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Dynamic analysis of civil engineering structures using joint time-frequency methods Black, Cameron John 1998

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D Y N A M I C ANALYSIS OF CIVIL ENGINEERING STRUCTURES USING JOINT TIME-FREQUENCY METHODS by C A M E R O N JOHN B L A C K B .A .Sc , The University of British Columbia, 1996 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Civil Engineering) We accept this thesis as conforming to the required standard/ THE UNIVERSITY OF BRITISH C O L U M B I A August 1998 © Cameron John Black, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of £l\JXL- E M ^ J / O g g g J ^ c ^ The University of British Columbia Vancouver, Canada DE-6 (2/88) A B S T R A C T The study of signals whose frequency content changes in time is prevalent in many academic fields. The objective of this thesis is to demonstrate that joint time-frequency analysis is suitable for the analysis of civil engineering vibration data. A discussion of different joint time-frequency analysis methods is presented with emphasis on the Wavelet Transform, the Wigner Distribution, Cohen's Class functions and the Short Time Fourier Transform. Most of Cohen's class functions are not directly applicable to the analysis of civil engineering vibration signals as they are not manifestly positive in the time-frequency plane. The Wavelet and the Short Time Fourier Transforms are manifestly positive and appear to be suitable for the analysis of civil engineering vibration data. This thesis explores the use of joint time-frequency analysis through 5 case studies. These include the analysis of ambient vibration data obtained from two bridges, data obtained from shake table testing and strong motion data collected from 2 instrumented buildings. The joint time-frequency analysis presented in the case studies makes use of the Short Time Fourier Trans-form. The dynamic behavior of 2 bridges is analyzed using ambient vibration data. It is shown that joint Abstract time-frequency analysis can be used to verify the stability of the dominant frequencies during the course of testing as well as explain anomalous results obtained from frequency domain analysis. During shake table testing of an unbonded concrete gravity dam model, upstream motion was observed at certain combinations of amplitude and frequency of base motion. Joint time-fre-quency analysis is used to improve the understanding of this phenomenon. The most promising application of joint time-frequency analysis is for the interpretation of strong motion data. The response of 2 instrumented buildings during the Northridge and San Fernando earthquakes is studied using frequency and joint time-frequency analysis techniques. A function called the Time Frequency Response Function is defined and used, to study many aspects of the dynamic behavior of structures not explained through typical frequency domain analysis of strong motion data. This includes the presence of coupling between modes of vibration and the temporal location of modal response. The case studies presented in this thesis demonstrate that joint time-frequency analysis is useful for the study of civil engineering vibration data and should be studied further. iii T A B L E O F C O N T E N T A B S T R A C T II T A B L E O F C O N T E N T S IV LIST O F F I G U R E S VII LIST O F T A B L E S X A C K N O W L E D G M E N T S XI DEDICATION XII C H A P T E R 1 INTRODUCTION 1 1.1 Objectives 2 1.2 Scope 2 1.3 Outline 3 C H A P T E R 2 B A C K G R O U N D 5 2.1 Structural Dynamics 5 2.2 Frequency Domain Analysis 13 2.3 Joint Time-Frequency Domain Analysis 20 2.4 Literature Review 31 C H A P T E R 3 EXPERIMENTAL DYNAMICS 33 3.1 Ambient Vibration 34 3.2 Strong Motion Data 38 C H A P T E R 4 INTRODUCTION T O C A S E STUDIES 43 4.1 Ambient Vibration Data 43 4.2 Shake Table Data 43 4.3 Strong Motion Data . . 44 4.4 Presentation of Joint Time-Frequency Analysis 46 C H A P T E R 5 AMBIENT VIBRATION O F A 3 SPAN BRIDGE 48 5.1 Introduction 48 Table of Contents 5.2 Description of the Bridge 49 5.3 Description of the Ambient Vibration Testing .50 5.4 Frequency Domain Analysis 53 5.5 Joint Time-Frequency Analysis 60 5.6 Conclusions 67 C H A P T E R 6 AMBIENT VIBRATION O F A SINGLE SPAN BRIDGE 68 6.1 Introduction 68 6.2 Description of the Bridge 69 6.3 Description of the Ambient Vibration Testing 71 6.4 Frequency Domain Analysis 74 6.5 Joint Time-Frequency Analysis 81 6.6 Conclusions 87 C H A P T E R 7 SLIDING OF A DAM MODEL 89 7.1 Introduction 89 7.2 Experimental Model Description 90 7.3 Shake Table Testing 94 7.4 Analysis 99 7.5 Joint Time Frequency Analysis 99 7.6 Conclusion 109 CHAPTER 8 ANALYSIS O F THE 1901 AVENUE O F THE S T A R S BUILDING 110 8.1 Description 110 8.2 Instrumentation 112 8.3 Frequency Domain Analysis 118 8.4 Joint Time-Frequency Analysis 121 8.5 Conclusion 133 C H A P T E R 9 ANALYSIS OF THE S H E R A T O N UNIVERSAL HOTEL 134 9.1 Description 134 9.2 Instrumentation 136 9.3 Frequency Domain Analysis 142 9.4 Joint Time-frequency Analysis 146 9.5 Conclusions 156 CHAPTER 10 CONCLUSIONS 157 10.1 Applications of Joint Time-Frequency Analysis 157 10.2 Limitations of Joint Time-Frequency Analysis 160 10.3 Suggested Future Research 161 Table of Contents CHAPTER 11 R E F E R E N C E S 163 APPENDIX A THE TIME F R E Q U E N C Y R E S P O N S E FUNCTION 167 A.1 The Time Frequency Response Function 167 A. 2 S D O F Example of the T F R F 173 APPENDIX B DATA ANALYSIS S O F T W A R E 176 B. 1 Ambient Vibration Data Analysis Software 176 B.2 Shake Table Data Analysis 177 B.3 Strong Motion Data Analysis Software 177 VI Figure 2.1: Mass-Spring-Damper Representation of a S D O F System 6 Figure 2.2: Dynamic Free Body Diagram 7 Figure 2.3: Three Degree of Freedom Model 9 Figure 2.4: Real and Imaginary Parts of the F R F 19 Figure 2.5: Hanning Window Function 21 Figure 2.6: Comparison of the Spectrogram and the Wigner Distribution 27 Figure 4.1: Location of the San Fernando and Northridge Earthquakes 45 Figure 4.2: Example of Time-Frequency Plot 47 Figure 5.1: Photo of the University Drive/Crowchild Trail Bridge 50 Figure 5.2: Schematic of Sensor Locations 52 Figure 5.3: Averaged Normalized Power Spectral Densities 55 Figure 5.4: Averaged Normalized Power Spectral Densities (Log-Normal Scale) 56 Figure 5.5: Mode Shapes of the University Drive/Crowchild Trail Bridge 58 Figure 5.6: Wide Band T-F Plot of the Calgary Bridge 62 Figure 5.7: Narrow Band T-F Plot of the Calgary Bridge 63 Figure 5.8: Narrow Band T-F Plot of the Calgary Bridge - Low Frequency Range 64 Figure 5.9: Narrow Band T-F Plot of the Calgary Bridge - Very Low Frequency Range . 65 Figure 6.1: Photo of the Lindquist Bridge 70 Figure 6.2: Photo of the Top of the Bridge 70 Figure 6.3: Photo Showing the Underside of the Bridge 71 List of Figures Figure 6.4: Sensor Locations and Directions of the Measurements 72 Figure 6.5: Log Normal Plot of the A N P S D s Obtained from March 31 Test 77 Figure 6.6: Log Normal Plot of the A N P S D s Obtained from April 1 Test 77 Figure 6.7: Mode Shapes of the Lindquist Bridge 79 Figure 6.8: Wide Band T-F Plot for the Lindquist Bridge 82 Figure 6.9: Narrow Band T-F Plot for the Lindquist Bridge 83 Figure 6.10: Narrow Band T-F Plot for the Lindquist Bridge - Low Frequency Range . . . . 86 Figure 7.1: Schematic of Dam Model 91 Figure 7.2: Photo of Dam Monolith Model 92 Figure 7.3: Photo Showing Plate Attachment to Dam Model and Table 93 Figure 7.4: Hydrostatic Load Assembly 94 Figure 7.5: Typical Dynamic Test Setup 96 Figure 7.6: The 5 Hz Harmonic Input Record 97 Figure 7.7: 5 Hz Earthquake Record 98 Figure 7.8: TFR Plot of 20 Hz Harmonic Input - No Upstream Motion 101 Figure 7.9: TFR Plot of 20 Hz Harmonic Input - Upstream Motion 102 Figure 7.10: TFR Plot of 22.5 Hz Harmonic Input - Upstream Motion 103 Figure 7.11: TFR plot of 15 Hz Earthquake Input - No Upstream Motion 105 Figure 7.12: TFR plot of 25 Hz Earthquake Input - Upstream Motion 106 Figure 7.13: Schematic Showing Upstream Motion at One Half the Forcing Frequency . 108 Figure 8.1: Photo of the 1901 Avenue of the Stars Building I l l Figure 8.2: Schematic of Sensor Location During San Fernando Earthquake 113 Figure 8.3: Schematic of Sensor Location During the Northridge Earthquake 114 Figure 8.4: Measured Accelerations During the San Fernando Earthquake 115 Figure 8.5: Measured Accelerations During the Northridge Earthquake 116 Figure 8.6: FRFs of the 1901 Avenue of the Stars Building 119 List of Figures Figure 8.7: Mode Shapes of the 1901 Avenue of the Stars Building 121 Figure 8.8: 1901 T-F Plot - Northridge Earthquake - Transverse Direction 124 Figure 8.9: 1901 T-F Plot - San Fernando Earthquake - Transverse Direction 125 Figure 8.10: Figure Showing Contribution of Heavy Shaking to F R F 127 Figure 8.11: 1901 T-F Plot - Northridge Earthquake - Longitudial Direction 129 Figure 8.12: 1901 T-F Plot - San Fernando Earthquake - Longitudial Direction 130 Figure 8.13: 1901 T-F Plot - San Fernando Earthquake - Longitudial Direction - 0 to 0.6 Hz 132 Figure 9.1: Photo of Sheraton Universal Hotel 135 Figure 9.2: Schematic of Sensor Location During the San Fernando Earthquake 137 Figure 9.3: Schematic of Sensor Location During the Northridge Earthquake 138 Figure 9.4: Measured Accelerations During the San Fernando Earthquake 139 Figure 9.5: Measured Accelerations During the Northridge Earthquake 140 Figure 9.6: Frequency Response Functions for the Sheraton 144 Figure 9.7: Mode Shapes of the Sheraton Universal 146 Figure 9.8: Sheraton T-F Plot - Northridge Earthquake -Transverse Direction 148 Figure 9.9: Sheraton T-F Plot - San Fernando Earthquake - Transverse Direction . . . . 149 Figure 9.10: Sheraton T-F Plot - Northridge Earthquake - Longitudial Direction 152 Figure 9.11: Sheraton T-F Plot - San Fernando Earthquake - Longitudial Direction . . . . 153 Figure 9.12: Sheraton T-F Plot - San Fernando Earthquake - Longitudial Direction - 0 to 0.8 Hz 155 Figure 11.1: S D O F Example Using the T F R F 175 IX LIST OF TABLE Table 5.1: Ambient Vibration Test Setup Locations and Directions 53 Table 5.2: Potential Natural Frequencies 57 Table 5.3: Summary of Mode Shapes 60 Table 6.1: Test Setup Locations and Directions for March 31, 1998 Test 73 Table 6.2: Test Setup Locations and Directions for April 1, 1998 Test 74 Table 6.3: Frequencies of Potential Modes 78 Table 6.4: Summary of Mode Shapes 81 Table 8.1: Frequencies of 1901 118 Table 9.1: Frequencies of Sheraton 143 X A C K N O W L E D G E M E N T First I would like to thank my parents Trish and Mike Black, my brother Fraser and my fiance Jennifer Royal for their unconditional support. I gratefully acknowledge the support of my supervisor Dr. Carlos Ventura. His experience in the field of dynamic testing and analysis has been invaluable to my research. His encouragement toward, and interest in, my academic career is much appreciated. I would like to thank Dr. Ricardo Foschi for being the second reader of this thesis and Dr. Helmut Prion for his guidance during my graduate studies at the University of British Columbia. Partial funding for this study was provided by ISIS Canada and the Natural Sciences and Engi-neering Research Council of Canada. The financial support of BC Hydro is appreciated. The technical assistance of Mr. Howard Nichol, U B C Earthquake Laboratory technician is appre-ciated. EDI Ltd. of Vancouver is acknowledged for the use of the software programs P2, V2 and U2. I would also like to thank Mr. Brian Schwartz and Dr. Mark Richardson of Vibrant Technology, Inc. for their help with Me'Scope. I would like to thank a number of current and former graduate students of the University of Brit-ish Columbia for their assistance throughout my graduate studies with special thanks to Dr. Andreas Felber, Mr. Vincent Lattendresse, Mr. Mahmoud Rezai, Mr. Tomas Horyna and Mr. Jachym Rudolf. I would like to thank Dr. Leon Cohen of the City University of New York for his valuable assis-tance in discussions of joint time-frequency analysis and his interest in the particular application of joint time frequency analysis to civil engineering. xi the memory of Nancy Lytle Black. C H A P T E R 1 Introduction The perception of our surroundings can be attributed to a large degree on our ability to distinguish frequencies which change in time. The detection of light of changing color and sound of varying pitch is an ability which, to many of us, shapes the way we interact with the world. Time-fre-quency analysis is thus fundamental to many academic fields including engineering, physics, chemistry, astronomy, geophysics and mathematics. Frequency domain methods are used widely in the field of civil engineering and in particular, earthquake engineering. The dynamic properties of structures are characterized by the frequency at which they vibrate. Earthquakes are described by their spectral, or frequency content. Building codes estimate the demand on a structure during a seismic event by prescribing the level of accel-eration to be overcome at different frequencies. With the importance of frequency in earthquake engineering it is surprising that little research has been done using joint time-frequency domain analysis. Many structures are assumed to behave linearly. That is, that the natural frequencies of structures are independent of loading and thus do not change in time. Although this assumption is false, the computational effort required to ana-lyze the dynamic behavior of structures with properties that change in time is substantial. l Chapter 1 Introduction 1.1 Objectives The objectives of this thesis are: • To explore different methods for joint time-frequency analysis • To demonstrate the applicability of joint time-frequency analysis to the analysis of civil engineering vibration data A large number of functions exist which can be used for joint time-frequency analysis. These range from the widely used and physically intuitive Short Time Fourier Transform to the arcane Wavelet Transform. An infinite class of functions known as Cohen's Class functions can also be used for joint time-frequency analysis. Each of the methods have positive and negative attributes which make them particularly useful to a specific field. This thesis briefly discusses the methods available for joint time-frequency analysis. To demonstrate the application of joint time-frequency analysis to the analysis of civil engineer-ing vibration data, a number of case studies making use of the STFT are presented. These include the analysis of data obtained from ambient vibration testing of a three span overpass in Calgary, Alberta and a single span logging bridge near Kamloops, British Columbia. An interesting phe-nomenon observed during shake table tests of a concrete gravity dam model is studied along with the seismic response of two instrumented buildings during the San Fernando and Northridge earthquakes. 1.2 Scope It is not the objective of this thesis to replace standard analysis procedures but rather show that a Chapter 1 Introduction joint time-frequency analysis can be useful to further the understanding of dynamic behavior. An attempt is made to interpret the results of the joint time-frequency analysis, but in many cases a more comprehensive study would be required to adequately explain the observed behavior. A comprehensive study would include the analyses of all of the measurements available for a certain structure not just portions of records and selected measurements. As well, a non-linear finite ele-ment model should be constructed to verify the observed results. 1.3 Outline Chapter 2 presents background information including a review of the fundamental concepts in structural dynamics and analysis in the frequency and joint time-frequency domains. Discussions on the Fourier Transform, the Short Time Fourier Transform, the Wigner Distribution, Cohen's Class functions and the Wavelet Transform are presented. Chapter 3 outlines the fundamental concepts of frequency and joint time-frequency domain analy-sis of ambient vibration and strong motion data. An introduction to the applications of joint time-frequency analysis presented in this thesis is given in Chapter 4. Chapters 5 and 6 analyze the ambient vibration of two bridges. Chapter 5 analyzes a 3 span bridge in Calgary, Alberta while Chapter 6 analyzes a single span bridge near Kamloops, British Columbia. For each bridge, a standard frequency domain analysis is conducted followed by a joint time-frequency analysis. 3 Chapter 1 Introduction Chapter 7 presents a study conducted on data collected during shake table tests of a concrete grav-ity dam monolith structure. An attempt is made to better understand behavior observed during testing. Chapters 8 and 9 explore the application of joint time-frequency analysis for the analysis of strong motion data. The records from two instrumented buildings during the California, San Fernando and Northridge earthquakes are studied. Chapter 8 analyzes the 1901 Avenue of the Stars build-ing while Chapter 9 analyzes the Sheraton Universal Hotel. A typical frequency domain analysis is conducted to obtain the mode shapes and natural frequencies followed by a joint time-fre-quency analysis to explain the presence of observed frequency shifts and the contributions of dif-ferent modes of vibration during the earthquake. Chapter 10 presents the conclusions of this thesis including suggestions for further research. A brief study of the Time Frequency Response Function is presented in Appendix A. This includes a derivation of the function, as well as an example of the application of this concept to the analysis of single degree of freedom system. Appendix B briefly describes the software programs used for the frequency and joint time-fre-quency analysis presented in this thesis. 4 C H A P T E R 2 Background This chapter presents the background information needed to understand the analysis presented in the following chapters. This includes an overview of the dynamic response of structural systems followed by a general discussion of frequency and joint time-frequency domain analysis. 2.1 Structural Dynamics To appreciate the theory and results presented later in this thesis, it is necessary to have an under-standing of the basic concepts in structural dynamics. The following sections present a cursory overview of the single and multiple degree of freedom systems. For a detailed investigation of structural dynamics refer to Chopra (1995). 2.1.1 Single Degree of Freedom System With proper assumptions, many structures can be idealized as Single Degree of Freedom (SDOF) systems. Simple structures with nearly concentrated mass m, supported by a nearly massless structure with stiffness k, in the lateral direction, can be analyzed as a SDOF system. An example of such a structure would be a walkway canopy, with a heavy concrete roof supported on light steel sections. To represent this structure as a SDOF system, the necessary assumptions would be that the light columns have negligible mass and the roof is rigid and constrained to move in one Chapter 2 Background direction. Figure 2.1 shows a spring, mass, dashpot SDOF model which could represent the above mentioned structure. c n —AAAAA-K u(t) M •0 0 Pit) — • Figure 2.1: Mass-Spring-Damper Representation of a SDOF System The easiest way to develop the equilibrium equation of motion for this system is to draw a free body diagram including the dynamic damping and inertia forces. D'Alembert's principle states that a structure is in equilibrium at every instant of time with the addition of the fictitous inertia force. This force acts in the opposite direction of the acceleration with a magnitude equal to the product of the mass and the acceleration. The dynamic free body diagram of the mass, spring, damper system is presented in Figure 2.2. 6 Chapter 2 Background u(t) u(t) Jh. m 1^ . 4 mu(t) ^ — — i ku(t) 0 0 Figure 2.2: Dynamic Free Body Diagram The equilibrium equation is: mu(t) + cu(t) + ku(t) = p(t). (2.1) Equation 2.1 assumes that the mass remains constant and the damping and stiffness are linear. The damping and stiffness forces acting on the structure are given by the relative motion of the mass and the ground where as, the inertia force is proportional to the absolute acceleration. For earthquake excitation, the acceleration term u(t), in equation 2.1, would be replaced by ux(t) + u(t). Where ug(t) is the ground motion relative to a reference point and u(t) is the acceleration of the mass relative to the ground. The resulting equation of equilibrium is: mu{t) + cu(t) + ku(t) = -mug(t). (2.2) 7 Chapter 2 Background Under free vibration, that is without any external force applied to the structure after initiation of movement, the mass will oscillate harmonically at the damped circular natural frequency of the structure co .^ This frequency can be expressed as co^  m - (c/2m) . For low levels of damping, the natural frequency of oscillation can be approximated by the relation co = Jk/m, where co is the undamped, circular natural frequency, in rad/sec. Once the equilibrium equation of motion has been formulated it is no longer a structural dynamics problem but rather a mathematical one. Methods for the solution of the equations of motion are described in detail by Chopra (1995). 2.1.2 Multi-Degree of Freedom Systems It is only in very special cases that a structure can be modeled as a single degree of freedom sys-tem. Generally the vibration of structures will require more degrees of freedom to properly char-acterize its motion. It is common practice to lump the mass of a structure at its nodes, or points were at least two members meet, or at the floor level. These nodes can have up to 6 degrees of freedom, depending on the assumptions made which may constrain the node in some way. An example of a constrained node is one where motion in the vertical and rotational directions are assumed to be negligible compared to the lateral directions. Figure 2.3 shows a three storey building model with these attributes. The mass of the structure is lumped at the floor levels, which are interconnected with columns of lateral stiffness k and dashpots with viscous damping constant, c. This planar, 2D structure can be modeled as a three degree of freedom shear beam type building; one degree of freedom to describe the horizontal motion of each floor. 8 Chapter 2 Background -*-u3(t) -*U2(t) -•u,(t) Figure 2.3: Three Degree of Freedom Model The equation of motion of the M D O F system in Figure 2.3 would have the same form as the SDOF equation with the constants m, k and c replaced with 3 by 3 matrices [M], [K] and /C], cor-responding to the three degrees of freedom and the time dependent variables u(t) and p(t) replaced by time dependent vectors { U(t)} and {P(t)} . The resulting equation for the M D O F system in Figure 2.3 is given by Equation 2.3. m 0 0 0 m 0 0 0 ? u\(t) u2(t) "3(f) 2c -c 0 u\(t) 4k -2k 0 «j(f) -c 2c -c u2(t) + -2k 4k -2k u2(t) = P2(t) (2.3) 0 -c c_ "3(f) _ 0 -2k 2k_ " 3 (0 P3(t) 9 1 Chapter 2 Background or, [M]{U(t)} + [C]{U(t)} + [K]{U{t)} = {/>(*)}. (2.4) Equation 2.4 is a system of coupled, 2nd order, differential equations in time. To solve this equa-tion one can make use of the orthogonality property of the mode shapes. Orthogonality means that one mode shape can not be represented by a combination of other mode shapes. The undamped, free vibration response of a classically damped structure will have the form: where {<])„} is mode shape vector for mode n, co„ is the circular natural frequency of mode n and an and (p„ depend on the initial conditions of motion of the system. For classically damped structures, the mode shape vectors of the undamped system form the basis for the solution of the equation of motion just as the x, y and z planes form a basis for describing a position in three dimensional space. That is, the structure will vibrate in a shape which is made up of a linear combination of the normal mode shape vectors. If equation 2.5 is substituted into equation 2.4, the result is the classical eigenvalue problem. Solution of this problem leads to the natural frequencies and natural mode shapes of the system. For a discussion on solving the eigen-value problem and the effects of non-classical damping see Humar (1990). {U(t)} = £a„{() )„}s in(0V + <p„) (2.5) n 10 Chapter 2 Background The mode shape vectors for a classically damped structure have the following orthogonality prop-erties: WifmWi} = m. and {^}T[M]{^} = 0 for i*j •WifmWi} = k. and W.fmty} = 0 ' for i*j (2.6) {<MT[C]{(|>.} = c. and WifiCMj} = 0 fori^y where { } T represents the transpose of a vector and the parameters m,, kt and c, are called the modal mass, stiffness and damping coefficients associated with the ith mode of the structure. For a non-classically damped structure the third set of properties, given in Equation 2.6, do not hold. A classically damped structure can therefore be defined as one that satisfies this property. This idealized and desirable situation results from assuming that the damping present in the struc-ture is mass or stiffness proportional (see Humar, 1990). The displacements of the force-excited system can be determined from the equation: where y,(r) are the normalized coordinates which are determined from the solution of the uncou-pled, viscously damped, single degree of freedom system represented by equation 2.8: (2.7) 11 Chapter 2 Background ^ • ( 0 +^,.(0 + ^ . ( 0 = Pi(t) (2.8) where pt(t) is the modal excitation given by: Pi(t) = Wifpit). (2.9) Solution of equation 2.8 involves the solution of an uncoupled, 2nd order ordinary differential equation similar to that given by equation 2.1. Therefore, a n-degree of freedom system can be broken down into a series of n, single degree of freedom systems, each with modal mass ra,, modal stiffness kt and modal damping c, . These single degree of freedom systems can be solved by a number of methods and then related back to the original structure using its associated mode shape {()),} and equation 2.7. Methods for the solution of the equation of motion are pre-sented in Chopra (1995). 2.1.3 Comment on Natural Frequencies A major focus of this thesis is the application of methods that would help determine if natural fre-quencies of a structural system change in time. However, for truly linear elastic systems, the nat-ural frequencies, by definition, do not change and thus a structure's natural frequencies are independent of the loading conditions and time invariant. A natural frequency which changes in time should be thought of as an instantaneous or temporary natural frequency and the observed change is relative to the actual natural frequency of the linear system. 12 Chapter 2 Background The term "natural" frequency is used throughout this thesis to describe the dominant frequencies of all systems, whether they are time invariant or not. 2.2 Frequency Domain Analysis When examining the dynamic response of structures it is convenient to talk in terms of frequency. As part of a typical analysis of structural vibration data, one is concerned with the frequency of the excitation and/or the natural frequency of the structure. As was seen in the proceeding sec-tion, the vibration of a structure at any point in time can be decomposed into a combination of its mode shapes oscillating at their associated natural frequencies. If a structure is excited by a forc-ing function with a frequency equal to a natural frequency of the structure, resonance occurs. At resonance, the energy present in the input force is added, in phase, to the energy of the structure and thus the response can grow to levels much larger than normal operating levels. The introduction of the Fourier Series and Fourier Transform have greatly facilitated frequency or spectral analysis. The basic mathematical representation of periodic signals is the Fourier Series, which is a weighted sum of harmonically related sinusoids or complex exponentials. Jean Baptist Joseph Fourier (1768-1830), a French mathematician, originally used such a series to describe the phenomenon of heat conduction and temperature distribution through bodies (Proakis and Mano-laksi, 1996). For an aperiodic signal of finite energy, analysis in the frequency domain is made possible by the Fourier Transform. The Fourier Transform is a special case of the Fourier series as the signal is periodic with infinite period. The frequency analysis of aperiodic, continuous time signals involves the following transform pair: 13 Chapter 2 Background Direct Fourier Transform: X((o) = \ x(t)e'mdt (2.10) — o o o o Synthesis Equation: x(t) = -z~- [ X(m)e~'m'd(a (2.11) IK J — o o Equation 2.10 is the Direct Fourier Transform and Equation 2.11 is the synthesis, or inverse, Fou-rier Transform. Most time signals which are analyzed today are discrete, or digitized, time sig-nals. This representation of the signal requires the use of a slightly different transform pair. The discrete value of x(t), at time t = r A , is written {xr} and the series {xr} is called a discrete time series. If T is the length of the signal in seconds and N is the number of discrete points then the time step A is given by A = T/N. The range of Fourier components Xk is limited to k = 0 to k = N - 1, which correspond to harmonics of frequency co* = 2nk/T = 2nk/NA. The Dis-crete Fourier Transform (DFT) pair is given by (Newland, 1993): N- 1 DFT: Xk = ^ £ x r e i ( 2 n k r / N ) k = 0,1,2,..., (N-l) (2.12) r = 0 N- 1 Inverse DFT: xr = J Xke~K2nkr/N) r = 0,1,2,..., (N-l) (2.13) k = 0 14 Chapter 2 Background Methods for analysis in the frequency domain have been around since the early 19th century but the computational effort required for complex, aperiodic signals made the use of this method practically impossible. In the 1960's an algorithm was developed which enabled the calculation of the Discrete Fourier Transform and Inverse Discrete Fourier Transform quickly and was hence named the Fast Fourier Transform or FFT (Cooley and Tukey, 1965). This algorithm makes use of the harmonic properties of the transform to greatly reduce the computation required. The intro-duction of the FFT lead to an explosion of research in, and application of, analysis in the fre-quency domain. Refer to Ramirez (1985) for a derivation of the Fast Fourier Transform. Analysis which makes use of the Fourier Transform is said to be a frequency domain analysis. Analysis in the frequency domain encompasses many different techniques depending on the nature of the experimental data. There are a few basic representations of a signal, or signals, in the frequency domain which are common to many applications. If Xa(co) is the Fourier Transform of a signal, then the auto power spectrum Gou(co) is given by: Gaa(o» = X f l(co)<(co) (2.14) where * denotes the complex conjugate. The auto power spectrum is a real valued expression which is proportional to the energy, or power, in the signal. 15 Chapter 2 For 2 time signals xa(t) and xh(t), the cross power spectrum is given by: Background G a b ^ =Xa(C0)XbW. (2.15) The coherence function "/^(co), is a measure of the likeness of two signals. It can be used to esti-mate the amount of noise present in the input-output signals and the presence of non-linearities. The coherence function between two signals "a" and "b" is given by: where Gah((a) is the cross spectrum and G a a(co) and Ghh((£>) are the auto spectrums of signals "a" and "b" respectively. The Frequency Response Function (FRF), is used extensively in frequency domain analysis of structures. It is defined as the response of a structure to an harmonic forcing function of unit amplitude. The dynamic equation of equilibrium with a unit amplitude harmonic forcing function is given by: mx + cx + kx = eim. (2.17) If the response x(t) is assumed to be given by x(t) = H(®)e"°', then Equation (2.17) can be rewritten as: yabW = (2.16) G a a ^ G b b ^ 16 Chapter 2 Background (- co2m + /cue + k)H(a)eiat = eim. (2.18) Now the Frequency Response Function, H(co) can be written as the ratio of output motion to input force: //(co) = - 1 , . (2.19) k - a r m + zcoc Using the frequency ratio P=co/con Equation (2.19) can be rewritten to yield: H(Q) = (2.20) The FRF H((3), is a complex valued expression and more commonly expressed in terms of its magnitude IH(P)I and phase angle <j)(P): |ff(P)| = I . 1 = (2.21) ^ ( 1 - P 2 ) 2 + (2P^) 2 «|>(P) = a t a n ^ ^ j (2.22) It is obvious from Equations 2.21 and 2.22 that, for small damping, the magnitude of the FRF tends to infinity at resonance (P=l) and the phase angle moves through 90 degrees. There are 2 ways in which the experimental FRF can be calculated. The choice depends on the anticipated location of significant noise. If the input signal is denoted 'a' and the output signal as 17 Chapter 2 ' b \ then the two methods are: Background * , « . ) - ^ (2.23) and Ga * ( ° > ) Most analysis programs have only one of the Frequency Response Functions; usually H{ (co) as it is easier to compute (Ewins, 1984). (co) is the better estimate when there is more noise in the output signal and H2(a>) is a better estimate when there is more noise in the input signal. The Frequency Response Function is a complex expression and therefore contains real and imag-inary components. At resonance, there is a peak in the imaginary part of the FRF and a corre-sponding zero value for the real part. Multiplying the numerator and denominator of Equation 2.20 by the conjugate of the denominator of Equation 2.20 yields: m ) _ K d - P 2 ) 2 - ^ ) ( P ) " fc(l-P2)2 + 4 p 2 ^ 2 V-*) which can be split into its real and imaginary components, as shown in Equation 2.26: H m -1 iLz£i f i m (2 26) W - ( 3 2 ) 2 + 4(32^2 k ( l _ p2)2 + 4 p2£2- ^ > 18 Chapter 2 Background It is obvious that at resonance the real part of Equation 2.26 is zero. Figure 2.4 shows the real and imaginary parts of Equation 2.26 for low levels of damping (5%). It shows that the imaginary part of the FRF is a maximum at (3=1. It should be noted however, that as damping increases, the peak moves closer to the origin. 15 10 5 0 -5 -10 •15 FREQUENCY RATIO — Real -^Imaginary Figure 2.4: Real and Imaginary Parts of the FRF One drawback of frequency domain analysis is that it can only be used for the analysis of linear, time invariant response. If the dynamic properties of the structure change in time, these changes will not be identified and will usually introduce anomalous results which complicate the analysis procedure. 19 Chapter 2 Background 2.3 Joint Time-Frequency Domain Analysis Joint time-frequency analysis involves the analysis of a signal in both the time and frequency domain simultaneously. There are two basic approaches to analysis in the joint time-frequency domain. The first approach is to initially cut the signal into slices in time and examine their fre-quency content. The second approach involves the filtering of discrete frequency bands which are in turn, sliced into discrete time bands and analyzed for their energy content. The first describes the Short Time Fourier Transform and Cohen's Class functions, while the latter of the two approaches describes the Wavelet Transform method of analysis. 2.3.1 Short Time Fourier Transform Analysis using the Short Time Fourier Transform, (STFT) involves emphasizing the signal at a desired time t and suppressing it all other times. This is accomplished by multiplying the time signal s{t) by a window function w(t), centered at time t. The window function is chosen to leave the signal unaltered at time t and near zero at distant times. The choice depends on the nature of the data and the desired results. A rectangular window leaves the signal unaltered at time t but it has poor frequency domain properties. The discontinuity of the window, in the time domain, introduces ripples in the frequency domain. These ripples are known as sidelobes. To alleviate the presence of large sidelobe oscillations, a window function which gradually decays toward zero should be used (Proakis and Manolakis, 1996). Therefore, the Hanning window was chosen for the analysis in this thesis. Chapter 2 For a Hanning window of M points, the time domain sequence has the form: w(0 = -I 1 - c o s — J Background (2.27) where T is the length of the window in seconds. A plot of Equation 2.27 for a window of 6.4 sec-onds is given in Figure 2.5. 3.2 Time (sec) Figure 2.5: Hanning Window Function The resulting windowed signal has the form: st(l) = s(T)w(T-t) (2.28) 21 Chapter 2 Background where x is the running time variable and w(x -1) is the window function centered at time t. If the time domain window size is taken to be small, the frequency band of the resulting spectrum will be large and hence is called a wide band spectrum. Conversely, if the window size is taken to be large in the time domain, the resulting frequency spectrum is denoted a narrow band spectrum. The STFT of the windowed signal s,(x) is given by Equation 2.29 and represents the distribution of frequency around time t. SX<$) = " i = f e m%s(x)w(x-t)dx 1 J2n J (2.29) The energy density spectrum, or spectrogram, is defined as (Cohen, 1995): P(t, CO) = St((0) = -ZCOT == f e *™"s(x)w(x-t)dx (2.30) where | | denotes the absolute value of the expression. To obtain the density of one variable in a joint density described by two variables, the other vari-able is integrated out. The resulting density is known as the marginal density or marginal. The summation of the energy distribution for all frequencies, at a particular time in a signal should give the instantaneous energy, and conversely, the summation over all times, at a particular fre-22 Chapter 2 quency, should give the energy density spectrum, trogram should satisfy: a) for the instantaneous energy Background Therefore, ideally, a joint time-frequency spec-o o J P(t, CG)dCO = \s(t)\2 (2.31) — o o and b) for the energy density spectrum o o J P(t, (0)dt = |S(C0)| 2 (2.32) — o o which are referred to as the time and frequency marginal conditions (Cohen, 1995). As well as satisfying the marginals, the total energy of the signal should be independent of the method used to calculate it. This is the case for distributions that satisfy Parceval's theorem, which is given by: o o o o E = j \s(t)\2dt = J \S((0)\2dw, (2.33) — o o — o o 23 Chapter 2 Background where E represents the energy of the signal. If a joint time-frequency spectrogram satisfies the marginals then it automatically satisfies the total energy requirement, although the inverse is not necessarily true (Cohen, 1995). One potential problem with the STFT is that the energy of the signal is scrambled with that of the window. This has the effect of introducing energy into the windowed signal which is not present in the original signal and therefore the time and frequency marginals are not satisfied. This failure to satisfy the marginals means that results obtained from the STFT may be slightly distorted with respect to the actual frequency content of the original signal in that, it will not give the correct answers for averages of functions of frequency or time (Cohen, 1995). Even though the STFT does not satisfy the marginals it can satisfy the total energy requirement if the window is normal-ized to have unit energy (Cohen, 1995). Another problem with the STFT, and all joint time-frequency analysis methods, is that the simul-taneous resolution in the time and frequency domains is limited by the uncertainty principle. That is, the frequency and time resolutions are interrelated. The uncertainty principle will be dis-cussed in a following section. Another problem with the STFT is the occurrence of leakage. A windowed spectrum is not local-ized to a single frequency and thus the power of the original signal sequence is spread by the win-dow to the entire frequency range of the window (Proakis and Manolakis, 1996). In a sense, the power of the signal is "leaked" into the entire frequency range of the window. To reduce leakage, a window with smaller sidelobes, such as the Hanning window is used (Proakis and Manolakis, 24 Chapter 2 Background 1996). When the length of the windowed section decreases the effect of leakage is more pro-nounced as the power of the window becomes significant compared to the signal. When using Short Time Fourier analysis, the choice of the window size is important as it controls which frequencies, or periods, can be identified by the analysis. Usually it is preferable to have a window which is longer, in time, than the longest dominant period in the signal analyzed. If this criteria is not met, the Fourier Transform of the windowed signal may not accurately identify the spectral content. In the case of long period oscillation, this consideration may seriously limit the time resolution of a joint time-frequency analysis. For example, the analysis of a structure which has a fundamental period of 2 seconds requires a windowed section with a minimum width of 2 seconds to accurately identify the fundamental period. As the length of the period increases, the window size used in the analysis must increase thus limiting the time resolution of the joint time-frequency analysis. 2.3.2 Wigner Distribution The Wigner Distribution, W(t, co), was originally introduced in the field of quantum mechanics and later introduced into signal analysis by Ville (Cohen, 1995). It is the prototype of a set of dis-tributions that are qualitatively different from the spectrogram. The Wigner Distribution is said to be bilinear in the signal as the signal enters twice in its calculation as shown in Equation 2.34. o o -/cox dx (2.34) — O O 25 Chapter 2 Background The Wigner Distribution does not use a window function to estimate the frequency content at a specific time but rather it folds the signal a time t and emphasizes overlapping frequencies. A positive attribute of the Wigner Distribution is that it satisfies the time and frequency marginals and hence Parceval's theorem. A serious consequence of this, at least for the analysis considered in this thesis, is that a bilinear distribution which satisfies the marginals cannot be positive throughout the time-frequency plane (Wigner, 1932). A time-frequency distribution which is not manifestly positive makes the identification of a structure's natural frequencies considerably more difficult and in some cases impossible. For signals with more than one dominant frequency, or multi-component signals, such as those found in civil engineering applications, a negative distri-bution can confuse the identification process. The physical significance of the joint time-fre-quency distribution is lost and many of the useful functions in the frequency domain, such as the Frequency Response Function, can not be represented clearly in the joint time-frequency domain with the presence of negative values. Another problem with the Wigner Distribution, and to a certain extent, all bilinear distributions, is the presence of cross terms in multi-component signals. This so called interference results from folding the signal. Due to interference, the Wigner Distribution is not necessarily zero at times when the signal is zero and, as well, it is not zero for frequencies that do not exist in the spectrum (Cohen, 1995). Figure 2.6 shows a comparison of the spectrogram and the Wigner Distributions for two multicomponent signals. The Wigner Distribution shows significant interference which makes the identification of distinct frequency components more difficult. 26 Chapter 2 Background (After: Cohen, 1995) Figure 2.6: Comparison of the Spectrogram and the Wigner Distribution 2.3.3 Cohen's Class Functions The Wigner Distribution was the first example of a joint time-frequency distribution which was qualitatively different then the Spectrogram. There exists however, a large family of joint time-frequency distributions which are similar to the Wigner Distribution. Cohen has shown that almost all time-frequency representations, of which there are many, can be obtained from: 27 Chapter 2 Background C(7, co) = —\\\s*(u-^x]s(u-\x](^(Q,x)e - iQt-ixm + (9M dudxdQ (2.35) where (j)(8, x) is a two dimensional function called the kernel. The Kernel determines the distri-bution and its properties. The variables u, x and 0 are different for each distribution. It is shown by Cohen (1995) that the Kernel for the Wigner Distribution is 1 while the Kernel for the spectro-gram is: The application of bilinear transforms, such as those belonging to the Cohen Class functions makes it possible to overcome the time-frequency resolution limitations encountered with the Short Time-Fourier Transform, since they are not based on signal segmentation (Bonato et al., 1997). As mentioned for the Wigner Distribution however, the Cohen Class Functions posses serious problems which make their use impractical for the analysis of civil engineering structures. These problems include the generation of spurious interference terms and the presence of nega-tive values in the time-frequency plane. 2.3.4 Wavelet Transform The Wavelet Transform has come into prominence in the last decade. Its application is primarily in the field of electrical engineering and is used for such things as image compression. The Wavelet Transform represents a time domain function f(t) as a linear combination of a fam-28 (2.36) Chapter 2 Background ily of basis functions. The basis functions are generated by scaling (dilation or expansion) and shifting of a single function fc(0 referred to as the mother wavelet (Daubechies, 1998). Mother wavelets must be oscillatory and have amplitudes that quickly decay to zero away from the center of the wavelet. A family of wavelet bases 4 / a / p ( f ) , is given by: where a is the scaling parameter and b is the shifting variable. The continuous Wavelet Trans-form, W(a, b), is defined as the inner products of f{t) and fc(r) and is given by equation 2.38: where *F* h{t) is the complex conjugate of ^ h(t). The parameters a and b are directly related to frequency and time respectively and, therefore, the Wavelet Transform can be used to analyze a signal in the joint time-frequency domain. The form of Equation 2.38 is similar to the Fourier Transform with the exponential term replaced by the wavelet function. The difference is that the Fourier transform is a weighted sum of sines and cosines while the Wavelet Transform is a weighted sum of wavelet bases functions. An advantage of the Wavelet Transform is that it is a multi-resolution transform. That is, it ana-lyzes high frequencies with good time resolution but poor frequency resolution and low frequen-29 (2.37) o o (2.38) — O O Chapter 2 Background cies with good frequency resolution but poor time resolution (Bonato, 1997). The poor time resolution of the Wavelet Transform, in the low frequency range, is due to the dilation of the mother wavelet at low frequencies in order to capture oscillations in that frequency range. In the high frequency range, the wavelet function is compressed to capture the high frequency oscilla-tions. This has the effect of producing a wide frequency spectrum which is localized in time. For more information on the Wavelet Transform see Newland (1996) or Rezai and Ventura (1995). 2.3.5 The Uncertainty Principle The uncertainty principle is an inescapable property of nature. It was first derived by W. Heisen-berg in 1927 in the field of quantum mechanics. The physicist C.G. Darwin (grandson of Charles Darwin) made the connection between the uncertainty principle in quantum mechanics to Fourier Transform pairs. A statement made by Skolnik (Skolnik, 1980) on the use of the name uncer-tainty principle, as applied to signal analysis, provides a useful introduction to the principle: "The use of the word 'uncertainty' is a misnomer, for there is nothing uncertain about the 'uncer-tainty relation' . . . . It states the well-known mathematical fact that a narrow waveform yields a wide spectrum and a wide waveform yields a narrow spectrum and both the time waveform and frequency spectrum cannot be made arbitrarily small simultaneously." A signal, or phenomenon that is narrow in one domain must be broad in the other domain, which is mathematically formulated: 30 Chapter 2 Background AtAf> P (2.39) The interpretation of P in Equation 2.39 is rather arbitrary and loses importance for practical con-siderations (Dossing, 1998). What is important is that time and frequency are interrelated and thus cannot both be made arbitrarily small. More specifically, if T, and B, are taken to be the duration and bandwidth of time and frequency of a normalized windowed segment, it can be shown that the uncertainty principle for the Short Time Fourier Transform is given by (Cohen, 1995): It is known that for an infinitely short duration signal, the bandwidth of the signal becomes infi-nite. With this fact in mind, it is intuitively obvious that as the duration of the signal decreases the frequency bandwidth approaches infinity which is confirmed by the inverse relation between the standard deviation of time and frequency in Equation 2.40. For further information on the uncertainty principle refer to Dossing (1998). A proof of the uncertainty principle and Equation 2.40 is given by Cohen (1995). Most of the literature on the use of joint time-frequency analysis is in the field of electrical engi-neering. There exists relatively few papers on the subject of the joint-time frequency analysis of (2.40) 2.4 Literature Review 31 Chapter 2 Background civil engineering vibration data. Bonato et al. (1997) investigated the use of Cohen's Class functions for the identification of mode shapes and natural frequencies of bridges from ambient vibration data. They use relatively advanced kernel filtering methods in attempt to overcome the problems with the interference terms resulting from the use of the Wigner Distribution. Huang et al. (1994) used the Wavelet Transform for the identification of the mass, stiffness and damping characteristics of simulated structural systems. Staszewski and Giacomin (1997) use the Wavelet transform for the analysis of mechanical engi-neering data. They define a wavelet based FRF similar to the TFRF defined in this thesis, to study road-vehicle interaction data. 32 C H A P T E R 3 Experimental Dynamics The use of analytical methods to calculate the natural frequencies and mode shapes of a structure requires knowledge of the mass, stiffness and damping characteristics of each of the elements within the structure. This information may be difficult to acquire, and in some cases impossible to predict given the potential uncertainty in the boundary conditions or the construction quality. Therefore, it may be necessary to determine the dynamic properties by experimental means. There are numerous ways in which a structure can be tested dynamically. The methods used depend on the size and characteristics of the structure. Typical forced vibration tests include: shaker, impact and pullback tests. Shake table tests are generally applicable to relatively small, or scaled structures while ambient vibration tests are suitable for obtaining dynamic characteristics from a wide range of structures. Another important source of data is the recorded response of instrumented structures during significant seismic events. This type of data is generally referred to as strong motion data. It is important to note that for this study nonparametric evaluation techniques are used. A non-parametric study is usually only considered qualitative because of the inherent errors in the proce-dure (Fenves and Desroches, 1994). A parametric study would include the matching or curve-33 Chapter 3 Experimental Dynamics fitting of an analytically derived expression for the Frequency Response Function with data obtained experimentally. As these modal parameter estimation techniques are out of the scope of this thesis they will not be discussed. For a detailed description of various frequency domain methods for parametric estimation techniques see Ewins, (1984). The nonparametric analysis of the strong motion data presented in this thesis is comprised of visual inspection of frequency and joint time-frequency domain information such as the fre-quency response functions and spectrograms. The inherent errors in this method stem from the subjectivity of the decision making process. The decision of which peak corresponds to natural frequencies, although sometimes obvious, is left to the researcher. 3.1 Ambient Vibration Ambient vibration testing refers to the measurement of the response of a structure to the surround-ing, or ambient, excitation. This excitation is provided by wind, human activity, micro-tremors, traffic and operating machinery. Unlike forced vibration testing, one does not measure the input into the structure. Natural frequencies and mode shapes are estimated by taking simultaneous measurements at predetermined locations of the structure. When using information obtained from ambient vibration measurements, the designer should be aware that the dynamic properties of a structure at low levels of vibration may be different, and in some cases significantly different, than those of the same structure subjected to greater levels of vibration. This should considered when the designer calibrates a computer finite element model of the structure being studied. 34 Chapter 3 Experimental Dynamics The most common method of analysis for ambient vibration data involves using frequency domain techniques. These techniques are well established and will be discussed in the following sections. There are however, other methods of analysis such as Random Decrement which are also capable of analyzing ambient vibration data. The Random Decrement method is a time domain method which uses the properties of the response of structures to white noise input to cal-culate the dynamic properties. One advantage of the Random Decrement technique is that it is capable of estimating the structural damping more accurately than frequency domain techniques. For more information on the Random Decrement method see Asmusen (1997). Joint time-frequency analysis enhances the standard analysis procedures as one is able to verify that the response of the structure does not change significantly during the test. It has been shown that temperature can significantly effect the frequencies of a structure (Farrar et al., 1998) and therefore, it is of interest to conduct a joint time frequency analysis on ambient vibration data col-lected over an entire day. 3.1.1 Frequency Domain Methods for the Analysis of Ambient Vibra-tion Data To use the techniques described in this section the structure under study must satisfy a number of conditions. In general, the structure must not violate the following assumptions: • The structure behaves as a linear system meaning that superposition is applicable. • The ambient excitation has white noise characteristics. • A l l of the modes of interest are excited. Practice shows that the above assumptions are usually satisfied for most structures. 35 Chapter 3 Experimental Dynamics Estimating Natural Frequencies The natural frequencies are estimated from the peaks in the auto spectra or the Power Spectral Density, (PSD), of ambient vibration measurements. If Xfl(co) is the Fourier Transform of the ambient vibration recorded, then the auto spectra GUfl(co) is given by: G a a ^ = ^ where * denotes the complex conjugate which is obtained by multiplying the imaginary part of the Fourier Transform by -1. To avoid missing modes of vibration by calculating the PSD at a node location, the PSDs from all points of the structure are averaged. This so called Averaged Normalized Power Spectral Density (ANPSD) function gives a better estimate of the natural fre-quencies (Felber, 1994). Estimating Mode Shapes Mode shapes are estimated from the relative motion between a reference location and other loca-tions chosen for measurement. Each measured location will allow that point to be defined and therefore, it is important to measure a sufficient number of locations to insure the accurate repre-sentation of the vibration shape. At a frequency corresponding to a natural frequency, a structure vibrates harmonically with a deflected shape equivalent to its mode shape. A l l the information needed to construct the deflected shape can be obtained from the transfer function Tsh((Q) between a measured point and a reference point. For two signals "a" and "b", the transfer function can be expressed as: 36 Chapter 3 Experimental Dynamics Tab™ = Xb((o)X*b((o) G « « ( ( 0 ) (3.2) The absolute value of the transfer function gives the amplitude of the point relative to the refer-ence sensor and the phase, between the two signals, gives the orientation. That is, if the phase between the two points is zero, the points are moving in the same direction. Similarly, if the phase is near 180 degrees the points are moving in opposite directions and the absolute value of Equa-tion 3.2 would be multiplied by -1. The resulting expression is denoted the Modal Ratio. For additional information on the theoretical derivation refer to Felber (1993), Diehl (1991) or Luz (1986). 3.1.2 Joint-Time Frequency Domain Analysis of Ambient Vibration Data A joint time-frequency analysis lends itself particularly well to ambient vibration data as the exci-tation is presumed to have white noise characteristics and thus the peaks in the spectrogram indi-cate potential natural frequencies. The spectrogram does not contain any phase information and thus it would not be possible to con-struct mode shapes using the techniques described in section 3.1.1. As stated earlier, the motivation for the joint time-frequency analysis is to determine if the domi-nant frequencies of vibration remain constant over the course of the test, or if factors such as tem-perature or level of input cause them to shift. Joint time-frequency analysis can also be useful in 37 Chapter 3 Experimental Dynamics the explanation of anomalous frequency domain results. 3.2 Strong Motion Data Strong motion data refers to the data collected from instrumented buildings during significant earthquakes. It is common for structures to be instrumented in seismically active zones. For example, during the 1994 Northridge earthquake in California, the California Strong Motion Instrumentation Program collected data from 77 instrumented structures (Finn et al., 1995). The structural response information collected during a seismic event is used to study the behavior of structures under seismic loading and their dynamic properties in general. For example, the maximum displacement, velocity and accelerations recorded during different events are used in the design of new structures. The records may be used to calculate the mode shapes and frequen-cies of the structure. If the building was damaged during the earthquake the strong motion records could be used to identify the severity and possibly the location of the damage. There exists a number of methods available for the determination of the dynamic properties of structures from strong motion data. The most prevalent being frequency and time domain analy-sis techniques. Before introducing the frequency domain and joint time-frequency domain analysis techniques, it is useful to discuss the general behavior of buildings as it provides a starting point from which to begin the analysis. It is known that buildings exhibit certain trends in the frequency ratios between the first mode and successive modes. Shear type buildings usually have 1/3/5 ratios 38 Chapter 3 Experimental Dynamics where as bending type buildings have 1/5/17 ratios (Chopra, 1995). After the first mode is iden-tified, the second mode frequency is probably between 3 and 5 times the first mode frequency. The National Building Code of Canada estimates the fundamental period of a building by n/10, where n is the number of stories (Associate Committee on National Building Code, 1995). If the building is 25 stories high the first mode is probably around 2.5 seconds or 0.4 Hz. 3.2.1 Frequency Domain Analysis of Strong Motion Data The peaks in the Frequency Response Function (FRF) are usually not sufficient to clearly estab-lish modal frequencies. The level of noise, material non-linearity, minor damage and vibration induced by the motion of non-structural components yield inconclusive FRF measurements. For this study, the natural frequencies and mode shapes are identified using several different tech-niques. Estimating Natural Frequencies The strong motion records are used to calculate the FRF and its components, the coherence func-tion and mode shapes. This information combined with a general knowledge of building behavior allows the natural frequencies to be estimated. The FRF is the response of a structure to a unit harmonic input. For a harmonic excitation at a particular frequency, it is intuitively obvious that the ratio of output motion to input force, at the resonant frequency, would be greater than at all other frequencies. Therefore, when looking at the FRF of building response, the peaks may indicate a possible mode of the structure. The frequency response functions are calculated by taking the record at the base as the input into the structure 39 Chapter 3 Experimental Dynamics and all other measurements as the outputs. Strictly speaking, when an acceleration record is used as the input rather than a force time history, the result is not a Frequency Response Function, but rather a dimensionless ratio of input to output motion. If the mass of the building was know, the dimensionless FRF could be divided by the mass to yield the standard FRF as you would have output motion over input base shear. In this thesis the dimensionless FRF and standard FRF are used interchangeably. As mentioned in the previous chapter, FRF is a complex function with both real and imaginary components. At a frequency which corresponds to a natural frequency, the real part of the FRF should be zero and the corresponding imaginary component will peak. The coherence function, y„ t(co), is a real valued function of frequency that indicates the fraction of response that is linearly related to the input force. For a structure that remains elastic and has no extraneous inputs, or noise the coherence function should be unity. Experimental modal anal-ysis shows that near resonance, the coherence function usually drops from unity. The lack of coherence can be attributed to bias errors, as the frequency resolution of the analyzer is not fine enough to describe the very rapidly changing functions encountered near resonance (Ewins, 1995). For the same reason, a lack of coherence is also usually seen at low and high frequencies. The properties described above are applicable to a single degree of freedom system. For a multi-degree of freedom system which has well separated modes, the response can be decomposed into a series of SDOF systems. This is due to the fact that at resonance, the response will be domi-nated by the mode in resonance and the contribution of all the other modes will be negligible. 40 Chapter 3 Experimental Dynamics Therefore, the methods presented for the analysis of SDOF systems are applicable to MDOF sys-tems with well separated modes. Estimating Mode Shapes The mode shapes of an instrumented building are determined by plotting the relative amplitude of the transfer functions (FRFs) between the input signal, measured at the base, and the output sig-nals measured throughout the building. This, however, can be misleading if there are not enough measurement points to accurately display the vibration shape. For instance, the animation of a building with 1 sensor at the basement and 1 at the roof level will only be capable of animating the first mode of the structure. In general, the highest mode which can be animated is one less then the number of floors with sensors. 3.2.2 Joint Time-Frequency Analysis If the modal characteristics of a structure are thought to be time variant a joint time-frequency analysis can be useful in improving the understanding of the structure's behavior. These tech-niques are used to investigate the "softening" phenomena observed during heavy shaking and for the detection of damage during a seismic event. Softening refers to the reduction in stiffness sometimes observed during periods of heavy shaking. A Time-Frequency Response Function (TFRF) is approximated by taking the ratio of the output time-frequency spectrogram to the input time-frequency spectrogram. The signal used for the input is the recorded acceleration time history at the base of a structure and the output signal is taken to be the acceleration time history at the roof of the structure. 41 Chapter 3 Experimental Dynamics As described previously, the time-frequency spectrogram P((£>,t) is obtained from P((0,t) = I S J O l 2 (3-3) where ^ ( O l 2 is the Short Time Fourier Transform of a time signal. If P(co, t)in represents the time-frequency spectrogram of the input and P(co, t)out the output time-frequency spectrogram, the TFRF is given by: TFRF((x), t) = ° U t (3.4) The TFRF can be used in much the same way as the FRF to estimate natural frequencies with one exception. To obtain a joint time-frequency distribution, the STFT of the signal must be squared, as shown in Equation 3.3, and thus the phase information is lost. This has the effect of limiting the analysis to estimating natural frequencies from peaks in the TFRFs. The TFRF is described in more detail in Appendix A. 42 C H A P T E R 4 Introduction to Case Studies Five examples of the application of joint time-frequency analysis to the civil engineering vibra-tion data are presented in this thesis. These examples include the ambient vibration measure-ments of 2 bridges, the response of a dam model during shake table testing and the response of two instrumented buildings during the Northridge and San Fernando earthquakes. 4.1 Ambient Vibration Data The dynamic characteristics of two bridges obtained from ambient vibration measurements are studied in this thesis. A 3 span bridge located in Calgary, Alberta is presented in Chapter 5, and a single span bridge near Kamloops, B.C. is presented in Chapter 6. For each bridge, a frequency domain analysis is conducted to determine the natural frequencies and mode shapes followed by a joint time-frequency analysis. 4.2 Shake Table Data During the shake table testing of an unbonded concrete gravity dam model, an interesting phe-nomenon was observed. At certain combinations of base excitation amplitude and frequency the model moved upstream. That is, the model moved in a direction opposite to that of the applied hydrostatic load. Joint time-frequency analysis is performed to better understand this behavior. 43 Chapter 4 Introduction to Case Studies 4.3 Strong Motion Data A typical frequency domain analysis procedure assumes that the structure behaves in a linear, time invariant manner and this assumption does not hold in some instances. During an earthquake the forces carried by the structure are high and in some cases may cause non-linear behavior or damage. Joint time-frequency analysis of recorded strong motions can be used to investigate non-linear behavior and determine the temporal changes in frequency content. To investigate the shift in the natural frequencies of structures often observed during seismic events, the response of two instrumented buildings is determined through analysis of strong motion records obtained from two earthquakes. The buildings analyzed are the The Sheraton Universal Hotel (Sheraton) and the 1901 Avenue of the Stars (1901). These buildings are located in Southern California, in the Los Angeles area. A similar study, involving instrumented building in California was conducted by Rezai and Ventura (1996) in which damage was detected. Each building has complete records from both the San Fernando and Northridge Earthquakes. The San Fernando earthquake occurred on February 9, 1971 at 7:00 am PST. The hypocenter was about 27 km from downtown L A with a focal depth of 8.5 km (Oakeshott, 1975). The earthquake lasted around 45 seconds and had a magnitude of M L = 6.5. The Northridge earthquake occurred on January 17, 1994 at 4:31 am PST. The hypocenter was about 32 km northwest of Los Angeles with a focal depth of 19 km (Hall, 1995). The Northridge earthquake lasted around 45 seconds and had a magnitude of M L = 6.6. A map showing the location of the San Fernando and Northridge earthquakes and the two buildings is given in Figure 4.1 44 Chapter 4 Introduction to Case Studies (Adapted from: Finn et al., 1994) Figure 4.1: Location of the San Fernando and Northridge Earthquakes 45 Chapter 4 Introduction to Case Studies Having more than one event record for each buildings allows for the comparison of the natural frequencies obtained from different earthquakes. A frequency domain analysis is conducted on each building to determine its natural frequencies and corresponding mode shapes. With this information, an attempt is made to identify large shifts in the natural frequencies of the structures. To further investigate the observed shift, a joint time-frequency analysis is conducted. Frequency domain analysis can indicate the presence of a shift, but it is not able to determine whether the shift was restricted to the heavy shaking or if the structure was damaged in the event. It is impor-tant to determine if the shift was localized to a specific time or if the natural frequency shifted per-manently after some point in time. The former would indicate a softening in the structural components, while the latter would indicate possible damage to the structure. 4.4 Presentation of Joint Time-Frequency Analysis In this thesis, the joint time-frequency analysis is presented in a tri-plot format such as the one shown in Figure 4.2. Depending on the application, either the time-frequency spectrogram or TFRF is shown as a two-dimensional contour plot. The time signals used in the analysis are shown adjacent to the time axis of the time-frequency plot. In addition, a frequency domain rep-resentation of the signals is presented along the frequency axis of the time-frequency plot. The advantage of this format is that the information is presented in the time, frequency and combined time-frequency domains simultaneously. 46 Chapter 4 Introduction to Case Studies TFRF Magnitude 0.003 • 0.03 6P-2 CO [0-1 0.0 60 50 40 30 20 10 -300 0 300 -15 0 15 Base Accel. Roof Accel. (cm/s2) (cm/s2) 0 o CD CD E 2 3 4 Frequency (Hz) Figure 4.2: Example of Time-Frequency Plot 47 C H A P T E R Ambient Vibration of a 3 Span Bridge 5.1 Introduction The purpose of this study was to determine key dynamic characteristics of the University Drive/ Crowchild Trail bridge immediately after construction as a bench mark for future comparison. This study was carried out in cooperation with ISIS (Intelligent Sensing for Innovative Structures) and the City of Calgary in order to investigate the dynamic characteristics of the bridge by way of ambient vibration testing. The dynamic characteristics of interest are the natural frequencies and corresponding mode shapes in the vertical and transverse directions, as well as the torsional modes of the bridge. Sections 5.1 through 5.4 of this chapter are based on a report prepared for ISIS Canada by Black et al. (1997a). The natural frequencies and mode shapes of the bridge were determined from ambient vibration testing. The ambient vibration tests were conducted on August 15, 1997 with the aid of the HBES (Hybrid Bridge Evaluation System) developed at the University of British Columbia (Felber, 1993). A joint time-frequency analysis is conducted to ensure that the natural frequencies remain con-stant throughout testing and to explain some anomalous frequency domain results. 48 Chapter 5 Ambient Vibration of a 3 Span Bridge 5.2 Description of the Bridge The original University Drive/Crowchild Trail bridge was not capable of carrying the load of today's heavier traffic and thus it was rebuilt using ISIS Canada technology. It is the first contin-uous span, steel-free bridge deck in the world (ISIS, 1997). Construction of the bridge, located in north-west Calgary, Alberta, was completed in mid August, 1997. It is a three span, 90 m over-pass which carries the South-bound traffic of Crowchild Trail over University Drive (see Figure 5.1). The deck is a 185 mm thick, polypropylene fibre reinforced concrete slab, without any inter-nal steel reinforcement. A steel free deck eliminates the problem of corrosion within the deck. The removal of steel rein-forcement is made possible by the addition of tension straps and polypropylene fibre reinforce-ment. The straps are regularly spaced across the tops of adjacent girders to provide lateral restraint and the fibre is included to control crack growth (Newhook and Mufti, 1996). The slab overhang and barriers on each side are reinforced with glass fibre rods. The 9030 mm wide deck is supported by five 900 mm deep steel girders. Figure 5.1 shows the North-South elevation view of the bridge. The bridge has been extensively instrumented by the University of Alberta in order to monitor the new steel-free bridge deck technology. The instruments monitor the performance with respect to the width and depth of the cracks and the fatigue performance of welded connections. 49 Chapter 5 Ambient Vibration of a 3 Span Bridge Figure 5.1: Photo of the University Drive/Crowchild Trail Bridge 5.3 Description of the Ambient Vibration Testing The test was conducted on Friday, August 15, 1997 from 9:00 a.m. to 11:00 p.m. The temperature was around 8 degrees Celsius with moderate to heavy rain for the majority of the test. The University Drive/Crowchild Trail bridge was not open to traffic at the time of testing and thus the ambient vibration was produced by wind, human activity and the traffic below. The maximum levels of recorded vibration in the vertical and transverse directions were 1.7 milli-g's. These val-ues were considered sufficient to proceed with further analysis of the data. 50 Chapter 5 Ambient Vibration of a 3 Span Bridge Eight accelerometers were used for the ambient vibration measurements. Forty-six locations were chosen for measurements, 44 on the bridge deck and 1 near the base of each pier. A total of 14 setups were needed to accurately capture the natural frequencies and mode shapes. Two accel-erometers were used as reference sensors for the first 11 setups and 3 accelerometers were used in the remaining 3 setups. Figure 5.2 shows the location and direction of the ambient vibration mea-surements taken during the test. Table 5.1 gives the location of each of the sensors during the test. The first column is the setup number. The second column lists the first five characters of the file name created. The remaining three characters identify the channel number. The eight columns (labelled Chan. 1 through Chan. 8) refer to the 8 channels of the data acquisition computer. Each number-letter combination refers to sensor location (see Figure 5.2) and the direction of measurement: vertical (V), longitudial (L) or transverse (T). Sensor location 18 was used as the reference location (R) with the transverse and vertical directions measured during all tests and the longitudial direction measured for tests 12-14. The attenuation of the signals are given in the last column of the table. The filter cut off was set to 50 Hz, which corresponds to the natural frequency of the accelerometers. A total of 65536 samples per channel, per test were recorded at a sampling rate of 100 samples per second (sps). 51 Chapter 5 Ambient Vibration of a 3 Span Bridge 29830 mm 32818 mm 30230 mm H M m w 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 Is -•5000 m Ill1 ill! t it 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 PLAN VIEW 44 45 ELEVATION VIEW Sensor positive direction convention: t = 0ut of plane of page, o = into plane of page, « - = in plane of page Figure 5.2: Schematic of Sensor Locations 52 Chapter 5 Ambient Vibration of a 3 Span Bridge Setup File Chan. 1 Chan. 2 Chan. 3 Chan. 4 Chan. 5 Chan. 6 Chan. 7 Chan. 8 Atten. (dB) 1 CTB01 R18V 19V 18V 19V 18V R18T 18T 18T 18 2 CTB02 R18V 23V 22V 25V 24V R18T 22T 24T 18 3 CTB03 R18V 21V 20V 27V 26V R18T 20T 26T 18 4 CTB04 R18V 17V 16V 29V 28V R18T 16T 28T 18 5 CTB05 R18V 15V 14V 31V 30V R18T 14T 30T 18 6 CTB06 R18V 13V 12V 33V 32V R18T 12T 32T 18 7 CTB07 R18V 11V 10V 35V 34V R18T 10T 34T 18 8 CTB08 R18V 9V 8V 37V 36V R18T 8T 36T 18 9 CTB09 R18V 7V 6V 39V 38V R18T 6T 38T 18 10 CTB10 R18V 5V 4V 41V 40V R18T 4T 40T 18 11 CTB11 R18V 3V 2V 43V 42V R18T 2T 42T 18 12 CTB12 R18V 1V OV R18L 42V R18T OT 42L 18 13 CTB13 R18V 44L 44V R18L 45V R18T 44T 45T ** 14 CTB14 R18V OL — R18L — R18T . . . 45L 18 Table 5.1: Ambient Vibration Test Setup Locations and Directions ** attenuation of 24 for channel 7 and 18 for others 5.4 Frequency Domain Analysis The natural frequencies and mode shapes in the vertical direction were estimated using 64 records of vertical ambient vibration measurements. Forty records from measurements in the transverse direction were used and 7 in the longitudial direction. The torsional response was estimated from the difference in the vertical motions obtained from opposite sides of the deck. This is possible as the deck is assumed to be rigid and therefore the difference in the motion, at each side of deck, gives a reasonable estimate of the torsion. 53 Chapter 5 Ambient Vibration of a 3 Span Bridge 5.4.1 Natural Frequencies The power spectral densities of the above measurements were averaged to give the Averaged Nor-malized Power Spectral Density (ANPSD). The ANPSD permits a convenient way to display, in one single plot, the most significant frequencies present in a series of recorded motions in a cer-tain direction of the structure (Felber, 1993). Figure 5.3 shows the ANPSDs for the vertical, transverse and torsional modes for two different frequency ranges, computed using 64 averages. The most significant peaks in the frequency range of 0-20 Hz have been identified in Part B of this figure. Other peaks in the 20-50 Hz range were also identified in the analysis, but were not considered for further study since the frequency range of interest was 0-20 Hz. Not all of the peaks in Figure 5.3 may correspond to a natural frequency (Bendat and Piersol, 1993) and additional tools are needed to confirm whether or not a peak of the ANPSD is associ-ated with a natural frequency. Furthermore, a natural frequency of the bridge is not necessarily at a peak of the ANPSD, but it may be in the vicinity of the peak. This is due to the presence of damping in the system, as well as the effect of averaging multiple records. More detailed information can be obtained from log-normal plots of the ANPSDs as shown in Figure 5.4. The presence of peaks with low amplitude can be easily determined from these plots. For example, a peak around 18 Hz can be easily seen in Figure 5.4, but not in Figure 5.3. This happens to be one of the natural frequencies of the bridge, as will be described in the following sections. Chapter 5 Ambient Vibration of a 3 Span Bridge 0 2 4 6 8 10 12 14 16 18 20 Frequency (Hz) b) 0 to 20 Hz Range Figure 5.3: Averaged Normalized Power Spectral Densities 55 Chapter 5 Ambient Vibration of a 3 Span Bridge Q CL 100.000 10.000 1.000 0.100 0.010 0.001 Frequency (Hz) Q CO 100.000 -§ 10.000 -4 1.000 -i 0.100 0.010 -g 0.001 6 8 10 12 Frequency (Hz) 14 16 18 20 Figure 5.4: Averaged Normalized Power Spectral Densities (Log-Normal Scale) 56 Chapter 5 Ambient Vibration of a 3 Span Bridge From the peaks in the ANPSDs it is seen that there are 15 possible modes below 20 Hz. A sum-mary of the frequencies identified from this analysis is given in Table 5.2. Peaks Frequency (Hz) 1 0.29 2 1.95 3 2.83 4 3.22 5 3.81 6 4.10 7 4.79 8 5.18 9 7.13 10 9.67 11 10.94 12 12.99 13 15.63 14 17.87 15 19.24 Table 5.2: Potential Natural Frequencies 5.4.2 Mode Shapes The ambient vibration records obtained were used to determine the transfer function, coherence and phase between the reference sensors and all other sensors. This information was used to con-firm if the peaks in the ANPSD corresponded to natural mode shapes or to a mode of vibration at Chapter 5 Ambient Vibration of a 3 Span Bridge that frequency. The peaks at 0.29 Hz and 1.95 Hz did not correspond to a natural mode of vibra-tion and were disregarded from further analysis. Selected mode shapes determined to be the natural modes of the vibration of the bridge are shown in Figure 5.5. Each mode is shown in three views: plan, elevation and 3-dimensional. The modes are not presented as sequential modes in a single direction, as numerous coupled modes exist. Rather, the modes are shown in order of frequency. Fundamental Vertical Mode (2.78 Hz) Fundamental Torsional Mode (3.13 Hz) Figure 5.5: Mode Shapes of the University Drive/Crowchild Trail Bridge 58 Chapter 5 Lightly Coupled Vertical/Torsion (3.76 Hz) Ambient Vibration of a 3 Span Bridge Torsional Mode (4.05 Hz) Figure 5.5: Mode Shapes of the University Drive/Crowchild Trail Bridge (Continued) 59 Chapter 5 Ambient Vibration of a 3 Span Bridge A total of thirteen modes were identified. Table 5.3 is a summary of the findings. The sequential number of the mode and the frequency and period of each mode is presented with a brief descrip-tion. The bracketed term indicates the sequential number in a certain direction and the * indicates the presence of coupling. Mode Fre-quency (Hz) Period (sec) Description 1 2.78 0.36 Fundamental Vertical Mode (1 ver) 2 3.13 0.32 Fundamental Torsional Mode (1 tor) 3 3.76 0.27 Coupled Mode [Vertical (2 ver*) / Torsion (2 tor*)] 4 4.05 0.25 Torsional Mode (3 tor) 5 4.64 0.22 Vertical Mode (3 ver) 6 5.18 0.19 Coupled Mode [Torsion (4 tor*) / Transverse (1 tra*) 7 7.13 0.14 Coupled Mode [Torsion (5 tor*) / Transverse (2 tra*)] 8 9.13 0.11 Vertical Mode (4 ver*) 9 10.74 0.09 Torsional Mode (6 tor) 10 12.84 0.08 Fundamental Transverse Mode (6 tra) 11 15.77 0.06 Torsional Mode (6 tor*) 12 17.68 0.06 Vertical Mode (5 ver) 13 19.29 0.05 Torsional Mode (7 tor) Table 5.3: Summary of Mode Shapes 5.5 Joint Time-Frequency Analysis A Short Time Fourier Analysis is used to obtain time information of the response. Due to the length of ambient testing and the limitations on the array size of the software used for the calcula-60 Chapter 5 Ambient Vibration of a 3 Span Bridge tion of the spectrogram, only a relatively small number of samples from each test can be analyzed simultaneously. A 10 second segment from one accelerometer, during each test, was extracted from each test and joined together to form a single file. The spectrogram, P((0, t), is calculated for each file with different window sizes to obtain the best time-frequency resolution for a given signal. The reference sensor was the obvious choice as the sensor because it remains in the same location for the entire duration and thus should give consistent readings. Each of the 14 tests recorded 65536 points sampled at 100 sps. From each test a 10 second sam-ple (1000 points) was extracted and joined together to create a 14000 point record. This record was analyzed using the STFT with two different window sizes; 64 point window yielding a wide band spectrogram and a 256 point window yielding a narrow band spectrogram. At a sampling rate of 100 sps, a 64 point window corresponds to a 0.64 second non-zero signal and the 256 point window corresponds to a 2.56 second non-zero signal. Both of the signals lengths are greater than the fundamental period of the structure. The wide and the narrow band Time-Frequency spectro-grams are shown as contour plots in Figures 5.6 and 5.7, respectively. Figure 5.8 shows the nar-row band time-frequency plot for frequencies close to the fundamental vertical mode (2.78 Hz). This plot is shown because the response, and hence the spectrogram of the response is dominated by the mode(s) near 10 Hz. Figure 5.9 shows the narrow band time-frequency plot for 0 to 3 Hz range. The time-frequency domain information is presented in time-frequency plots as described in Chapter 4. 61 Chapter 5 Ambient Vibration of a 3 Span Bridge Chapter 5 Ambient Vibration of a 3 Span Bridge Chapter 5 Ambient Vibration of a 3 Span Bridge Chapter 5 Ambient Vibration of a 3 Span Bridge Chapter 5 Ambient Vibration of a 3 Span Bridge The consequence of the uncertainty principle, discussed in Chapter 2, is clearly evident in these figures as the wide band time-frequency plots lack the frequency resolution of the narrow band time-frequency plots. This is shown as "spreading" along the frequency axis. On first inspection of the time-frequency plot presented in Figures 5.6 and 5.7 there would appear to be a slight shift at or near 10 Hz as the test progresses. The center of the contours shift slightly to a lower frequency. This is a result of a lack of excitation at the frequencies in the area above 10 Hz rather than a shift. There exists two possible reasons for this lack of excitation. First, the time segments chosen to represent the later tests may not have contained this excitation. To be certain that this was not the case, a number of other segments were chosen and analyzed in a similar fash-ion, yielding similar results. The other reason for the apparent lack of excitation is that the testing was completed over a 14 hour span, with the last 4 tests being conducted in the late evening. This may explain the decrease in excitation which is visible in the time history and inferred from the contour plot. From Figure 5.8, it is seen that there is no discernible shift in the fundamental vertical and funda-mental torsional modes, represented by the ANPSD peaks at 2.78 and 3.13 Hz, respectively. The ANPSD contains a peak at 1.95 Hz which was later discarded after analyzing the vibrating shape at, or near, that frequency. It is of interest therefore, to study the joint time-frequency response of that peak in an attempt to explain its origins. The time-frequency plot shown in Fig-ure 5.9, shows little or no response at 1.95 Hz except during the first three tests (0-30 seconds). It can be inferred from the plot that this frequency was only present for a brief time and is therefore 66 Chapter 5 Ambient Vibration of a 3 Span Bridge is not likely a natural mode of vibration. 5.6 Conclusions The frequency domain analysis of ambient vibration data obtained at the University Drive/Crow-child Trail bridge identified thirteen modes below 20 Hz. The fundamental frequencies in the ver-tical and transverse directions were identified as 2.78 Hz and 12.84 Hz respectively while the fundamental torsional mode was identified as 3.13 Hz. The bridge has a number of coupled modes including several highly coupled modes at 3.76 Hz, 5.18 Hz and 7.13 Hz. The joint time-frequency analysis was useful in that it showed that there were no significant fre-quency shifts during the testing. The peaks near 10 Hz and those corresponding to the fundamen-tal vertical and torsional modes of the bridge remained relatively stable during the test. There are a few possible reasons for the lack of observable shifts. Testing was completed on a very cold, rainy day, when the temperature remained relatively constant, as opposed to a sunny day when the temperature may change by 10 or more degrees Celsius from mid-day to late evening. Another reason is that the resolution of the joint time-frequency domain analysis is not capable, especially for the case around 10 Hz, of identifying slight shifts in frequency. This lack of resolution is not a significant problem as a shift of that magnitude would not affect the outcome of testing. A joint time-frequency analysis of the peak near 1.95 Hz showed that this frequency was only present for a very brief time and thus not likely a mode of the bridge. Although this conclusion could be drawn from the standard frequency domain analysis alone, it is reassuring to see that the peak in frequency is very localized in time. 67 C H A P T E R 6 Ambient Vibration of a Single Span Bridge 6.1 Introduction The results of ambient vibration testing of the Lindquist Bridge located near Kamloops, British Columbia are presented in this chapter. This study was carried out in order to investigate the dynamic characteristics of the bridge by way of ambient vibration testing (Black and Ventura, 1998a). It is a complimentary study to one conducted for Reid Crowthers, in which the bridge was instrumented with 6 strain gages, four on the girders, at the third and center positions, and 2 on the straps at the same location (Cook et al. 1998). The Lindquist bridge is of particular interest as it carries relatively high loads. It serves as a logging bridge transporting logging trucks over Darlington Creek. Sections 6.1 through 6.4 of this chapter are based on a report prepared by Black and Ventura (1998a). Similar to the University Drive/Crowchild Trail Bridge described in Chapter 5, the purpose of this study was to determine key dynamic characteristics of the Lindquist as a bench mark for future comparison. The dynamic characteristics of interest were the natural frequencies and correspond-ing mode shapes in the vertical and transverse directions, as well as the torsional modes of the bridge. 68 Chapter 6 Ambient Vibration of a Single Span Bridge Two complete ambient vibration tests were conducted, one on March 31 and one on April 1, 1998 with the aid of the HBES (Hybrid Bridge Evaluation System) developed at the University of Brit-ish Columbia (Felber, 1993). A joint time-frequency analysis is conducted to determine if the dominant frequencies shifted through the testing and to explain anomalous frequency domain results. 6.2 Description of the Bridge The original wooden bridge spanning Darlington creek was recently replaced with a steel-free bridge deck supported by two steel girders. It is a 24 m single span bridge which carries logging traffic. The deck is a 185 mm thick, polypropylene fibre reinforced concrete slab without any internal steel reinforcement. A steel free deck eliminates the problem of corrosion within the deck. The removal of steel rein-forcement is made possible by the addition of tension straps and polypropylene fibre reinforce-ment. The straps are regularly spaced across the tops of adjacent girders to provide lateral restraint and the fibre is included to control crack growth (Newhook and Mufti, 1996). Figure 6.1 shows the North-South elevation view of the bridge; Figure 6.2 is a photo looking over the top of the bridge and Figure 6.3 is a view from below the bridge showing the girders and the straps. 69 Chapter 6 Ambient Vibration of a Single Span Bridge Chapter 6 Ambient Vibration of a Single Span Bridge Figure 6.3: Photo Showing the Underside of the Bridge 6.3 Description of the Ambient Vibration Testing The vibration measurements were performed twice, once on March 31 and again on April 1, 1997. The temperature was around 6 degrees Celsius with mainly sunny conditions. Excitation of the bridge was provided by wind and human activity. Eight accelerometers were used for the ambient vibration measurements. Thirty locations were chosen for measurements, 26 on the bridge deck and 4 on the abutments. A minimum of 8 setups on each day were needed to accurately capture the natural frequencies and mode shapes. Two accelerometers were used as reference sensors and six were used as roving sensors. 71 Chapter 6 Ambient Vibration of a Single Span Bridge Figure 6.4 shows the location and direction of the ambient vibration measurements recorded dur-ing the test. N 2 4 6 c i t, 1 : 5 10 12 1 t 1 , 1 ] 4 16 18 20 22 24 26 t i , t 1 1 ± = £ .3 5 7 9 11 13 1 5 17 19 21 23 25 27. 28 1 29 t Accelerometer direction • Accelerometer direction out of page o Accelerometer Figure 6.4: Sensor Locations and Directions of the Measurements 72 Chapter 6 Ambient Vibration of a Single Span Bridge Tables 6.1 and 6.2 list the location of each of the sensors during the tests on March 31 and April 1, respectively. The first column of each table corresponds to the setup number and the second column lists the first five characters of the file name created. The eight columns (labelled Chan. 1 through Chan. 8) refer to the 8 channels of the data acquisition computer. Each number-letter combination refers to sensor location in Figure 6 and the direction of measurement, vertical (V), longitudial (L) or transverse (T). A l l measurements were sampled at 200 sps for 327.67 seconds giving 65536 points per channel, per test. Setup File Chan. 1 Chan. 2 Chan. 3 Chan. 4 Chan. 5 Chan. 6 Chan. 7 Chan. 8 Atten. (dB) 1 LB01 R15V 15V 17V 14V 16V R15T 15T 17T * 2 LB02 R15V 13V 19V 12V 18V R15T 13T 19T ** 3 LB03 R15V 11V 21V 10V 20V R15T 1 IT 2 IT 0 4 LB04 R15V 9V 23V 8V 22V R15T 9T 23T 0 5 LB05 R15V 7V 25V 6V 24V R15T 7T 25T 0 6 LB06 R15V 5V 27V 4V 26V R15T 5T 27T 0 7 LB07 R15V 3V 29V 2V 28V R15T 3T 29T 0 8 LB08 R15V IV 19V ov 18V R15T IT 19T 0 Table 6.1: Test Setup Locations and Directions for March 31,1998 Test *segments 1-4: V=12, H=6; segments 5-8: V=6, H=6 **segments 1-5: V=24, H=12; segments 6-8: V=6, H=6 73 Chapter 6 Ambient Vibration of a Single Span Bridge Setup File Chan. 1 Chan. 2 Chan. 3. Chan. 4 Chan. 5 Chan. 6 Chan. 7 Chan. 8 Atte n. (dB) 1 LB01 R U T 15V 17V 14V 16V R11V 15T 17T 0 2 L B 0 2 R U T 13V 19V 12V 18V R 1 1 V 13T 19T 0 3 LB03 R U T 11V 21V 10V 20V R11V 11T 2 IT 0 4 L B 0 4 R U T 9 V 23V 8V 22V R11V 9T 23T 0 5 L B 0 5 R U T 7 V 25V 6V 24V R11V 7T 25T 0 6 L B 0 6 R U T 5 V 27V * 4 V 26V R11V 5T 27T 0 7 L B 0 7 R U T * 3 V 29V * 2 V 28V IR11V * ! 3 T !29T ## 8 L B 0 8 R U T I V *15V OV *14V IR11V !1T !15T ** 9 R U T 3 V 4 V 2V * 6 V R11V 3T * 4 T 0 Table 6.2: Test Setup Locations and Directions for April 1,1998 Test ** channels 6,7,8 at attenuation of 6; channels 1-5 at 0 attenuation * redundant records determined to be of low quality 6.4 Frequency Domain Analysis The vertical and torsional natural frequencies and mode shapes were estimated using 28 vertical ambient vibration measurements on each day. The torsional response was estimated from the dif-ference in the vertical motions obtained from opposite sides of the deck. This is possible as the deck is assumed to be rigid and therefore the difference in the motion, at each side of deck, gives a reasonable estimate of the torsion. Twenty-Eight records from measurements in the transverse direction were used to obtain the natural frequencies and mode shapes in the transverse direction. The testing on March 31 was conducted with the reference sensor at the center location. As a result, the symmetric modes are well defined but the asymmetric modes are not. The problem 74 Chapter 6 Ambient Vibration of a Single Span Bridge with the asymmetric modes is due to the normalization of the mode shape to the small value at the center. The testing on April 1 had the reference sensor at the third point of the bridge and thus it provides nicely defined asymmetric modes. Ideally, one would have two reference sensors, one at the center and one at another point, not likely to be a node location, during each test. This how-ever may be difficult due to hardware and time limitations. If only one test is conducted the refer-ence position would be chosen in a location where a node is not likely to be present in order to maximize the number of modes that can be properly identified. Other testing activity was present during the ambient vibration testing and thus the data may include spurious frequencies. These may include the revolutions of the truck or car engines in the vicinity or a generator which was used for a power supply. 6.4.1 Natural Frequencies The power spectral densities of the above measurements were averaged to give the Averaged Nor-malized Power Spectral Density (ANPSD). The ANPSD permits a convenient way to display, in one single plot, the most significant frequencies present in a series of recorded motions in a cer-tain direction of the structure (Felber, 1993). Figures 6.5 and 6.6 show the ANPSDs for the verti-cal, transverse and torsional modes computed using 64 averages. The magnitude of the transverse ANPSD should not be compared with the vertical and torsional magnitudes as it is normalized to itself. It is rare to have 2 complete ambient vibration tests for the same bridge, under similar ambient conditions. It is of interest therefore, to look at the differences in the ANPSDs. The peaks at 13.3 75 Chapter 6 Ambient Vibration of a Single Span Bridge Hz, 17.3 Hz, 18.2 Hz and 22.9 Hz are more pronounced for the March 31 test than the April 1 test. The ANPSD calculated from the March 31 test has many peaks near 30 Hz, as compared to the single dominant peak in the April 1 test. Another interesting difference is the shift in dominant frequency observed near 40 Hz. The peak at 41.4 Hz is present in both ANPSDs but the peak seen at 40.5 Hz in the April 1 test is absent in the March 31 test. It is evident from the observed variation that results obtained from ambient vibration testing are only approximate. The uncertainty arises from the variability in ambient conditions which affect the measured frequencies. The modal frequencies were estimated from the measurements taken on March 31 as the peaks are generally more pronounced. Not all of the peaks in Figure 6.5 correspond to a natural fre-quency (Bendat and Piersol, 1993) and additional tools are needed to confirm whether or not a peak of the ANPSD is associated with a natural frequency. For more information on the normal-ization scheme see Felber (1993). From the peaks in the ANPSDs it is seen that there are 9 possi-ble modes below 50 Hz. A summary of the frequencies identified from this analysis is given in Table 6.3. 76 Chapter 6 Ambient Vibration of a Single Span Bridge 100.0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 Frequency (Hz) Figure 6.5: Log Normal Plot of the ANPSDs Obtained from March 31 Test 100.0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 Frequency (Hz) Figure 6.6: Log Normal Plot of the ANPSDs Obtained from April 1 Test 77 Chapter 6 Ambient Vibration of a Single Span Bridge Mode Frequency (Hz) 1 6.3 2 7.3 3 13.2 4 17.3 5 18.2 6 22.9 7 26.1 8 30.2 9 41.4 Table 6.3: Frequencies of Potential Modes 6.4.2 Mode Shapes The ambient vibration records were used to determine the transfer function, coherence and phase between the reference sensors and all other sensors. This information was used to confirm if the peaks in the ANPSD corresponded to natural mode shapes or to a mode of vibration at that fre-quency. The vibration shapes which were determined to be natural modes of the bridge are shown in Fig-ure 6.7. Each mode is shown in three views: plan view at the top of the figure, elevation view at the middle and 3 dimensional view at the bottom. The modes are not presented as sequential modes in a single direction, as coupled modes exist. Rather, the modes are shown in order of fre-quency. 78 Chapter 6 Fundamental Vertical Mode (6.3 Hz) Ambient Vibration of a Single Span Bridge Fundamental Torsional Mode (7.32 Hz) Chapter 6 Ambient Vibration of a Single Span Bridge Torsional/Abutment (18.2 Hz) Torsional Mode (22.85 Hz) Vertical Mode (30.2 Hz) Torsional Mode (41.4 Hz) Figure 6.7: Mode Shapes of the Lindquist Bridge (Continued) 80 Chapter 6 Ambient Vibration of a Single Span Bridge A total of eight modes were identified. Table 6.4 gives a summary of the findings. The sequential number of the mode, the frequency and period of each mode is presented with a brief description. The bracketed term indicates the sequential number in a certain direction. Mode Frequency (Hz) Period (sec) Description 1 6.3 0.16 Fundamental Vertical Mode (1 ver) 2 7.3 0.14 Fundamental Torsional Mode (1 tor) 3 13.2 0.076 Fundamental Transverse (1 tra) w/ torsional coupling 4 17.3 0.058 Vertical Mode (2 ver) 5 18.2 0.055 Torsional-Abutment Mode (3 tor) 6 22.9 0.044 Torsional Mode (4 tor) 7 30.2 0.033 Vertical Mode (3 ver) 8 41.4 0.024 Torsional Mode (5 tor) Table 6.4: Summary of Mode Shapes 6.5 Joint Time-Frequency Analysis Each of the 8 tests conducted on March 31 recorded 65536 points sampled at 200 sps. From each test a 10 second sample (2000 points) was extracted and joined together to create a 14000 point record. This record was analyzed using the STFT to yield the joint time-frequency spectrogram. At a sampling rate of 200 sps, the 64 point window corresponding to the wide band spectrogram has a length of 0.32 seconds which is longer than the fundamental period. The narrow band spec-trogram has calculated with a window length of 256 points, or 1.28 seconds. As described in Chapter 4, the time-frequency information is presented in a time-frequency plot. Figures 6.8 and 6.9 present the wide and narrow band time-frequency plots respectively. 81 Chapter 6 Ambient Vibration of a Single Span Bridge Amplitude Frequency (Hz) Figure 6.8: Wide Band T-F Plot for the Lindquist Bridge 82 Chapter 6 Ambient Vibration of a Single Span Bridge Chapter 6 Ambient Vibration of a Single Span Bridge Both the wide and narrow band time-frequency plots shown in Figures 6.8 and 6.9, respectively indicate a marked shift in the dominant frequency near 30 Hz. This shift is most pronounced dur-ing the 20 to 30 second segment of the data, corresponding to the third of eight setups. These plots reveal the importance of presenting the time-frequency information in the tri-plot for-mat. It is clearly evident in the time-frequency plot that the segment with the abrupt change in frequency, as seen in the contour plot, also saw an increase in the amplitude of vibration as shown in the acceleration time history. The time-frequency plot also shows that in the frequency domain, the signal has multiple peaks near 30 Hz. With the use of joint time-frequency analysis it is clear that the multiple peaks near 30 Hz are in fact a single mode which has shifted. If the anal-ysis was conducted solely in the frequency domain, the researcher would have a difficult time explaining, or explaining away, the multiple peaks seen near 30 Hz. Even though the ANPSD does not show multiple peaks near the fundamental vertical mode (6.3 Hz) it is of interest to determine if there was a corresponding shift, like that seen near 30 Hz). Figure 6.10 shows the time-frequency plot for the frequency range 0 to 20 Hz. A shift in the fun-damental mode is not evident from the time-frequency plot. During the time segment in which the mode near 30 Hz shifted, there was little response in the fundamental vertical mode. It appears that the significant shaking, seen in the time history, was dominated by the mode near 30 Hz. Figure 6.10 shows the existence of frequency content near 13 Hz which does not have a corre-sponding peak in the ANPSD. This is the fundamental transverse mode. The reason that it shows 84 Chapter 6 Ambient Vibration of a Single Span Bridge up in the vertical direction is that the transverse mode has torsional coupling as discussed in sec-tion 6.4.2. The vertical reference sensor used for the joint time-frequency analysis also serves as the torsional reference sensor and thus it contains torsional response that is not present in the ver-tical ANPSD. 85 Chapter 6 Ambient Vibration of a Single Span Bridge Amplitude Frequency (Hz) Figure 6.10: Narrow Band T-F Plot for the Lindquist Bridge - Low Frequency Range 86 Chapter 6 Ambient Vibration of a Single Span Bridge 6.6 Conclusions The ambient vibration of the Lindquist bridge was sufficient to identify 8 modes below 50 Hz. The fundamental frequency in the vertical direction was identified as 6.3 Hz and in the transverse direction to be 13.2 Hz. The fundamental torsional mode was observed at 7.3 Hz. The results show consistent coupling of the torsional and transverse modes. In Modes 2, 3, 5, 6 and 8 both transverse and torsional components exist. This can be explained by the fact that when the bridge deck deforms in a torsional mode the shape can not be formed without an associated displacement in the transverse direction. For example, as the center of the bridge deck moves in the transverse direction one side naturally lifts and one dips. Mode 5 warrants particular attention as it shows large movement in the abutment. The abutment at the north end is supported by longer piles than the south end as a result of the local soil condi-tions. It appears that the long piles, in conjunction with the laterally flexible soil, have a rocking mode of vibration at or near 18.2 Hz. This rocking motion, in turn, drives the bridge in a torsional mode similar to the torsional mode at 22.85 Hz. Joint time-frequency analysis of the ambient vibration data identified a significant shift in the fre-quency of the 7th mode. This joint time-frequency information is very valuable as the standard frequency domain analysis yields two peaks of approximately the same magnitude near 30 Hz. An attempt to determine the natural frequencies near 30 Hz from a standard frequency domain analysis is nearly impossible as one would not be able to explain the significance of both peaks. This inability to verify the behavior of the bridge near 30 Hz may lead to skepticism of the results. 87 Chapter 6 Ambient Vibration of a Single Span Bridge The joint-time frequency analysis of the data showed that it is not two modes but in fact one mode that shifted in the third test. The reason for the observed shift is not clear. It is possible that the mode near 30 Hz was driven by ambient vibration with a dominant frequency equal to the shifted frequency. Due to the ampli-fication of a structures response near one of its natural modes the response could be dominated by a forcing function in close proximity to a natural mode of vibration. As the input into the struc-ture is unknown it is not possible to calculate the Frequency Response Function (FRF) of the bridge which would, theoretically, remove the response of the structure to a forcing function. If the bridge's response was as a result of ambient vibration dominated by the frequency equal to the shifted frequency one would still, however, expect to see the actual unshifted frequency. The shift could be due to non-linearities in the bridge. If the stiffness was not linear the greater level of excitation observed in the time history could cause a shift in the frequency. These non-linearities however appear to be localized. This is evident from the fact that similar shifts were not observed in other frequencies. 88 C H A P T E R 7 Sliding of a Dam Model 7.1 Introduction In the spring of 1998 a study was conducted at The University of British Columbia in order to investigate certain aspects of the dynamic response of concrete gravity dams. The research was carried out in collaboration with British Columbia Hydro and Power Authority (BC Hydro), Maintenance, Engineering and Projects Division. This research is Phase 2 of a multi-phased research project which aims to develop suitable techniques for determining equivalent static lat-eral load coefficients for seismic assessment of existing dams. The results of these studies can be found in the reports presented to B C Hydro (Horyna et al., 1997 and Black et al., 1998b). Sec-tions 7.1 through 7.4 of this chapter are based on a report prepared for B C Hydro by Black et al. (1998b). A small scale model of a dam, to which different friction surfaces could be applied, was subjected to harmonic and earthquake excitations of different frequency content and amplitude. The rela-tive displacement between the model and the shake table was measured. The results from these tests allowed the estimation of the static and kinetic coefficients of friction as well as the compu-tation of the ratio of the dynamic force to net static force required to cause a prescribed level of motion. 89 Chapter 7 Sliding of a Dam Model During the course of the test an interesting phenomenon was observed. At certain combinations of base excitation amplitude and frequency the model moved upstream. That is, under base exci-tation the model moved in a direction opposite to that of the applied hydrostatic load. In an attempt to explain this behavior a joint time-frequency analysis was performed. A description of the experiment and the analysis is presented in the following sections. 7.2 Experimental Model Description The model consists of a dam model, sliding surfaces and a hydrostatic load assembly. Sliding occurs between a sliding surface attached to the bottom of the dam and one attached to the shake table. The hydrostatic load assembly is designed to provide a constant pulling force in the down-stream direction. 7.2.1 Dam Monolith The dam monolith is approximately 1500 mm high and 480 mm wide (see Figures 7.1 and 7.2). It was cast in July 1996 as part of Phase 1 of the project. The material used for the monolith was a mix consisting of Portland cement type 10, perlite, Styrofoam, silica fume and water. The weight composition of the mix was 42.2% cement, 40% water, 12.5% perlite, 3.6% silica fume and 1.7% Styrofoam (Horyna et al. 1997). This mix was chosen in an attempt to satisfy similitude require-ments between the model and an actual dam monolith, although it is understood that the intent of this study was not to predict the performance of an existing dam but rather to compare static and dynamic properties of the model. For a discussion of the similitude considerations see Horyna et al. (1997). 90 Chapter 7 Sliding of a Dam Model 225 1250 1250 Figure 7.1: Schematic of Dam Model 91 Chapter 7 Sliding of a Dam Model Figure 7.2: Photo of Dam Monolith Model 7.2.2 Sliding Surfaces The upper and lower sliding surfaces measure 1150 mm by 480 mm and 1500 mm by 660 mm respectively. The upper plate is clamped to the bottom of the dam and the lower plate is attached to the shake table by way of a steel frame (as shown in Figure 7.3). These plates are unbonded and the friction measured is a result of the upper plate attached to the dam moving over the lower Chapter 7 Sliding of a Dam Model plate attached to the shake table. Two different friction surfaces were used: a smooth surface and a rough surface. The smooth sur-face was created by sanding the "cement milk" from the friction surface of the plates to give a uniform surface. The rough surfaces were created by using an ultra-high pressure water jet to remove the cement matrix leaving an exposed aggregate surface. Figure 7.3: Photo Showing Plate Attachment to Dam Model and Table 7.2.3 Hydrostatic Load Simulation The hydrostatic load was simulated by applying a pulling force in the downstream direction on the downstream side of the dam model. This force was provided by a cable attached to the model at the height corresponding to the resultant of the simulated hydrostatic load. This cable was 93 Chapter 7 Sliding of a Dam Model attached to a hanging weight equivalent to the required force. The mass was attached to two ver-tical rods which constrained it to move only in the vertical direction. Friction between the mass and the sliding rods was minimized by bronze-oilite bearings which provided a smooth contact surface. A photo of the hydrostatic load assembly is given in Figure 7.4. Figure 7.4: Hydrostatic Load Assembly 7.3 Shake Table Testing The concrete gravity dam model was instrumented to measure its response in five different tests for four different surfaces. The tests included: static, harmonic, earthquake, cyclic and impact. A l l of the tests were conducted with the model placed on the shake table at the Earthquake Engi-94 Chapter 7 Sliding of a Dam Model neering Research Facility at UBC. 7.3.1 Shake Table The shake table measures 3m by 3m. It is moved by up to 5 actuators in various configurations. It has a payload capacity of 156 kN, with a maximum displacement of +/- 7.6 cm in the horizontal direction. The actuators are controlled by a state of the art Multi Exciter Vibration Control Soft-ware program which performs closed loop control. For the tests described in this thesis, the table was setup to produce only unidirectional motion in the horizontal direction. 7.3.2 Tests with Harmonic Input A series of tests were conducted on each surface at nine harmonic frequencies ranging from 5 Hz to 25 Hz with increments of 2.5 Hz. The input signal for the 5 Hz harmonic test is shown in Figure 7.6 for illustration. This signal has 6 segments with different amplitudes, ranging from 1.5g to 2.0g, in increments of O.lg. This record was used as the input signal to the shake table where it was reduced to desired amplitude levels. For example, if the record was run at 20% then the maximum acceleration on the table was 0.4g. Each test at a particular frequency was run multiple times with increasing amplitudes in order to capture the amplitude of motion which caused the model to slide a prescribed amount. Typically, the record for a certain frequency was run 5 to 10 times providing 30 to 60 different 10 second records at different amplitudes. Chapter 7 Sliding of a Dam Model Figure 7.5: Typical Dynamic Test Setup 96 Chapter 7 Sliding of a Dam Model 0.00 18.75 37.50 56.25 75.00 Time (sec) Figure 7.6: The 5 Hz Harmonic Input Record 7.3.3 Tests with Earthquake Input Records The setup and data collection for the earthquake tests were the same as for the harmonic tests. A series of simulated earthquake tests were conducted at 9 dominant frequencies. Each test had three different earthquakes with approximately the same dominant frequency. The records were created by taking the Power Spectral Density (PSD) of three earthquakes and shifting the spectral shape to obtain the desired dominant frequency. This was accomplished using the program SIM-Q K E (Gasparini, D.A. and Vanmarcke, E.H., 1976). For this study, an earthquake test which is denoted by a single frequency, for example the 20 Hz earthquake test, actually has a frequency band of approximately 6 Hz. The frequency band runs from the dominant frequency minus 2 Hz to the dominant frequency plus 4 Hz. For example, the earthquake test denoted 12.5 Hz has an 97 Chapter 7 Sliding of a Dam Model approximate frequency content of 10.5 Hz to 16.5 Hz with a dominant frequency of 12.5 Hz. The input earthquake record for 5 Hz is shown in Figure 7.7 for illustration. The first earthquake is a simulated earthquake with the same power spectral density (PSD) shape as the 1992 Califor-nia Landers earthquake measured at the Joshua Tree Fire Station in the E/W direction. The actual PSD was shifted in frequency to give the desired dominant frequency, which in this case is 5 Hz. The second is a simulated earthquake with the same PSD shape as the 1994 California Northridge earthquake recorded at the Tarzana Nursery in the E/W direction. The third earthquake is a simu-lated earthquake with the same PSD shape as the 1979 California Imperial Valley earthquake. 10 20 30 40 Tim e ( sec ) Figure 7.7: 5 Hz Earthquake Record 98 Chapter 7 Sliding of a Dam Model 7.4 Analysis More than 500 shake table tests were conducted on the dam model. The analysis of the data was comprised mainly of time domain data processing as the frequency of the excitation was con-trolled and hence known. The main objective of the experimental part of this study was to deter-mine the magnitude of prescribed base excitations which caused a specified level of motion between a concrete gravity dam monolith model and its unbonded base. The experimentally determined magnitude of the base acceleration required to move the model was converted to a pseudo-static force equal to the mass of the model multiplied by the accelera-tion of the model. This pseudo-static force was compared with the measured static force to give the ratio of the dynamic to static forces required to displace the model a prescribed amount. This ratio was determined for both harmonic and earthquake excitation and a comparison between the ratios was made for each excitation and each different surface. For more information on this study refer to Black et al. (1998b). 7.5 Joint Time Frequency Analysis During the course of the study, an interesting phenomenon was observed. At certain combina-tions of frequency and amplitude of base excitation, the dam model moved upstream overcoming the hydrostatic load. This was generally observed during tests where the amplitude of excitation approached lg and the frequency of base motion was greater than 20 Hz. It is not obvious why the dam model would behave in such a manner. 99 Chapter 7 Sliding of a Dam Model In an attempt to better understand the upstream behavior of the dam a joint time-frequency analy-sis was conducted. The measured acceleration at the base of the dam model was analysed using the STFT. The objective of the analysis was to identify dominant frequencies other than the forcing fre-quency. A Hanning window of 128 points was used for all of the time-Frequency plots. With a sampling rate of 200 sps, a window of 128 points yields a non-zero segment 0.64 seconds. The dominant periods in this study are much smaller than the segment length and hence the spectro-gram will easily capture the period range of interest. Figures 7.8 and 7.9 show the time-frequency plots for two tests with 20 Hz harmonic base excita-tion on the rough surface. Upstream motion is indicated by negative values of base motion. The time-frequency plot of the first test shows that when the model moves downstream, the only dom-inant frequency is the forcing frequency. During the second round of tests, upstream motion occurred during the second through fifth segments. The time-frequency plot shows two other fre-quencies, in addition to the forcing frequency, during upstream motion. The first at one half the forcing frequency and the other at one and a half times the forcing frequency. To verify this result another harmonic base excitation test was analyzed. The time-frequency plot presented in Figure 7.10 shows the response of the model on a smooth surface excited at 22.5 Hz. This plot confirms that during upstream motion, the model is vibrating at half and one and a half times the forcing function. 100 Chapter 7 Sliding of a Dam Model Contour Legend :::::::> 5000 100000 CD Q CO CL 5000 3000 1000 90 -1.20.0 1.2 0 5 10 15 Base Accel. (g) Model Displ. (cm) 75 60 45 30 15 ;l •i !i •r r, 'I 'A lj •1 i'L -/.—-1! I-!! i-j i i; -/ { •1 :i --vf ;l 1: ;i c ~ > .7 \ 1 * \ i ii 1: :i > 1 ) 1 -1. ;l ;i |i J -t 4 1 1 i 0 10 20 30 Frequency (Hz) 40 50 Figure 7.8: TFR Plot of 20 Hz Harmonic Input - No Upstream Motion 101 Chapter 7 Sliding of a Dam Model Contour Legend 10000 "> 5000 ..J 100000 CO Q Q _ 6000 2000 90 75 60 45 30 •1.2 0.0 1.2 -10 -5 0 5 15 I B a s e Acce l . (9) Model Displ . (cm) --F.;> I! ^ 4 -1'-o CD CD E 0 10 20 30 Frequency (Hz) 40 50 Figure 7.9: TFR Plot of 20 Hz Harmonic Input - Upstream Motion 102 Chapter 7 Sliding of a Dam Model Contour Legend LV.V.VJ 5000 100000 CD CO 8000 5000 Q co Q- 2000 90 75 60 45 30 •1.2 0.0 1.2 -15 -5 15 Base Accel. Model Displ. (g) (cm) A * i f is 5-IS «!• IT o CD CD E i-0 10 20 30 Frequency (Hz) 40 50 Figure 7.10: TFR Plot of 22.5 Hz Harmonic Input - Upstream Motion 103 Chapter 7 Sliding of a Dam Model The vibration observed during earthquake excitation was considerably more violent than the har-monic tests and thus the time-frequency analysis shows a much larger bandwith of response. Figures 7.11 and 7.12 show time-frequency plots for the response of the model to earthquake excitation. The first plot is for a test at 15 Hz in which no upstream motion occurred. During the test shown in Figure 7.12 the earthquake input had a dominant frequency of 25 Hz and the model moved upstream. While the model moved downstream, no frequency is observed at half the forc-ing frequency. At one and a half times the forcing frequency, a few relatively small peaks are observed. During upstream motion, the dam model vibrates at many frequencies with the most prominent being the forcing frequency. The frequency band of excitation extends to half the forc-ing frequency at times corresponding to large upstream motion. 104 Chapter 7 Sliding of a Dam Model ' 1 1 1 • Q I I I I I I 1 1 1 1 1 -1.20.0 1.2 0 10 20 o 10 20 30 40 50 Base Accel. Model Displ. Frequency (Hz) (g) ( c m) Figure 7.11: TFR plot of 15 Hz Earthquake Input - No Upstream Motion 105 Chapter 7 Sliding of a Dam Model Chapter 7 Sliding of a Dam Model An explanation for this behavior is that the dam model is rocking and possibly leaving the table at certain times producing a net upstream displacement. A possible rocking motion which would produce a frequency half that of the forcing frequency, for upstream motion, is shown in Figure 7.13. The figure shows the motion of the dam model and the shake table for two cycles of base motion and the corresponding single cycle of upstream motion. The upstream motion is shown by the dashed arrow. 107 Chapter 7 Sliding of a Dam Model Chapter 7 Sliding of a Dam Model 7.6 Conclusion A joint time-frequency analysis of the upstream motion of a dam model showed that during upstream motion, the dam vibrates at frequencies other than the forcing frequency. This informa-tion led to increased understanding and further study into the behavior. A standard frequency domain analysis would not indicate the presence of the other frequencies for times when the dam model moved upstream. A possible explanation for the frequency observed at one half times the forcing frequency was presented. This motion involves rocking about the downstream edge while the table moves in the downstream direction. 109 C H A P T E R Analysis of the 1901 Avenue of the Stars Building 8.1 Description The Avenue of the Stars building is located in the Century City area of Los Angeles at 1901 Ave-nue of the Stars. This moment resisting, steel frame, office building has 19 stories above the ground level with four parking levels below ground. The floor plan of the 1901 building mea-sures 73 m long by 34 m wide (Murphy, 1973). A Photo of the building is shown in Figure 8.1. The soil conditions at the site are generally fine sand. The building's foundation is comprised of driven steel I-beam piles under the main structural tower and spread footings elsewhere. The lat-eral load resisting system consists of four ductile steel moment-resisting frames in the major axis direction and 5 X-braced steel frames in the minor axis direction (Murphy, 1973). The epicenter of the Northridge earthquake was located 20 km north-west of the 1901 Avenue of the Stars building and the peak ground acceleration at the site was 0.32g. The epicenter of the San Fernando earthquake was approximately 39 km north-west of the building, with a peak ground acceleration at the site of 0.15 g's. Damage during the San Fernando earthquake was not severe. During the Northridge earthquake bracing elements at the penthouse buckled and signs of distress and motion along the brace lines 110 Chapter 8 Analysis of the 1901 Avenue of the Stars Building in upper floors were observed. Minor non-structural and content damage was observed but oper-ation resumed within one day (Naeim, 1997). $ l l § ! B p (Adapted from Naeim, 1997) Figure 8.1: Photo of the 1901 Avenue of the Stars Building 111 Chapter 8 Analysis of the 1901 Avenue of the Stars Building 8.2 Instrumentation The 1901 Avenue of the Stars building was instrumented with 7 strong motion accelerometers during the San Fernando earthquake. These were positioned 3 at each of the basement, 10th and roof levels, with an additional vertical sensor at the basement. A schematic of the sensor layout is given in Figure 8.2. During the Northridge Earthquake, the 1901 Avenue of the Stars building was instrumented with 15 strong motion accelerometers with 3 at each of the basement, 1st, 2nd and roof as well as two on the 8th floor and an additional vertical accelerometer at the basement level. A schematic of the sensor layout during the Northridge earthquake is shown in Figure 8.3. The sampling rate during the San Fernando and Northridge earthquakes was 100 sps. During the Northridge earthquake, 2 transverse records were obtained on each instrumented floor. The transverse signals used in this analysis were obtained by adding the 2 transverse records and dividing by a factor of two. Similarly, the torsional signals used for the analysis were obtained by subtracting the 2 transverse records and dividing by 2. A l l other signals used in this analysis are the original recorded accelerations. Figures 8.4 and 8.5 show the San Fernando and Northridge earthquake signals used for the analysis presented in this chapter. 112 Chapter 8 Analysis of the 1901 Avenue of the Stars Building O ) T > CD CD O O o +-> <n i CD 10 © CD O J C < w O Z o < CJ o rr o en z LU 0) - 4 -c o o o z o O A .ow.t-.eieor C CO a! + o o C P J ' N " 0 4) £ « S i « - r 3 5 o c W O K M l c IP CO «s a " 1 . —s—* • • • • a a • « E • 2 o o • <5 • J • 9-es a • s • o i l 0 ? u «0 5 o a I C N ' O • .etc cO ix Q > (Adapted from: Shakal et al.,1994) Figure 8.2: Schematic of Sensor Location During San Fernando Earthquake 113 Chapter 8 Analysis of the 1901 Avenue of the Stars Building r,oc an c ; an Plan ' OH • a -' <"T -J o • PH • » Plan ' >H , CO u . m 1 • *H' « o n S 2 Plan o . o • -si «*-. • o • Flo • o x c . o • Flo : E : : | • O S - t -• ; j H-> : o o : , c. n *—> o 1 «T3 si OS g v S • I-58 n CO »—« 0-*H o o DO O I co CM JS ° co 2 c o w C a. < § o 2 CO z o Y~ < o o _ l o to z UJ to (Adapted from: Shakal et al., 1994) Figure 8.3: Schematic of Sensor Location During the Northridge Earthquake 114 Chapter 8 Analysis of the 1901 Avenue of the Stars Building Transverse - Basement O) 0.2 Accel. ( 6 o to b — Transverse - 9 th Floor D) 0.2 "CD o.o o o < -0.2 Transverse - Roof a> 0.2 "CD 0.0 o <t -0.2 Longitudial - Basement S 0.2 "CD 0.0 < -0.2 — —-———-—-———— Longitudial - 9th Floor 3 0.2 ill 1 ll 8 0 0 i r . n o — — - — ~ — -Longitudial - Roof 3 0.2 "CD 0.0 o A V V w Y v v v v w ^ -0 10 20 30 40 50 60 Time (sec) Figure 8.4: Measured Accelerations During the San Fernando Earthquake 115 Chapter 8 Analysis of the 1901 Avenue of the Stars Building Transverse - Basement O) 0.3 CD 0.0 O O < -0.3 Transverse - 2nd Floor CD 0.3 CD 0.0 O O < -0.3 Transverse - 8th Floor CD 0.4 Transverse - Roof CD 0.5 CD 0.0 O O < -0.5 Longitudial - Basement o5 0.3 CD 0.0 O O < -0.3 Longitudial - 2nd Floor "3 0.4 30 40 50 Time (sec) 60 70 80 90 Figure 8.5: Measured Accelerations During the Northridge Earthquake 116 Chapter 8 Analysis of the 1901 Avenue of the Stars Building Logitudial - 8th Floor CO 0.4 CD 0.0 o o < -0.4 Longitudial - Roof CD 0.4 CD o.o o o < -0.4 Torsional - Basement CD 0.2 CD 0.0 o o < -0.2 Torsional - 2nd Floor cn 0.2 CD 0.0 O CJ < -0.2 Torsional - 8th Floor CD 0.2 CD 0.0 o o < -0.2 Torsional - Roof CD 0.2 CD 0.0 O o < -0.2 JUL 0 10 20 30 40 50 T ime (sec) 60 70 80 90 Figure 8.5: Measured Accelerations During the Northridge Earthquake (Continued) 117 Chapter 8 Analysis of the 1901 Avenue of the Stars Building 8.3 Frequency Domain Analysis 8.3.1 Natural Frequencies The modal frequencies of the 1901 Avenue of the Stars building are presented in table 8.1. These frequencies were obtained from the methods described in Chapter 3. The magnitude of the fre-quency response functions for the roof to basement pairs are presented in Figure 8.6. As indicated in the figure, the plot at the top shows the FRFs for the transverse direction (shown as Y) followed by the longitudial direction (shown as X) in the middle and the torsional FRFs at the bottom. There are no torsional frequencies calculated during the San Fernando earthquake as there were not enough measurements taken to calculate the rotation at the base. Mode San Fernando (Hz) Northridge (Hz) TRA1 0.29 0.32 TRA2 1.27 1.22 TRA3 2.51 2.68 LON1 0.29 0.22 LON2 0.78 0.71 LON3 1.25 1.25 ROT1 N A 0.24 ROT2 N A 0.98 ROT3 N A 1.90 Table 8.1: Frequencies of 1901 The fundamental natural frequency in the longitudial direction shifted considerably between the San Fernando and Northridge analysis. 118 Chapter 8 Analysis of the 1901 Avenue of the Stars Building a) Y direction Frequency (Hz) Figure 8.6: FRFs of the 1901 Avenue of the Stars Building 119 Chapter 8 Analysis of the 1901 Avenue of the Stars Building 8.3.2 Mode Shapes The first two mode shapes in each direction corresponding to the above frequencies are given in Figure 8.7. The torsional mode shapes presented in Figure 8.7 were obtained by taking the rotational values of displacement and plotting them as a linear displacement. For example, the second mode would be turning one way at the bottom and the opposite way at the top corresponding to the left and right side of the undeformed shape. The San Fernando mode shapes are not well defined, particularly in the transverse direction. This may be the result of only having 2 sensors above ground level. The Northridge mode shapes are interesting because of the relative displacement between the basement and first floor. For the mode shape analysis, the first floor was often assumed to be the base of the structure and the vibration shape above ground was taken to be the mode shape. 120 Chapter 8 Analysis of the 1901 Avenue of the Stars Building N o r t h r i d g e E q S a n F e r n a n d o E q Figure 8.7: Mode Shapes of the 1901 Avenue of the Stars Building 8.4 Joint Time-Frequency Analysis A joint time-frequency analysis was conducted using the methods described in Chapter 3. The time-frequency information is displayed in joint time-frequency plots which contain the input and 121 Chapter 8 Analysis of the 1901 Avenue of the Stars Building output time signals, the standard FRF and the TFRF. The TFRF is shown as a contour plot. The contours have been limited to 3 for simplicity. A legend of the contour levels is shown in the top left corner of each plot. At a sampling rate of 100 sps, the length of a windowed signal with a window size of 256 points is 2.56 seconds. The longest fundamental period is 3.4 seconds. The windowed signal does not capture one full oscillation of the structure's fundamental period and thus the results may be slightly distorted. Only the narrow band TFRFs are presented for the 1901 Avenue of the Stars building as the wide band windowed signal has a non-zero duration of one-fifth the length of the fundamental period. The following points should be considered when viewing time-frequency plots. The TFRF of strong motion is made up of a series of peaks as the frequency content of the acceleration mea-sured at the basement level is not constant. If the input was continuous in frequency and time, the peak would be in the form of a ridge. This ridge would be present, at a particular frequency, for all time. The TFRF is cut at the highest contour level which does not correspond to the peak magnitude of the TFRF. The width of the contour however, conveys information about the peak. A contour with small width means the magnitude is close to the peak, or that the peak itself is very wide. The spacing of the contours indicates the sharpness of the peak. The amplitude of the TFRF does not indicate the presence of heavy shaking. It indicates which frequency components are present at a certain time and their relative strengths. 122 Chapter 8 Analysis of the 1901 Avenue of the Stars Building The information contained in the TFRF is fundamentally different than the FRF. The FRF aver-ages the frequency content over the entire time of the record. More specifically, it is the ratio of the total energy contained in the output, at a particular frequency, over that in the input. The TFRF conveys information on the relative energy levels, at a particular frequency, during a small window of time. For this reason, the magnitude of the TFRF may be misleading. When the mag-nitude of shaking at the roof level, for a given time and frequency, is greater than that at the base, a peak will be shown. Hence, a large peak may be present during free decay of the vibration when the input is very small. Each time-frequency plot contains a vast amount of information on the structure's response to earthquake excitation. Due to the scope of this thesis, a comprehensive study can not be con-ducted for each time-frequency plot. A discussion of the important features, with respect to the building's response, is presented instead. The general characteristics of the Time-Frequency Response Function are discussed, as required to facilitate understanding of the information con-tained in the TFRF. Figures 8.8 and 8.9 show the narrow band time-frequency response plots for the Northridge and San Fernando earthquakes in the transverse direction. The figure caption states the earthquake and direction for each plot 123 Chapter 8 Analysis of the 1901 Avenue of the Stars Building TFRF Contour Magnitude -300 0 300-600 0 600 Base Accel. Roof Accel. (cm/s2) (cm/s2) 80 70 60 50 40 30 20 10 0 • i o CD CD E i-0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Frequency (Hz) Figure 8.8: 1901 T-F Plot - Northridge Earthquake - Transverse Direction 124 Chapter 8 Analysis of the 1901 Avenue of the Stars Building TFRF Contour Magnitude co 60 I 1 I ' I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 -200 0 200 -300 0 300 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Base Accel. Roof Accel. Frequency (Hz) (cm/s2) (cm/s2) Figure 8.9:1901 T-F Plot - San Fernando Earthquake - Transverse Direction 125 Chapter 8 Analysis of the 1901 Avenue of the Stars Building The TFRFs show stable frequencies for both earthquakes. This is expected as the frequency domain analysis showed little variation in the calculated frequencies. One interesting feature of the TFRF is its ability to provide insight into the behavior of the FRF. For example, the frequency fluctuations observed in the TFRFs around the second natural fre-quency may explain the spreading in the peak of the FRF. As well it is seen that the first mode does not dominate the response during heavy shaking. The magnitude of the peaks in the TFRF shown in Figure 8.8, are less during the heavy shaking portion of the record than during the free vibration portion. For the case of a steady-state forced input, a structure will vibrate at the same frequency as the driving force. An earthquake however is transient and steady state motion would not be achieved. For this reason, there will be some modal response of the building during heavy shaking. The low level of modal response seen in the TFRF does not necessarily mean that the building does not vibrate in its modes during heavy shaking, just that the magnitude of the response, at a particular frequency, is eclipsed by the magnitude of the input. To explain this behavior further a typical frequency domain FRF is calculated using one segment of data from heavy shaking (0 - 20 seconds) and one during free vibration (40-60 seconds). The data used in the analysis is the recorded accelerations at the basement and roof, measured during the Northridge earthquake in the transverse direction. The results are shown in Figure 8.10. 126 Chapter 8 Analysis of the 1901 Avenue of the Stars Building 80 70 60 50 o CD £• CD E 40 30 20 10 a) F R F for 40 to 60 sec . Segment CO CO rr 0.5 1.0 1.5 Frequency (Hz) b) F R F for 0 to 20 sec . Segment co CO 0 -500 0 500-500 0 500 Base Accel. Roof Accel. (cm/s2) (cm/s2) 0.5 1.0 1.5 Frequency (Hz) 2.0 2.0 Figure 8.10: Figure Showing Contribution of Heavy Shaking to FRF 127 Chapter 8 Analysis of the 1901 Avenue of the Stars Building Plot a) of Figure 8.10 shows that during the free vibration of the structure the modes are well defined as the magnitude of the FRF during this time is high. Plot b) of Figure 8.10 shows little contribution of the first mode to the response. There is no peak at 0.32 Hz, which corresponds to the first natural frequency and the second mode contributes only slightly. The magnitude of the TFRF, at a particular time, may be several orders of magnitude greater than the FRF. It is shown in Appendix A that the TFRF has the same form as the FRF but its magni-tude is squared and thus is greater than the FRF. Another reason for the large magnitude is that the FRF is averaged, for a particular frequency, over the entire record. The small modal response of the fundamental mode, relative to the input, observed during the heavy shaking is averaged with the relatively greater response during free vibration. The TFRF is localized in time and hence during free vibration the output, at a natural frequency, far exceeds the input which is nearly zero. This is confirmed by the relative magnitudes of the FRFs in Figure 8.10. Figures 8.11 and 8.12 present the narrow band time-frequency plots for the 1901 Avenue of the Stars building in the longitudial direction. 128 Chapter 8 Analysis of the 1901 Avenue of the Stars Building TFRF Contour Magnitude Ji 100 1 900 j 1700 -400 0 400 -400 0 400 Base Accel. Roof Accel. (cm/s2) (cm/s2) 80 70 60 50 40 30 20 10 < i • i if i) * O CJ) CD E 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Frequency (Hz) Figure 8.11:1901 T-F Plot - Northridge Earthquake - Longitudial Direction 129 Chapter 8 Analysis of the 1901 Avenue of the Stars Building TFRF Contour Magnitude '.::::::> 75 I I 400 r_r_r_rj 750 -200 0 2 0 0 - 3 0 0 0 300 Base Acce l . Roof Acce l . (cm/s 2) (cm/s 2) 60 50 40 30 20 10 v. i i8 • " 1 1 > - . 1 • i n i> 1 ii '•v. o CD CD E 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Frequency (Hz) Figure 8.12:1901 T-F Plot - San Fernando Earthquake - Longitudial Direction 130 Chapter 8 Analysis of the 1901 Avenue of the Stars Building Section 8.3 showed a shift in the frequency in the fundamental longitudial direction between the two earthquakes. The narrow band TFRF calculated from the Northridge earthquake, shown in Figure 8.11, shows that the fundamental longitudial mode has a relatively constant peak at 0.22 Hz. The TFRF for the San Fernando earthquake however, does show a shift. Figure 8.12 shows a gradual decrease from 0.29 Hz to 0.22 Hz. The FRF appears to be dominated by the strong peak during the first 5 seconds of shaking. Figure 8.13 presents a close up of the time-frequency plot for the San Fernando earthquake in the frequency range 0 to 0.6 Hz. The TFRF appears to have two dominant frequencies, one being around 0.32 Hz and one 0.25 Hz. The FRF however, only indicates one peak at 0.29 Hz. The FRF averages the peaks over time and thus it is possible that the two dominant frequencies combined in such a way that only a single peak is visible in the frequency domain. The fundamental mode in the transverse direction is 0.32 Hz. If coupling exists between the transverse and longitudial directions, the fundamental mode in the transverse direction could be influencing the FRF in the longitudial direction. The combina-tion of two frequencies which yield a single peak is sensitive to the duration of the two signals, the sampling rate and the amplitude of the signals. With the right combination of the above factors, it can be shown that a time signal with two distinct frequency components can average to form a single peak in the frequency domain. 131 Chapter 8 Analysis of the 1901 Avenue of the Stars Building -200 0 200-300 0 300 Base Accel. Roof Accel. (cm/s2) (cm/s2) 60 50 40 30 20 10 0 / 1 / \ 1 «'/•' M • 1 / \\ 1 \ \ • " v I l i i 1; i i 1 M V il> . ' A ; f 1 / 1 1 • IN 1 \ 11 i ; j i / 1 1 1 1 > 1 1 i i i \ i - -1 1 \ \ i I i \, / / • • y I i \ i . i i i -- i i -i i # * i f 0 _ \ \ *\ V / , / _ \ V \ 1 *\ 1 / 1 » \ t | \ 1 I f 1 i 1 , *' I M I _ 1 ' \ 1 * A l i A , 1 / \ 1 i / i / • A V /f\\< i /. V ' \ : i ll J ji i li J It i 1 , ! •' M> -ft . o CD E i— 0.0 0.1 0.2 0.3 0.4 0.5 Frequency (Hz) 0.6 Figure 8.13:1901 T-F Plot - San Fernando Earthquake - Longitudial Direction - 0 to 0.6 Hz 132 Chapter 8 Analysis of the 1901 Avenue of the Stars Building 8.5 Conclusion The frequency domain analysis identified the first three natural frequencies in the longitudial and transverse directions for both the San Fernando and Northridge earthquakes as well as the tor-sional natural frequencies during the Northridge earthquake. The mode shapes of the building are kinked at the ground level due to a discontinuity in stiffness. The frequency domain analysis identified a shift in the fundamental longitudial mode. The reason for this shift is not decipherable from frequency domain analysis alone. A joint time-frequency analysis shows that the shift is not due to heavy shaking but that the peak at 0.29 Hz is actually a combination of two modes; the fundamental longitudial mode vibrating at 0.22 Hz and the funda-mental transverse mode at 0.32 Hz. 133 C H A P T E R 9 Analysis of the Sheraton Universal Hotel 9.1 Description The Sheraton-Universal Hotel is located in the Universal City area of Los Angeles at 3838 Lank-ershim Boulevard. The building is a 20-storey concrete structure that serves as a hotel and con-vention center. The construction of the Sheraton-Universal was completed in 1968 at a cost of $7.5 million. It is a podium style building with a central tower, which runs from the fourth floor to the roof. The dimensions of the tower are approximately 56 m long by 18 m wide. The base-ment through the 3rd floor measures approximately 61m long by 30 m wide. The tower portion is separated from the rest of the building by expansion joints, which serve to seismically isolate the tower from the rest of the building (Murphy, 1973). A photo of the Sheraton is shown in Fig-ure 9.1 The structure is supported by reinforced concrete spread footings. The underlying soil deposits consist primarily of bedded sandstone with deposits of clay and shale (Murphy, 1973). The lateral load resisting system, for each direction, is a reinforced concrete moment resisting frame. The epicenter of the Northridge earthquake was 19 km north-west of the Sheraton and a peak ground acceleration of 0.33g was felt at the site. The epicenter of the San Fernando earthquake 134 Chapter 9 Analysis of the Sheraton Universal Hotel was approximately 30 km north of the Sheraton, with a peak ground acceleration of 0.17g at the site. While no structural damage was apparent following the Northridge earthquake, serious nonstruc-tural damage occurred in all floors. Six or seven window glasses were broken. The cooling tower shifted off the bracket and extensive damage of the wall coverings including the bathroom tiles and dry wall were observed (Naeim, 1997). (Adapted from: Naeim, 1997) Figure 9.1: Photo of Sheraton Universal Hotel 135 Chapter 9 Analysis of the Sheraton Universal Hotel 9.2 Instrumentation During the San Fernando earthquake the Sheraton was instrumented with seven strong motion accelerometers. Two at each of the basement, 11th floor and the 21st levels, plus a vertical sensor at the basement. A schematic diagram showing sensor location is given in Figure 9.2. During the Northridge earthquake the Sheraton Universal was instrumented with 16 strong motion acceler-ometers. Three at each of the: basement, 3rd, 9th, 16th and roof levels with an additional vertical accelerometer at the basement. A schematic diagram of the sensor layout is shown in Figure 9.3. The sampling rate during both earthquakes was 50 sps. During the Northridge earthquake, 2 transverse records were obtained on each instrumented floor. The transverse signals used in this analysis were obtained by adding the 2 transverse records and dividing by a factor of two. Similarly, the torsional signals used for the analysis were obtained by subtracting the 2 transverse records and dividing by 2. A l l other signals used in this analysis are the original recorded accelerations. Figures 9.4 and 9.5 show the San Fernando and Northridge earthquake signals used for the analysis presented in this chapter. 136 Chapter 9 Analysis of the Sheraton Universal Hotel C CD C N c cu E CO CO CO CQ • a 2 .. « 2 41 cc H « o w *-» 3 (0 <*•» +^  O C » o to z g < u o _ l DC o to LU to c Pla b o 21st FIc c o u_ a o x > O +-» CO 6 C M • X) o o o X JZ tl o z o in M i .m .t.Ui .2 '•P CO > LU LU I i 5 (Adapted from: Shakal et al., 1994) Figure 9.2: Schematic of Sensor Location During the San Fernando Earthquake 137 Chapter 9 Analysis of the Sheraton Universal Hotel I 0 o c JO D_ i_ o o u_ T > C O . J 2 9 CO « n,l c CU E cu w CO C D 8 „ £ 2 « £ ~ o c to O Z o < o o rr o oo z 00 IT) •« ' ' c c Pla JO n o J 2 Roof I J 3th Flc CN c JO D L o o sz CD CU +-> o X > i— o +-* CO I O C M C D . 5 o 6 O 2 > s ro «-* (/> 0. S to u o X "§ o Z _ o cc oi t CM O CD 4*r a t c o (0 > JO) LU LU (Adapted From: Shakal et al., 1994) Figure 9.3: Schematic of Sensor Location During the Northridge Earthquake 138 Chapter 9 Analysis of the Sheraton Universal Hotel Transverse - Basement 3 0.2 CD o.o *^^ | f l j | f l |wV^—-~ < -0.2 Transverse -11th Floor 3 0.1 CD 0.0 O O < -0.1 Transverse - Roof CO 0.2 CD 0.0 O O < -0.2 Longitudial - Basement 3 0.2 CD o.o — o < -0.2 Longitudial - 11th Floor 3 0.1 CD o.o o o < -0.1 Longitudial - Roof 3 0.2 CD o.o o o < -0.2 0 10 20 30 40 50 60 T ime (sec) Figure 9.4: Measured Accelerations During the San Fernando Earthquake 139 Chapter 9 Analysis of the Sheraton Universal Hotel Transverse - Basement CD 0.3 CD 0.0 o o < -0.3 Transverse - 3rd Floor CD o.3 r o < -0.3 CD 0.0 Transverse - 9th Floor CD 0.4 CD 0.0 o o < -0.4 Transverse -16th Floor CD 0.4 CD 0.0 o o < -0.4 Transverse - Roof CD 0.4 Longitudial - Basement CD 0.3 o o < -0.3 0 o.o •—"Hffi||^^^ • 0 10 20 30 T ime (sec) 40 50 Figure 9.5: Measured Accelerations During the Northridge Earthquake 140 Chapter 9 Analysis of the Sheraton Universal Hotel Longitudial - 3rd Floor CD 0.3 CD 0.0 O O < -0.3 Longitudial - 9th Floor CD 0.3 CD 0.0 O o < -0.3 Longitudial -16th Floor D) 0.3 o5 o.o I— i^iif!^ ^ o < -0.3 Longitudial - Roof CD 0.4 CD 0.0 o o < -0.4 Torsional - Basement 3 0.1 -"CD o.o o < -0.1 Torsional - 3rd Floor 3 0.2 "CD o.o o o < -0.2 0 10 20 30 T ime (sec) 40 50 Figure 9.5: Measured Accelerations During the Northridge Earthquake (Continued) I4l Chapter 9 Analysis of the Sheraton Universal Hotel Torsional - 9th Floor 3 0 2 "a> o.o o — i w ^ i f ( f t y v f M | l / \ l | A ^ — — ~ — ~ < -0.2 Torsional -16th Floor 3 0 2 "CD 0.0 •~~~-~^t^jljl^M<tl^rfil(^t\*j4f<r^^ — —~ ~~ o < -0.2 Torsional - Roof 3 0.3 (I "CD 0.0 — i ( J w w y w J \ / l / i j ^ — — • — • o 1 < uyn < -0.3 Figure 9.5: Measured Accelerations During the Northridge Earthquake (Continued) 9.3 Frequency Domain Analysis 9.3.1 Natural Frequencies The frequency domain procedure outlined in Chapter 3 was used to identify the first three natural frequencies in each direction. The modal frequencies for the Sheraton Universal Hotel are given in Table 9.1. Figure 9.6 shows the magnitude of the FRFs between the basement and roof signals during the San Fernando and Northridge earthquakes. As indicated in the graph, the plot at the top shows the FRFs for the Transverse direction (shown as Y) followed by the Longitudial direc-tion (shown as X) in the middle and the torsional modes at the bottom. There are no torsional 142 Chapter 9 Analysis of the Sheraton Universal Hotel modes calculated during the San Fernando earthquake since there were not enough measurements made to calculate the rotation at the base. Mode San Fernando (Hz) Northridge (Hz) TRA1 0.51 0.34 TRA2 1.42 1.10 TRA3 2.20 2.10 LON1 0.46 0.39 LON2 1.46 1.17 LON3 2.05 1.85 ROT1 N A 0.42 ROT2 N A 1.42 ROT3 N A 2.88 Table 9.1: Frequencies of Sheraton 143 Chapter 9 Analysis of the Sheraton Universal Hotel a) Y direction b) X direction Northridge Eq San Fernando Eq c) Torsional 80 50 20 Frequency (Hz) Figure 9.6: Frequency Response Functions for the Sheraton 144 Chapter 9 Analysis of the Sheraton Universal Hotel The natural frequencies of the Sheraton, calculated from the Northridge earthquake records, are consistently lower than those obtained from the analysis of the San Fernando earthquake records. The reason for this shift cannot be determined from frequency domain analysis alone. It is possi-ble that the larger amplitude of shaking during the Northridge earthquake caused the building to behave in non-linear manner. This non-linearity may be due to damage incurred during the earth-quake or the result of temporary reduction of the structural stiffness during heaving shaking. 9.3.2 Mode Shapes The first two mode shapes corresponding to the above frequencies are given in figure 9.7. The resolution of the mode shapes obtained for the Sheraton during the Northridge earthquake are par-ticularly good as five levels of sensors measured the building response. Similar to the 1901 Avenue of the Stars building, the torsional mode shapes presented in Figure 9.7 were obtained by taking the rotational values of displacement and plotting them as a linear displacement. 145 Chapter 9 Analysis of the Sheraton Universal Hotel Northridge Eq San Fernando Eq Figure 9.7: Mode Shapes of the Sheraton Universal 9.4 Joint Time-frequency Analysis The Sheraton Universal Hotel was analyzed using the same methods as those used for the 1901 Avenue of the Stars building. When sampling at 50 sps, a 256 point window gives a non-zero sig-nal of 5.12 seconds. This signal length is sufficient as it is nearly twice that of the fundamental 146 Chapter 9 Analysis of the Sheraton Universal Hotel transverse mode. Figures 9.8 and 9.9 present the narrow band time-frequency plots for the Northridge and San Fernando earthquakes in the transverse direction. 147 Chapter 9 Analysis of the Sheraton Universal Hotel TFRF Contour Magnitude ':::::::< 50 I I 350 r_~J"_ _ J 650 CD CO DC 8 -300 0 300 -400 0 400 Base Accel. Roof Accel. (cm/s2) (cm/s2) 35 30 25 20 15 10 0 Hi o CD O E i-0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Frequency (Hz) Figure 9.8: Sheraton T-F Plot - Northridge Earthquake -Transverse Direction 148 Chapter 9 Analysis of the Sheraton Universal Hotel TFRF Contour Magnitude 35 30 25 20 15 10 -200 0 200 -200 0 200 Base Accel. Roof Accel. (cm/s2) (cm/s2) 0 • • i o CD CD E 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Frequency (Hz) Figure 9.9: Sheraton T-F Plot - San Fernando Earthquake - Transverse Direction 149 Chapter 9 Analysis of the Sheraton Universal Hotel As seen earlier in the 1901 Avenue of the Stars building, the modal response relative to the input is not pronounced during heavy shaking. In Section 9.3 it was shown that the frequency of the fundamental transverse mode, calculated from the Northridge earthquake, is substantially lower than that calculated from the San Fernando earthquake; 0.34 and 0.51 respectively. The TFRFs shown in Figures 9.8 and 9.9 show relatively stable ridges throughout the duration of shaking. If the building experienced significant structural damage during the earthquake, the natural fre-quencies of the building would change. If there was a temporary reduction in the structural stiff-ness, the TFRF would indicate this change over time. It can be inferred therefore, that the Sheraton did not suffer significant damage or structural softening during either earthquake. The fundamental natural frequencies in the transverse mode can be approximated by: for the San Fernando earthquake, where kn and ks represent the modal stiffness and mn and ms (9.1) for the Northridge earthquake and (9.2) 150 Chapter 9 Analysis of the Sheraton Universal Hotel represent the modal mass of the Northridge and San Fernando earthquakes respectively. The ratio of Equation 9.1 and 9.2 yields: T 5 — * 0 . 5 . (9.3) ksmn If the structural stiffness is assumed to be the same during both earthquakes, Equation 9.3 sug-gests that the modal mass of the Sheraton Universal Hotel was twice as great during the Northridge earthquake as the San Fernando earthquake. Although hotels are more susceptible to changes in mass, as compared to office buildings, it is not likely that the mass doubled in the years between the two earthquakes. Although no major structural damage occurred during the Northridge earthquake, there was severe non-structural damage (Naeim, 1997). This level of damage would not cause the stiffness of the system to decrease by a factor of 2 and thus some other factor must be responsible for the decrease in frequency. It is possible that soil-structure interaction could have reduce the frequencies during the Northridge earthquake. A more compre-hensive study is required to verify this. Figures 9.10 and 9.11 give the narrow band time-frequency plots for the longitudial response of the Sheraton during the both earthquakes. 151 Chapter 9 Analysis of the Sheraton Universal Hotel TFRF Contour Magnitude -200 0 200 -300 0 300 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Base Acce l . Roof Acce l . Frequency (Hz) (cm/s 2) (cm/s 2) Figure 9.10: Sheraton T-F Plot - Northridge Earthquake - Longitudial Direction 152 Chapter 9 Analysis of the Sheraton Universal Hotel TFRF Contour Magnitude -200 0 200 -300 0 300 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Base Acce l . Roof Acce l . Frequency (Hz) (cm/s 2) (cm/s 2) Figure 9.11: Sheraton T-F Plot - San Fernando Earthquake - Longitudial Direction 153 Chapter 9 Analysis of the Sheraton Universal Hotel A joint time-frequency analysis of the longitudial direction did not show the presence of any major shifts in frequency. The most interesting feature of the longitudial response is the presence of simultaneous peaks near 0.5 Hz in the San Fernando earthquake shown in Figure 9.11. The TFRF shows 2 distinct ridges between 8 and 20 seconds. Figure 9.12 shows the frequency range 0 to 0.8 Hz. The peak at 0.46 Hz corresponds to the fundamental longitudial frequency. The con-current peak at 0.51 Hz however cannot be the fundamental longitudial mode as a single mode cannot have two simultaneous frequencies. The fundamental mode in the transverse direction has a frequency of 0.51 Hz and thus it can be inferred from the TFRF that coupling exists between the transverse and longitudial directions. 154 Chapter 9 Analysis of the Sheraton Universal Hotel TFRF Contour Magnitude co (0 DC 8 -200 0 200 -300 0 300 Base Acce l . Roof Acce l . (cm/s 2) (cm/s 2) 35 30 25 20 15 10 0 1 1 1 1 i 1 : i f ; '7' ; i ' / J 1 i ' / li • - • •/ / ii / 1 i --i ( i i i , /* 11 i i i -i i i • i > it it o CD CD £ i-0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Frequency (Hz) Figure 9.12: Sheraton T-F Plot - San Fernando Earthquake - Longitudial Direction - 0 to 0.8 Hz 155 Chapter 9 Analysis of the Sheraton Universal Hotel 9.5 Conclusions The frequency domain analysis of the Sheraton Universal Hotel identified the first 3 natural fre-quencies and corresponding mode shapes in each direction. A large shift was observed in the fun-damental transverse mode between the two earthquakes. During the Northridge earthquake the calculated frequency was 33% lower than in the San Fernando earthquake. A joint time-frequency analysis of the Sheraton Universal shows no significant shift in the TFRFs calculated for the transverse direction during the Northridge and San Fernando earthquakes. This suggests that the Sheraton Universal Hotel did not suffer any structural damage during either earthquake. 156 C H A P T E R 10 Conclusions Joint time-frequency analysis is prevalent in many fields. This thesis has shown that it is also applicable to the analysis of vibration data obtained from civil engineering structures. Several methods for joint time-frequency domain analysis were presented. The Wavelet Trans-form, in theory, appears to be a suitable analysis tool although no analysis was conducted using the transform. Many of the Cohen's Class functions, including the Wigner distribution however, do not appear to be easily applied to the analysis of civil engineering data. They are more accu-rate than other methods in terms of energy, but any bilinear distribution which satisfies the mar-ginals is not manifestly positive. Negative values in the time-frequency plane complicate the analysis and limit the physical significance of the results. The Short Time Fourier Transform, which can also be expressed by the Cohen Class function, is a manifestly positive distribution. It also has the advantage of being based on a fundamental analysis tool, the Fourier Transform. 10.1 Applications of Joint Time-Frequency Analysis The Short Time Fourier Transform was used to illustrate the applicability of joint time-frequency analysis to the analysis of civil engineering vibration data. It serves mainly as a supplemental analysis technique to improve standard system identification analysis. Time-frequency analysis, 157 Chapter 10 Conclusions however, also provides insight into the dynamic behavior of structures not available through fre-quency domain analysis alone. The ambient vibration of a 3 span bridge in Calgary, Alberta was analyzed using the standard fre-quency domain approach to estimate natural frequencies and mode shapes. Joint time-frequency analysis was subsequently used to demonstrate that the properties of the bridge did not change significantly during the test. The joint time-frequency analysis showed that an unexplained peak in the averaged power spectral density of the records was only present during a small portion of the testing and thus not a natural frequency. A similar study was conducted on a single span bridge near Kamloops, British Columbia. A stan-dard frequency domain approach yielded an averaged power spectral density which had a number of peaks near a certain frequency. The frequency domain analysis identified one natural mode of vibration near that peak, but failed to explain the occurrence of the other adjacent peaks. A joint time-frequency analysis, however showed that the numerous peaks were in fact one natural mode whose frequency shifted in time. Joint time-frequency analysis was used to investigate upstream sliding observed during shake table testing of a concrete gravity dam model unbonded at the base. The analysis showed that, during upstream motion, the dam model vibrates not only at the base excitation frequency, but also at one half and one and a half times the forcing frequency. This behavior may have been the result of the model rocking as it moved upstream. 158 Chapter 10 Conclusions The most promising application of joint time-frequency analysis is for the interpretation of strong motion data. The Time-Frequency Response Function, (TFRF), presented in this thesis allows for the study of many aspects of the dynamic behavior of structures not easily studied with frequency domain analysis. This includes the coupling between modes, the temporal location of modal response and the identification of the temporal location of shifts seen between earthquakes. The use of joint time-frequency analysis to explain anomalous results in the frequency domain greatly enhances the standard frequency domain analysis. A joint time-frequency analysis can explain anomalous shifts in frequency. One such shift was seen in the longitudial direction of the 1901 Avenue of the Stars building. A joint time-frequency analysis showed that the shift is not due to heavy shaking but that the peak at 0.29 Hz is actually a combination of two modes; the fundamental longitudial mode vibrating at 0.22 Hz during free vibration and the fundamental transverse mode vibrating at 0.32 Hz during heavy shaking. The frequency of the fundamental transverse mode of the Sheraton Universal Hotel was consider-ably lower during the Northridge earthquake. A joint time-frequency analysis showed that the frequencies were stable during both earthquakes and thus the shift in frequency must have been caused by some other factor in the years other than damage or structural softening. The TFRF confirms that the fundamental mode of the structure controls its response during free vibration. 159 Chapter 10 Conclusions 10.2 Limitations of Joint Time-Frequency Analysis The practical application of joint time-frequency analysis is limited to relatively few functions. The bilinear Cohen's Class functions are more accurate in terms of energy content than the Short Time Fourier Transform, but they yield negative values in the time-frequency plane making the application of the functions difficult. The Short Time Fourier Transform is manifestly positive but if suffers from slightly distorted energy values as the energy of the window is combined with the energy of the original signal. The resolution of the STFT is also limited by the uncertainty principle. The time and frequency distributions of a signal are directly related as is the standard deviation of the time and frequency distributions. The standard deviations of the time and frequency distributions can not both be made arbitrarily small and therefore there is a trade off between time and frequency accuracy. Another problem with the Short Time Fourier Transform is leakage. The window function leaks energy into the surrounding frequencies of a windowed signal. Leakage is minimized by choos-ing window functions, such as the Hanning window, which slowly decay to zero. In the dynamic structural analysis of civil engineering structures, the lower modes of vibration are of primary interest. The window function used for the Short Time Fourier Analysis must be cho-sen sufficiently wide to capture these long period oscillations. For this reason the time-resolution of joint time-frequency analysis may be limited. Chapter 10 Conclusions 10.3 Suggested Future Research The Wavelet transform should be explored in more detail with similar case studies as those pre-sented in this thesis for the STFT. An in depth study of the TFRF is required to better understand its behavior. This study may include the use of different window types and sizes to explore their effect on the energy contained in the spectrogram. A study on the effect of the initial conditions of the windowed signal and the effect of leakage should be made. The TFRF function could be used to study many interesting aspects of the response of a structure to strong motion input including the prediction of damage during a seismic event and the response of base isolated buildings. The TFRF should be calculated between the basement and other floors of a building to emphasize other dynamic characteristics of the structure. A study should be conducted on how a function, similar to the TFRF, could be formed by taking the ratio of the Fourier Transform of a windowed output signal over the Fourier Transform of a windowed input signal. This function would yield magnitudes similar to the Frequency Response Function as well as retain the phase information required to construct the structure's mode shapes. It would be useful, and relatively simple, to write a customized computer program which would calculate the TFRF directly from 2 time signals. This program should include different windows and window sizes. The option to normalize windows to unit energy should be available, as this would produce spectrograms with the correct total energy content. The ability to add and subtract signals prior to the calculation of the time-frequency spectrogram and the TFRF should be incor-porated in the program to allow for the calculation of the torsional TFRF. The program should I 6 l Chapter 10 Conclusions also include a smoothing function to allow for easier presentation and initial investigation of the TFRF. 162 C H A P T E R 11 References Asmussen, J.C. (1997) Modal Analysis Based on The Random Decrement Technique -Application to Civil Engineering Structures. Ph.D. Thesis. Department of Building Technology and Structural Engineering, Aalborg University. Associate Committee on National Building Code (1995) National Building Code of Canada 1995, National Research Council, Ottawa. Bendat, J.S. and Piersol, A .G . (1993) Engineering Applications of Correlation and Spectral Analysis. John Wiley & Sons, New York, NY. Black, C.J., Ventura, C E . and Tsai, P. (1997a) Ambient Vibration Measurements of the University Drive/ Crowchild Trail Bridge in Calgary, Alberta. Technical report submitted to ISIS. Project Number: EQ 97-005. Black, C.J. (1997b) Seismic Response of Four Instrumented Buildings During the Northridge and San Fernando Earthquakes. Civi l 598 Report prepared for C E . Ventura. Black, C.J., Ventura, C E . (1998a) Ambient Vibration Measurements of the Lindquist Bridge. Technical report. Project Number: EQ 98-003. Black, C.J., Horyna, T., Ventura, C E . and Foschi, R.O. (1998b) Shake Table Testing of a Concrete Gravity Dam Model Unbonded at Base - Phase 2. Technical report submitted to B C Hydro. Project Number: EQ 98-001. Black, C J . and Ventura, C E . (1998c) Seismic Response of Four Instrumented Buildings During two Earthquakes. Proceedings of the XVIth International Modal Analysis Conference, Santa Barbara, California. Bonato, P., Ceravolo, R., and De Stefano, A . (1997) Time-Frequency and Ambiguity Function Approaches in Structural Identification, Journal of Engineering Mechanics. December. Chopra, A . K . (1995) Dynamics of Structures: Theory and Applications to Earthquake 163 Chapter 11 References Engineering, Prentice Hall, New Jersey. Cohen, L . (1995) Time-Frequency Analysis. Prentice Hall PTR, Englewood Cliffs, New Jersey. Cook, S., Ventura, C.E., Jackson, S. and Black, C.J. (1998) Static and Dynamic Strain Measurements of the Lindquist Bridge. Technical report prepared for Reid Crowther and Partners Ltd. Project Numer: EQ 98-002. Cooley, J.W. and Tukey, J.W. (1965) An algorithm for the Machine Calculation of Complex Fourier Series, Mathematics of Computation, 19. Daubechies, I. (1988) Orthonormal Bases of Compactly Supported Wavelets. Comm. on Pure andAppl. Math, Vol. 4, Nove. Diehl, J.G. (1991) Ambient Vibration Survey: Application Theory and Analytical Techniques Application Note No. 3. Kinemetrics: Pasadena, C A . Dossing, O. (1998) Uncertainty in Time/Frequency Domain Representations, Sound and Vibration, January. EDI, Experimental Dynamic Investigations Ltd. (1995) U2, V2 & P2 Manual. Ewins, D.J. (1984) Modal Testing: Theory and Practice, Research Studies Press Ltd. John Wliley & Sons Inc. New York. Farrar, C.R., Doebling, S.W. and Cornwell, P.J. (1998) A Comparison Study of Modal Parameter Confidence Intervals Computed Using the Monte Carlo and Bootstrap Techniques, Proceedings of the XVIth International Modal Analysis Conference, Santa Barbara, California. Felber, A.J . (1993) "Development of a Hybrid Evaluation System", Ph.D. Thesis, University of British Columbia, Department of Civi l Engineering, Vancouver, 1993. Fenves, G.L. and Desroches, R. (1994). Response of the Northwest Connector in the Landers and Big Bear Earthquakes, Report No. UCB/EERC-94/02, Earthquake Engineering Research Center, The University of California at Berkeley. Finn, W.D.L., Ventura, C.E. and Schuster, N.D. (1995) Ground Motions During the 1994 Northridge Earthquake, Canadian Journal of Civil Engineering: Vol. 22, Number 2, April 164 Chapter 11 References Gasparini, D.A. and Venmarcke, E.H. (1976) SIMQKE - A Program for Artificial Motion Generation, User's Manual. Massachusetts Institute of Technology, Department of Civil Engineering, Cambridge, Massachusetts. Horyna, T., Ventura, C E . and Foschi, R.O. (1997) Shake Table Testing of a Concrete Gravity Dam Model Unbonded at the Base. Technical report submitted to B C Hydro. Project Number: EQ 96-006. Huang, S.Y., Qi, G.Z. and Yang, J.C.S. (1994) Wavelet For System Identification, Proceedings of the XHth International Modal Analysis Conference, Tennessee. Humar, J.L. (1990) Dynamics of Structures, Prentice Hall, New Jersey. ISIS (1997) Innovator Newsletter of ISIS Canada, April. Luz, E., Gurr-Beyer, C. and Stoecklin, W. (1984) Experimental Investigations of Natural Frequencies and Modes of the HDR Nuclear Power Plant by Means of Microtremor Excitation, Proceedings of the 8th World Conference on Earthquake Engineering, San Francisco, California. Murphy, L . M . (1973) San Fernando, California, Earthquake of February 9, 1971. US Department of Commerce. National Oceanic and Atmosphere Administration. Washington, D . C National Instruments (1995) Joint Time Frequency Analyzer 3.1a Toolkit, Labview Executable. Newhook, J.P. and Mufti, A . A . (1996) A Reinforcing Steel-Free Concrete Deck Slab for the Salmon River Bridge, Concrete International, June 1996. Newland, D.E. (1996) An Introduction to Random Vibrations, Spectral and Wavelet Analysis, 3rd Ed, Addison Wesley Longman Limited, Essex, United Kingdom. Proakis, J.G. and Manolakis, D.G. (1996) Digital Signal Processing, Principles, Algorithms, and Applications, 3rd Ed, Prentice Hall, New Jersey. Ramirez, R.W. (1985) The FFT Fundamentals and Concepts. Englewood Cliffs, New Jersey: Tektronix. Rezai, M . K . and Ventura, C E . (1996) Structural Dynamic Characteristics of a 12 Storey Steel Frame Building During the 1971 San Fernando and the 1994 Northridge Earthquake, CSCE 1996 Annual Conference Proceedings: Volume lib. Rezai, M . K . and Ventura, C E . (1995) Wavelet Transform Analysis of Acceleration Data Recorded 165 Chapter 11 References at Treasure Island Site During Loma Prieta Earthquake, Proceedings of the Second International Conference on Seismology and Earthquake Engineering, Tehran, Islamic Republic of Iran. Schuster, N.D. (1994) Dynamic Characteristics of a 30 Storey Building During Construction Detected From Ambient Vibration Measurements. M.A.Sc. thesis, University of British Columbia, Canada. Shakal, A. , Huang, M . , Darragh, R., Cao, T.Q., Sherburne, R.W., Malhotra, P., Cramer, C , Sydor, R., Graizer, V., Maldonado, G., Peterson, C. and Wampole, J. (1994) CSMIP Strong Motion Records from the Northridge, California Earthquake of January 17, 1994. California Department of Conservation, Division of Mines and Geology, Office of Strong Motion Studies, Report OSMS 94-07, February. Skolnik, M.I. (1980) Introduction to Radar Systems, Mcgraw-Hill Book Co. Staszewski, W.J. and Giacomin, J. (1997) Applications of the Wavelet Based FRFs to the Analysis of Nonstationary Vehicle Data, Proceedings of the XVth International Modal Analysis Conference, Orlando, Florida. Vibrant (1996) Me'Scope Modal. Vibrant Technology, Inc. Jamestown, California. Wigner, E.P. (1932) On the Quantum Correction for Thermodynamic Equilibrium, Physical Review, vol. 40 166 APPENDIX A The Time Frequency Response Function If the input into a structure is known, the Time Frequency Response Function can be used to esti-mate the natural frequencies of the structure. It is analogous to the frequency domain FRF, in that, it is the ratio of the output to input motion. The TFRF however, provides information in both the time and frequency domains simultaneously. A proof of the TFRF is given along with a single degree of freedom example. A.1 The Time Frequency Response Function As shown in Chapter 2, the Frequency Response Function, (FRF), is defined as the response of a structure to a unit amplitude, harmonic function and is given by: H ( 0 ) J = fM (A.,) Therefore, the FRF can be calculated by dividing the Fourier Transform of the output signal by the Fourier Transform of the input signal. To prove the TFRF, it is easiest to show that it has the same form as the FRF. That is, show that the windowed signal will yield a correct value of frequency and then extend it to all window loca-167 Appendix A The Time Frequency Response Function tions introducing the time component of the TFRF. As described in Chapter 2, the time-frequency spectrogram, P(co, t), is obtained from: P(co, 0 = I ^ C O ] 2 (A.2) where |Su,(f)| is the Short Time Fourier Transform of a time signal. If P(co, 0;« represents the time-frequency spectrogram of the input and P((H, t)out represents the output time-frequency spectrogram, then the TFRF is given by: P(to, t) v ' 'in The proof of Equation 3 begins with the dynamic equation of equilibrium for a single degree of freedom system with a forcing function p{t) given by: my{t) + cy{t) + ky = p{t) (A.4) Multiplying both sides of Equation 4 by a window function w(t) yields: my(t)w(t) + cy(t)w(t) + ky(t)w(t) = p(t)w(t). (A.5) 168 Appendix A The Time Frequency Response Function If: x(t) = y(t)w(t), (A.6) then: x(t) = y(t)w(t) + y(t)w(t) (A.7) and x(t) = y(t)w(t) + 2y(t)w(t) + y(t)w(t). (A.8) If a rectangular window w(t), in the time interval (ta, th), is chosen for the analysis, then: l f o r t <t<th w(t) = ; a b 0 elsewhere The first and second derivatives of the window function are given by: w(t) = $(t-ta)-b(t-tb) (A.9) and w(t) = &(t-ta)-8(t-tb), (A.10) where §(t-ta) and 8(t-th) represent a Dirac delta function centered at time ta and tb, respec-169 Appendix A The Time Frequency Response Function tively and 8' is the first time derivative of the Dirac delta function. Substituting Equations 9 and 10 into Equations 7 and 8 yields the following expressions for x{t) and x(t) : x(t) = y(t)w(t) + Yad(t-ta)-YbHt-tb) and ( A . l l ) x(t) = y (0w(0 + 2 | y a 5 ( r - r a ) - ^ 5 ( f - ^ ) | - r a 5 ' ( r - ? a ) - F ^ 8 ' ( r - ^ ) , (A. 12) where Ya = y(ta), Yh = y{th), Ya = y(ta) and Yh = y{th). If equations 11 and 12 are rearranged for y(t)w(t) and y(t)w(t) and substituted into Equation 5 the result is: mx(t) + cx(t) + kx(t)-mB(t)-cA(t) = f(t) (A. 13) where f(t) = p(t)w(t), A(t) = Yab(t-ta)-Ybb(t-tb) (A.14) 170 Appendix A The Time Frequency Response Function and B(t) = 2 | F f l 5 ( . - . a ) - y ^ - ^ ) | - F f l 5 ' a - y - r ^ ( r - y . (A.15) Notice, that A(t) and B(t) are zero for all locations other than ta and tb Taking the Fourier Transform of both sides of Equation 13 yields: 2 (&-co m + /coc)X(co) - mB(d)) - cA(co) = F(co) (A.16) where: X(cd) and F(co) are the Fourier Transforms of x(t) and f(t) respectively and. i(Htn i(i)th A(co) = Yae a-Ybe (A.17) and B((0) = (2Ya + i(0Ya)e a-(2Yb + mYb)e . (A.18) By defining an effective forcing function F((fl) = F(co) - mB((a) + cA(co) the windowed H((£>) function at time t is given by: 171 Appendix A The Time Frequency Response Function H((ti) = z± . (A.19) Ft((0) Equation 19 has the same form as Equation 1 for the Frequency Response Function with the Fou-rier Transform of the forcing function replaced by the Fourier Transform of a modified forcing function which includes the effect of the end conditions, that is, the displacement and velocity of y(t) at t = ta and t = th. Equation 19 should, in theory, yield the same FRF for each time shift of the window. Therefore, if the center of the window, w(t), at any given time is defined by tc, then Htc((d) =>'H((0, tc) as t is translated through the record. Equation 19 can therefore be rewritten as: H((0,tc) = zz . (A.20) The TFRF given by Equation 3 is obtained by sqauring the magnitude of tc) and F(co, tc) in Equation 20. The resulting expression is the spectrogram of the output divided by the spectro-gram of the input with the corrections for the end conditions and is given by: 172 Appendix A The Time Frequency Response Function TFRF((0, tc) = c out |*eMc>|2 (A.21) 2 Therefore, if the input is corrected for the end conditions of the window function, the TFRF can be defined as the square of the amplitude of the response of a structure to a unit harmonic input function. In practical signal analysis, the numerical evaluation of the spectrogram is done with a Hanning window function which minimizes undesirable leakage effects. If a Hanning window is used for the computation of the spectrogram in the numerator and denominator of Equation 21, it can be shown that the product of A(f) and B(t) with the Hanning window produces zero values in the interval (t(„ th). As a consequence, there is no need to account for the end conditions in (ta, th), and therefore the TFRF calculated using the Hanning window function is a good approximation to the square of FRF. It is only an approximation, as the Hanning window slightly distorts the spec-trogram. The TFRF was calculated for a single degree of freedom system subjected to the Northridge earthquake. The SDOF system has a natural frequency of 2 Hz and a damping value of 5%. The resulting time-frequency plot is presented in Figure 11.1. The TFRF shows a ridge at 2 Hz for the entire length of the record confirming the accuracy of the A.2 SDOF Example of the TFRF 173 Appendix A The Time Frequency Response Function TFRF. It can be inferred from the contour levels in Figure 11.1 that the amplitude of the TFRF is around 0.09 which is the square of the amplitude of the FRF as predicted by Equation 21. 174 Appendix A The Time Frequency Response Function TFRF Magnitude L V . V . V . " 0 . 0 0 3 I I 0 . 0 3 d)0.2 CO ;0.1 0.0 60 50 40 30 20 10 -300 0 300 -15 0 15 B a s e A c c e l . Roof A c c e l . (cm/s 2 ) (cm/s 2 ) 0 0 i J o JO, CU E 2 3 4 Frequency (Hz) Figure 11.1: SDOF Example Using the TFRF 175 APPENDIX B Data Analysis Software This appendix briefly describes the frequency domain and joint time-frequency domain software programs used for the analysis presented in this thesis. B.1 Ambient Vibration Data Analysis Software Frequency Domain Analysis The custom data acquisition program A V D A (Schuster, 1994) was used to record the ambient vibrations of both bridges studied in this thesis. The computer programs P2, U2 and V2 (EDI, 1995) were used to identify the natural frequencies and mode shapes of each bridge. Program P2 was used to compute the Averaged Normalized Power Spectral Density (ANPSD) for a series of ambient vibration records. Program U2 was used to view signals quickly in order to decide whether the data obtained was satisfactory. It was also used to calculate individual power spectral densities and the potential modal ratios needed for the frequency and mode shape estimations. V2 was developed in conjunction with the program U2 to illustrate and animate mode shapes obtained from ambient vibration data. Joint Time-Frequencv Analysis The Lab view JTFA Toolkit 3.1a (National Instruments, 1995) was used to compute the time-fre-176 Appendix B Data Analysis Software quency spectrograms. Due to the input characteristics no further analysis was required. B.2 Shake Table Data Analysis Joint Time-Frequency Analysis The time-frequency spectrograms were calculated using the JTFA Toolkit (National Instruments, 1995). No further analysis programs were required. B.3 Strong Motion Data Analysis Software Frequency Domain Analysis The dynamic properties (mode shapes and frequencies) of the buildings studied in this thesis are calculated using the frequency domain techniques described above. To facilitate this, the program Me'Scope Modal (Vibrant, 1997) was used. The time domain acceleration data recorded during the seismic events is read into Me'Scope. Me'Scope allows the user to animate a structure by building a model and assigning records to certain joints. By assigning the base records as inputs and the others as outputs, Me'Scope calculates the frequency response matrix for the given struc-ture. The frequency response information can be viewed in a number of ways: FRF magnitude, co-quad plot (real vs. frequency and imaginary vs. frequency), phase, coherence and individual power spectral densities. This information along with animation in the frequency domain allows the user to determine the structure's natural frequencies and mode shapes. Joint Time-Frequency Analysis The strong motion data measured at the basement and roof levels are used to calculate the Short Time Fourier Transform spectrogram. The spectrogram was calculated using the JTFA Toolkit 177 Appendix B Data Analysis Software (National Instruments, 1995). The TFRF is constructed using a spreadsheet program to divide the roof spectrogram by the basement spectrogram. To avoid any undefined points in the TFRF all points with nearly zero magnitude are given a magnitude of 0.0001 prior to the construction of the TFRF. 178 

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