 Description of Welch PSD analysis. (Trauth, 2007) 2.3.3 Auto-regressive moving-average (ARMA) method An ARMA model is a combination of an auto-regressive (AR) model on one part and a moving-average (MA) model on another. The basis for the operation of each of these is the assumption that a dynamic response function {x(t)} is the random single output of a linear system which is driven by a random single input. This parametric approach is commonly applied to systems where only the output is measured and the unknown input is taken as white noise (Pollock, 1999; Maia and Silva, 1997). - 30 - 2 – Literature Review The AR component expresses the response function as a linear function of its past values where the order of the AR model {n} indicates the number of lagged points that are included. It is defined through the function, n x t - ∑ φ r x t −r = e t 2.29 r =1 , where {xt} is the value of the function at time {t}, {φr} is the AR coefficient, {xt-r} is the value of the function at {r} past points and {et} is the error residual from the prediction process. The MA component expresses the function as an unevenly weighted, moving average of the {et} series, demonstrated by, m x t = e t - ∑θ s e t − s 2.30 s =1 , where {xt} is the value of the function at time {t}, {θs} is the MA coefficient, {et-s} is the value of the function at {s} past points and {et} is the error residual from the prediction process. Similarly, the order of the model {m} is indicative of the number of lagged points that are included in the estimation. By combining the features of AR and MA modelling, the ARMA method executes a system identification algorithm through the function, n m r =1 s =1 x t - ∑ φ r x t −r = e t - ∑θ s e t − s where n > m 2.31 The idea of the method is to identify the system and predict both present and future responses from past inputs and outputs (He and Fu, 2001; Pollock, 1999). Furthermore, by increasing the order of either - 31 - 2 – Literature Review component of the model (‘n’ or ‘m’), the computational demand of the model will grow and the accuracy of the estimation will improve. 2.4 Signal processing issues in AVT Signal processing forms an integral part of the analysis of ambient vibration measurements. Therefore, it is important to appreciate and address some pertinent concerns. The key signal processing issues relate to the data acquisition process as well as to the analysis methods. Some of these issues regard sampling, averaging, spectral leakage, and resolution. 2.4.1 Sampling An alias is an error resulting from the processing of time signals. One such error is commonly introduced into analyses by improper sampling of the data. This aliasing can be reduced substantially by adhering to Shannon’s sampling theorem where the sampling rate (SR) of the data must be at least twice the maximum frequency of interest (fmax) in the signal (Equation 2.32). This ensures that the sampling interval (Δt) (Equation 2.33) is small enough to obtain at least two sample points in each cycle of the largest frequency of interest. Inman (2006) noted this alias results from misinterpretation of the analogue signal by digital recorders during conversion. Otnes and Enochson (1972) further indicated that at least 2.5 samples per cycle was a better choice. In addition, antialiasing filtering is employed to remove components of the analogue signal with a higher frequency than can be properly resolved by recording devices. Low pass filters cut off frequencies that are larger than roughly half of the maximum frequency of interest; this is the Nyquist frequency - 32 - 2 – Literature Review (fNYQ) (Equation 2.34) (Inman, 2006). The Nyquist frequency and the maximum frequency of interest influence the selection of the sampling rate. SR ≥ 2f max 2.32 SR = 1 Δt 2.33 f NYQ ≈ f max 2 2.34 2.4.2 Averaging Since Fourier Transforms do not strictly exist for random processes, the frequency content of time signals is estimated using power spectral densities and correlation functions. The frequency content is estimated from the Fourier Transforms of the signals introducing spurious results. Ewins (2000) described the sequential and overlap averaging techniques for sampling data from within the time signal to minimise such artefacts in the results (Figure 2-10). Overlap averaging was employed in the Welch’s approach to spectral analysis. 2.4.3 Spectral leakage Spectral plots exhibit the phenomenon of energy leakage at frequencies immediately below and above true resonant frequencies. It is the result of the assumption of periodicity of the finite time history measurement which allows the relative amplitude of spectral side lobes to be erroneously large. Figure 2-11 presents two time histories where the only difference between them is the finite length of the measurements. To - 33 - 2 – Literature Review process the time domain measurements into the frequency domain, both signals were assumed to be cyclic with a period (T) equal to their lengths. In the first scenario (a), the start and end of the measurement allow uninterrupted continuity between consecutive cycles; consequently, the spectral plot correctly indicates a single line at the frequency of the signal. However, a marked discontinuity between consecutive cycles in the second instance (b) results in significant energy leakage in the spectrum; power is leaked around the true frequency of the signal. Short of attempting the almost impossible task of adjusting the recorded signal length to reflect the period(s) within the random signal, the impact of leakage is more ordinarily diminished with a windowing approach. Here, one of the many constructs of windows is imposed upon the time signal prior to its transformation to the frequency domain. The finite time signal is strategically scaled between factors zero and one (0 – 1.0) to promote continuity between consecutive segments of the signal stemming from the assumption of signal periodicity. Windowing is important to the generation of an average of segmented signals, especially when the segments overlap. The Rectangular window is the simplest type. It is a function with a constant magnitude of one (1.0); therefore the subjected signal or segment is wholly repeated. When the segments of the signal overlap, the repetition of portions of the signal can be depreciated by using windows that attenuate the extremities and maintain the interior values of the segments. The Hanning, Hamming and Blackman windows produce such an effect on signals and are used frequently in signal processing. The Rectangular window is inappropriate for this task. Figures 2-12, 2-13 and 2-14 illustrate some of the types of windows used commonly to alleviate the influence of spectral leakage and the result of using each of them. Averaging the segments of a signal reduces the level of noise that can contaminate the resulting spectra, allowing the interesting features of the signal to be identified (Smith, 1997). - 34 - 2 – Literature Review Figure 2-10 Interpretations of multi-sample averaging: (a) sequential and (b) overlap. (Ewins, 2000) Figure 2-11 Finite-length sample and spectral leakage. (Ewins, 2000) - 35 - 2 – Literature Review Figure 2-12 Window functions used frequently in spectral analysis. (Stoica and Moses, 1997) - 36 - 2 – Literature Review Figure 2-13 The effect of the window functions from Figure 2-12 in the frequency domain. The windows allow differing levels of spectral leakage from processed signals. (Stoica and Moses, 1997) - 37 - 2 – Literature Review Figure 2-14 Demonstration of how original time histories (left and right) are treated with select windows (centre). Window types are (a) Boxcar (Rectangular); (b) Hanning; (c) Cosine-taper, and (d) Exponential. (Ewins, 2000) 2.4.4 Resolution Processing signals through the DFT has been found to yield spectra with inadequate frequency resolution or spectral resolution. The problem originates from a combination of the limited number of discrete points, size of the frequency range of interest and/or length of the time sample required to generate acceptable results. Ewins (2000) indicated two typical means of addressing this problem: controlling the size of the transform and padding the signal with zeros. The size of the transform (number of data points in the transform) can be increased to abate the resolution problem, especially with modern computational advancements and the efficient FFT by Cooley and Tukey (Trauth, 2007, Pollock, 1999). However longer transforms provide increased resolution properties while little is done to remove noise levels - 38 - 2 – Literature Review in the spectra. A solution is to impose a window on many segments of the original sample with a smaller size of transform. Alternatively, a larger transform may be used with a low pass filter, which cuts off higher frequencies from the signal (Ewins, 2000). Smith (1997) indicated these two solutions provide similar spectral resolutions though the latter is more computationally intensive. Advancements in computing capabilities diminish the significance of this constraint. Ewins (2000) suggested increasing the length of the measurement and increasing the sampling rate to improve spectral resolution. This may not be pragmatic; however, a simple way to increase the spectral resolution is by padding the measurement with zeros. Here, the number of data points describing the record is increased to the size of the transform by appending zero-valued coordinates. The increased number of points in the signal facilitates spectra with a finer resolution. - 39 - Chapter 3 Case Study This chapter describes the bridge structure selected for ambient vibration testing. The structural characteristics of the bridge pertinent to modal analysis are presented and details of the testing procedure are discussed. Subsequently, the spectral analysis techniques used on the ambient vibration records are presented. Finally, analytical models of the bridge that were created for comparative purposes are described. 3.1 Bridge structure The bridge is an above-grade crossing of St. Jean Boulevard over Donegani Avenue and the railway lines from the Canadian National (CN) and Canadian Pacific (CP) railway companies in the city of Pointe-Claire (Quebec). Figure 3-1 shows the location and a picture of the bridge. The 2006 edition of the Canadian Highway Bridge Design Code (CHBDC 2006) (CAN/CSA-S6, 2006) classifies this structure as an emergency-route bridge which carries and crosses over routes that need to be open, at minimum, to emergency and security/defence services immediately following the design earthquake. The CHBDC 2006 recommends the use of a design earthquake with a 10% probability of exceedance in 50 years (equivalent to 15% probability of exceedance in 75 years, with a return period of 475 years). The bridge has a total length of 68.89 m (226 ft) with three spans (21.34 - 26.21 - 21.34 m). The deck is 25.3 m wide and accommodates traffic flow in two directions (northbound and southbound) which are separated by a central median. The pavement width is 11 metres in each - 40 - 3 – Case Study ni ega Don e u n Ave N i gan one eD u n Ave CP and CN y lines a w l rai Monitored bridge le Bou enir) Souv u d ( 20 route Auto vard nir) ouve (du S 0 2 route Auto ea St-J A-N n Assumed field North (a) (b) Figure 3-1 Views of the monitored bridge: (a) plan view sketch map indicating the location of the bridge (Not drawn to scale); (b) western elevation of the bridge. (De La Puente Altez, 2005) - 41 - 3 – Case Study direction consisting of three 3.65 m lanes and no emergency shoulder. Figure 3-1(b) shows a full view of the three spans of the bridge with plan and elevation details indicated in Figures 3-2 and 3-3. The composite bridge deck system is a reinforced concrete deck (190 mm thick) underpinned by twelve structural steel girders (WF846x210 mounted with 20mm-φ-shear stud connectors) aligned along the longitudinal direction. The deck is supported at four locations – two internal moment-resisting, reinforced concrete frames and an abutment at either end. Each of the identical moment-resisting frames (Figure 3-4) comprises twelve 0.4572 m-square columns (Figure 3-5) that stand 4.88 m tall on a 3.66 m high reinforced concrete wall having cross-sectional dimensions of 0.762 m and 25.654 m. The moment-resisting frames are completed by a 0.4572 m-square reinforced concrete beam which connects all twelve columns at the top. The steel girders were spliced within the internal span - to ensure structural continuity along the entire length of the bridge - and framed by a steel joist (WF457x75) above each support location (Figure 3-3). Within each bay are three lines of transversely aligned channel sections (C381x51) that help to maintain the girders’ positions. Along with shear stud connectors atop each girder, these transverse brace elements increase girder resistance to lateral buckling through reduced effective span. They also contribute toward the transverse, in-plane stiffness of the deck system. - 42 - 3 – Case Study i 0.190 m ii 0.8458 m viii ix iii ix 0.4572 m vii iv 4.8768 m v 3.6576 m vi 3.6576 m 21.340 m 26.210 m 21.340 m (a) i 0.190 m ii 0.8458 m ix viii x iii vii iv 4.8768 m v 3.6576 m vi 3.6576 m 21.340 m 26.210 m 21.340 m (b) ID Structural Details 0.4572 m ID Structural Details i R.C. Slab (190 mm thick) vi R.C. Foundation (mass strip) ii Steel Girder (WF 845 x 210) vii Moment splice (for continuity) iii R.C. Cap beam (457 x 457 mm) viii Hinge support (atop column) iv R.C. Column (457 x 457 mm) ix Roller support (atop abutment) v R.C. Wall (762 x 25,654 mm) x Hinge support (atop abutment) Figure 3-2 Sketch views of the bridge deck framing system: (a) profile view: Case 1; (b) profile view: Case 2. (Not drawn to scale) - 43 - 3 – Case Study 25.300 m 21.340 m 26.210 m 21.340 m WF845x210 structural steel beam C381x51 structural steel channel WF457x75 structural steel beam (a) Figure 3-3 Sketch view of the bridge deck framing system: plan view. (Not drawn to scale) Centre line (a) (b) Figure 3-4 Sketch views of the moment-resisting, reinforced concrete frame: (a) sectional view; (b) frontal view. (Not drawn to scale) (De La Puente Altez, 2005) - 44 - 3 – Case Study 457.2 #8 Ø 25.4 mm #10 Ø 32.3 mm 457.2 #10 Ø 32.3 mm #8 Ø 25.4 mm #4 Ø 12.7 mm #8 Ø 25.4 mm (a) 10 Ø 32.3 mm #10 Ø 32.3 mm 54 114.3 120.7 114.3 54 #8 Ø 25.4 mm 304.8 Stirrups #4 Ø 12.7 mm 304.8 (b) Figure 3-5 Sketch sections through a typical column in the moment-resisting, reinforced concrete frame: (a) plan cut-section; (b) profile cut-section. (Not drawn to scale) (De La Puente Altez, 2005) The connection of the girders to the top of each moment resisting frame is detailed to preclude separation of and moment transfer between the two elements. A 25 mm-wide expansion joint exists at each end of the bridge between the bridge deck system and the abutments. They accommodate the thermal-induced, longitudinal, volumetric changes of the structure. The connection of the girders to the abutments is also detailed to preclude separation of and moment transfer between the two. Details of - 45 - 3 – Case Study the connection between the bridge deck system to the moment-resisting frame is shown in Figure 3-6 and to the abutment in Figure 3-7. (a) (b) Figure 3-6 Sketch views of the girders’ connection to the top of the moment-resisting, reinforced concrete frame: (a) profile view (transverse direction); (b) frontal view (longitudinal direction). (Not drawn to scale) (De La Puente Altez, 2005) road level 2 layers of asphalt paper or the equivalent beam support device abutment Figure 3-7 Profile view (transverse direction) sketch of the beam connected to the top of the abutment. (Ministry of Transport of Quebec) - 46 - 3 – Case Study 3.2 Construction materials The 2006 edition of the CHBDC 2006 (CAN/CSA-S6, 2006) suggests several means by which the properties of the construction materials comprising an existing bridge can be determined. Two of these methods recommend estimating the material properties from reviewing asbuilt construction plans and approximating material usage from the date of construction. The as-built construction drawings that were provided by the Ministry of Transport of Quebec (MTQ) were dated 1961 and 1962. From 1955 to 1965, the minimum yield strength (fy) of structural steel ranged from 250 to 350 MPa with an estimated Modulus of Elasticity (Es) of 2.0 x 105 MPa (Canadian Institute of Steel Construction, 2004). The 28-day compressive strength of concrete cylinders (fc’) varied from 20 to 25 MPa (De La Puente Altez, 2005) during the same period. Figure 3-8 illustrates the corresponding design stress-strain plots of the main materials at the date of construction (De La Puente Altez, 2005). The compressive strength of concrete generally increases with long-term hydration of cement followed by some retrogression. Brinkerhoff (1993) noted that much concrete construction from the 1960’s, attain compressive strengths exceeding 35 MPa (5000 psi). Neville (1996) found American Portland cements made at the start of the 20th century led to an increase in the strength of concrete stored outdoors where the 50-year strength was generally 2.4 times the 28-day strength. Alternately, cements made since the 1930’s attained peak strengths between 10 and 25 years and then exhibited some retrogression (Neville, 1996). Furthermore, German Portland cements made in 1941 led to 30-year strengths of 2.3 times the 28-day strengths. Design concrete compressive strengths of 25 MPa (3500 psi), 35 MPa (5000 psi) and 45 MPa (6500 psi) were used in consideration of these variations in the unknown, present compressive strength. - 47 - 3 – Case Study Steel Reinforcement Material Information 500 Concrete Material Information 30 Stress [MPa] Stress [MPa] 450 25 400 350 20 300 Yield Stress: 300 MPa Initial Elastic Modulus: 200000 MPa Strain at strain hardening: 7.0 mm/m Strain at maximum stress: 100.0 mm/m Ultimate Stress: 450 MPa 250 200 Cylinder Strength: 25.0 MPa Tensile Strength (auto): 1.63 MPa Cracking Strain: 0.069 mm/m Strain at peak stress (auto): 1.90 mm/m Initial tangent Stiffness: 23499.9 MPa Tension stiffening factor: 0.00 15 10 150 100 5 50 Strain [mm/m] Strain [mm/m] 0 0 0 20 40 60 80 100 120 0 0.5 (a) Steel 1 1.5 2 2.5 3 (b) Concrete Figure 3-8 Assumed stress-strain plots for the steel and concrete at the time of construction of the bridge. (De La Puente Altez, 2005) Table 3-1 indicates the material strengths that were assumed for the analytical modal analysis. Assumed material strengths Material Structural Steel Concrete Property Value (MPa) minimum yield stress (fy) 300 minimum ultimate stress (fu) 450 modulus of elasticity (Es) 2.0 x 105 minimum 28-day, compressive cylinder 25, 35, 45 strength (fc’) modulus of elasticity (Ec) 25.0 x 103, 29.6 x 103, 33.5 x 103 Table 3-1 Assumed material strengths for the analytical modal analysis. - 48 - 3 – Case Study 3.3 Equipment Field investigations were conducted using two velocimeter seismometers manufactured by Lennartz Electronic (Figure 3-9). Each seismometer simultaneously detects motion of the structure in three orthogonal directions - two perpendicular horizontal axes and one vertical axis. Coaxial cables link each sensor to a Cityshark II data acquisition station (DAS) with conditioning and analog-to-digital convertors. The measurements are stored in ASCII format on 32 MB flashcards. Figure 310 shows an extract from one of the measurement files. The operational range of parameters and other specifications of the geophones and recorders are provided in Table 3-2. Global positioning system (GPS) capability is used to synchronise the internal clocks of each battery-powered DAS. Several analysis methods required the use of simultaneous records. A remote control system (RCS) and wireless communications (walkie-talkie) devices were used to obtain simultaneous recordings. Details of the sampling parameters of each record are presented in the following section addressing the testing procedure. (a) Figure 3-9 (a) Experimental equipment: seismometer (left) and DAS (right). - 49 - 3 – Case Study (b) Figure 3-9 (b) Experimental equipment: (b) RCS antenna attached to DAS and GPS sensor mounted on railing. O riginal file nam e: 09110310.023 O riginaled fileinto: nam070911_0310.023 e: 09110310.023 T ransform T ransform ed into: R eadC ity version: 3.2070911_0310.023 R eadCserial ity version: S tation num ber:3.2 027 Station serial ber: 027 S tation softw arenum version: 0623 Stationnum softw are3 version: 0623 C hannel ber: C hannel num ber: 3 S tarting date: 11.09.2007 Starting 11.09.2007 S tarting timdate: e: 03:10:03.223 Starting tim11.09.2007 e: 03:10:03.223 E nding date: E nding 11.09.2007 E nding timdate: e: 03:10:03.223 E nding tim100 e: 03:10:03.223 S am ple rate: Hz Sam rate: Hz S am pleple num ber:100 90000 Sam ple num ber: 90000 R ecording duration: 15 m n R ecordingfactor: duration: 15 m n C onversion 52428.6 C onversion factor: 52428.6 G ain: 16 G ain:ic16range: 5 V D ynam D ynamsam ic range: 5V C lipped ples: 0.00% C lipped: sam ples: 0.00% L atitude 45 26.632 N L atitude : 734548.904 26.632WN L ongitude: L ongitude: A ltitude : 66 m73 48.904 W : 663 m N o.A ltitude satellites: N o. satellites: 3 M axim um am plitude: 0/1 M axim um am plitude: -689 -171 0 / 1 51 -689 -171 51 -661 -177 -114 -661 -177 -580 -148 23-114 -580 -510 62 -148 24 23 -510 -548 48 62 -5424 48 -54 … -548 … … … … … … … ... … … … … … … … … ... Figure 3-10 Extract of a data file indicating record details followed by five rows of data points. - 50 - 3 – Case Study Specifications of data extraction system Sensor Type LENNARTZ ELECTRONIC Triaxial velocity seismometer (LE-3D/5s) Eigenfrequency 0.2 Hz (5 s eigenperiod) Frequency Range 0.5 – 50 Hz Power supply 12 V (DC) Number 2 Data acquisition station Type CITYSHARK II Microtremor Acquisition Station Analog input Single 3D channel Gain Amplifier 14 selectable values (1-8192) Dynamic Range 108 dB at 100 Hz; 90 dB at 250 Hz Sampling Rate 18 selectable values (1-1000 Hz) Temperature Range -10 0C / +50 0C Compatibility Lennartz Number 2 Other Global positioning system (GPS) sensors Coaxial cables Remote control system (RCS) device Adaptors Wireless communications (walkie-talkies) Table 3-2 Specifications of the seismometers and data acquisition stations used in the ambient vibration testing. - 51 - 3 – Case Study 3.3.1 Testing After identifying the bridge for monitoring, the next step was to establish an appropriate measurement protocol. The purpose of the ambient modal testing is to determine the natural frequencies of the structure from strategically located ambient vibration measurements. In this study, the sources of ambient vibrations are vehicular and pedestrian traffic, wind and micro-tremors, and excitation from the trains travelling beneath the structure. Tests were conducted during two field trips: September 10th 2007 and November 30th 2007. Since seismometers are sensitive equipment, it was important to avoid unnecessary oversaturation of the data records from noisy input. To minimise oversaturation, sampling was planned at night when vehicular and train activities are minimal. A large proportion of ambient vibrations are from vehicular traffic and the gain parameter was adjusted to an appropriate low level to allow for it. In spite of this, each record had to be reviewed to eliminate portions that were oversaturated (Smith, 1997; Felber et al., 1995). Figure 3-11 shows a typical time history measurement. Post-processing steps were used to synchronise pairs of records for further analysis. All of the records were de-trended to correct errors that are introduced by the analog-to-digital conversion in sampling and storage (Trauth, 2007; Farrar et al., 1994). Subsequently, experimental modal analyses were performed on the adjusted ambient vibration measurements. - 52 - 3 – Case Study Figure 3-11 Typical measurement time history in three orthogonal directions (Record 6: File 27 from DAS 28 on November 30th 2007). - 53 - 3 – Case Study A B C D E (a) F G Position of reference sensor Position of roving sensor A B C (b) Position of reference sensor Position of roving sensor B A (c) Position of sensors Figure 3-12 Testing configurations: (a) A - along the central longitudinal axis; (b) B along a transverse line; and (c) C - along one exterior longitudinal axis. - 54 - 3 – Case Study 3.3.2 Test locations Test locations were selected at various positions across the bridge (Figure 3-12). Since only two sensors were available for testing, a maximum of two simultaneous recordings was possible for each sampling session; a reference sensor was stationed at one position while a roving sensor was deployed to various positions. In configuration A, measurements were obtained along the central longitudinal axis of the bridge. The reference sensor was positioned at mid-span of the outer span of the bridge. The roving sensor was deployed to the end of the bridge, over inner supports and the mid-span of the bridge’s middle span. Additional measurements were obtained from configuration B along the transverse direction, above an inner support. The reference sensor was positioned at the outer edge of the bridge deck and the roving sensor was deployed to the central and the other outer edge points along that transverse line. In configuration C, measurements were sampled at two points along an outer longitudinal line of the bridge. Nineteen (19) records were obtained from all tests (Table 3-5). Of these, five (5) record pairs were measured. 3.3.3 Sampling parameters The sampling parameters are the sampling rate, duration of recording, and sampling sensitivity (gain). The operational range of the sensors and acquisition units were described in the previous section. Sampling parameters were identified for each test and are summarised in Table 3-3. A sampling rate of 100 Hertz (100 data points per second) was employed during the tests on September 10th 2007. Synchronised records were required for some of the experimental modal analysis methods. They - 55 - 3 – Case Study were first obtained using the remote control system (RCS). However, the RCS device was malfunctioning and a higher sampling rate of 1000 Hertz (1000 data points per second) was selected and combined with a manual synchronisation of pairs of records using the internal clock of each device. These tests were performed on November 30th 2007. The synchronisation procedure is detailed in the following subsection. Sampling parameters Sampling rate (Hz) 100, 1000 Duration (minutes) 10, 15 Gain 16, 32, 64 Table 3-3 Sampling parameters used in testing. The choice on appropriate record duration is a debatable issue. Past research used record durations ranging from several seconds up to a number of minutes (He et al., 2005; Felber et al., 1995; Rainer and Van Selst., 1976). The selection of record duration is influenced by the sampling rate and the desired level of output resolution (Smith, 1997). For this research, measurements were sampled for 15 minutes at 100 Hz and for 10 minutes at 1000 Hz. These are similar to record durations used by Rosset et al. (2004), De La Puente Altez (2005) and Madriz (2004) in ground ambient noise monitoring and by Huang (2007) in ambient vibration testing of buildings, in Montreal. Measurement sensitivity is controlled by the gain in each acquisition unit. Increasing the gain amplifies the observed signal whereas decreasing the gain attenuates it. Appropriate sensitivity level depends on the sampling location and anticipated amplitude of vibrations during the recording session. In general, less sensitivity is needed when the sensors - 56 - 3 – Case Study are deployed at mid-span locations whereas greater sensitivity is required above supports. 3.3.4 Tests The tests were conducted on two dates and relevant ambient conditions such as weather and traffic level were noted (Table 3-4). Traditionally, sensors are connected to the structure or positioned on concrete blocks (Ren et al., 2004; Farrar and James III, 1997). In this case, the sensors were deployed onto the sidewalk and central divider due to safe access issues and time constraints (Figure 3-12). Each sensor was aligned vertically and horizontally. The north-south axis of the sensor was aligned with the transverse direction of the bridge while the east-west axis of the sensor was aligned with the longitudinal direction of the bridge (Figure 3-1). Table 3-5 shows the measurement catalogue with notes including dates, sampling rates, gain values, sampling locations and disturbances during each measurement sampling. 3.3.5 Synchronisation Synchronisation of the two data acquisition systems became necessary after failure of the RCS unit during testing. Before testing, the internal clocks of both data acquisition units were synchronised using the GPS. This allowed the delay (Δ) between each pair of measurements to be identified using the start and end times that were stamped into the data files. A section of the record corresponding to the delay was removed from the start of an early record and from the end of the late record. Figure 3-12 illustrates the records pre- and post-synchronisation. - 57 - 3 – Case Study Ambient conditions during testing Date 10-Sep-2007 30-Nov-2007 Times 23:00 - 24:00 00:00 - 04:00 Traffic Conditions Light to moderate Light Weather Cloudy Mainly clear Temperature (0C) 17.1 -2.4 Wind speed (km/hr) 0 33 Wind direction ( ) n/a 260 Humidity (%) 71 66 Record duration (min) 15 10 Configurations * C A, B 0 Table 3-4 Ambient conditions during testing. *Refer to Figure 3-12 structural response Record 1 structural response Record 1 time / s Δ structural response time / s Δ Δ structural response Record 2 Δ Record 2 time / s time / s (a) (b) Figure 3-13 Synchronisation of a pair of measurements: (a) original time history pair with a time difference (Δ) attributed to manual initiation; (b) modified records after post-testing synchronisation of the time histories in (a). - 58 - - 59 - Measurement catalogue. Table 3-5 3 – Case Study 3 – Case Study 3.4 Analysis of experimental results Experimental modal analysis (EMA) was performed with the ambient vibration measurements. Nineteen (19) measurements were obtained from the tests (Table 3-5). Of these, five (5) were measurement pairs where two records were obtained simultaneously. The MATLAB platform (The MathWorks, 2007) was used to process the data files and execute spectral analyses. In conducting experimental modal testing, the following assumptions were used: 1. Excitation of the structure is provided by ambient sources inclusive of pedestrian, and vehicular traffic on the bridge, oncoming wind and micro-tremors; 2. The excitation is described by a forcing function that was assumed to be white noise (Gaussian) where the frequency domain representation of the excitation is observed as a flat line; 3. The response of the structure is a linear combination of inputs, due to system linearity. 4. The response measurements are ergodic and stationary random processes and representative of a linear vibrating system. 5. P-Delta effects and material nonlinearities are negligible. Four analysis methods were used and compared: (1) Fourier spectral approach; (2) Welch’s PSD spectral approach; (3) Welch’s CSD spectral approach; (4) Auto-regressive, Modified Covariance spectral approach. 3.4.1 Fourier method The FFT algorithm was executed with the number of data points in each record used as the corresponding length of the FFT (approximately 132,000 and 1,000,000 points). None of the records were segmented and a high resolution Fourier spectrum was constructed from each record in - 60 - 3 – Case Study terms of its Fourier amplitude. Resonant frequencies were read directly from each spectrum. Eighteen records were treated with this approach (except record No. 13 on Table 3-5). 3.4.2 Welch’s PSD method First, the time history was cut into 8 segments of equal length with 50% overlap between adjacent segments. A Hamming window (Figure 215) was then applied to each segment to attenuate the overlapping portions. Windowing reduces spectral leakage from the ensuing analysis where the segments are assumed periodic. A FFT, as long as the segments (ranging between 11,250 and 75,000 points), was then applied to each segment and corresponding periodograms were created. Finally, the average of the periodograms from all segments was computed. This periodogram describes the power spectral density spectrum in terms of power per frequency. All of the nineteen records were treated by this approach (Table 3-5) and resonant frequencies were read directly from each averaged spectrum. 3.4.3 Welch’s CSD method The cross spectral density estimate of simultaneous readings was computed with Welch’s modifications. Initially, each record was cut into 100 equal segments with 50% overlap between adjacent segments (Figure 2-13). A Hanning window was then applied to each segment (Figures 2-15). After applying a FFT to each segment (approximately 8,200 points) and computing the corresponding periodograms, an average periodogram was constructed to obtain the estimated cross spectral density spectrum in terms of power per frequency. Resonant frequencies - 61 - 3 – Case Study were read directly from the spectra. Five measurement pairs were treated with this algorithm (records No. 1, 2, 3, 4 and 6 listed in Table 3-5). 3.4.4 Auto-regressive, Modified Covariance method This parametric method uses an auto-regressive fitting of each sequence of measurements to a prediction model that is dependent upon the assumed order of the model. Following preliminary runs and computational limitations, the order of the prediction model was set to 150. Lower order models generate spectral plots in which resonance is not easily discernable. Higher orders increase the resolution of the spectra with minimal improvements to identifying the resonant frequency peaks. The Modified Covariance form of the auto-regressive algorithm minimises forward and backward errors, in the least squares sense, to produce estimated power spectral density spectra in terms of power per frequency (The Mathworks, 2007). Resonant frequencies were read directly from each spectrum. All nineteen measurements were treated with this algorithm (Table 3-5). 3.5 Analysis of analytical models Two finite-element models were constructed on the structural analysis platform SAP2000 (Computers and Structures, Inc, 2007). The input properties were extracted from construction plans. A modal analysis was performed on each model, using eigenvalue analysis to obtain modal parameter estimates for the bridge. - 62 - 3 – Case Study 3.5.1 Models The two models were built using the lumped-mass idealisation concept to represent the as-built configuration of the bridge (Tedesco et al., 1999; Wilson, 1998). The structural components of the bridge were idealised by an assemblage of nodes connected by single, mass-less framing members. Each member was defined between adjacent nodes and described by sectional and material properties of structural components between the nodes. Only gross concrete sections were assumed for reinforced concrete sections since the stiffness of the steel is typically neglected when conducting an elastic dynamic analysis (Wilson, 1998). The mass of the structure was determined and proportionately assigned to each node as representative reactive masses. Each reactive mass defined a mass DOF for the ensuing modal analysis. Only elements with relevant stiffness properties were modelled and the significant nonstructural components were included in computing the masses (Wilson, 1998). Member fixity and support conditions were guided by the construction details. Notable assumptions used for the two analyses were: 1. The skew of the superstructure and the camber of the bridge deck were considered minimal and neglected; 2. Dynamic soil-structure interactions were not considered; 3. Material and structural nonlinearities were neglected; 4. P-Delta effects were not considered; 5. Deterioration of structural members was not considered. The two linear models idealise the bridge for analysis in three orthogonal directions (X-Y-Z). The Z direction was aligned with the vertical axis of the bridge, the Y direction with the transverse axis of the bridge (i.e. across the width of the bridge) and the X direction was aligned with the longitudinal axis of the bridge (i.e. along the length of the bridge from - 63 - 3 – Case Study abutment to abutment). The second model describes a more detailed three-dimensional representation of the bridge. Reactive masses were determined from all structural elements and permanently secured fixtures. Table 3-6 itemises the main components which contribute toward the reactive mass calculations. Component Number Descriptive dimensions Unit Density (mm) (kg/m3) Reinforced Concrete . Deck slab --- 190 mm (thick) 2447 . Columns 24 457 x 457 x 2,440 mm 2447 . Walls 2 762 x 25,900 x 1,830 mm 2447 . Cap beams 2 457 x 457 x 25,750 mm 2447 . Girders 36 W840x295x209.8 7849 . Joists 99 W381x51 7849 . Connectors 4 W457x75 7849 . Connections --- 5% of total steel mass --- 114 mm (average thickness) Structural Steel Asphalt --2400 Table 3-6 Contributors to the reactive mass. 1. Analytical Model 1 (AM1): This model was formulated purely in one plane, using line elements along the longitudinal axis of the bridge. Two-node frame members were used to represent combined sections with translational and rotational DOF at each node. A total of 17 internal nodes with reactive masses and 4 support nodes were used. Three types of massless, equivalent sections idealised the framing components, equivalent - 64 - 3 – Case Study composite deck of reinforced concrete slab on steel girders, columns and walls. Of the 20 framing members in the model, 12 are horizontal and 8 are vertical. The hinge-rocker supports at the abutments, which allow for 25 mm of thermal induced movements, were modelled in two configurations (Figure 3-2). In Case 1, the abutment supports are modelled as hinges at both ends in the transverse direction, and modelled by a hinge at one end with a roller at the other end in the longitudinal direction. Case 2 assumed degradation of the bearings at the abutments and these supports were modelled by hinges at both ends of the bridge in the longitudinal and the transverse directions. The foundations of the two lines of columns were represented by fixed bases because of their assumed rigidity. Moment transfer between the bridge deck system and the supporting moment-resisting reinforced concrete frame was not allowed in accordance with the connection details. This model comprises 21 reactive mass DOFs. Figure 3-14 illustrates model AM1. 2. Analytical Model 2 (AM2): This larger model is an extension of AM1. Twelve identical frames were perpendicularly spaced at 2.24 m between the columns comprising the moment-resisting frames underpinning the bridge deck. The modelled bridge deck system is a composite floor arrangement of steel girders rigidly connected to overlaying reinforced concrete slab sections. Shell sections, representative of the reinforced concrete slab, were meshed along the top of the girders at nine points. This modelling approach was used to better reflect the structural characteristics and the connectivity of the components comprising the bridge deck. Six typical mass-less sections (deck shell; wall shell; channel, steel beam; reinforced concrete beam; reinforced concrete column) were used to describe the bridge. Two-node frame members were used for beams - 65 - 3 – Case Study and columns while four-node sections were used for deck and wall sections. Geometrical and member properties were modified to reflect equivalent sectional details. Of the 313 frame sections, 265 are horizontal and 48 are vertical. The model consists of 132 horizontal deck sections and 44 vertical wall sections. Abutment supports and internal foundations were described similar to those in AM1; hinges at the abutments and fixed foundations beneath the walls were used in Case 1, and hinges at one abutment with a roller at the other abutment were used in Case 2 (Figure 3-2). Moment transfer between the bridge deck system and the supporting moment-resisting, reinforced concrete frame was not permitted in accordance with the construction plans. In total, this model is built with 204 structural nodes and 48 support nodes. Figure 3-15 presents AM2. - 66 - 3 – Case Study (a) (b) (c) Figure 3-14 AM1: (a) solid view of the model; (b) schematic of the structural model for AM1, Case 1; (c) schematic of the structural model for AM1, Case 2. - 67 - 3 – Case Study (a) (b) (c) Figure 3-15 AM2: (a) solid view of the model; (b) schematic of the structural model for AM2, Case 1; (c) schematic of the structural model for AM2, Case 2. - 68 - Chapter 4 Discussion and Recommendations This chapter presents a discussion of the findings obtained from the analyses that were presented in Chapter 3. Firstly, the results for the four procedures (FFT, Welch’s PSD, Welch’s CSD and AR-MCOV) used to analyse the ambient vibration measurements are outlined. This is followed by the presentation of the results for the two analytical models of the bridge (AM1 and AM2) for different support conditions (two support cases) and material properties (three values of compressive strength of concrete). The results from all of the analysis techniques are discussed. Also, recommendations for the modal testing of similar bridges are proposed. For similarly constructed bridges with narrower decks and taller internal supports, the fundamental frequency ranges between 0.5 and 2.5 Hertz. The fundamental frequency for the monitored bridge is expected to be more than 2.5 Hertz because of its shorter internal supports and broader deck; both of these characteristics will improve the stiffness of the bridge. In keeping with this expectation, all discernable resonant frequencies were scrutinised within this range and up to 10 Hertz since the fundamental frequency for most civil engineering structures occurs within these limits. 4.1 Notes on the experimental analysis The full set of response spectra was scrutinised within the range of 0.5 Hertz up to 10 Hertz to identify the fundamental frequency of the bridge and any other important frequencies at the lower extremity of the - 69 - 4 – Discussion and Recommendations spectra. Resonant frequencies are identifiable in all orthogonal directions. The directional components for each measurement were aligned with the principal axes of the bridge (Figure 3-1); the Z component was aligned with the vertical axis of the bridge, the Y component with the transverse axis of the bridge (across the width of the bridge) and the X component was aligned with the longitudinal axis of the bridge (along the length of the bridge from abutment to abutment). Figure 4-1 displays typical response spectra with the four experimental methods while the full set of spectra for each approach can be found in Appendices A through D. The probability of occurrence of a resonant frequency within each spectrum and along each axis of motion is expressed as the repeatability of the resonant frequency. The repeatability of each resonant frequency was computed in terms of percentage and is illustrated in Figures 4-2, 4-3, 4-4 and 4-5 for each procedure used. Figures 4-6, 4-7 and 4-8 present the resonant frequencies estimated by all of the procedures in each orthogonal direction. The Fourier (abbreviated by FFT) method produced three resonant frequencies with at least 80% repeatability in the full set of spectra (Figure 4-2). Interestingly, these values occur solely in one of the orthogonal directions (3.20 Hz in the vertical direction, 6.12 Hz in the transverse direction and 3.20 in the longitudinal direction). The Welch’s PSD (abbreviated by PSD) method produced three resonant frequencies with more than 80% repeatability in all of the output spectra (Figure 4-3). Similar to the results of the Fourier method, these values occur solely in one of the orthogonal directions (3.22 Hz in the vertical direction, 3.54 Hz in the transverse direction and 3.22 in the longitudinal direction). The Welch’s CSD (abbreviated by CSD) method produced twelve resonant frequencies with at least 80% repeatability (Figure 4-4). In the vertical direction, resonance is indicated at 3.22, 4.94, 5.95, 6.21 and 7.01 Hertz. In the transverse direction, there are signs of resonance at 3.53, - 70 - 4 – Discussion and Recommendations 5.17 and 6.13 Hertz; while in the longitudinal direction, resonance is observed at 3.22, 4.58, 5.00 and 6.18 Hertz. The Auto-Regressive, Modified Covariance (abbreviated by MCOV) method indicated resonance at seven frequencies with approximately 80% repeatability or greater (Figure 4-4). Resonance is indicated at 3.21, 6.08 and 7.02 Hertz in the vertical direction and at 3.53 and 6.11 Hertz in the transverse direction. In the longitudinal direction, 3.22, 6.08 and 7.00 are suggested resonant frequencies. Other resonant frequencies are identified on the spectra but are evident with much lower repeatability levels (Table 4-1). Despite their lower incidence rate, these frequencies can be important and must be analysed further. The amplitude of the peaks at some resonant frequencies overshadows the amplitudes of other peaks within the same spectrum. This could explain the disparity in the levels of repeatability of resonant frequencies. Noise also exhibits an adverse effect on the clarity of resonant frequencies with difficulty discerning distinct peaks within the spectral plots. Interference from components of the measuring system, switching power lines or electromagnetic fields can be ruled out because their influence occurs in the frequency ranges of 10 to 100, 25 to 40 and approximately 60 Hertz, respectively (Smith, 1997). However, it still remains that resonance is suggested at the indicated frequencies although their lack of repeatability is not fully understood (Table 4-1). Other works noted that peaks in the spectra often result from nonstationary inputs which are then interpreted as resonant responses of the structure (Farrar et al., 1999). This observation could also explain the frequencies occurring with low levels of repeatability in all the spectra since all sources of the unknown excitation were assumed to be stationary inputs. As well, some of the resonant frequency estimates from the four experimental methods are clustered around select values (Figures 4-6, 4-7 and 4-8; indicated with boxes). The frequencies marked by this - 71 - 4 – Discussion and Recommendations observation in the vertical direction of motion (Z) include 3.20-3.22 Hertz, 4.55-4.62 Hertz, 4.93-4.94 Hertz, 5.95 Hertz, 6.08-6.21 Hertz, and 7.017.02 Hertz. In the transverse direction of motion (Y), similar observations occur at 3.20-3.21 Hertz, 3.53-3.54 Hertz, 5.15-5.19 Hertz, and 6.11-6.13 Hertz. In the longitudinal direction of motion (X), similar observations exist at 3.20-3.22 Hertz, 4.58-4.60 Hertz, 4.95-5.00 Hertz, 5.97-6.18 Hertz and 6.97-7.00 Hertz. While this occurrence suggests corroboration among the experimental methods, some uncertainty is introduced by the variability in the repeatability of those resonant frequency estimates; as much as 45% variation is noted across all three plots. Furthermore, some of these clusters include estimates generated by only two of the four experimental methods. This too requires further investigation. Table 4-2 summarises the prominent frequencies for all four of the methods used in the analysis of ambient vibration measurements from the bridge. The table presents the frequencies that were observed in the full set of spectra at least 80% of the time, and notes the directional measurement from which the estimates were extracted. - 72 - - 73 (d) (b) Date:2007-11-30, DAS:, File:28, Vertical (Z) component. CSD spectrum of simultaneous measurements; Date: 2007-11-30, DAS:28 and 27, File:24, Vertical (Z) component; (d) AR-MCOV spectrum; 11-30, DAS:28, File:24, Vertical (Z) component; (b) Welch’s PSD spectrum; Date:2007-11-30, DAS:28, File:, (24) component; (c) Welch’s Typical response spectra from the four experimental approaches using the ambient vibration measurements: (a) FFT spectrum; Date: 2007- Figure 4-1 (c) (a) 4 – Discussion and Recommendations 4 – Discussion and Recommendations Resonant frequencies* CSD MCOV PSD CSD MCOV 3.22 3.21 3.21 1.95 3.53 3.53 3.20 1.96 3.22 3.22 100** 44 100 100 50 41 80 63 100 56 80 95 6.20 3.22 3.99 4.55 3.54 3.20 5.17 5.17 --- 3.22 4.58 4.60 67 100 40 42 67 71 100 58 100 80 74 7.01 4.57 4.62 4.94 5.15 3.54 6.13 6.11 3.94 5.00 6.08 61 56 60 47 50 88 100 79 56 80 79 --- 4.93 4.94 6.08 6.12 5.19 6.85 --- --- 4.59 6.18 7.00 39 80 84 89 65 40 61 100 68 --- 5.95 5.95 7.02 7.96 5.99 --- --- --- 4.95 6.97 --- 72 80 84 44 41 56 40 --- 6.20 6.21 --- --- 6.13 --- --- 56 100 --- 7.01 7.01 --- --- --- --- --- 72 100 --- --- --- --- --- FFT PSD 1.95 FFT 3.20* FFT MCOV Longitudinal Direction (X) CSD Transverse Direction (Y) PSD Vertical Direction (Z) --- --- --- --- 5.97 --- --- --- 6.17 71 44 56 --- --- --- --- --- --- 6.97 56 * ** Frequencies in Hertz. Repeatability in percent: the percentage of observable resonance at that frequency in the full set of spectra from this experimentation. Table 4-1 Summary of resonant frequency estimates and repeatability rates by direction and processing method. - 74 - 4 – Discussion and Recommendations Resonant frequencies* (up to 10 Hz) Algorithm 3.20*, 3.20, 6.12 FFT (z**) (x) (y) 3.22, 3.22, 3.54 PSD (z) (x) (y) 3.22, 3.53, 4.58, 4.94, 5.00, 5.17, 5.95, 6.13, 6.18, 6.21, 7.01 CSD (z) (y) (x) (z) (x) (y) (z) (y) (x) (z) (z) 3.22, 3.22, 6.08, 6.08, 6.11, 7.02 MCOV (z) (x) (x) (z) (y) (z) * Frequencies in Hertz. ** Direction in which the resonant peak was detected with 80% repeatability in the full set of spectra. Table 4-2 Summary of resonant frequency estimates for all four algorithms in EMA and corresponding directions of occurrence. Probability of resonant peak (%) 100 80 60 40 20 0 1 2 3 4 5 6 7 8 9 Frequency (Hz) FFT - Z FFT - Y FFT - X Figure 4-2 Frequency estimates and repeatability rates using the FFT approach. - 75 - 10 4 – Discussion and Recommendations Probability of resonant peak (%) 100 80 60 40 20 0 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) PSD - Z PSD - Y PSD - X Figure 4-3 Frequency estimates and repeatability rates using the Welch’s PSD approach. Probability of resonant peak (%) 100 80 60 40 20 0 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) CSD - Z CSD - Y CSD - X Figure 4-4 Frequency estimates and repeatability rates using the Welch’s CSD approach. - 76 - 4 – Discussion and Recommendations Probability of resonant peak (%) 100 80 60 40 20 0 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) MCOV - Z MCOV - Y MCOV - X Figure 4-5 Frequency estimates and repeatability rates using the MCOV approach. Probability of resonant peak (%) 100 80 60 40 20 0 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) FFT - Z PSD - Z CSD - Z MCOV - Z Figure 4-6 Frequency estimates and repeatability rates in the vertical direction (Z). - 77 - 4 – Discussion and Recommendations Probability of resonant peak (%) 100 80 60 40 20 0 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) FFT - Y PSD - Y CSD - Y MCOV - Y Figure 4-7 Frequency estimates and repeatability rates in the transverse direction (Y). Probability of resonant peak (%) 100 80 60 40 20 0 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) FFT - X PSD - X CSD - X MCOV - X Figure 4-8 Frequency estimates and repeatability rates in the longitudinal direction (X). - 78 - 4 – Discussion and Recommendations 4.2 Notes on the analytical analysis Finite element models were used in the analytical approach to modal analysis of the bridge. Regarding both AM1 and AM2 models, Case 1 considered a hinge support at one abutment and a roller at the other, and Case 2 considered both supports as hinges. The results for AM1 show little difference between Case 1 and Case 2 and estimates of the natural frequencies are mostly identical (Table 4-3). For AM2, the results for Case 2 are consistently 1.6% greater than for Case 1 (Table 4-4). Furthermore, the estimates between AM1 and AM2 are considerably different with AM1 suggesting a stiffer structure with a fundamental frequency as much as 13% larger than for AM2. As mentioned previously, when describing the analytical AM1 was constructed with 21 reactive masses while AM2 was developed with 252 reactive masses. Since frequencies are associated with each mass DOF the frequency estimation for AM2 is more accurate than with AM1. Also, AM2 is less rigid than AM1 because of a greater use of three-dimensional modelling. Current data for the compressive strength of concrete in the bridge is unavailable. Based on the practice at the time of construction, the assumed compressive strength of concrete is 25 MPa at 28 days (De la Puente Altez, 2005). Now, 50 years later, the strength is expected to reach 45 MPa (Brinkerhoff, 1983; Neville, 1996). The two analytical models (AM1 and AM2) with support conditions Case 1 and fc’ of 25 MPa simulate the bridge at the time of construction. The two analytical models, with support conditions Case 2 and fc’ of 45 MPa, better represent the current state of the bridge. Figure 4-9 reveals that the fundamental frequency increases by as much as 16% from 2.44 Hertz to 2.83 Hertz between immediately-after-construction and at-present conditions using model AM1; for AM2, the fundamental frequency - 79 - 4 – Discussion and Recommendations increases by 4.5% from 2.42 Hertz to 2.53 Hertz for the same two conditions. Figure 4-9 also illustrates the relationship between the compressive strength of the concrete, fc’, and the fundamental frequency of vibration of the bridge: the frequency of vibration increases when the concrete strength is increased. This observation is evident for both analytical models; AM1 and AM2. While Figure 4-9 presents results for the first mode of vibration, this proportional relationship is also evident for higher modes of vibration. 4.3 Fundamental frequency estimates Figure 4-9 indicates estimates of the frequency for the first mode of vibration from all modal analysis approaches. EMA produced an estimate of 3.21 Hertz. At the same time, the estimates between AM1 and AM2 are considerably different with AM1 suggesting a stiffer structure with a fundamental frequency of vibration as much as 13% larger than estimates for AM2. AM1 is a conservative model of the bridge that is based on simplified assumptions. AM2 is more complex, incorporating the threedimensional nature of the bridge to a greater extent. The current compressive strength of the concrete (fc’) in the bridge is considered to be 45 MPa. Using AM1, the fundamental frequency was estimates at 2.83 Hertz for Case 1 and Case 2. Using AM2, the fundamental frequency was estimated at 2.49 Hertz for Case 1 and at 2.53 Hertz for Case 2 All four EMA approaches yielded higher estimates of the bridge’s fundamental frequency than any of the analytical approaches; EMA results are at least 13% larger than AM1 results and 27% larger than AM2 findings. EMA estimated the fundamental frequency of 3.21 Hertz by FFT method, 3.22 Hertz by PSD method, 3.22 Hertz by CSD method and 3.21 Hertz by MCOV method. - 80 - 4 – Discussion and Recommendations Resonant frequencies* Mode of vibration * Case 1 Case 2 fc’ (MPa) fc’ (MPa) 25 35 45 25 35 45 1 2.44* 2.65 2.83 2.44 2.66 2.83 2 3.63 3.95 4.21 3.63 3.95 4.21 3 4.40 4.79 5.10 4.40 4.79 5.10 4 8.80 9.58 10.20 8.80 9.58 10.20 5 10.68 11.61 12.37 12.14 13.22 14.07 Frequencies in Hertz. Table 4-3 Summary of natural frequency estimates for AM1. Resonant frequencies* Case 1 Mode of vibration * Case 2 fc’ (MPa) fc’ (MPa) 25 35 45 25 35 45 1 2.42* 2.46 2.49 2.46 2.50 2.53 2 2.61 2.66 2.70 2.65 2.70 2.74 3 3.33 3.40 3.45 3.35 3.42 3.48 4 3.43 3.48 3.51 3.50 3.55 3.59 5 3.67 3.73 3.78 3.76 3.82 3.86 6 3.94 3.99 4.03 3.98 4.03 4.07 7 4.21 4.28 4.32 4.35 4.31 4.36 8 4.32 4.41 4.47 4.38 4.47 4.53 9 4.79 4.88 4.94 4.83 4.91 4.97 10 5.04 5.18 5.29 5.05 5.19 5.30 Frequencies in Hertz. Table 4-4 Summary of natural frequency estimates for AM2. - 81 - - 82 analyses. FFT 3.20 PSD 3.22 CSD 3.22 MCOV 3.21 Modal Analysis Approach Fundamental frequency estimates (frequency at first mode of vibration) for all experimental and analytical modal 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 Figure 4-9 Frequency of 1st Mode of Vibration (Hz) AM1, Case1, fc'=25MPa 2.44 AM1, Case1, fc'=35MPa 2.65 AM1, Case1, fc'=45MPa 2.83 AM1, Case2, fc'=25MPa 2.44 AM1, Case2, fc'=35MPa 2.66 AM1, Case2, fc'=45MPa 2.83 AM2, Case1, fc'=25MPa 2.42 AM2, Case1, fc'=35MPa 2.46 AM2, Case1, fc'=45MPa 2.49 AM2, Case2, fc'=25MPa 2.46 AM2, Case2, fc'=35MPa 2.50 AM2, Case2, fc'=45MPa 2.53 4 – Discussion and Recommendations 4 – Discussion and Recommendations 4.4 Reviewing the analysis The experimental modal results are consistent with each other but not with the analytical results. Differences between the experimental and analytical modal analyses can be explained by dissimilarities between the actual structure and the idealised analytical models. The analytical models may not accurately and completely reflect the structural properties of the bridge. Knowledge on the geometrical specifications of the structure, material properties and force-transferring mechanisms is required to develop realistic models of the structure since finite element modelling of full-scale structures is limited by this information. Better instrumentation and more accurate construction details could provide better agreement between the experimental and analytical results. Some influential parameters include material properties, extent and location of material deterioration and conditions of connection and joint assemblies. The results of EMA are traditionally used to calibrate and tune analytical models of bridges so that structural issues can be identified (Ewins, 2000; Maia and Silva, 1997). Here, the disagreement between the EMA and analytical results may be due to possible problems with the assumptions for the connection of the bridge deck to the column supports, the accuracy of as-built drawings or other issues associated with field measurements. The models AM1 and AM2 were modified to consider a full rigid connection of the support columns to the continuous deck above it. Under this assumption AM1 has an increased fundamental frequency of 3.01 Hertz while AM2 increases to 2.73 Hertz, representing increases of 6% and 8% respectively for results from Case 2 investigations with both models. This means a 7% discrepancy from the EMA estimate for the fundamental frequency of 3.21 Hertz. - 83 - 4 – Discussion and Recommendations Further calibration of the experimental and analytical models would permit even better agreement of the results and should include the analysis for higher modes. Vibration mode shapes and damping ratios can be estimated from the analysis of the experimental results. These types are beyond the scope of the current thesis. 4.5 Recommendations for ambient vibration testing of bridges For EMA using ambient vibration measurements, it is essential to understand the factors that influence the experimentation. Here are some notes and recommendations to assist with performing efficient modal testing of bridges. 1. Problem Definition: Define the results of the modal testing clearly from the outset of the investigation. Typically, the parameters of interest include natural frequencies, mode shapes, damping ratio estimates, a combination of these and other parameters. 2. Structural Details: Review current details of the structure. Important features include, i. sectional and material properties, ii. connections between structural components, iii. support types, iv. distribution and estimation of masses. 3. Experimental Modal Analysis Techniques: Be aware of current modal analysis techniques. Algorithms are available in time and frequency domains. Do not rely solely on one technique. 4. Research: Review reports of bridges subjected to vibration tests for guidance on the use of various measurement details and analysis techniques. Noteworthy measurement details include equipment type, configurations and sampling parameter settings. Extra focus on bridges - 84 - 4 – Discussion and Recommendations of similar construction and scale will suggest an anticipated range of results. It is also useful to be well-informed of signal processing issues. 5. Instrumentation: Be aware of available equipment. Some equipment can be too sensitive for the investigation while others can be insufficiently so. The operational features (gain, sampling rate, sample duration, data storage capacity, range of weather-resistance, among others), practicality of measuring with the unit(s) and any other relevant limitations are issues which must be considered. European Commission (2002) evaluated the suitability of 17 sensors and 13 digitisers for ambient noise measurements. A list of tested accelerometers was found insensitive for capturing lower frequency data; only a few velocimeter transducers were endorsed by the organisation. An additional recommendation is the mandatory 10minute ‘warm-up’ of sensors and digitisers before sampling to allow for stabilisation of the equipment. 6. Testing Procedure: Some recommendations for modal testing of bridges similar to the monitored structure include: i. Instrumentation: a. Use a measuring device with a fundamental frequency lower than the minimum frequency of interest from the bridge. Otherwise, correcting the contaminated measurements could be difficult. b. Use a minimum of two instruments when testing bridges. The required number of units must be guided by the goals of the testing. More instruments may be installed based on the computational needs, instrument availability and practicality of the experimentation. ii. Sampling a. The sampling rate must satisfy the anticipated range of resonant frequencies of the bridge and equipment capabilities (as presented in Table 3-2) and it must - 85 - 4 – Discussion and Recommendations meet Shannon’s sampling rate requirement (Equations 2.32 to 2.34). For bridges similar to the structure tested in this research, a sampling rate of 100 Hertz is adequate. This is ten times greater than Shannon’s recommendation. More information can be sampled by using a sampling rate of 1000 Hertz or more; the greater precision is especially helpful where post-testing synchronisation is required. b. The duration of each measurement must be long enough to capture the structure’s response. In combination with an appropriate sampling rate, use a minimum duration for bridges between 5 minutes and 15 minutes. c. Saturation of measurements is detrimental since much of the data can be lost during sampling. Vibration amplitudes are sensitive to the location of the sensors. Larger vertical amplitudes occur at mid-span and greater longitudinal or transverse amplitudes occur at deck support positions. Select the gain to accommodate sampling in each individual direction or in all three directions simultaneously without encouraging saturation of the recording. iii. Testing: For typical 3-span bridges with moment-resisting, reinforced concrete support frames and composite decks of reinforced concrete slab on steel girders, and with spans limited to 90 metres and deck width up to 15 metres, the fundamental frequency ranges between 0.5 Hertz and 2.5 Hertz. This research found that for a similar 3-span bridge with spans limited to 30 metres, deck width of 22 metres and total length of 70 metres, the fundamental frequency was 3.21 Hertz. - 86 - 4 – Discussion and Recommendations iv. Testing Configurations: Sample measurements along the central longitudinal axis of the bridge. Also, measure at positions along one or both edges of the bridge deck and along transverse lines. These measurements can be analysed to estimate mode shapes and damping ratios as well as increase the full set of records. The objective of the investigation, type of instrumentation and the expected response (mode shapes) of the structure are important inputs for designing the sampling scheme. The instruments should be deployed on the deck above supports as well as at mid-span. Intermediate positions should be considered to increase the density of the measurements. v. Number of measurements: The quality of estimates can be improved by computing an average of several sets of results. Record a minimum of three sets of measurements. Greater numbers of measurements will increase the quality of estimates by minimising the presence of erroneous data in the measurement records. Characteristics of the configurations used will further guide the number of measurements that are required. 7. Estimating Natural Frequencies: All four algorithms (FFT, PSD, CSD and MCOV) generated similar estimates for the fundamental frequency of the bridge. Results for higher vibration modes vary considerably between the methods. This suggests that several approaches should be used to ensure that pertinent estimates are not overlooked and are fully identified by experimental analysis of vibration measurements. The CSD method produced at least twice as many estimates of natural frequencies as the other methods. - 87 - Chapter 5 Conclusions and Future Recommendations This chapter presents the conclusions from the experimental and analytical modal analyses of the bridge. A review of the analysis is discussed briefly and a summary of recommendations for modal testing of bridges is provided. As well, future research directions in experimental modal analysis of bridges using ambient vibration measurements are outlined. 5.1 Conclusions 1. Field investigation by ambient vibration testing (AVT) estimated the fundamental frequency of the bridge at 3.21 Hertz. 1.1. FFT method estimated 3.20 Hertz in vertical measures, 3.20 Hertz in longitudinal measures and 6.12 Hertz in transverse measures. 1.2. PSD method estimated 3.22 Hertz in vertical measures, 3.22 Hertz in longitudinal measures and 3.54 Hertz in transverse measures. 1.3. CSD method estimated 3.22 Hertz in vertical measures, 3.22 Hertz in longitudinal measures and 3.53 Hertz in transverse measures. 1.4. MCOV method estimated 3.21 Hertz in vertical measures, 3.22 Hertz in longitudinal measures and 6.11 Hertz in transverse measures. - 88 - 5 - Conclusions and Research Prospects 1.5. All methods are relatively equivalent in obtaining the fundamental mode. Differences are apparent mainly for higher modes. 1.6. Various configurations of devices were investigated for measuring ambient noise and provided consistent and repeatable results for the fundamental frequency. 2. Analytical modal analysis (AMA) using two models of the bridge yielded fundamental frequency estimates between 2.42 and 3.01 Hertz. 2.1. Various assumptions for restraints at joints and supports and material properties were used to determine the theoretical fundamental frequency of the bridge. 2.2. A parametric study of the analytical models indicates natural frequency estimates for the bridge is proportional to the compressive strength of concrete. 2.3. Sensitivity of fundamental frequency to the level of detail of a model, the restraint conditions, material properties and mass (total and spatial distribution) was investigated. For this bridge, results were relatively insensitive to support and joint conditions (deck-dominated or longitudinal response). Material properties and mass had the greatest influence on fundamental frequency. 3. Estimates of fundamental frequency by field investigation exceeded that from analytical analysis by 7 to 13 percent. 4. Ambient noise has been demonstrated as an efficient response property for estimating fundamental frequency of bridges. 4.1. Despite using best estimates of material properties, restraints and mass, a 7 to 13 percent discrepancy between experimental and theoretical results remains. This is good agreement compared with results from other researchers. - 89 - 5 - Conclusions and Research Prospects 4.2. The most plausible explanation for this discrepancy may be underestimation of the total mass of the deck, construction details of the deck and connections or undocumented retrofits that provide more stiffness than considered by the analytical models. 5.2 Reviewing the analysis Extensive knowledge of the present characteristics of the bridge is required to develop a realistic model of the structure. This would permit better agreement between experimental and analytical results. Of particular importance are material properties, extent and location of material deterioration and conditions of connection and joint assemblies. 5.3 Recommendations for ambient vibration testing of bridges The following suggestions are recommended for future modal testing of similar bridges. 1. Use velocimeter sensors to measure translational displacements. 2. Use a measuring device with a fundamental frequency that lies outside the range of frequencies being investigated to lessen contamination of ambient vibration measurements. 3. Use at least two setups for modal testing of bridges. 4. Implement a ‘warming’ of sensors and digitisers before each sampling session to permit stabilisation of the equipment. 5. Use a sampling rate of 100 Hertz or greater for data acquisition, satisfying Shannon’s sampling rate requirement. 6. Use a measurement duration between 5 and 15 minutes. - 90 - 5 - Conclusions and Research Prospects 7. Monitor and adjust gain parameter to abate saturation of measurements. 8. Obtain measurements at positions along longitudinal centreline and edge lines of the bridge. 9. Obtain measurements at positions along transverse lines over supports and at mid-span and intermediate positions. 10. Obtain several measurements for each recording to compute averaged estimates. 11. Use several methods to ensure resonant frequencies are not overlooked and will be fully identified by experimental analysis of the vibration measurements, especially for bridges with closely spaced vibration modes. 5.4 Proposal for future research AVT is beneficial to the dynamic study of bridges. Unlike forced vibration testing, AVT is non-invasive and does not disturb occupancy; the equipment is relatively portable and inexpensive; testing procedures are easily implemented and less time is required for testing. Applications of AVT include the following: structural health monitoring; hazard analysis; soil-structure interaction; serviceability monitoring; improved numerical modelling; design validation. 1. Structural health monitoring: In Canada alone, major crossings such as the Lions Gate Bridge, Port Mann Bridge and Queensborough Bridge in British Columbia have already been subjected to vibration testing. Structural health monitoring of bridges is now widely practiced. With instrumentation in place, it is possible to repeatedly conduct dynamic assessments of a monitored bridge whether it is in construction or an existing structure. The impact of different types of time varying forces - 91 - 5 - Conclusions and Research Prospects (resulting from human and environmental activities) and the deterioration of material properties on the behaviour of the structure can be investigated at anytime and be more fully understood. Furthermore, problems with support mechanisms can be identified. 2. Hazard analysis: Structural vulnerabilities of bridges can be identified by using ambient vibration testing when performing hazard analysis. The City of Montreal and researchers at McGill University are currently implementing a network of permanently instrumented infrastructure for this purpose. The Jacques Bizard Bridge which connects Montreal Island to L'Île Bizard is earmarked for such instrumentation. 3. Soil-structure interaction: In general there exists on-going research aimed at defining soil-structure interactions and its influence on the dynamic response of foundations and structures. Already, ambient vibration measurements have proven to be useful to the microzonation mapping of regions, thus furthering efficiency of the study of seismic hazard analysis. 4. Serviceability monitoring: Ambient noise measurements can be used to evaluate engineering structures from a serviceability perspective. While resonance can lead to structural failure, vibrations can also adversely affect occupancy and equipment. 5. Improved numerical modelling: AVT helps to characterise the dynamic behaviour of bridges through estimates of natural frequencies, mode shapes and modal damping and can be been used to improve analytical models. When combined with soil-structure interaction simulation and careful modelling of bridge supports and member connections, these tuned models produce more reliable estimates of structural responses. 6. Design validation: The results of EMA can be used to validate and update design assumptions, identify structural deficiencies and increase structural efficiency and project economy. This practice of updating design models is a component of structural health monitoring. - 92 - 5 - Conclusions and Research Prospects 5.5 Fulfilment of objectives The goals that were established at the onset of the dissertation have been achieved as follows: 9 1. Ambient vibration testing of bridges was reviewed. 9 2. Experimental and analytical modal analyses were conducted on a typical bridge in the City of Montreal in Canada. 9 3. The results for the experimental and analytical modal analyses were discussed and compared. 9 5. Recommendations for future testing of similar bridges were provided. 9 6. 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R-7 APPENDIX A - OUTPUT SPECTRA FROM FFT ESTIMATION - Appendix A - Output spectra from FFT estimation - A-2 Appendix A - Output spectra from FFT estimation - A-3 Appendix A - Output spectra from FFT estimation - A-4 Appendix A - Output spectra from FFT estimation - A-5 Appendix A - Output spectra from FFT estimation - A-6 Appendix A - Output spectra from FFT estimation - A-7 Appendix A - Output spectra from FFT estimation - A-8 Appendix A - Output spectra from FFT estimation - A-9 Appendix A - Output spectra from FFT estimation - A - 10 APPENDIX B - OUTPUT SPECTRA FROM WELCH’S PSD ESTIMATION - Appendix B - Output spectra from Welch’s PSD estimation - (b) B-2 Appendix B - Output spectra from Welch’s PSD estimation - B-3 Appendix B - Output spectra from Welch’s PSD estimation - B-4 Appendix B - Output spectra from Welch’s PSD estimation - B-5 Appendix B - Output spectra from Welch’s PSD estimation - B-6 Appendix B - Output spectra from Welch’s PSD estimation - B-7 Appendix B - Output spectra from Welch’s PSD estimation - B-8 Appendix B - Output spectra from Welch’s PSD estimation - B-9 Appendix B - Output spectra from Welch’s PSD estimation - B - 10 APPENDIX C - Output spectra from Welch’s CSD estimation - Appendix C - Output spectra from Welch’s CSD estimation - C-2 Appendix C - Output spectra from Welch’s CSD estimation - C-3 Appendix C - Output spectra from Welch’s CSD estimation - C-4 APPENDIX D - Output spectra from AR-Modified Covariance estimation - Appendix D - Output spectra from AR-MCOV estimation - D-2 Appendix D - Output spectra from AR-MCOV estimation - D-3 Appendix D - Output spectra from AR-MCOV estimation - D-4 Appendix D - Output spectra from AR-MCOV estimation - D-5 Appendix D - Output spectra from AR-MCOV estimation - D-6 Appendix D - Output spectra from AR-MCOV estimation - D-7 Appendix D - Output spectra from AR-MCOV estimation - D-8 Appendix D - Output spectra from AR-MCOV estimation - D-9 Appendix D - Output spectra from AR-MCOV estimation - D - 10