DYNAMIC ANALYSIS OF BRIDGES WITH LAMINATED WOOD GIRDERS by Tomas Horyna Dipl. Engineer, The Czech Technical University in Prague, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER 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 COLUMBIA AUGUST 1995 © Tomas Horyna, 1995 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 CIVlU ENGINEERING, The University of British Columbia Vancouver, Canada D a t e OCTOBER l o t h , l«W5" DE-6 (2/88) ABSTRACT The response of bridges due to loading by heavy vehicles is one of the important aspects of bridge design. The effects of traffic on the response are significant if the bridge is a short and light-weight structure and the vehicle is very heavy. This is the case of a loaded logging truck passing over a logging bridge. The passage of a vehicle over a bridge is a complex dynamic problem and, for simplicity, bridge design codes treat it in a pseudostatic manner. Generally, the codes require a static analysis of the bridge using a specified design truck. The static response of the bridge is then multiplied by a dynamic load allowance factor in order to account for the dynamic response. The timber industry is one of the most important in the province of British Columbia and transportation of its products is a significant part of this industry. Logging bridges made from wood form an important element of this transportation network and their optimal design is, therefore, of importance to the province. However, provisions of bridge design codes are mostly supported by investigations conducted on highway bridges. Such is the case of the new Canadian Highway Bridge Design Code (CHBDC), which is now being prepared in Canada for release in 1996. It is desirable, therefore, to investigate i f the recommendations of C H B D C for wooden bridges correspond with the actual response of a logging bridge loaded with a passing logging truck. The objective of this study was to develop a tool to perform dynamic analysis of the system logging bridge - logging truck and to verify, that the dynamic load allowance factor from the C H B D C represents adequately the dynamic amplification factor obtained from dynamic analysis of the system logging bridge - logging truck. A numerical model to simulate the ii passage of a vehicle over a bridge was developed. The model was calibrated with results of ambient and forced vibration testing of an existing bridge. Finally, dynamic amplification factors were obtained from numerical simulations and compared with CHBDC provisions. The results of this study showed that these provisions are generally adequate for wooden logging bridges. iii TABLE OF CONTENTS Abstract ii Table of Contents iv List of Tables vi List of Figures viii Acknowledgments xii Dedication xiii Chapter 1 - Introduction 1 1.1 Overview of the Project 1 1.2 Previous Research 2 1.3 Definition of the Dynamic Amplification Factor 6 1.4 Approach in the Canadian Highway Bridge Design Code 7 1.5 Objectives and Scope of this Study 9 1.6 Outline of the Thesis 9 Chapter 2 - Numerical Model for the System Bridge-Vehicle 11 2.1 Derivation of the Stiffness Matrix 11 2.2 Derivation of the Mass Matrix 19 2.3 Derivation of the Damping Matrix 21 2.4 Roughness of the Bridge Deck 23 2.5 Numerical Model for the Vehicle 24 2.6 Derivation of Matrices of the Coupled System 26 2.7 Procedure for Solution of the Equations of Motion 35 Chapter 3 - Dynamic Testing of the Lillooet River Bridge 40 3.1 Description of the Structure 40 3.2 Description of the Data Acquisition System 45 3.3 Ambient Vibration Testing 48 3.4 Forced Vibration Testing 51 Chapter 4 - Calibration and Verification of the Numerical Model 68 4.1 Determination of Actual Stiffness of the Bridge 68 4.2 Verification of the Mass Matrix 69 4.3 Verification of the Model From Results of the Forced Vibration Testing 70 Chapter 5 - Numerical Simulations With a Loaded Truck 102 5.1 Summary of the Input Data 102 5.2 Results of Simulations With Vehicle L5441 103 Recommendations for Future Research 117 Conclusions 118 Nomenclature 120 Abbreviations 122 References 123 Appendix A - Bridge-Vehicle Interaction Software 126 A. 1 Overview of the Programs 126 A.2 Program B V I - Operating Instructions 128 A.2.1 Running the Program 128 A.2.2 Description and Example of the Input File 129 A.2.3 Structure of the Output File 134 A.3 Program B V I E - Operating Instructions 135 A.3.1 Running the Program 135 A.3.2 Description of the Graphic Display of the Results 135 A.4 Program B V I D A F - Operating Instructions 136 A.4.1 Running the Program 136 A.4.2 Description of the Output File 136 A. 4 Program B V L N V - Operating Instructions 137 Appendix B - Drawings of the Lillooet River Bridge 138 B . l General Layout 138 B . 2 Superstructure Details 139 Appendix C - Details of the Dynamic Tests 140 C. l Details of the Ambient Vibration Testing 141 C. 2 Details of the Forced Vibration Testing 145 Appendix D - Program Frequency Response Function - Operating Instructions 152 D. 1 Description 152 D.2 Installation, System Requirements and Program Execution 153 D. 3 Example 155 Appendix E - Program Time History Viewer - Operating Instructions 157 E . l Description 157 E.2 Installation, System Requirements and Program Execution 157 v LIST OF TABLES Table 3.1. Natural frequencies of the Lillooet River Bridge 51 Table 3.2. Measured dynamic amplification factors at the midspan of the long span (point 19) due to the passage of the testing vehicle T3162 60 Table 3.3. Measured dynamic amplification factors at the midspan of the long span (point 20) due to the passage of the testing vehicle T3162 61 Table 3.4. Measured dynamic amplification factors at the long span (point 23) due to the passage of the testing vehicle T3162 62 Table 3.5. Measured dynamic amplification factors at the long span (point 24) due to the passage of the testing vehicle T3162 63 Table 3.6. Measured dynamic amplification factors at the long span (point 27) due to the passage of the testing vehicle T3162 64 Table 3.7. Measured dynamic amplification factors at the long span (point 28) due to the passage of the testing vehicle T3162 65 Table 3.8. Measured dynamic amplification factors at the midspan of the short span (point 45) due to the passage of the testing vehicle T3162 66 Table 3.9. Measured dynamic amplification factors at the midspan of the short span (point 46) due to the passage of the testing vehicle T3162 67 Table 4.1. Experimental and numerical natural frequencies of the Lillooet River Bridge. .... 70 Table 4.2. Wave length characteristics of the roughness function 73 Table 4.3. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 1, location: short span at midspan 78 Table 4.4. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 1, location: long span at midspan 79 Table 4.5. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 2, location: short span at midspan 80 Table 4.6. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 2, location: long span at midspan 81 Table 4.7. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 3, location: short span at midspan 82 Table 4.8. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 3, location: long span at midspan 83 Table 4.9. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 4, location: short span at midspan 84 Table 4.10. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 4, location: long span at midspan 85 Table 4.11. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 5, location: short span at midspan 86 Table 4.12. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 5, location: long span at midspan 87 Table 4.13. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 6, location: short span at midspan 88 Table 4.14. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 6, location: long span at midspan 89 Table 4.15. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 7, location: short span at midspan 90 Table 4.16. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 7, location: long span at midspan 91 Table 4.17. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 8, location: short span at midspan 92 Table 4.18. Dynamic amplification factors due to the passage of the testing truck T3162, roughness of the bridge deck: type 8, location: long span at midspan 93 Table 5.1. Dynamic amplification factors due to the passage of the vehicle T5441, location: midspan of the long span 115 Table 5.2 Dynamic amplification factors due to the passage of the vehicle T5441, location: midspan of the short span 116 Table C . l . Ambient vibration testing of the Lillooet River Bridge - details of the setups. .. 142 Table C.2. Forced vibration testing of the Lillooet River Bridge - details of the setups 149 Table C.3. Locations and types of the sensors for the forced vibration testing 146 LIST OF FIGURES Figure 1.1. Definition of the dynamic amplification factor 6 Figure 2.1. Kinematic relations in the cross-section of the beam 12 Figure 2.2. Numbering of degrees of freedom of the 6-d.o.f. element 14 Figure 2.3. Inertia forces acting on an element of volume dV 19 Figure 2.4. Two-dimensional model of the vehicle 25 Figure 2.5. Contact of the i-th axle with the bridge 29 Figure 2.6. Values of the 1-st natural frequency of the Lillooet River Bridge versus the position of the vehicle T3162 (the testing truck) 37 Figure 2.7. Values of the 2-nd natural frequency of the Lillooet River Bridge versus the position of the vehicle T3162 (the testing truck) 37 Figure 2.8. Values of the 1-st natural frequency of the Lillooet River Bridge versus the position of the vehicle T5441 38 Figure 2.9. Values of the 2-nd natural frequency of the Lillooet River Bridge versus the position of the vehicle T5441 38 Figure 3.1. Elevation view of the Lillooet River Bridge from the up-stream (West) side 41 Figure 3.2. View of the bridge with passing fully loaded logging truck 41 Figure 3.3. The South abutment of the bridge with the beginning of the short span 42 Figure 3.4. View of the pier between the approaching and the long spans 43 Figure 3.5. View of the long span from below 44 Figure 3.6. View of the bridge deck and approaching testing vehicle T3162 44 Figure 3.7. View of a plate with two accelerometers on the bridge deck 50 Figure 3.8. Testing vehicle T3162 - the empty logging truck used for testing 53 Figure 3.9. The bridge with the testing vehicle T3162 in the long span 53 Figure 3.10. Window in frequency domain used for filtering of the acceleration records 55 Figure 3.11. Time histories of measured acceleration and calculated dynamic displacement; Location: midspan of the long span (point 23), setup 7, speed 28.94 km/h 56 Figure 3.12. Time histories of measured acceleration and calculated dynamic displacement; Location: midspan of the short span (point 45), setup 7, speed 28.94 km/h 56 Figure 3.13. Measured dynamic amplification factors at the midspan of the long span (point 19) due to the passage of the testing vehicle T3162 60 Figure 3.14. Measured dynamic amplification factors at the midspan of the long span (point 20) due to the passage of the testing vehicle T3162 61 Figure 3.15. Measured dynamic amplification factors at the long span (point 23) due to the passage of the testing vehicle T3162 62 Figure 3.16. Measured dynamic amplification factors at the long span (point 24) due to the passage of the testing vehicle T3162 63 Figure 3.17. Measured dynamic amplification factors at the long span (point 27) due to the passage of the testing vehicle T3162 64 Figure 3.18. Measured dynamic amplification factors at the long span (point 28) due to the passage of the testing vehicle T3162 65 Figure 3.19. Measured dynamic amplification factors at the midspan of the short span (point 45) due to the passage of the testing vehicle T3162 66 Figure 3.20. Measured dynamic amplification factors at the midspan of the short span (point 46) due to the passage of the testing vehicle T3162 67 Figure 4.1. The window for scaling amplitudes of the roughness process 72 Figure 4.2. Roughness of the bridge deck - types 1,2,3 and 4 74 Figure 4.3. Roughness of the bridge deck - types 5,6,7 and 8 75 Figure 4.4a. Range of the dynamic amplification factors from Table 4.3 78 Figure 4.4b. Averaged dynamic amplification factors from Table 4.3 78 Figure 4.5a. Range of the dynamic amplification factors from Table 4.4 79 Figure 4.5b. Averaged dynamic amplification factors from Table 4.4 79 Figure 4.6a. Range of the dynamic amplification factors from Table 4.5 80 Figure 4.6b. Averaged dynamic amplification factors from Table 4.5 80 Figure 4.7a. Range of the dynamic amplification factors from Table 4.6 81 Figure 4.7b. Averaged dynamic amplification factors from Table 4.6 81 Figure 4.8a. Range of the dynamic amplification factors from Table 4.7 82 Figure 4.8b. Averaged dynamic amplification factors from Table 4.7 82 Figure 4.9a. Range of the dynamic amplification factors from Table 4.8 83 Figure 4.9b. Averaged dynamic amplification factors from Table 4.8 83 Figure 4.10a. Range of the dynamic amplification factors from Table 4.9 84 Figure 4.10b. Averaged dynamic amplification factors from Table 4.9 84 Figure 4.11a. Range of the dynamic amplification factors from Table 4.10 85 Figure 4.1 lb. Averaged dynamic amplification factors from Table 4.10 85 Figure 4.12a. Range of the dynamic amplification factors from Table 4.11 86 Figure 4.12b. Averaged dynamic amplification factors from Table 4.11 86 Figure 4.13a. Range of the dynamic amplification factors from Table 4.12 87 Figure 4.13b. Averaged dynamic amplification factors from Table 4.12 87 ix Figure 4.14a. Range of the dynamic amplification factors from Table 4.13 88 Figure 4.14b. Averaged dynamic amplification factors from Table 4.13 88 Figure 4.15a. Range of the dynamic amplification factors from Table 4.14 89 Figure 4.15b. Averaged dynamic amplification factors from Table 4.14 89 Figure 4.16a. Range of the dynamic amplification factors from Table 4.15 90 Figure 4.16b. Averaged dynamic amplification factors from Table 4.15 90 Figure 4.17a. Range of the dynamic amplification factors from Table 4.16 91 Figure 4.17b. Averaged dynamic amplification factors from Table 4.16 91 Figure 4.18a. Range of the dynamic amplification factors from Table 4.17 92 Figure 4.18b. Averaged dynamic amplification factors from Table 4.17 92 Figure 4.19a. Range of the dynamic amplification factors from Table 4.18 93 Figure 4.19b. Averaged dynamic amplification factors from Table 4.18 93 Figure 4.20. Comparison of experimental and numerical dynamic amplification factors, Location: midspan of the short span, vehicle: T3162, roughness: type 2 94 Figure 4.21. Comparison of experimental and numerical dynamic amplification factors, Location: midspan of the long span, vehicle: T3162, roughness: type 2 94 Figure 4.22a. Representative example of the roughness profile used for simulations 96 Figure 4.22b. Positions of significant points along the Lillooet River Bridge 96 Figure 4.23. Acceleration records from experiment and from simulations, position of the vehicle; Location: short span at midspan, vehicle: T3162, speed: 13.36 km/h. 100 Figure 4.24. Dynamic displacement from experiment and from simulations, pos. of the vehicle; Location: short span at midspan, vehicle: T3162, speed: 13.36 km/h 101 Figure 5.1. Numerical simulations - example of the results, pos. of the truck during simulation; Location: midspan of the short span, vehicle: T5441, speed: 10 km/h 107 Figure 5.2. Numerical simulations - example of the results, pos. of the truck during simulation; Location: midspan of the long span, vehicle: T5441, speed: 10 km/h 108 Figure 5.3. Numerical simulations - example of the results, pos. of the truck during simulation; Location: midspan of the short span, vehicle: T5441, speed: 40 km/h 109 Figure 5.4. Numerical simulations - example of the results, pos. of the truck during simulation; Location: midspan of the long span, vehicle: T5441, speed: 40 km/h 110 Figure 5.5. Numerical simulations - example of the results, pos. of the truck during simulation; Location: midspan of the short span, vehicle: T5441, speed: 70 km/h I l l Figure 5.6. Numerical simulations - example of the results, pos. of the truck during simulation; Location: midspan of the long span, vehicle: T5441, speed: 70 km/h 112 Figure 5.7. Numerical simulations - example of the results, pos. of the truck during simulation; Location: midspan of the short span, vehicle: T5441, speed: 100 km/h 113 x 1 Figure 5.8. Numerical simulations - example of the results, pos. of the truck during simulation; Location: midspan of the long span, vehicle: T5441, speed: 100 km/h 114 Figure 5.9a. Range of the dynamic amplification (phase sequences 1 to 10); location: midspan of the short span, vehicle: T5441 115 Figure 5.9b. Comparison of the averaged dynamic amplification factors from Table 5.1 with the 1996 C H B D C code value 115 Figure 5.10a. Range of the dynamic amplification (phase sequences 1 to 10); location: midspan of the long span, vehicle: T5441 116 Figure 5.10b. Comparison of the averaged dynamic amplification factors from Table 5.2 with the 1996 CHBDC code value 116 Figure B. 1. Lillooet River Bridge - General Layout (drawing) 138 Figure B.2. Lillooet River Bridge - Superstructure Details (drawing) 139 Figure C . l . Instrumentation of the Lillooet River Bridge for ambient vibration testing 141 Figure C.2. Ambient vibration testing of the Lillooet River Bridge - an example of the acceleration records measured on the long span (setup 2) 143 Figure C.3. Ambient vibration testing of the Lillooet River Bridge - an example of the acceleration records measured on the short span (setup 15) 144 Figure C.4. Instrumentation of the Lillooet River Bridge for the forced vibration testing. . 148 Figure C.5. Forced vibration testing of the Lillooet River Bridge - an example of acceleration records measured during the passage of the vehicle T3162 (setup 1) 150 Figure C.6. Forced vibration testing of the Lillooet River Bridge - an example of acceleration records measured during the passage of the vehicle T3162 (setup 16) 151 Figure D. 1. Program FRF - an example of input data and results; a) and b) - the input signals; c), d) and e) - results of the analysis on the input signals 156 xi ACKNOWLEDGMENTS I would like to express my sincere thanks to my thesis advisor, Dr. Ricardo Foschi of the Department of Civil Engineering at U B C . His vast experience in structural analysis and stochastic mechanics contributed a lot to the success of this project. I would also like to acknowledge financial support from his Research Grant, which enabled us to work on this topic and to perform vibration testing of the Lillooet River Bridge. Finally, his comments and suggestions during the preparation of this thesis, are sincerely appreciated. I would also like to thank Dr. Carlos Ventura, my thesis co-advisor. His experience with instrumented structures and signal processing was very helpful. His help in organizing the field testing of the bridge is highly appreciated. He also co-authored part of the software developed for purposes of this project. I am indebted to Dr. Andreas Felber and Mr. Norman Schuster, both former graduate students of the Department of Civil Engineering, U B C . They developed the data acquisition system and data analysis software which was used in this study. I would like to thank Mr. Howard Nichol, earthquake lab technician, for maintaining the data acquisition system, for his interest in the tests and for suggestions and valuable hints for the testing. I would like to thank Mr. Ken Paterson, the Resource Officer Engineer, Squamish Forest District. He granted us permission to test the bridge and assisted us in getting drawings of the Lillooet river bridge. I would also like to thank all of the people from U B C who provided valuable assistance during the field testing. These people include Dr. Helmut Prion and graduate students Brad Kemp, Isabelle Villemure, Vincent Latendresse, Anuz Khan and Mahmoud Rezai. I would like to thank Dr. R Sexsmith and Dr. C. Ventura who, along with Dr. R. Foschi, reviewed this thesis. The last, but not the least, I would like to thank my wife Petra for her patience and support she had of me while I was working on this project. xii 7b my grandfather Jan xiii C H A P T E R 1 INTRODUCTION This chapter presents an overview of the research project. A general description of the problem is discussed in the first section (§ 1.1). This is followed by a literature review (§ 1.2). Definition of dynamic amplification factor is given in section § 1.3. The approach taken by the Canadian Highway Bridge Design Code is summarized in section § 1.4. The next section, §1.5, describes the scope and objectives of the study and it is followed by an outline (§ 1.6) of this thesis. 1.1 O V E R V I E W O F T H E P R O J E C T Bridges are civil engineering structures having fundamental significance for every country, and today they are longer and more slender than they were in the past. The demand of traffic grows very rapidly and new bridges have to be designed for very heavy vehicles and trains. Vibration limitation requirements are getting more strict in order to provide bridge users with particular safety and comfort. The use of new materials and technologies has also added new problems in bridge engineering. A l l these demands bring new challenges to both bridge designers and researchers who generate new data to support or change provisions of bridge design codes. However, research on design of bridges has been focused on highway bridges resulting in corresponding provisions in the bridge design codes. The forest industry is one of the leading industries in the province of British Columbia (B.C.). There are thousands of kilometers of forest service roads with hundreds of logging bridges. 1 Logging bridge design is similar to that of highway bridges but there are some notable departures. Logging bridges are usually single lane, wooden-deck structures carrying relatively large weights of logging trucks. The response of these bridges may vary from the response of highway bridges. In particular, the dynamic response of the logging bridges may be different, since a strong bridge-vehicle interaction can be expected, given the relative high ratio between the mass of the vehicle and the mass of the bridge. The focus of this study was on the response of logging bridges under moving logging trucks. First, a numerical model of the structural system bridge-vehicle was developed and implemented in a computer program to solve the response of a bridge due to the passage of a vehicle. Then, ambient vibration and forced vibration tests were conducted on an existing logging bridge. The experimental results were used for the evaluation and calibration of the numerical model of the system bridge-vehicle implemented in the computer program. This was then used to analyze the response of the logging bridge tested under a moving, fully loaded, logging truck. Finally, results of this analysis were used to assess the provisions for dynamic load allowance in the new 1996 Canadian Highway Bridge Design Code, which was available to the author in a draft form dated February 1995. 1.2 PREVIOUS RESEARCH The dynamic behavior of structures under moving loads has been studied using theoretical and experimental methods for about 140 years. The large number of reports published reflects this long time span. Within the framework of this report it is hardly possible to discuss at length 2 even the small portion known to the author. Nevertheless, it will be attempted to mention briefly some of the main efforts from the last two decades. The problem of the dynamic response of a bridge, while a vehicle travels across it at a constant speed, is neither theoretically nor experimentally simple to study. If dynamic equilibrium is formulated for a point on the structure, one obtains a partial differential equation for vertical motions of the structure which is dependent on the load position as well as on the time. Appropriate solutions to this problem, that is, those using appropriate models of the bridge, the surface of the bridge deck, and the vehicle, have only been found since it became possible to integrate the differential equations of motion numerically, using powerful computers. Parallel to the development of theoretical methods, a considerable advance has also been made in the development of experimental techniques. Experimental investigations must be carried out in field tests. Only in this way can the boundary conditions regarding the bridge, surface of the bridge deck and the vehicle be considered correctly. Although an appropriate laboratory model of a bridge can certainly be constructed, it is not possible to simulate accurately a vehicle such as a heavy truck. Although techniques for measuring and recording dynamic bridge deflections were implemented more than one hundred years ago by Frankel, efficient methods of digital signal processing in the time and frequency domains, determination of the longitudinal profile of the surface of the bridge deck and measurement of the dynamic wheel loads have only been developed in last two decades. There has been a great deal of research in the area of bridges under moving loads. However, this research has focused on highway and railway bridges. Researchers from many countries contributed to the better understanding of the dynamic response of bridges under moving loads. Some of the countries in which highway bridge investigations were conducted in last 3 decades are Canada, the Czech Republic, Germany, Great Britain, Japan, Switzerland and the United States. In Canada, the dynamic amplification factor was first presented as a function of the fundamental frequency of the bridge by Csagoly, Campbell and Agarwal in 1972. A large experimental basis was generated to support provisions of the Canadian bridge design codes. The experimental testing of the short and medium span highway bridges was described in a paper by Billing and Agarwal in 1990. The analysis of the dynamic response of the slab-type bridges were presented by Humar and Kashif in 1995. The experimental and numerical analysis of short-span highway bridges under heavy vehicles with leaf-spring and air-spring suspensions was presented by Green, Cebon and Cole in 1995. Research in the Czech Republic was always directed equally toward theory and experiment. For instance, the work by Fryba was oriented to both experimental and analytical behavior of railway and highway bridges (Fryba, 1972). The research team of Bata used three-dimensional bridge-vehicle model and also conducted countless dynamic tests on highway bridges (Bily, 1990; Bata and Plachy, 1977; Bata, Plachy and Travnicek, 1987). The research activities in Germany has consisted basically of comprehensive numerical modeling the highway bridges and vehicles (Drosner, 1989). In Great Britain, research activities have been concentrated at the University of Cambridge and at the Transport and Road Research Laboratory (Mitchell, 1987). Bridge-vehicle interaction was also studied by Green (1990). It can be assumed that only a small portion of the research reports published in Japan have been translated into other languages. It is seen from the work of Kajikawa and Honda (1987), 4 that theoretical methods, particularly stochastic methods, are highly developed in Japan. The above mentioned authors did not show comparisons between their theoretical models and experimental results. Investigation of dynamic behavior of bridges has long tradition in Switzerland. The E M P A institute is highly involved in bridge testing. The Deibuel Bridge test contributed towards better understanding of the bridge-vehicle interaction process (Cantieni, 1992). In the late 1950's the A A S H O Road Test was performed in the United States. The goal of this test was to study the behavior of pavements and bridges under a wide variety of known dynamic loads, in order to gain information towards optimization of their design. Several types of bridges were investigated: reinforced concrete bridges with simply supported beams, composite bridges (steel I-beams with reinforced concrete deck) and bridges of prestressed concrete. Altogether 15 bridges were studied and 14 vehicles were used in the testing. The research in the United States in the last decades has concentrated on the development of refined bridge models (Eberhardt, 1972) and on employing probability concepts in bridge design codes (Hwang and Nowak, 1989). The dynamic response of the logging bridges due to loading by heavy logging trucks may vary from that of the highway bridges because the former have specific, distinct characteristics (Nicol-Smith, 1988): • normally, they have only a single lane for both directions of traffic, and the width of the bridge deck is from 4.3 m to 4.9 m; • the bridge deck is usually made of timbers, and the characteristics of the roughness of the bridge deck are different than those of highway bridges with concrete or steel deck; • sidewalks are not constructed on logging bridges; 5 • logging trucks start at the size of a semi-trailer; the smallest logging trucks, designed for the highway use, weight about 440 kN, and "off highway" trucks can weight more than 1000 kN. No previous works have been found by the author on the dynamic behavior of wooden bridges used for heavy logging traffic. 1.3 DEFINITION OF THE DYNAMIC AMPLIFICATION FACTOR The purpose of introducing a "dynamic amplification factor" is to relate the structural response due to a load dynamically applied on a structure to the response of the same load applied statically. In attempting to make this relation, one is immediately faced with the basic definition of the dynamic amplification factor. Several other names are used for dynamic amplification factor: impact factor, dynamic coefficient, dynamic load factor, dynamic load allowance, dynamic magnification factor, etc. Different terms may also have different definitions. The name Dynamic Amplification Factor will be used in this thesis and it will be denoted as DAF. 1E-02 8E-03 ? 6E-03 c | 4E-03 a I 2E-03-OE+00 -2E-03 0 2 4 6 8 10 Time (s) static displacement — total displacement | Figure 1.1. Definition of the dynamic amplification factor. 6 Dynamic amplification factor is defined here as the ratio between the maximum total value (static plus dynamic components) of a quantity and the maximum static value of the same quantity. The quantity can be a displacement, a rotation, a bending moment, a shear force, a strain or a stress. For instance, Figure 1.1 shows the time history of the total displacement at a point of a bridge due to a truck moving over the bridge at a constant speed, and the static component of this displacement. Using Figure 1.1, one can define the dynamic amplification factor D A F as: DAF = -4eL. (1.1) where: Atot is the maximum value of the total displacement; Astat is the maximum value of the static displacement. 1.4 APPROACH IN THE CANADIAN HIGHWAY BRIDGE DESIGN CODE The analysis of the passage of a heavy vehicle over a bridge is a complex structural problem. There are many characteristics of the system bridge-vehicle having an influence on the structural response; among the most important are: modal characteristics of the bridge, roughness of the bridge deck, modal characteristics of the vehicle and its speed. Provisions of bridge design codes convert the dynamic to a pseudostatic problem and require that a "design truck", which is a model of a truck with certain weight, number of axles and axle weight distribution, be used for bridge analysis. In the pseudostatic analysis, the weight of the truck is multiplied by the dynamic amplification factor and the resulting load is statically 7 applied on the bridge. The dynamic amplification factor is used in order to take into account the dynamic character of the load caused by actual trucks in a simplified manner. The dynamic amplification factor is generally determined as a function of either the length of the bridge or the fundamental natural frequency of the bridge. This is the approach that most design codes use to take into account the bridge-vehicle interaction. Canada has a long tradition of considering bridge-vehicle interaction in its bridge design codes. The Dynamic Load Allowance (DLA) coefficient was already expressed as a function of the fundamental natural frequency of the bridge in the 1983 version of the Ontario Highway Bridge Design Code (OHBDC). This coefficient simulated the dynamic character of a passing vehicle over the bridge and was used with the design truck. The relation of D L A coefficient and the dynamic amplification factor D A F defined in the previous section is: DLA = DAF-\ (1.2) The total weight of the design truck was 600 k N distributed on four axles. The same approach was used in the 1988 Canadian Design of Highway Bridges Code (CAN/CSA-S6-88). The OHBDC changed the methodology in its 1991 version. The dynamic load allowance was made dependent on the number of axles of the design truck in the design lane. The total weight of the design truck was increased to 740 k N distributed on six axles. A new issue of the Canadian bridge design code S6 is being prepared for release in 1996. Only a February 1995 draft of this Code was available to the author. The design truck in this new code is the same as that in OHBDC from 1991. The D L A for the design truck decreases as the number of axles of the design truck increases. The D L A goes from DLA=0.4 (when one axle of the truck is used) to DLA=0.25 (when three or more axles of the truck are used). It implies that the D L A is greater for local effects and smaller for global effects of the live load. This Code, in a manner 8 similar to the previous codes, reduces the D L A for wood components to 70% of the standard values. 1.5 O B J E C T I V E S A N D S C O P E O F T H I S S T U D Y The objective of this study was to develop a numerical model of the structural system bridge-vehicle and to implement this model into a computer program. This computer program was then validated and calibrated using the results of dynamic testing of an existing bridge. Following verification of the program, it was used to determine the response of the same bridge due to the passage of a heavy vehicle. Results of this analysis were to be used to verify the provisions for dynamic load allowance in the new 1996 Canadian Highway Bridge Design Code. The scope of the project was limited to study the global dynamic behavior of logging bridges with glued-laminated girders. Consequently, only the Code provisions for the global effects of the live load, that is, only the case i f three or more axles of the design truck are used, were assessed. 1.6 O U T L I N E O F T H E T H E S I S This thesis is comprised of five chapters and five appendices. Chapter 2 presents a numerical model of the system bridge-vehicle. Chapter 3 describes the bridge of interest together with the dynamic tests conducted on the bridge. Chapter 4 deals with the validation of the numerical model using the results of the experiments. Chapter 5 presents results of numerical 9 simulations of passages of a fully loaded logging truck over the Lillooet River Bridge. Finally, conclusions and recommendations for CHBDC, 1996, are presented. Appendix A contains descriptions and operating instructions of computer programs that were developed through the course of this research. Appendix B shows selected drawings of the Lillooet river bridge. Appendix C contains details about the dynamic testing of the Lillooet River Bridge. Appendix D and E include operational instructions for the computer programs Frequency Response Function and Time History Viewer. 10 C H A P T E R 2 NUMERICAL MODEL OF THE BRIDGE-VEHICLE SYSTEM This chapter presents the background theory for the development of a Bridge-Vehicle Interaction (BVI) analysis and the computer program. It describes the numerical model for: 1) the bridge structure (§2.1 to §2.4); 2) the vehicle (§2.5); 3) the combined system bridge-vehicle (§2.6). The procedure for solving the forced vibrations of the coupled system is briefly described at the end of this chapter (§2.7). 2.1 DERIVATION OF THE STIFFNESS MATRIX Bridges are complex civil engineering structures consisting of many elements. Numerical models to simulate the response due to various kinds of loads mostly take into account only the main structural elements of the bridge. In the case of logging bridges, the main structural elements are the girders, which are located symmetrically with respect to the longitudinal axis of the bridge. These main girders can be modeled as beams. Logging trucks are the main load on the logging bridges. This load is transferred through the bridge deck and the cross-ties to the main girders. Since the logging bridges have mostly a single lane and the width of the trucks is comparable with the width of this lane, the load can be assumed to be distributed evenly or in the ratio 60/40 on each main girder. This assumption permits the modeling of the main structural system of the bridge as a single beam. 11 Kinematic relations resulting from deformation of the cross-sections of the beam are shown in Figure 2.1. The axes y and z are centroidal and principal axes of inertia of the cross-section of the beam. It is assumed that the deformed cross-section of the beam remains plane, but that the angle y/ between the cross-section and the tangent to the deflected x axis is not 90°. This introduces the.effect of shear, considered substantial in short wooden beams. It is also assumed that the vertical displacement w(x,t) depends only on the x-coordinate (the position of the point on the bridge), and on the time t, but it is independent of the distance z from point A to the centroid of the cross-section O. Figure 2.1. Kinematic relations in the cross-section of the beam. The symbols in Figure 2.1 represent: • A is a point in the cross-section at distance z from x axis, or centroid O; • w - w(x,t)A is the vertical displacement at point A, which is constant over the cross-section; 12 • uA is the axial displacement of point A; • u is the axial displacement of the centroid O; • y/ is the total rotation of the cross-section, this includes the rotations due to the bending and due to the shear. From Figure 2.1 and using the small deflections beam theory, the axial displacement of point A is: uA=u-\\rz = u-(— + Q)z (2.1) dx where 9 is the shear rotation which is assumed constant over the cross-section. The axial strain at point A is: duA du d2w 36 zA =—- = z—z—z— (2.2) A dx dx dx2 dx K ' Assuming the centroidal axial displacement u is constant throughout the beam (no axial load is applied), Equation 2.2 can be simplified to: zA = - 4 - = -z—T - z— = -z(w"+Q') (2.3) dx dx dx The corresponding shear strain y can be expressed as: dw duA dw dw _ . .„ „. y = — + — - = 9 = -G (2.4) dx dz dx dx Since the highest order derivatives in Equation 2.3 are w" and 0', it is necessary to establish the continuous functions w,w' and 6 at each point along the beam. It is convenient to discretize the beam into finite length elements, each having two nodes i and j, and three degrees of freedom per node w,w' and 9. This results in a six-degree-of-freedom element (6-d.o.f. element), as shown in Figure 2.2. The cross-sectional characteristics are assumed to be constant throughout the element. 13 Figure 2.2. Numbering of degrees of freedom of the 6-d.o.f. beam element. The symbols in Figure 2.2 represent: • w, and Wj are the displacements of nodes /' and j, respectively; • dxJ, 1 \dxJ, nodes /' and j, respectively; are the first derivatives with respect to x of displacements at • 0; and Qj are the shear rotations at nodes /' and j , respectively; • / is the length of the element. The deflected shapes resulting from applying a unit value of each degree of freedom of the element, while constraining the other degrees of freedom, are called shape functions (Clough and Penzien, 1993). The displacement at any point of the element can be expressed in terms of the nodal variables and the shape functions: 14 w(x,t) = w - Ml a (2.5) where a is the vector of the nodal variables, all functions of x and t: aT = {wi w\ 0t Wj w'j. Oj) and MQ is a vector of suitable shape functions, cubic polynomials: l - 3 £ 2 + 2 £ 3 ' 0 3?-2? 0 M0 = (2.6) (2.7) where £ is a nondimensional variable E,=x/\ • Here, x is the local coordinate (x = 0 at point /', x - I at point j.) <M The first derivative of M with respect to x is - ; it can be denoted as M x and has the 0 dx following form: 1 dx I - 6 £ + 6 £ 2 / ( l - 4 £ + 3 | 2 ) 0 m2 - 2 0 0 (2.8) d2M The second derivative of M with respect to x is z- 5 - ; it can be denoted as M7 and has 0 dx the following form: 15 M2 = d2M0 dx2 J _ I2 -6 + 12^' /(-4 + 6£) 0 6-12| 1(6^-2) 0 (2-9) The shear rotation at any point of the element can be expressed by the nodal variables and shape functions as: 0 = LT0a (2.10) where a is the vector of nodal variables (see Equation 2.6) and L0 is a vector of suitable shape functions, linear polynomials: L0 = 0 0 0 0 7 J 0 0 i - < r 0 0 4 (2.11) dLn The first derivative of L0 with respect to x is — - ; it can be denoted as L\ and has the dx following form: dx I (2-12) 16 The used shape functions ensure continuity of w,w' and 9 along the beam. They also provide the exact solution for an unloaded beam element and a good approximation to represent the deflected shape of the element loaded by inertia forces. Using Equations 2.5 and 2.10, Equation 2.3 for axial strain from bending can be rewritten: eA = -z(w" + 6') = -Z[M% + Zfja (2.13) A virtual strain can be then written as: 8eA = -z\Ml + i [ ] & r (2.14) Similarly, using Equations 2.10, Equation 2.4 for the shear strain can be rewritten: Y = -0 = -lJoa (2.15) A virtual shear strain can be then written as: dy = -LT0Sa (2.16) The beam is modeled as a structure with linear elastic behavior. Therefore, the stress from bending can be written as: oA = EeA = -Ez\MT2 + Z[ ]a (2.17) Similarly, the shear stress can be expressed: r = Gy = -GLT0a (2.18) The internal virtual work then can be written as: SW, =\yaSsdV +jyr bydV (2.19) where jy is an integral over the volume of the element and dV = dxdydz. Substituting Equations 2.14, 2.16, 2.17 and 2.18 into Equation 2.19, the following expression for internal virtual work is obtained: 17 SWt = Jv(-ZF\MZ +L\]a)(-z[Ml +lJ[]Sa)dV + jv(-GLla)(-LT0Sa)dV (2.20) or SWT = SaT (JVZ2E[M2 + L\ ][M2r +lJ[]dV+ \ y GL0LldV)a (2.21) Introducing the moment of inertia of the cross-section about the y axis (I = jAz2dydz) and the area of the cross-section (A = \Adydz ), Equation 2.21 can be rewritten as: SWi = SaT ll(El[M2 +L\\MT2 + GALjJ^ixa = 6aTKea (2.23) where Ke represents the stiffness matrix of the 6-d.o.f beam element, which results from evaluating the integral in Equation 2.23. The form of Ke is as follows: UEI 6EI 0 -Y2EI 6EI 0 I3 I2 I3 I2 6EI 4EI EI -6EI 2EI -EI I2 I T I2 I I 0 EI I EI , AGl I 3 0 -EI I -EI . AGl I 6 -12.EJ / 3 -6EI I2 0 UEI I3 -6EI I2 0 6EI 2EI -EI -6EI 4EI EI I2 I I I2 I I 0 -EI I -EI . AGl I 6 0 EI I EI , AGl I 3 wt w; WJ w; where: E is Young's modulus of elasticity; G is the shear modulus; / is the moment of inertia of the cross section; A is the area of the cross section; / is the length of an element. (2.24) 18 Nodal variables are indicated below each particular column of the stiffness matrix. The global stiffness matrix of the beam is assembled from local stiffness matrices of the elements following standard finite element method (FEM) procedures. 2.2 DERIVATION OF THE MASS MATRIX The derivation of the mass matrix of the 6-d.o.f. element was based on two types of motion of the beam cross-section: • up/down planar motions; • rotations of the cross-section about y axis (horizontal axis perpendicular to the longitudinal x axis of the element). w 7 z 7T / A X A/A il l Figure 2.3. Inertia forces acting on an element of volume dV. Figure 2.3 shows the inertia forces /, and I2 associated with these two types of motions. These inertia forces are acting on an element of volume dV at points. 19 The symbols in Figure 2.3 represent: • A is a point in the plane of the cross-section; • z is the distance between the point A and x axis; • w = is the vertical acceleration of the element; d r • w' + Q = d +^) j s t j j e rotational acceleration of the element; dr • p is the volume density of the material of the beam; • Zj = pw dV, is the inertia force produced by the vertical acceleration; • I2= pz(w' + $)dV, is the inertia force produced by the rotational acceleration. Using the finite element notation from Equations 2.5 and 2.10 the inertia forces can be written: I^pMladV (2.25) I2 =pz^Ml +Ll\adV (2.26) The virtual work of the inertia forces can then be expressed as: 8I=\v{jMld\Ml8aW^\v{p\Ml +z£]a)(z[M1r +LT0fa)dV (2.27) or, after integrating over the area of the cross-section: SI = SaT(\lo(pAM0MT0 + pllM, +LQ\M^ +Z^])&)a = SaTMea (2.28) where Me represents the mass matrix of the 6-d.o.f. beam element. 20 The consistent mass matrix, after evaluating the integrals in Equation 2.28, has the form shown in Equation 2.29. Nodal variables are indicated below the columns of the matrix: 156c,+36c2 22/q +3/c2 -6c 3 54cj - 36c2 -13/c,+3/c 2 -6c 3 22/c, +3lc2 4 / 2 ( c 1 + c 2 ) /c 3 13/c, - 3/c2 - 3 / 2 c , - / 2 c 2 - / c 3 -6c 3 /c 3 4/c3 6c 3 - / c 3 2/c3 54c, -36c 2 13/c, -3 /c 2 6c3 156c, +36c2 -22/c, -3 /c 2 6c 3 -13/c,+3/c 2 - 3 / 2 c , - / V - / c 3 -22/q -3 /c 2 4 / 2 ( c 1 + c 2 ) /c 3 -6c 3 - / c 3 2/c3 6c 3 /c 3 4/c3 w; (2.29) where: p is the density per unit volume; / is the moment of inertia of the cross section; A is the area of the cross section; / is the length of the element; constants c, , c 9 and c, have values: c, = ; c, = ; c, = — . 1 2 3 1 420 2 30/ 3 12 The global mass matrix of the beam is assembled from local mass matrices of the elements following standard F E M procedures. 2.3 DERIVATION OF THE DAMPING MATRIX Damping of structures is a complex phenomenon. Although there are several sources contributing to the resulting structural damping, damping in numerical applications is 21 simulated by a simple mechanism. One of the most usual is viscous damping. The assumption of viscous damping is also used in this thesis. One procedure to evaluate the damping matrix of a structure with viscous damping is described by Clough and Penzien (1993). The damping matrix C can be evaluated as a linear combination of the mass matrix M and stiffness matrix where co m and co n are m-th and n-th natural circular frequencies of the structure (co „ is greater than co m ) and £ is the modal damping ratio. It is assumed the modal damping ratio has the same value £ for both m-th and n-th modes. This assumption results in modal damping greater than £ for natural modes with circular frequencies smaller than com or with circular frequencies greater than a)n This assumption also results in modal damping smaller than £ for natural modes with circular frequencies between com and a>n. The disadvantage of this approach is, that the modes with very low and very high frequencies are assigned very large values of damping, which may result in unrealistic response of the structure. However, this can be eliminated by equating the value of © m to the first natural circular frequency of the structure and the value of (On to the natural circular frequency of a higher natural mode. It is supposed, that the contribution of this higher natural mode, e.g. the fourth or the fifth mode, to the total response of the structure is not significant and, therefore, the error introduced by assuming this damping mechanism is negligible. K. C = aoM + a\K (2.30) where the constants ao and tfi can be evaluated from: (2.31) 22 2.4 ROUGHNESS OF THE BRIDGE DECK The response of a bridge when a vehicle is passing over it depends on the excitation of the coupled system bridge-vehicle. This excitation is significantly influenced by the roughness of the bridge deck, and this is therefore considered to be one of the major factors influencing the dynamic behavior of the system bridge-vehicle (see Bata et al., 1987 or Cantieni, 1992). The roughness profile of an actual bridge deck can be described as a zero mean stationary stochastic process; the characteristics of which can be measured with surveying equipment and used as a direct input for numerical analysis. However, this approach is not followed here. Instead, the characteristics of the roughness profile are estimated from an assessment of the detailing of the deck construction as described in section 4.3. This assessment is used to estimate the power spectral density function of the process, S(co), with which individual profiles can be simulated as a sum of harmonic series with random phase shifts, as follows: h(x) = YAiCOs(a>, x + (f>t) (2.32) where: h(x) is the value of the roughness function at distance X from the beginning of the bridge; At is an amplitude of a cosine component at circular frequency Gh\ <fk is a random phase shift in the interval (0,2 7t). The amplitudes A i are determined from values of the power spectral density at each circular frequency ah, as follows: Ai = fiS(an)ki> (2.33) 23 where S ( c o i ) is the one-sided power spectral density of the process h(x) at circular frequency fi)» and Aco is step of circular frequency. In the simulation of the roughness, the values of power spectral density S(co) can be scaled in order to satisfy a condition for crossing a given barrier R with a given probability p(R). This probability can be evaluated for a stationary random process using Rice's formula (Rice, 1944). Assuming a zero mean Gaussian process the following formula for this probability can be obtained: p(R) = 2no, exp ( R2\ (2.34) where a h is the standard deviation of roughness function h(x) as a stationary random process and <r H is the standard deviation of first derivative of h(x) with respect to X. Expressing ah and crfc. in terms of the power spectral density S ( c o ) , the probability p(R) can be written in the following form: exp -R 2 ^ (2.35) 2.5 NUMERICAL MODEL FOR THE VEHICLE The vehicle, as a complex mechanical system, can be simplified and idealized for purposes of bridge-vehicle interaction. Generally, the vehicle can be modeled as a plane or space system of masses, springs, torsional springs, dashpots and other structural elements. Vehicles have been 24 modeled as a single or two degree of freedom systems. Numerous examples of vehicle models are described by Bata et al.(1987) or Cantieni, (1992). Figure 2.4. Two-dimensional model of the vehicle. The model for the vehicle used for simulations has to be compatible with the numerical model of the bridge itself. Therefore, using a two-dimensional model of the bridge, only a two-dimensional model of the vehicle should be used. The two basic types of motions of the body of the vehicle are: up/down motion and rocking about its horizontal axis perpendicular to the longer dimension of the vehicle. Considering the interaction of short logging bridges with long logging trucks, the model must also describe the force from each axle of the vehicle and how it is transferred to the bridge. Each axle of the vehicle can be considered with its own mass as one degree of freedom. The model used here for an empty logging truck is shown in Figure 2.4. This represents a vehicle with three axles. 25 The symbols in Figure 2.4. represent: • n is the number of axles of the vehicle; • C.G. is the center of gravity of the body of the vehicle; • VY,..., Vn+2 are the dynamic degrees of freedom of the vehicle; • mT is the mass of the body of the vehicle; • IT is the moment of inertia of the body of the vehicle about the horizontal axis perpendicular to the longitudinal axis of the vehicle; • n\,..., mn are the masses of the axles; • C j , . . . , cn are the constants of the dashpots; • ..., k"n are the spring constants of suspensions (upper springs); • k[,..., k'n are the spring constants of tires (lower springs); • e, , . . . , en are the distances (eccentricities) of the axles from the center of gravity C.G. 2.6 DERIVATION OF MATRICES OF THE COUPLED SYSTEM The structural system bridge-vehicle (the coupled system) can be described in terms of its stiffness, mass and damping matrices - K,M a n d C . These matrices are called, in the following, the stiffness matrix of the coupled system, the mass matrix of the coupled system and the damping matrix of the coupled system or, briefly, the coupled stiffness matrix, the coupled mass matrix and the coupled damping matrix. These coupled matrices consist of several components: 26 the "bridge" part - these are the stiflhess, mass and damping matrices of the bridge as they were derived earlier in this chapter. They are denoted as KB,MB and CB in the following equations. These matrices are always located in the upper-left corner of the corresponding coupled matrices. the "vehicle" part - these are the stiffness, mass and damping matrices of the vehicle on a rigid foundation. These matrices are denoted KV,MV andQ, . Because of the relatively small number of dynamic degrees of freedom of the vehicle it is easy to derive these matrices directly when considering motions of the coupled system bridge-vehicle. These matrices are always located in the lower-right corner of the corresponding coupled matrices. the "bridge-vehicle" part - these are the components of the coupled matrices that are always off the main diagonal and they represent the connection of the bridge and the vehicle. There are two of these submatrices for each coupled matrix, symmetric about the main diagonal of the coupled matrix. These submatrices are denoted KBy,MBV and CBV (below the main diagonal, in the lower-left part of the coupled matrix) or KvB'Myg a n d C r a (above the main diagonal, in the upper-right part of the coupled matrix). The connection of the vehicle with the bridge is realized only by the tires which are represented by their spring constants (lower springs - k\, ...,kln in Figure 2.4). Therefore, the "bridge-vehicle " parts of the coupled matrices have nonzero elements only in the case of the coupled stiffness matrix, and have only zero elements in the case of the coupled mass or the coupled damping matrices. Locations of the nonzero elements in the coupled stiffness matrix depend on location of the axles of the vehicle on the bridge. These 27 locations change within the simulation because different parts of the bridge or, different beam elements are in contact with the tires of the vehicle. The coupled stiffness matrix can then be written in the following form: K = KB Kyg KBy Ky (2.36) Similarly, the coupled mass and coupled damping matrices can be written as: M = ' MB ~MB 0 " _MBV My _ 0 My_ c ~CB 0 " 0 Cy (2.37) (2.38) where 0 represents a submatrix with only zero elements. The "beam parts " KB,MB,CB of the matrices K,M,C were derived previously. The derivation of the remaining parts of the matrices K,M,C is summarized below. Figure 2.5 shows the structural connection of the body of the truck and the deck of the bridge. The bridge is represented by a finite element between nodes j and k. The structural connection for i-th axle (/' = 1,2,...,«) is realized through the spring of the suspension k", the dashpot ct and the spring from the tire kj. The displacement of the i-th axle is Vt and its mass is mt. The contact of i-th axle with the bridge is at point Pt, where the vertical displacement of the bridge is wPi. The value of the roughness function at point Pt is h(Pt). The springs of suspensions and tires are assumed elastic and linear. Forces in dashpots are assumed proportional to the velocity of motion in the dashpot. 28 Figure 2.5. Contact of the i-th axle with the bridge. The symbols in Figure 2.5 represent: mT is the mass of the body of the vehicle; IT is the moment of inertia of the body of the vehicle about the horizontal axis of the body perpendicular to the longitudinal axis of the body; C.G. is the center of gravity of the body of the vehicle; Vn+1 is the vertical displacement of the body of the truck; Vn+2 is the rotation of the body of the truck; k" is the spring constant of suspension (upper spring) of i-th axle; c, is the damping constant of the dashpot at i-th axle; mi is the mass of i-th axle; Vi is the displacement of i-th axle; 29 • e, is the eccentricity of i-th axle from C.G.; • kj is the spring constant of the tire of i-th axle; • 5 is the distance from the point j to the point Pt; • / is the length of the beam element between nodes y'and&; • h(Pj) is the value of the roughness function at point Pt. Let now d,d,d be defined as the global vectors of nodal displacements, velocities and accelerations, including the vehicle's degrees of freedom: d = w N W N e N n+1 n+2j d = w. w N w N 9 N v. n+l d = w N W N 9 N n+l n+2 (2.39) where N is the number of the degrees of freedom of the bridge, n is the number of the axles of the vehicle and n + 2 is the number of the degrees of freedom of the vehicle. 30 Also, let the vectors Qt(/' = l,2,...w,w + l,w + 2) be defined as having only zero elements except for one single unit-valued element. The location of the nonzero element is the (N+i)-th row of the vector, as shown for example in Equation 2.40 for i=l: 0 0 ••• 0 0 0 1 0 — t (N + 1) - th row 0 0 0} Further, let the vector Nt be defined as: 0 0 M M ) . M M t ) 0 0 0 N,= where the nondimensional coordinate is: 6 = (2.40) (2.41) (2.42) xi being the distance from the left node of the element in contact with i-th axle to the point Pi and / is the length of this element. 31 If the i-th axle is located on a beam element, four elements of the vector Nt are nonzero, the others have zero values. The nonzero elements of iv~, are located in the rows where variables w and w' of the corresponding beam element are located. These nonzero elements have values of the cubic shape functions evaluated at the coordinate . The cubic shape functions were defined by Equation 2.7 where vector M0 was defined. Explanation of indexes in Equation 2.41 is shown by following example: Considering the element M o m ( £ , ) o f the vector Nt, the value of index m is the number of the row of the shape function in vector M0. Recalling Equation 2.7, it is obvious, only values m = 1,2,4,5 are possible because for m = 3 or m = 6 the value of M0m is equal to zero. The vector JV, has only zero elements i f the i-th axle is located off the bridge. The total kinetic energy of the system bridge-vehicle, for the degrees of freedom considered, can be written as: L = f ( v M f +±(V„J + t * ( l t f (2.43) where K B E m is the kinetic energy of the bridge, which would lead to the mass matrix of the bridge, derived earlier. Therefore, KBEAM does not have to be dealt with here. The total strain energy of the system bridge-vehicle can be written: u = - v ) 2 +^-(vt -w„ +*P/))2 j+cw (2.44) where UBEAM is the strain energy of the bridge, which would lead to the stiffness matrix of the bridge, derived earlier. Therefore, UBEAM does not have to be dealt with here. The velocities in Equation 2.43 can be written in vector form: 32 v, = Qjd; v„ + 1 = Ql^d; v„+2 = Q*+2d and similarly, the displacements in Equation 2.44 can be written in vector form: Vi = Qjd; v„+l = Q*+ld; vn+2=Q*+2d; wPi=Njd Equation 2.43 can be rewritten substituting from Equation 2.45: L = ^ d'Q^Ql.d+^d'Q^Ql.d + t^d'Q^Jd+K^ Similarly, Equation 2.44 can be rewritten substituting from Equation 2.46: (2.45) (2.46) (2.47) i=l — dT(Q , + 2 ^ . - Q \ ( Q +Q „e - o) + k1 - ^ f d ' ^ - N ^ - N f d ^ +kl.(Q.-N.)Th(P.)d + kl.h2(x) * + UBEAM <2-48> The mass and the stiffness matrices can be obtained using Lagrange's equations (Flugge 1962) in the following form: d_ dt fdl> Kdd) dU n + = 0 dd (2.49) from which the resulting matrices of the system bridge-vehicle can be written as: • the mass matrix: M= mTQ„+1QT„+1 +ITQ„+2Ql2+TmiQiQ! + MB where MB is the mass matrix of the bridge derived earlier. (2.50) the stiffness matrix: 33 K = jjfi(Q**+Q***t-ftX&n+Qm**i-QiY + */(&-NIXQI-*I) t)+KB (2-51) 1=11 I where is the stiffness matrix of the bridge derived earlier. • the damping matrix of the system bridge-vehicle can be written similarly with the stiffness matrix: C = ict(QM + 0 , + 2e, - aXfi^ i + Qnrfi - Of + CB (252) where CB is the damping matrix of the bridge derived earlier. Finally, the term - h(P.)d in Equation 2.48 makes a contribution to the right hand side of the system of equations of motion. The right hand side, considering also the weight of the truck and its axles, has the following form: R = l(mTQ„+l +E*/(A^ -0,>(i>) (2.53) V /=1 J (=1 The final system of the equations of motion has the form: Md+Cd+Kd = R (2.54) The mass matrix of the combined system M is given by Equation 2.50, the damping matrix of the combined system C is given by Equation 2.52, the stiffness matrix of the combined system K is given by Equation 2.51, the right hand side R is given by Equation 2.53, and the vectors d, d and d are given by Equation 2.39. The procedure to solve this system of the equations of motion is shown in the next section. 34 2.7 PROCEDURE FOR SOLUTION OF THE EQUATIONS OF MOTION A l l necessary components of the numerical model of the system bridge-vehicle were obtained in previous sections of this chapter. The final step is to select a method for solving the problem of forced vibrations. Modal analysis and the integration of the equations of motion are two the most usual methods to solve forced vibrations. Integration of the equations of motion is a general method but it is computationally very demanding. Modal analysis is computationally less demanding but it has some limitations. One of the assumption of the modal analysis method is that the modal characteristics of the structural system remain constant during the analysis. In order to decide on the suitability of the modal analysis for the problem considered, the modal characteristics of the system bridge-vehicle were calculated for the system of interest. This system consisted of the Lillooet River Bridge (described in Chapter 3), and one of the following vehicles: an empty logging truck and the same truck fully loaded. The empty logging truck called here T3162 (Truck, 3 axles, weight 162 kN), is described in Chapter 3. The fully logging truck is called here T5441 (Truck, 5 axles, weight 441 kN), and its characteristics are given in the example of the input file in Appendix A, § A.2.2. The models of the trucks were developed using Figure 2.4. The passage of the above-introduced vehicles over the Lillooet River Bridge was simulated and the modal characteristics of the coupled system were calculated during the passage. Then, the changes of values of selected natural frequencies were studied. Since the most significant response of the bridge was expected in the first and the second modes of the bridge, the natural frequencies corresponding to these modes were selected. It has to be mentioned that 35 the bridge has two independent spans assumed here as simply supported beams. The first mode of the bridge corresponds to the first flexural mode of the long span and the second mode of the bridge corresponds to the first flexural mode of the short span. The modal characteristics were calculated using the program BVTNV (Bridge-Vehicle Interaction - Natural Vibration), described in Appendix A. The results of these analyses are shown in Figures 2.6 to 2.9. Notation "Position of C.G. of the truck" in these figures means the distance from the beginning of the bridge to the center of gravity of the body of the truck. Figures 2.6 and 2.7 show how the first and the second natural frequencies of the bridge change with the position of the vehicle T3162. The first natural frequency valued 2.43 Hz decreased to 2.07 Hz when the vehicle was in the middle of the long span of the bridge. The second natural frequency increased from 5.49 Hz to 6.08 Hz when the vehicle was in the middle of the short span. Figures 2.8 and 2.9 show how the first and the second natural frequencies of the bridge change with the position of the vehicle T5441. The first natural frequency initially increased to 2.81 Hz, then decreased to 2.2 Hz during the passage of the vehicle. The second natural frequency increased from 5.49 Hz to 6.05 Hz when the vehicle was in the middle of the short span. It can be observed from Figures 2.6 to 2.9 that the selected natural frequencies change very significantly with the position of the truck and it is difficult to predict the character of these changes. The above considerations lead to the conclusion that modal analysis could not be used effectively, given that it is efficient only when the modal characteristics of the structural system remain constant during the analysis. Therefore, the direct time integration of the 36 Figure 2.6. Values of the 1st natural frequency of the Lillooet River Bridge versus the position of the vehicle T3162 (the testing truck). Figure 2.7. Values of the 2nd natural frequency of the Lillooet River Bridge versus the position of the vehicle T3162 (the testing truck). 1st frequency versus position of truck Vehicle: T5441 2.9 -, , 2.1 +—i 1 1 1 . , i - l 1 1 1 1 1 1 1 1 1 1 -10 0 10 20 30 40 50 60 70 position of C.G. of the truck (m) Figure 2.8. Values of the 1st natural frequency of the Lillooet River Bridge versus the position of the vehicle T5441. Figure 2.9. Values of the 2nd natural frequency of the Lillooet River Bridge versus the position of the vehicle T5441. equations of motion is then selected as the method for solving the problem of forced vibrations. Two alternatives of the time integration method were implemented in the program BVT: the average constant acceleration and the linear acceleration methods. Algorithms of both methods are available (Clough and Penzien, 1993) and are not presented here. The average constant acceleration method is unconditionally stable and the linear acceleration method is conditionally stable. The condition for its stability limits the time step At: where T is the shortest natural period of the system to be considered in the analysis. Both the average constant acceleration and the linear acceleration methods implemented in the analysis, lead to the time histories of displacement, velocity and acceleration of each degree of freedom of the system. (2.55) 39 C H A P T E R 3 DYNAMIC TESTING OF THE LILLOOET RIVER BRIDGE This chapter presents a description of the dynamic testing of the Lillooet River Bridge performed for this project. The description of the Lillooet river bridge is given in the first section (§3.1). The second section (§3.2) deals with data acquisition and data evaluation systems used for this testing. The next section (§3.3) describes ambient vibration testing. The final section (§3.4) contains a description of the forced vibration testing of the bridge. 3.1 DESCRIPTION OF THE STRUCTURE The bridge of interest is situated about 25 km North-West of Pemberton, B.C. , over the Lillooet river, on Hurley Forest Service Road. The bridge is called the Lillooet River Bridge. It was designed in 1967 by the Forest Service, Engineering Services Division in Victoria, B.C. and completed in 1967 by its owner, the Province of British Columbia, Ministry of Forests (Squamish District). The drawings of the bridge can be found in the Forest Service of the Province of British Columbia in Victoria, file No. 0240678. The bridge is 69.1 m long and 4.88 m wide and is composed of three spans. The width of the deck permits only a single lane for traffic. Figure 3.1 shows a general view of the structure from the right bank of the Lillooet river. A fully loaded logging truck, crossing the bridge, is shown in Figure 3.2. The lengths of the three spans are (from South to North): • 22.1 m - called the "short" span in this thesis; • 40.1 m - called the "long" span in this thesis; • 6.7 m - called the "approaching" span in this thesis. 40 Figure 3.1. Elevation view of the Lillooet River Bridge from the up-stream (West) side. Figure 3.2. View of the bridge with passing fully loaded logging truck. 41 The South abutment, shown in Figure 3.3, is supported directly on wood piles, while the intermediate piers are made of concrete. The pier between the short and long spans situated in the middle of the river is based on piles. The pier between the long and the approaching spans (see Figure 3.4.) is based on the concrete foundation. The North abutment supporting the approaching span is a small wooden structure. Figure 3.3. The South abutment of the bridge with the beginning of the short span. 42 Figure 3.4. View of the pier between the approaching and the long spans (on the North side). The main structural system of the long span consists of two pairs of glued-laminated I beams (see Figure 3.5). The short span has only two single beams of the same material as the long span. The approaching span is made of timber beams. Al l the three spans are simply supported and they are not connected at the intermediate supports. This permits independent rotations of neighboring spans. 43 Figure 3.5. View of the long span from below. Figure 3.6. View of the bridge deck and approaching testing vehicle T3162 from the South. 44 The deck of the bridge is supported by cross ties, spaced about 0.5 m, spanning the distance between main girders. The deck is made of two layers of planks - diagonal and longitudinal. Longitudinal planks, the layer in contact with tires of vehicles, are about 4 m long. They are visible in Figure 3.6 showing a logging truck approaching the bridge. The Lillooet river bridge has only single lane for traffic. Selected drawings of the bridge, the General Layout and the Superstructure Details, are included in appendix B . The Lillooet River Bridge was selected for testing for several reasons. The span of 22.1 m is a typical span length for logging bridges with glulam girders. The 40.1 m long span is one of the longest that have been built in British Columbia using glulam girders. The selection of the Lillooet River Bridge permitted the measurement of the response of two independent, simply supported spans with different lengths, all in one dynamic test. The Lillooet River bridge is also easily accessible from Vancouver, B.C. 3.2 DESCRIPTION OF THE DATA ACQUISITION SYSTEM Dynamic testing of the Lillooet river bridge was conducted in order to obtain experimental results for comparison with numerical analyses. Two types of tests were done: • ambient vibration testing for determination of modal characteristics of the bridge. Results of this testing were used later for verification of the numerical model of the bridge. • forced vibration testing for determination of dynamic amplification factors of the Lillooet river bridge due to the passage of an empty logging truck. 45 The Hybrid Bridge Evaluation System - HBES (Felber, 1993; Schuster, 1994) was used for the data acquisition in both tests. This system is used at the Department of Civil Engineering, U B C , as a tool for dynamic measurements on structures. The main components of the HBES system are as follows: • sensors and cables: The HBES includes ten Kinemetrics force balanced accelerometers (model FB-11) capable measuring accelerations of up to ± 0 . 5 g with a resolution of 0.2 /Jg. The dynamic range of these accelerometers is 130 dB in the frequency range 0 - 50 Hz (Kinemetrics, 1991). The sensors are mounted at various locations throughout the structure and are used to transform structure's vibrations into electronic signals. Cables are used to transmit the electronic signals from sensors to a central recording unit. The cables are shielded to minimize noise during the transmission. • signal conditioner: This component improves the quality of the data by filtering and amplifying signals. The amplifiers and filters for all channels are mounted in a portable signal conditioner. A Kinemetrics signal conditioner, model AM-3I, is used to condition the signals from up to eight sensors. Each signal is filtered and amplified independently. Twelve different amplification levels ranging from 1 to 2000 are available. Low pass filters can remove unwanted frequencies above 2.5 Hz, 5 Hz, 12.5 Hz, 25 Hz, or 50 Hz. High pass filters can remove all components below 0.1 Hz or 5 Hz. The portable signal conditioner is battery powered. It can be used continuously for approximately 20 hours before the batteries have to be recharged. 46 analog to digital (A/D) converter: The amplified and filtered analog signals are converted to digital form before they are stored on a data acquisition computer. The A/D converter in this case is a Keithley model 575 that is able to sample up to sixteen signals with sampling frequency sufficient for the requirements of the dynamic measurements of civil engineering structures. The A/D converter is controlled from the data acquisition computer using a custom program called A V D A - Schuster, 1994. data acquisition computer: The whole process of data acquisition is controlled from the data acquisition computer using the program A V D A . The current data acquisition computer is a PC - Compaq 286 portable computer. Its rugged design and size make it ideal for field work. Signals in digital form are stored as binary files on the hard disk. The computer has a storage capacity of approximately 30 M B . This is sufficient storage, since during a regular day of testing, approximately 20 M B of data is collected, data analysis computer: After the data acquisition computer has obtained a set of recordings, the data is transferred to a data analysis computer. This can be done in between the measurements, using the commercial software package LapLink. This arrangement allows data to be collected on one computer while a second, and faster, computer can be used to process the data in-situ to check its quality as well as to perform some preliminary analyses. The data analysis computer utilizes data reduction software, in this case program U L T R A (Felber, 1993) and data interpretation software, in this case V I S U A L (Felber, 1993). Program U L T R A is used for the graphical display of signals, Fast Fourier transform analysis, calculating power 47 spectral densities (PSD), transfer functions, coherence functions, potential modal ratio functions (PMR), etc. Program U L T R A enables to check the data very quickly during measurements and repeat data acquisition when the signals are distorted. Program V I S U A L is used for visualization of the structure's motions, i.e., for visual identification of natural modes. The above-described software did not permit visualization of all signals from one setup at once. Therefore, the program Time History Viewer (THV) was developed during the course of this study. This program is capable of displaying up to 16 time histories on the screen of the data analysis computer and allows the user quick observation of the signals. The operating instructions of the first version of the program T H V can be found in Appendix E . 3.3 AMBIENT VIBRATION TESTING Modal characteristics of the bridge were investigated because of the necessity to verify the modal characteristics of the numerical model, used later for numerical simulations of passing the vehicle over the bridge. The ambient vibration technique was used for determination of the modal characteristics of the bridge. Theory of ambient vibration testing has been summarized by Felber (1993) or Schuster (1994) and will not be dealt with in this thesis. In ambient vibration testing, a structure is excited by wind, machinery, traffic, environmental noise, etc. Unlike forced vibration testing, one does not control the force that is applied to vibrate the structure. Natural frequencies and mode shapes are obtained by measuring the vibrations of the structure simultaneously at several locations on the structure. Basic 48 assumptions for obtaining reliable estimates of modal characteristics by ambient vibration testing are: • the structure behaves as a linear system, that is, superposition method can be used; • the structure is classically damped and, therefore, only has real valued modes; • the ambient excitation has uniform frequency content, that is, the excitation has white noise characteristics; • excitation has spatial distribution, that is, all modes of interest are excited. It was considered, that these assumptions were satisfied for the Lillooet river bridge at the levels of vibration caused by ambient excitation. Major sources of excitations were wind, seismic microtremors and the stream of the Lillooet river. Ambient vibration testing was conducted on the bridge on August 30, 1994. It was a sunny summer day, the temperature range was approximately 15 - 24 °C during the day. Vertical and transverse responses of the structure were measured with eight FBA-11 accelerometers. Figure 3.7. shows fastening of a plate with two accelerometers to the bridge deck. The traffic on the bridge was always stopped for the duration of each measurement setup. Nineteen setups were taken during the day with the duration of each setup of about 6 minutes. The sampling rate for ambient vibration testing was 100 samples per second. The locations of measured points, summary of the setups and examples of recorded signals are included in Appendix C. 49 Figure 3.7. View of a plate with two accelerometers on the bridge deck. Time histories of acceleration at each measured point were obtained. The data was reduced using program U L T R A and, finally, natural frequencies and mode shapes were determined using program VISUAL. Only vertical mode shapes were important for this project because only dynamic amplification factors of the bridge in the vertical direction were investigated. Five natural modes in the vertical direction were identified in the frequency range 0 - 2 5 Hz. 50 Table 3.1 presents the values of the natural frequencies of the bridge together with the explanation of the character of the corresponding natural modes. Table 3.1. Natural frequencies of the Lillooet River Bridge. Mode No. Frequency (Ffz) Character of natural mode 1 2.51 Long span, 1st vertical flexural mode 2 5.56 Short span, 1st vertical flexural mode 3 7.88 Long span, 2nd vertical flexural mode 4 15.18 Short span, 2nd vertical flexural mode 5 15.60 Long span, 3rd vertical flexural mode 3.4 FORCED VIBRATION TESTING The purpose of the forced vibration testing was to obtain data for evaluation of the dynamic amplification factors for the Lillooet River Bridge due to the passage of a heavy vehicle. The forced vibration testing was conducted on August 31, 1994. The data acquisition system allowed the measurement of the response of the bridge at eight points. The response was measured on both (East and West) sides of the bridge. Sensors were located at the midspans of both the long and the short spans and at the thirds of the long span. The location of measured points can be found in Appendix D. The testing vehicle, an empty logging truck, was provided by C.R.B. Logging in Squamish, B.C. The truck was a type 4964F made by Orion Western Star Trucks L T D . in Clearbrook, B.C. The truck was used empty because the driver would not be able to turn a loaded truck 51 close to the bridge after each run. The total weight of the truck was about 162 k N spread over three axles The axle loads from the front to the rear axle were: 46 kN, 58 k N and 58 kN, respectively. The testing truck shown in Figure 3.8, is called in this thesis T3162 (Truck - 3 axles, 162 kN). Figure 3.9 shows the vehicle T3162 during the test in the long span. Three types of tests were conducted during the experiments. First, the static displacements at the midspans of the short and the long spans due to the loading by the vehicle T3162 were measured with surveying equipment. The testing vehicle was located in the spans so that the maximum deflections occurred. The measurements were taken ten times. The averaged static displacements at midspans had values of 14.2 mm and 7.5 mm for the long span and the short span, respectively. Static displacements at the other measured points were determined after the testing from the static analysis, calibrated by the results of the static displacement measurements at the midspans. Second, acceleration time histories at the selected bridge locations due to the passage of the testing vehicle T3162 were obtained. The testing vehicle ran over the bridge three times at nominal speeds of 10, 20, 30, 40, 50 and 60 km/h. The actual speed of the vehicle was always measured and compared with the corresponding nominal speed and it was found, that the actual speeds were very close to the corresponding nominal speeds in all setups. The data from eighteen test setups was recorded and stored. The locations and numbering of measured points, summary of the setups and examples of recorded signals can be found in Appendix D. Third, three setups of a complementary hammer testing of the Lillooet River Bridge were conducted. Information about these setups is included in Appendix D. However, the data obtained from this testing was not used in this thesis. 52 Figure 3.8. Testing vehicle T3162- the empty logging truck used for testing. After the forced vibration testing, the dynamic amplification factors of the Lillooet River Bridge due to the passages of the testing vehicle T3162 were calculated. The data used for obtaining dynamic amplification factors included static displacements and acceleration records. The acceleration records were used for further analysis as "raw records" without instrument corrections. This was possible because the error in measured accelerations is negligible in the frequency range of interest 0 to 25 Hz (Kinemetrics, 1991). The sequence of the steps which were taken in order to obtain the dynamic amplification factors follows: • first, the static displacements due to the vehicle T3162 were evaluated at the points where the static displacements were not measured. These values were obtained using a static analysis calibrated to the results of the measured points. Thus, the maximum static displacements due to the vehicle T3162 at all measured points were then known; • second, acceleration records obtained from the forced vibration testing were filtered and integrated twice in order to obtain time histories of dynamic displacement at measured points due to the passages of the testing vehicle T3162. A computer program was developed to perform this analysis. The program converts the acceleration record from the time domain to the frequency domain using the Fast Fourier Transform. Then, the displacement is obtained by the double integration performed in the frequency domain by dividing the Fourier transform of the acceleration by square value of the circular frequency. The resulting values are multiplied by a pass-band window. This window, shown in Figure 3.10, has zero values for frequencies smaller than 1 Hz and greater than 26 Hz. Its values between frequencies 1.5 Hz and 25 Hz are equal to 1. The window function changes its values from 0 to 1 in the frequency interval (1.0, 1.5) Hz, and from 1 to 0 in the frequency 54 interval (25, 26) Hz. A harmonic function is used in these intervals in order to ensure smooth character of the window. The purpose of applying this type of window is to remove components of signals with frequencies smaller than 1.5 Hz and greater than 25 Hz. Using this window is justified because the results of ambient vibration testing did not show any natural modes of the bridge outside the frequency range 1.5 - 25 Hz. Finally, after applying the window, the displacement time history is obtained using the Inverse Fast Fourier Transform. 1.2 -i 1 1 1 » JD.8 |).6 — : c 0.4 0.2 o 4—1—h 1—;—i 1 1 1 1 1 1 1 * 1 I 0 5 10 15 20 25 30 frequency (Hz) Figure 3.10. Window in frequency domain used for filtering of the acceleration records. Examples of acceleration records and the dynamic displacement time histories obtained by integration of these acceleration records are shown in Figures 3.11 to 3.12. Both figures show the measured acceleration records (part a) and the calculated dynamic displacement (part b) from the setup 7. Figure 3.11 presents the acceleration and the dynamic displacement at the midspan of the long span, and Figure 3.12 presents the same quantities at the midspan of the short span. 55 _ o n a) measured acceleration: a (m/sA2) u.ou —l CM < CO 0.00 -0.80 0.001 0 5 10, % 15 20 time (s) b) dynamic component of displacement calculated from measured acceleration: d (m) 0.000 -0.001 10 time (s) 15 20 Figure 3.11. Time histories of measured acceleration and calculated dynamic displacement; Location: midspan of the long span (point 23), setup 7, speed: 28.94 km/h. CM < 1 CO 1.50 0.00 -1.50 a) measured acceleration: a (m/sA2) 0.001 0.000 -0.001 b:) dynamic component of displacement calculated from measured acceleration: d (m) 10 time (s) Figure 3.12. Time histories of measured acceleration and calculated dynamic displacement; Location: midspan of the short span (point 45), setup 7, speed: 28.94 km/h. 56 The procedure to obtain the dynamic displacement time history from the acceleration record was tested on single and multiple frequency harmonic signals where the expressions for the double-integrated signals were known. The results of these tests were satisfactory. In order to ensure the proper transformation of the signal by the FFT, the program adds sufficient number of zero values at the end of the original signal. The proper transform also requires zero value at the beginning of the signal in order to avoid undesirable effects of discontinuities. This was satisfied in the case of the acceleration records from the forced vibration testing because the bridge was at rest before each run of the testing truck. third, the maximum values of the dynamic displacement were obtained from the dynamic displacement time histories; finally, the dynamic amplification factors were calculated. This calculation was not based on the definition of the dynamic amplification factor, Equation 1.1. This definition could not be followed because the total displacements of the bridge were not measured. The explanation why the total displacements could not be measured can be found in Appendix C. The dynamic amplification factors were calculated as follows: A +A stat,max dyn.max ^ j \ A stat,max where: A is the maximum static displacement and stat,max A^ is the maximum dynamic displacement. dyn.max It can be shown, that Equation 3.1 gives an upper bound for the dynamic amplification factor calculated according to its definition, Equation 1.1. The result of Equation 3.1 is 57 exact only when the maximum dynamic and static displacements occur at the same time. However, the used approach can be justified by the following considerations: Let us consider only the static component of the total displacement the midspan of the long span of the Lillooet River Bridge. When the truck runs over the span, the midspan moves down from its initial position of a zero deflection, then the motion reaches its maximum and finally, the point moves back to its initial position when the truck leaves the span. This motion can be considered as a "semi-periodic" motion and the time from the first zero deflection to the second zero deflection can be considered as a half of the period of this motion. It can be shown that the period of this motion is more than ten times the period of the first natural mode of the long span. The shape of the kinetic component of the deflection at the midspan is given by the influence line of this deflection. This influence line has a single maximum and it is very flat around its maximum. It was observed from the numerical analysis of the Lillooet River Bridge that at least one peak of the first mode response occurs in that part of the kinetic component of the deflection where its values are from 97% to 100% of the maximum value. This generates a relative error of the dynamic amplification factor up to 3%. This error analysis was done numerically for the highest speed of the truck, which is the case when the error is maximum. The error is negligible for lower speeds of the testing vehicle T3162. The dynamic amplification factors obtained by the above-described sequence of steps are presented for all measured points in Tables 3.2 to 3.9. Each table lists all measured setups and the dynamic amplification factors are presented in the last column of the tables. Figures 3.13 to 3.20 present the dynamic amplification factors in graphical form. In each figure, the 58 dynamic amplification factors are indicated in the vertical axis, and the truck speed is indicated in the horizontal axis. It can be concluded from the Figures 3.13 to 3.20, that the results for the points on the West side of the bridge (points 19, 23, 27, 45) are very similar to the results for the corresponding points on the East side of the bridge (points 20, 24, 28, 46). This is in good agreement with the assumption that the bridge is loaded by the passing truck symmetrically with respect to its longitudinal axis. The distributions of the dynamic amplification factors, as functions of the truck speed, are very similar for the points on the long span (points 19, 20, 23, 24, 27, 28). This can be observed from Figures 3.13 to 3.18. The larger values of the factor at the lowest speeds decrease to minimum factors at speeds between 20 and 30 km/h and then the factors increase again at higher speeds. The distributions of the dynamic amplification factors for the points on the short span (points 45 and 46) are also very similar. The only difference was observed for the nominal speed 60 km/h when the factors from the point 46 are about 1.3 compared to those from the point 45 which have values about 1.21. The values of the amplification factors from different setups at the same nominal speeds have very similar values for all the measured points. The maximum observed differences of the corresponding dynamic amplification factors are 0.03. The dynamic amplification factors resulting from the forced vibration testing were found satisfactory and it was concluded that these could be used later for calibration and verification of the numerical model of the system bridge-vehicle. 59 Table 3.2. Measured dynamic amplification factors at the midspan of the long span (point 19) due to passage of the testing vehicle T3162. Max. Dynamic Static Dynamic Setup Speed Displacement Displacement Amplification (km/h) (mm) (mm) Factor 1 13.36 1.53 12.2 1.13 2 13.36 1.64 12.2 1.13 3 13.10 1.66 12.2 1.14 4 19.80 1.05 12.2 1.09 5 19.78 0.96 12.2 1.08 6 19.74 0.91 12.2 1.07 7 28.94 0.91 12.2 1.07 8 31.73 0.96 12.2 1.08 9 29.03 1.08 12.2 1.09 10 38.60 1.56 12.2 1.13 11 38.78 1.34 12.2 1.11 12 39.71 1.36 12.2 1.11 13 49.16 2.16 12.2 1.18 14 49.65 2.28 12.2 1.19 15 48.82 2.07 12.2 1.17 16 58.39 2.60 12.2 1.21 17 58.53 2.55 12.2 1.21 1.4 1.35 J 1 3 f 1 1-25 I 1-2 + o E 1.15 ro & 11 Q 1-1 1.05 f 1 10 Dynamic amplification - point 19 Vehicle: testing truck T3162 20 30 40 speed (km/h) 50 60 Figure 3.13. Measured dynamic amplification factors at the midspan of the long span (point 19) due to passage of the testing vehicle T3162. Table 3.3. Measured dynamic amplification factors at the midspan of the long span (point 20) due to passage of the testing vehicle T3162. Setup Speed (km/h) Max. Dynamic Displacement (mm) Static Displacement (mm) Dynamic Amplification Factors 1 13.36 1.65 12.2 1.14 2 13.36 1.75 12.2 1.14 3 13.10 1.66 12.2 1.14 4 19.80 1.15 12.2 1.09 5 19.78 1.13 12.2 1.09 6 19.74 1.15 12.2 1.09 7 28.94 1.06 12.2 1.09 8 31.73 1.09 12.2 1.09 9 29.03 1.10 12.2 1.09 10 38.60 1.61 12.2 1.13 11 38.78 1.67 12.2 1.14 12 39.71 1.75 12.2 1.14 13 49.16 2.38 12.2 1.19 14 49.65 2.47 12.2 1.20 15 48.82 2.26 12.2 1.19 16 58.39 2.33 12.2 1.19 17 58.53 2.26 12.2 1.19 1.4 1.35 I 1-3 J 1-25 C L § 1-2 o E 1.15 f <o § 1 . 1 -1.05 1 10 Dynamic amplification - point 20 Vehicle: testing truck T3162 20 30 40 speed (km/h) 50 60 Figure 3.14. Measured dynamic amplification factors at the midspan of the long span (point 20) due to passage of the testing vehicle T3162. Table 3.4. Measured dynamic amplification factors at the long span (point 23) due to passage of the testing vehicle T3162. Max. Dynamic Static Dynamic Setup Speed Displacement Displacement Amplification (km/h) (mm) (mm) Factor 1 13.36 1.71 14.2 1.12 2 13.36 1.81 14.2 1.13 3 13.10 1.84 14.2 1.13 4 19.80 1.16 14.2 1.08 5 19.78 1.14 14.2 1.08 6 19.74 1.01 14.2 1.07 7 28.94 1.01 14.2 1.07 8 31.73 1.08 14.2 1.08 9 29.03 1.29 14.2 1.09 10 38.60 1.78 14.2 1.13 11 38.78 1.54 14.2 1.11 12 39.71 1.52 14.2 1.11 13 49.16 2.41 14.2 1.17 14 49.65 2.58 14.2 1.18 15 48.82 2.35 14.2 1.17 16 58.39 2.96 14.2 1.21 17 58.53 2.86 14.2 1.20 1.4 1.35 Dynamic amplification - point 23 Vehicle: testing truck T3162 & 1.3 f -13 | 1-25 j-I 1-2 +-o E 115 CO ^ 11 1.05 1 10 20 30 40 speed (km/h) 50 60 Figure 3.15. Measured dynamic amplification factors at the long span (point 23) due to passage of the testing vehicle T3162. Table 3.5. Measured dynamic amplification factors at the long span (point 24) due to passage of the testing vehicle T3162. Setup Speed (km/h) Max. Dynamic Displacement (mm) Static Displacement (mm) Dynamic Amplification Factor 1 13.36 1.78 14.2 1.13 2 13.36 1.88 14.2 1.13 3 13.10 1.82 14.2 1.13 4 19.80 1.27 14.2 1.09 5 19.78 1.25 14.2 1.09 6 19.74 1.27 14.2 1.09 7 28.94 1.14 14.2 1.08 8 31.73 1.18 14.2 1.08 9 29.03 1.25 14.2 1.09 10 38.60 1.83 14.2 1.13 11 38.78 1.86 14.2 1.13 12 39.71 1.94 14.2 1.14 13 49.16 2.66 14.2 1.19 14 49.65 2.74 14.2 1.19 15 48.82 2.51 14.2 1.18 16 58.39 2.65 14.2 1.19 17 58.53 2.48 14.2 1.17 1.4 1.35 c •g 1.3 "Hi !E 125 I 1-2 \~ 0 1 115 Dynamic amplification - point 24 Vehicle: testing truck T3162 >• 1 1 1.05 1 10 20 30 40 speed (km/h) 50 60 Figure 3.16. Measured dynamic amplification factors at the long span (point 24) due to passage of the testing vehicle T3162. Table 3.6. Measured dynamic amplification factors at the long span (point 27) due to passage of the testing vehicle T3162. Max. Dynamic Static Dynamic Setup Speed Displacement Displacement Amplification (km/h) (mm) (mm) Factor 1 13.36 1.41 12.2 1.12 2 13.36 1.47 12.2 1.12 3 13.10 1.50 12.2 1.12 4 19.80 0.96 12.2 1.08 5 19.78 1.03 12.2 1.08 6 19.74 0.83 12.2 1.07 7 28.94 0.92 12.2 1.08 8 31.73 0.93 12.2 1.08 9 29.03 1.19 12.2 1.10 10 38.60 1.55 12.2 1.13 11 38.78 1.35 12.2 1.11 12 39.71 1.33 12.2 1.11 13 49.16 2.11 12.2 1.17 14 49.65 2.27 12.2 1.19 15 48.82 2.10 12.2 1.17 16 58.39 2.58 12.2 1.21 17 58.53 2.50 12.2 1.20 1.4 1.35 j-1.3 S 1.25 f a. E CO o E CO c >. Q 1.2 --1.15 •-1.1 1.05 10 Dynamic amplification - point 27 Vehicle: testing truck T3162 20 30 40 speed (km/h) 50 60 Figure 3.17. Measured dynamic amplification factors at the long span (point 27) due to passage of the testing vehicle T3162. Table 3.7. Measured dynamic amplification factors at the long span (point 28) due to passage of the testing vehicle T3162. Setup Speed (km/h) Max. Dynamic Displacement (mm) Static Displacement (mm) Dynamic Amplification Factor 1 13.36 1.44 12.2 .1.12 2 13.36 1.50 12.2 1.12 3 13.10 1.48 12.2 1.12 4 19.80 1.03 12.2 1.08 5 19.78 1.03 12.2 1.08 6 19.74 1.06 12.2 1.09 7 28.94 0.93 12.2 1.08 8 31.73 0.97 12.2 1.08 9 29.03 1.03 12.2 1.08 10 38.60 1.49 12.2 1.12 11 38.78 1.56 12.2 1.13 12 39.71 1.62 12.2 1.13 13 49.16 2.24 12.2 1.18 14 49.65 2.33 12.2 1.19 15 48.82 2.10 12.2 1.17 16 58.39 2.50 12.2 1.20 17 58.53 2.30 12.2 1.19 1.4 1.35 Dynamic amplification - point 28 Vehicle: testing truck T3162 •S 1.3 f - -(M 1.25 f • C L £ 1-2 0 1 1.15 n c >» 1 1 Q 1.05 1 10 20 30 40 speed (km/h) 50 60 Figure 3.18. Measured dynamic amplification factors at the long span (point 28) due to passage of the testing vehicle T3162. Table 3.8. Measured dynamic amplification factors at the midspan of the short span (point 45) due to passage of the vehicle T3162. Setup Speed (km/h) Max. Dynamic Displacement (mm) Static Displacement (mm) Dynamic Amplification Factor 1 13.36 1.48 7.5 1.20 2 13.36 1.50 7.5 1.20 3 13.10 1.36 7.5 1.18 4 19.80 0.91 7.5 1.12 5 19.78 0.74 7.5 1.10 6 19.74 0.94 7.5 1.13 7 28.94 0.94 7.5 1.13 8 31.73 1.02 7.5 1.14 9 29.03 0.90 7.5 1.12 10 38.60 1.43 7.5 1.19 11 38.78 1.34 7.5 1.18 12 39.71 1.29 7.5 1.17 13 49.16 0.81 7.5 1.11 14 49.65 0.85 7.5 1.11 15 48.82 0.79 7.5 1.11 16 58.39 1.62 7.5 1.22 17 58.53 1.53 7.5 1.20 1.4 1.35 I 1 3 13 | 1.25 f-I 1-2 + u E1.15 f £ 11 1.05 f 1 10 Dynamic amplification - point 45 Vehicle: testing truck T3162 20 30 40 speed (km/h) 50 60 Figure 3.19. Measured dynamic amplification factors at the midspan of the short span (point 45) due to passage of the vehicle T3162. Table 3.9. Measured dynamic amplification factors at the midspan of the short span (point 46) due to passage of the vehicle T3162. Setup Speed (km/h) Max. Dynamic Displacement (mm) Static Displacement (mm) Dynamic Amplification Factor 1 13.36 1.25 7.5 1.17 2 13.36 1.27 7.5 1.17 3 13.10 1.15 7.5 1.15 4 19.80 0.81 7.5 1.11 5 19.78 0.87 7.5 1.12 6 19.74 0.77 7.5 1.10 7 28.94 1.04 7.5 1.14 8 31.73 0.90 7.5 1.12 9 29.03 0.93 7.5 1.12 10 38.60 1.31 7.5 1.17 11 38.78 1.14 7.5 1.15 12 39.71 1.02 7.5 1.14 13 49.16 1.01 7.5 1.13 14 49.65 1.02 7.5 1.14 15 48.82 0.96 7.5 1.13 16 58.39 2.33 7.5 1.31 17 58.53 2.23 7.5 1.30 1.4 Dynamic amplification - point 46 Vehicle: testing truck T3162 1.35 +--1.3 +--1 125 C L g 1-2 u E 1.15 is £ 1-1 1.05 1 10 20 30 40 speed (km/h) 50 60 Figure 3.20. Measured dynamic amplification factors at the midspan of the short span (point 46) due to passage of the vehicle T3162. C H A P T E R 4 CALIBRATION AND VERIFICATION OF THE NUMERICAL MODEL This chapter deals with the calibration and verification of the numerical model for the system bridge-vehicle, which was described in Chapter 2 of this thesis. The actual stiffness of the bridge needed for numerical simulations is determined from results of measurements of static displacements of midspans of the Lillooet river bridge due to the loading by the testing truck T3162 (§4.1). Results from the dynamic testing are used for verification of the numerical model. Modal characteristics obtained from the ambient vibration testing are used to verify the mass matrix (§4.2). Results of the forced vibration testing are used for verification of the damping ratio of the bridge and determination of characteristics of the roughness of the bridge deck (§4.3). 4.1 DETERMINATION OF ACTUAL STIFFNESS OF THE BRIDGE The suitability of the stiffness matrix used for modeling the bridge structure in the program Bridge-Vehicle Interaction (BVI) was first tested with known solutions on a simple beam, with very good agreement. Predictions of the static analysis of the coupled system bridge-vehicle were then compared to the experimental measurements of the static displacements at the midspans of the Lillooet River Bridge due to the loading by the testing truck T3162. The result of this comparison enabled calibration of the value of Young's modulus of elasticity of the beams. The original value considered was E = 13,000 MPa, a nominal value for Douglas 68 fir. This value was increased to 14,375 MPa in order to obtain the same displacements at the midspans from numerical analysis as those from experiments. This calibrated value of E is well within the expected range of 11000 to 15000 MPa, assuming a 7.5% coefficient of variation in E. The value of shear modulus was assumed as a fraction of modulus of elasticity, at G = E/17. 4.2 VERIFICATION OF THE MASS MATRIX The verification of the mass matrix was based on a comparison of the natural modes and the natural frequencies of the Lillooet River Bridge from experiments to those from the numerical analysis. This verification could be done because the natural frequencies and modes are a solution of the eigenproblem of the mass and the stiffness matrices. Since the stiffness matrix was already calibrated, only the mass matrix remained to be verified by the natural frequency comparison. The experimental natural modes and natural frequencies of the Lillooet River Bridge were compared to the corresponding natural modes and frequencies obtained from the program B V I N V . Summary of this comparison is given in Table 4.1. This table presents five identified experimental natural frequencies of the bridge with corresponding frequencies from the numerical analysis. Each natural mode is characterized in the last column of this table. Higher natural frequencies and modes obtained from the numerical analysis are not given in Table 4.1. The agreement of the experimental and the numerical values of the natural frequencies in Table 4.1 is generally good. The maximum relative error of analytical results with respect to experimental ones is 14.1%, which corresponds to the case of the 5th natural frequency. 69 Relative errors at the other natural frequencies are smaller. The comparison of experimental and numerical modal characteristics was found satisfactory, implying that the mass matrix of the bridge was properly modeled. Table 4.1. Experimental and numerical natural frequencies of the Lillooet River Bridge. Numerical Analysis Experimental Values Mode No. Frequency (Hz) Frequency (Hz) Relative Error Character of the natural mode 1 2.43 2.51 -3.2% Long span, 1st vertical flexural mode 2 5.49 5.56 -1.3% Short span, 1st vertical flexural mode 3 8.71 7.88 +10.5% Long span, 2nd vertical flexural mode 4 18.4 16.2 +13.5% Short span, 2nd vertical flexural mode 5 18.6 16.3 +14.1% Long span, 3rd vertical flexural mode 4.3 VERIFICATION OF THE MODEL FROM THE RESULTS OF THE FORCED VIBRATION TESTING Having calibrated and verified the stiffness and the mass matrices of the numerical model of the bridge, the only remaining parameters of the model of the bridge requiring calibration are damping of the bridge and the roughness characteristics of the bridge deck. The results of the forced vibration tests allowed the calibration of the damping ratio used for evaluation of the damping matrix, presented in Chapter 2. The damping of the bridge was evaluated from five time histories of free vibration response of both the short and the long 70 spans. The average value of modal damping ratio was 1.9% for the first natural mode of the short span and 1.7% for the first natural mode of the long span of the Lillooet River Bridge. As the B V I software evaluates damping matrix of the bridge only from one modal damping ratio, the average value of 1.8% was selected. The roughness of the bridge deck can be characterized by the length of bumps and by the maximum height the bumps. These parameters are represented by the maximum value of the random process and by its spectral characteristics, when the roughness of the bridge deck is considered in numerical simulations as it was described in Chapter 2. Since the roughness of the deck of the Lillooet River Bridge was not measured with precision, the value of the maximum bump on the bridge deck was estimated from a quick surveying measurement as ±0.01 m. The spectral characteristics of the roughness were determined from detailing of the bridge deck. It was estimated that the spectrum had narrow band characteristics with components at essentially two frequencies or corresponding wave lengths. The length of running planks of about 4 m gave a first significant wave length of 8 m. The spacing of ties under running planks of about 0.5 m gave a second significant wave length of 1 m. The above values of the wave lengths were based on the assumption that the shape of a running plank is a half-sine wave of the length equal to 4 m. Similar assumption has been made for the shape of the planks between two ties. The two values of 1 m and 8 m were modified by ±30% to form two bands, that is, the intervals (0.7 m, 1.3 m) and (5.6 m, 10.4 m). This information, transferred from the wave length domain to the frequency domain, was used for the simulation of the roughness of the bridge deck. Within these intervals, the values of the power spectral densities were assumed constant for all frequencies and then scaled using the window shown in Figure 4.1. 71 0 H 1 1 i 1 ' I i I I I i 1 0 0.5 1 1.5 2 2.5 3 frequency f (Hz) Figure 4.1. The window for scaling amplitudes of the roughness process. Window selection was based on the assumption that the high frequency component amplitudes of the roughness function are smaller than the amplitudes of the components of low frequencies. The final values of the power spectral density were obtained using Equation 2.35, so that the probability of the roughness process exceeding the value of 0.01 m per unit length was 5%. Besides the above-described combination of wave lengths of l m and 8 m, seven other combinations were considered in order to study the differences in the response of the bridge with changing spectral densities for the roughness process. These combinations were: 1 m and 4 m, 1 m and 12 m, 1 m and 16 m, 2 m and 4 m, 2 m and 8 m, 2 m and 12 m, 2 m and 16 m. In each case, the roughness profile was generated for each of the above combinations using two bands with boundaries at ±30% from the specified values of each wave length. The above combinations of the wave lengths, together with the original combination 1 m and 8m, gave 72 eight types of the roughness of the bridge deck. Table 4.2 shows the wave length characteristics of the eight types of the roughness spectra for the bridge deck. Table 4.2. Wave length characteristics of the roughness function. Type of the roughness 1st wave length (m) 2nd wave length (m) 1 1 4 2 1 8 3 1 12 4 1 16 5 2 4 6 2 8 7 2 12 8 2 16 Examples of roughness profiles generated from these eight types are shown in Figures 4.2. and 4.3. Values of the roughness function /*(/?) are plotted on the vertical axis. Coordinate p is the position on the bridge, that is, the distance from the beginning of the bridge and it is plotted on horizontal axis. Numerical simulations of the passage of the testing truck T3162 over the bridge were performed considering all eight types of the roughness characteristics. This was done in order to study the response of the bridge with changing roughness type. The program B V I generates for each simulation a new sequence of random phase shifts to construct a new roughness profile. To eliminate their influence on results of the analysis, the same sequence of random phase shifts was used with the different spectra. Since the results with using only one sequence of random phase shifts could not give comprehensive picture of 73 Roughness of the bridge deck - type 2 10 20 30 40 position p (m) 50 60 70 0.02 Roughness of the bridge deck - type 3 30 40 position p (m) 50 60 70 Roughness of the bridge deck - type 4 30 40 position p (m) 50 60 70 Figure 4.2. Roughness of the bridge deck - types 1, 2, 3 and 4. 0.02 £ 0.00 --0.02 Roughness of the bridge deck - type 5 10 20 30 40 position p (m) 50 60 70 Roughness of the bridge deck - type 6 10 20 30 40 position p (m) 50 60 70 0.02 0.00 -0.02 Roughness of the bridge deck - type 7 i p r 30 40 position p (m) 50 60 70 0.02 0.00 -0.02 Roughness of the bridge deck - type 8 30 40 position p (m) 50 60 70 Figure 4.3. Roughness of the bridge deck - types 5, 6, 7 and 8. 75 the response of the bridge, ten sets of random phase shifts were pregenerated and stored. They were used for simulations with each of the eight types of the roughness spectra. The resulting dynamic amplification factors were calculated as the average of the ten simulations using the ten sets of random phase shifts. Equation 1.1 was used for calculation of the dynamic amplification factors. The response of the bridge was studied at midspans of the long and the short spans, and the following speeds of the vehicle were used for analysis: 10, 20, 30, 40, 50 and 60 km/h. The dynamic amplification factors were calculated from the results of these simulations and they are presented in Tables 4.3 to 4.18. The tables with the odd numbers (4.3, 4.5, 4.17) present results for the midspan of the short span, and the tables with the even numbers (4.4, 4 . 6 , 4 . 1 8 ) present the results for the midspan of the long span. The dynamic amplification factors are presented columnwise for each sequence of the random phase shifts in Tables 4.3 to 4.18, and the basic statistics obtained from the factors for phase shifts sequences 1 to 10 are shown in the last two columns of each table. The figures below these tables present similar information, but in graphical form. Figures 4.4a to 4.19a (parts "a" of the figures) show the ranges of the dynamic amplification factors from the corresponding table, that is, the maximum and the minimum factors. These factors were used to form the upper and the lower bounds of the dynamic amplification for each speed, calculated using the random phase shifts 1 to 10. These figures show the distribution of the dynamic amplification factors with respect to the speed of the vehicle T3162. Figures 4.4b to 4.19b (parts "b" of the figures) present the average amplification factors from the corresponding table. Verification of the roughness process characteristics consisted in finding the best match of the results presented in Figures 4.4 to 4.19 (both parts "a" and "b") to the results of experiments. The results of experiments were shown in Figures 3.15 and 3.16 for midspan of the long span, 76 and in Figures 3.19 and 3.20 for midspan of the short span. By comparing Figures 3.15 and 3.16 with those generated using numerical results for the long span (Figures 4.5, 4.7, 4.9, 4.11, 4.13, 4.15, 4.17, and 4.19), it was found that the best match was obtained for the results in Figure 4.7. It was also concluded from the numerical results for the short span (Figures 4.4, 4.6, 4.8, 4.10, 4.12, 4.14, 4.16, and 4.18), that the best match with the experimental results (Figures 4.19 and 4.20) was obtained for the results in Figure 4.6. The comparison then showed that the characteristics of the roughness process given by the roughness type 2, are the best approximation for the actual roughness profile on the Lillooet River Bridge from the eight tested types. The agreement of the experimental dynamic amplification factors with those from the numerical simulations, using this type of the bridge deck roughness, was found satisfactory. The final comparison of the experimental and the numerical dynamic amplification factors is presented in Figure 4.20 for the short span and in Figure 4.21 for the long span of the Lillooet River Bridge. The selected roughness type 2 was used in the simulations to calculate the numerical dynamic amplification factors. The experimental dynamic amplification factors were obtained as the average values over the measured setups (3 for each nominal speed) and over the side of the bridge (2 points at each midspan). The agreement of the experimental and the numerical results is good, and therefore, the roughness characteristics type 2 will be used for further numerical analysis. 77 •o 'a o J3 en C O 1 P .14 I B O a ••8 I o s a 1 I a 1 E2 Standard deviation O I N O M A C D 0 0 0 0 0 0 Mean value 1.330 1.278 1.175 1.1 ol 1.156 1.094 Phases 10 Factor 1.228 1.25 1.164 1.193 1.177 1.121 Phases 9 | Factor j 1.228 1.303 1.179 1.028 1.103 1.099 Phases 8 Factor | 1.4 1.374 1.132 1.163 1.079 1.059 Phases 7 Factor | Phases 6 | Factor | 1.342 1.194 1.13 1.142 1.105 1.086 Phases 5 Factor CO CO CM 00 IT) 5 CO 0) v 00 T - CD Phases 4 Factor | 1.346 1.236 1.156 1.162 1.318 1.073 Phases 3 Factor | 1.272 1.148 1.105 1.256 1.065 1.075 Phases 2 Factor Phases 1 Factor | 1.294 1.281 1.205 1.081 1.2 1.127 Random Phases Sequence I Speed (km/h) I 2 8 8 § 5? 8 o if 2. co s. in « w 0 •-• •-0 •-• •-— in m c* ^ jojoej uojieoyi|duiv O J I U B U A Q co •<* v £ e J2 c Jo its I 3 3 e ° J 3 II a a eo 78 I 1 § •a a •a I •9 S § •a H •o c CO T3 C CD i s CO o. CO .n 0. CO 0. (0 0_ (0 0. CO - C 0. ra .c a CO X 0. or o. »' 6 6 d 6 d d 8 8 £ 8 g i- T- CM i - T • * CM CM in — CM <N t 8 S * ^ CN (N t- T-3 § 8 3 § « CO jj » 10 j* CM °> ^ g • ^ «- o (N CO m m 8 ^ S CN J 5 T - r>~ CM T - T - CM S 8 § 8 8 79 81 82 83 85 87 § o. w •o 'a c est O, ts o EQ c •I 1 O c oo C I S—' •a e O U 7 a 1 2 V 3 I 1 1 •a Standard deviation 0 0 0 0 0 0 Mean value 1.057 1.468 1.447 1.429 1.238 1.304 Phases 10 Factor 1.015 1.303 1.187 1.216 1.176 1.138 Phases 9 I Factor | 1.051 1.341 1.665 1.359 1.197 1.448 Phases 8 Factor 1.033 1.34 1.405 1.312 1.264 1.395 Phases 7 I Factor | 0 35 "9F T CM Phases 6 | Factor | 1.04 1.501 1.441 1.426 1.35 1.426 Phases 5 | Factor | Phases 4 Factor nun Phases 3 Factor | 1.05 1.236 1.303 1.443 1.201 1.36 Phases 2 I Factor I 1.116 1.76 1.716 1.758 1.416 1.218 Phases 1 I Factor I 1.03 1.421 1.425 1.208 1.216 1.28 Random Phases Sequence I Speed (km/h) I o n co c s. CO & >> t— in in cv c JS en 3 s 8' u> in v n IN T - p jope-j uoj)eayj|dujv O I I U B U A Q 3 a 12 I op 2 a 1 is 18 a 3 eo 88 1 § ••g -a •a J3 a •a p o ao a s O a •a 1 § •a I E2 ™ 8 is i c (0 S - i (0 IS a CO a. (0 a. (0 a. (0 o. to a. CM CO x: a. CO E co S -§ $ s 8 c 3 8 3 S 8 doodad ^ n t - o o (N | g E £ CN O T— S N ^ g S S M N 00 . S S R 8 S 8 8 H 1 E 1 •8 3 8 3 ao E 3 I Is '1 8 i § u J3 V fx op tr s ,° 2 3 ao E 89 90 91 Dynamic Amplification Factors Location: short span at midspan 1-3 T 1 -1 1 1 1 1 1 1 . 1 1 1 10 20 30 40 50 60 speed (km/h) | » experiment simulation - roughness Type 2 Figure 4.20. Comparison of experimental and numerical dynamic amplification factors, Location: midspan of the short span, Vehicle:T3162, Roughness of the bridge deck:Type 2. Figure 4.21. Comparison of experimental and numerical dynamic amplification factors, Location: midspan of the long span, Vehicle:T3162, Roughness of the bridge deck:Type 2. The numerical and experimental results were compared in terms of the corresponding average dynamic amplification factors. It might also be useful to compare the results of simulations with those from testing in terms of the corresponding time histories of acceleration and dynamic component of displacement. This comparison can only be approximate, since the simulated response of the bridge due to the moving vehicle was calculated using a bridge deck roughness profile which may be different from the actual profile of the bridge. The simulated profile had the same spectral characteristics and the peak value as the actual one, but the phase shifts of the individual frequency components of this profile may have been different from the actual ones. However, it was found useful to compare the corresponding time histories of acceleration and dynamic component of displacement from simulations with those from the testing. The amplitudes and the frequency content of these signals were considered in this comparison, which was done for the response at midspan of the short span and for the speed of the 13.36 km/h of the testing truck T3162. The results of the setup 1 of the forced vibration testing and the results of simulations with the bridge deck roughness type 2 were used for this comparison. The example of the roughness profile, generated from the roughness characteristics type 2, is shown in Figure 4.22a. Figure 4.22b presents a simple elevation of the Lillooet River Bridge with significant points - supports and midspans of the spans. The positions of these points along the bridge are also shown in Figure 4.22b. These positions correspond to the coordinate p, plotted on the horizontal axis in the Figure 4.22a. The comparison of the numerical and experimental time histories is shown in Figures 4.23 for accelerations and in Figure 4.24 for dynamic component of displacement. Each of these figures consists of five parts - parts "a" to "e". The time from the beginning of the analysis is 95 a) 0.02 E 0.01 -\ i f in d> c o.oo H O) p -0.01 -0.02 -20 I ' I 20 40 position on the bridge: p (m) 60 80 b) AT short span long span B o k 1 c/3 2 x-> •S9 II G o< T3 CN CM Ik I C/3 5 >/-> IS «4H II O G Is" G in = CN CN 1 on Figure 4.22. a) Representative example of the roughness profile used for numerical simulations; b) Positions of significant points along the Lillooet River Bridge. 96 plotted on the horizontal axis and the scale of this axis is the same for all these parts. Part "a" contains the time histories obtained from the experiments. Parts "b" to "d" present the time histories obtained from the numerical simulations. Three sequences of the phase shifts of the roughness were considered during these simulations in order to show the differences in the response time histories for the roughness profile with the same amplitudes and spectral characteristics but with different phase shifts. Part "e" shows the positions of the axles of the vehicle T3162 from the beginning of the bridge versus the time from the beginning of the analysis. The top straight line in this part represents the front axle of the truck while the two straight lines below this line, one close to the other, represent the two rear axles. The significant points along the bridge, introduced in Figure 4.22b, are also shown in the part "e" of each figure. The analysis of Figures 4.23 and 4.24 was based on a visual inspection of the corresponding time histories, on an approximate comparison of their amplitudes, and on their frequency content. It was observed from both figures that each time history is composed of two parts: a forced vibration and a free vibration. The forced vibration part occurs when the vehicle or part of it is on the short span. This is the time interval from about 1 s to 9 s in Figures 4.23 and 4.24. The free vibration part starts when the rear axle of the vehicle leaves the short span, which is the time interval from 9 s to the end of the simulation in these figures. These two parts of the response can be identified using parts "e" of the figures. Let us consider in more detail Figure 4.23, showing acceleration time histories. The general character of these acceleration time histories is the same. If the amplitude of the experimental acceleration is considered as a reference, then the peak amplitudes of the computed accelerations are about 210% (part b), 120% (part c) and 130% (part d) of the peak 97 experimental value. The differences of the amplitudes are very significant. The average value of the three amplitudes of the accelerations presented here is not equal to the experimental amplitude, since the phase shifts for the roughness profile were selected randomly and the number of simulations is not sufficient to give a reliable mean value. The frequency content of the forced vibration part of the signals is characterized by significant components at 3.2 Hz for the experimental acceleration and at 3.0 Hz and 5.5 Hz for all the three numerical accelerations. The 3.0 Hz to 3.2 Hz component of both the experimental and the numerical accelerations can be justified as a frequency range of the response due to the excitation by the bumps with the wave length of 1 m. The 5.5 Hz component, which is much less significant than the 3.2 Hz component, is the first bending mode component. The only significant component of the free vibration part of all the four studied signals is a 5.5 Hz component, which represents the first mode response. For the truck speed of 13.36 km/h, the comparison of experimental and numerical accelerations showed that the frequency content of particular signals is similar. Let us now consider Figure 4.24, showing the dynamic components of displacement. The general character of all dynamic components of the displacement is, again, the same. If the amplitude of the experimental dynamic displacement is considered as a reference, then the peak amplitudes of the computed dynamic displacements are about 180% (part b), 120% (part c) and 140% (part d). Considering the amplitudes of the signals, the same conclusion can be drawn as that in the previous paragraph, that several numerical simulations have to be done with different phase shifts in order to get a sufficient number of values to calculate the average amplitude. The frequency content of the forced vibration part of the signals is characterized by significant components at 2.8 Hz for the experimental dynamic displacement and at 3.2 Hz for 98 all the three computed dynamic displacements. The reason for the response at this frequency is the same as for the acceleration signals. The only significant component of the free vibration part of all the four studied signals is a 5.5 Hz component, which is the same as in the case of acceleration records. For the truck speed of 13.36 km/h, the comparison of experimental and numerical dynamic components of displacement showed that the frequency content of these signals is similar. In conclusion, the numerical model of the system bridge-vehicle has been thus verified and calibrated to the Lillooet River Bridge deck roughness. It is necessary to perform a series of numerical simulations to obtain a "mean value" response of the bridge i f the actual profile of the roughness is not known. This averaging would not be required, of course, i f the roughness profile is known and used as part of the input for the simulations. 99 a) Acceleration - experiment: a (m/sA2) CM < CO E, CO -1.0 1.8 CN < to CO CM < (0 CO CM < co CO -1.8 1.0 0.0 -1.0 1.5 0.0 Q. n | , | , | , p b) Acceleration - simulation (using random phases - sequence 1): a (m/sA2) - i | i | i | i | i | i | 1 1 1 1 r c) Acceleration - simulation (using random phases - sequence 2): a (m/sA2) -i | i | i r- 1 i 1 i 1 i 1 i — r d) Acceleration - simulation (using random phases - sequence 3); a (m/sA2) e) Position of the axles from the beginning of the bridge: p (m) Support 3: p=62.2 m \ Rear axle Support 2: p=22.1 m Midspan of the short span _Sur4>oilJ^p^QjtnJ 8 10 12 14 16 18 20 time (s) Figure 4.23. Acceleration records from experiment and from simulations; position of the vehicle; Location: midspan of the short span, vehicle: T3162, speed: 13.36 km/h. 100 0 002 a^ *- >y n a m" ; displacement - experiment (calculated from acceleration record): d (m) £ 0.000 -0.002 0.003 1 i | i | i I i | i [ — i I i I r b) Dynamic displacement - simulation (using random phases - sequence 1): d (m) -0.003 -| , 1 . 1 1 r-0 002 c) Dynamic displacement - simulation (using random phases - sequence 2): d (m) «§ 0.000 -0.003 60 -40 -20 -0 e) Position of the axles from the beginning of the bridge: p (m) Support 3: p=62.2 m Midspan of the short span _ J i i i p r x H l J j 4 ) ^ f i m J Q. —] i | i | r — i i | i | i 1 1 1 r— 4 6 8 10 12 14 16 18 20 time (s) Figure 4.24. Dynamic displacement from experiment and from simulations; position of the vehicle; Location: midspan of the short span, vehicle: T3162, speed: 13.36 km/h. 101 C H A P T E R 5 NUMERICAL SIMULATIONS WITH A LOADED TRUCK The first section of Chapter 5 presents a summary of the input data for simulations with a fully loaded logging truck T5441, a vehicle with a weight of 441 kN. This is followed by results of these simulations. 5.1 SUMMARY OF THE INPUT DATA The input data for the simulations can be divided into three groups: • characteristics of the bridge; • characteristics of the vehicle; • characteristics of time integration process. The Lillooet River Bridge was modeled as a system consisting of two independent, simply supported spans. As in the dynamic testing of this bridge, the vehicle approached the bridge from the South, that is, traversed first the "short", 22.1 m span. After the short span, the vehicle ran over the "long", 40.1 m span. The short span of the bridge was modeled by eight beam elements, the long one by sixteen elements. Only characteristics of the beam cross section were used for evaluation of the stiffness matrix. The mass matrix of the bridge was generated using characteristics of the beams, the bridge deck and the railings. The equivalent first mode damping ratio, used for evaluation of the damping matrix, was taken as 1.8%. The characteristics of the roughness of the bridge deck were of the type 2 described in Chapter 4. The numerical values of the input 102 data for the bridge vehicle are given in the example of the input file for the program B V I in Appendix A. A numerical model of a fully loaded logging truck was used for simulations. This was the model called T5441 that represents the truck with a total weight of 441 k N and 5 axles. This model had seven degrees of freedom. The characteristics of this vehicle are given in the example of the input file for the program B V I in Appendix A. The constant average acceleration method of integration was used for the simulations. The time step varied from 0.05 s for the truck's speed of 10 km/h to 0.001 s for the speed of 100 km/h. The speeds of the truck were varied from 10 km/h to 100 km/h, in steps of 10 km/h. The simulations began with the truck at a sufficient distance from the beginning of the bridge in order to "generate" initial conditions for the truck's degrees of freedom when entering the bridge. The roughness process on the approach had the same characteristics as that on the bridge. Results in form of displacements, velocities and accelerations at the midspans of both the short and the long spans were computed and stored in digital form. The total number of simulations with this model was 100, using ten sequences of random phase shifts for the roughness function for each speed of the truck. Simulations took from three to seven minutes, depending on the speed of the truck, on a PC-486/66 Mhz. 5.2 RESULTS OF SIMULATIONS WITH THE VEHICLE T5441 The response of the Lillooet River Bridge under the moving vehicle was determined by numerical simulations. The examples of the results are shown in Figures 5.1 to 5.8. These 103 figures show the response of the bridge at midspan of the short (Figures 5.1, 5.3, 5.5, and 5.7) and at midspan of the long spans (Figures 5.2, 5.4, 5.6, and 5.8). The results are shown for truck speeds of 10, 40, 70 and 100 km/h. Each figure consists of five parts; time is plotted on the horizontal axes and the scale is the same for all the parts. Parts "a" to "d" show the response at the point in terms of its acceleration, dynamic component of displacement, static component of displacement and total displacement. Part "e" shows the position of the truck on the bridge during the simulation. The notation of significant points along the bridge corresponds to the Figure 4.22b. The top straight line in part "e" represents the position of the front axle of the truck during the simulation. The remaining four straight lines represent positions of the remaining four axles of the truck. It can be observed from the Figures 5.1 to 5.8, that the response consists of two parts: a forced vibration and a free vibration part. The forced vibration part is characterized by the vehicle inside the considered span, while the free vibration part occurs when the vehicle has passed. For example, in Figure 5.5, the forced vibration part is from time about 1 s to 3 s, and the free vibration part from 3 s till the end of the time history. The significant components of the forced vibration parts have various frequencies. As these frequencies are, within one simulation, about the same for the response of the short and the long spans, it can be concluded that the forced vibration response is governed by the character of the excitation, that is, the character of the roughness of the bridge deck. This conclusion is also supported by the results of frequency domain analysis: • The free vibration response has always one significant harmonic component with a frequency corresponding to the first natural flexural mode of the span. It is about 5.50 Hz for the short and 2.45 Hz for the long span. 104 • The significant frequency of the forced vibration part depends on the speed of the truck and on the wave length of the significant bumps of the roughness profile of the bridge deck. This frequency is the same for the short and the long spans. It can also be seen that the maximum dynamic displacement may not occur at the same time as the maximum static displacement (see Figures 5.1, 5.2, 5.4, and 5.6). However, the influence of this difference on the dynamic amplification factor is not significant, as it was discussed in section 3.4. The dynamic amplification factors (DAF) were calculated at the end of each simulation. Ten values of D A F were calculated for each truck speed as ten sets of the random phase shifts for the roughness process were used. Resulting dynamic amplification factors were obtained as the corresponding mean values. The dynamic amplification factors at midspan of the short span are presented in Table 5.1, and those for the long span in Table 5.2. The ranges of the dynamic amplification factors are shown for the short span in Figure 5.9a and for the long span in Figure 5.10a. The mean dynamic amplification factors from the simulations were compared to those proposed by the 1996 CHBDC code. This code prescribes the dynamic load allowance coefficient for the case when three or more axles of the design truck are on the bridge, as: D L A = 0.25x0.7 = 0.175 (5.1) This value is the same for both spans of the Lillooet River Bridge because more than three axles of the design truck are present in the span in both cases. The dynamic load allowance DLA=0.175 is, according to Equation 1.2, equivalent to the dynamic amplification factor DAF= 1.175. The comparison of the mean dynamic amplification factors from the simulations 105 with the code value is shown in Figure 5.9b for the short span and in Figure 5.10b for the long span. It can be observed from Figure 5.9b, that the maximum value of D A F for the short span was obtained for the truck speed of 50 km/h when DAF=1.179. This value is slightly higher than 1.175, which is the value of D A F according to the proposed 1996 C H B D C code. The remaining values of D A F are smaller than the value prescribed by the code. The smallest values were obtained for speeds 10 and 30 km/h when D=1.07. It can also be observed from Figure 5.10b, that the maximum value of D A F for the long span was obtained for the truck speed of 80 km/h when DAF= 1.169. This is smaller than the code value of 1.175. The smallest value of D A F was obtained for a speed of 20 km/h when DAF=1.04. It can be concluded that the proposed Code value represents an upper bound for the average value of D A F from the numerical analysis for both the short and the long spans, for all the speeds considered. It can also be said that the Code value is not an overestimation of the actual response for this bridge. The recommended Code dynamic load allowance coefficient may then be said to be adequate for this type of wood bridges, given that the bridge studied had two spans with different lengths and it can be considered as representative of logging wooden bridges. 106 CM < co CO 1.00 0.00 -1.00 0.003 0.000 -0.003 0.02 a) Acceleration: a (m/sA2) I ' I r b) Dynamic component ofdisplacement: d (m) c) Static component of displacement: s (m) 0.02 E 0.01 - \ d) Total displacement: t (m) 0.00 E, Q. e) Position of the axles from the beginning of the bridge: p (m) Midspan of the short span Support 1 • p=0 ml i | i | i | , 15 20 25 30 time (s) Figure 5.1. Numerical simulations - example of the results, position of the truck during simulation; Location: midspan of the short span,vehicle: T5441, speed: 10 km/h. 107 a) Acceleration: a (m/sA2) c) Static component of displacement: s (m) i 1 i 1 r d) Total displacement: t (m) 0 5 10 15 20 25 30 time (s) e) Position of the axles from the beginning of the bridge: p (m) 0 5 10 15 20 25 30 time (s) Figure 5.2. Numerical simulations - example of the results, position of the truck during simulation; Location: midspan of the long span,vehicle: T5441, speed: 10 km/h. 108 2.0 CM < (A CO 0.002 & 0.000 -0.002 0.02 0.02 E 0.01 0.00 E, CL a) Acceleration: a (m/sA2) b) Dynamic component of displacement: d (m) c) Static component of displacement: s (m) d) Total displacement: t (m) time (s) e) Position of the axles from the beginning of the bridge: p (m) Midspan of the short span SupporUL p=Q_ml 10 0 2 4 6 8 time (s) Figure 5.3. Numerical simulations - example of the results, position of the truck during simulation; Location: midspan of the short span,vehicle: T5441, speed: 40 km/h. 109 a) Acceleration: a (m/sA2) b) Dynamic component of displacement: d (m) 0.000 c) Static component of displacement: s (m) 0.04 -0.02 -0.00 -d) Total displacement: t (m) 0.04 -0.02 -0.00 -0 2 4 6 8 time (s) e) Position of the axles from the beginning of the bridge: p (m) 0 2 4 6 8 10 time (s) Figure 5.4. Numerical simulations - example of the results, position of the truck during simulation; Location: midspan of the long span,vehicle: T5441, speed: 40 km/h. 110 0.02 -0.01 -0.00 d) Total displacement: t (m) I 4 time (s) e) Position of the axles from the beginning of the bridge: p (m) T 4 time (s) Figure 5.5. Numerical simulations - example of the results, position of the truck during simulation; Location: midspan of the short span,vehicle: T5441, speed: 70 km/h. 11 CN < co E co 8.00 0.00 -8.00 a) Acceleration: a (m/sA2) 0.012 0.000 -0.012 b) Dynamic component of displacement: d (m) c) Static component of displacement: s (m) d) Total displacement: t (m) Q . e) Position of the axles from the beginning of the bridge: p (m) 4 time (s) Figure 5.6. Numerical simulations - example of the results, position of the truck during simulation; Location: midspan of the long span,vehicle: T5441, speed: 70 km/h. 112 a) Acceleration: a (m/sA2) b) Dynamic component of displacement: d (m) 0 2 4 Q Q2 c) Static component of displacement: s (m) d) Total displacement: t (m) e) Position of the axles from the beginning of the bridge: p (m) Support 3: p=62. Support 2: p=22.1 m Midspan of the short span Support 1: p=0m Figure 4 6 8 time (s) 5.7. Numerical simulations - example of the results, position of the truck during simulation; Location: midspan of the short span,vehicle: T5441, speed: 100 km/h. 113 a) Acceleration: a (m/sA2) b) Dynamic component of displacement: d (m) c) Static component of displacement: s (m) d) Total displacement: t (m) e) Position of the axles from the beginning of the bridge: p (m) T 4 time (s) Figure 5.8. Numerical simulations - example of the results, position of the truck during simulation; Location: midspan of the long span,vehicle: T5441, speed: 100 km/h. 11 SIT TI do" c 3 11 8,1. il U 8 g 8 $ I § 2 8 P Q -» p o o p O g - Q t O ^ 8 6 2 8 1 8 8 g - ' M O O M O - ' - ' O cn cn -» S- 0 0 - * 0 - * Q . O - * 0 c n ^ i K o ) i 5 i * - » - ' 0 > <* 8 K B ft 8 ^ 8 s H i S O ->• o _» O) CO • M ( D M U O O p p p p p p p p p 8 8 IT 01 0) 3" 0) 3" CO "0 3" CO 3" CO s cn •o 3" CO 8 in oo "0 3" 0) CO CO 3" co CD 3 a co I sr fl>' Q. » 3 "H. I s t? a . e a s s-§ O 911 en c 3 o T IS I. o 3 f S 00 M _ •NI ro oi oo M - » 0 0 <J yi s J>. co - -: ^ o o ^ u c f l c 5 c o - - - 4 r o o o W - * 0 0 u l W - > J - ' C J > - * 0 0 N M W - ' - ' O Q O O O ^ ^ ( 3 i o - N i o o a i N 3 - » t n M - » - ' ( D O 1 0 ) U I O O l W o g - ' - ' o o i - . o o Q - g o o o i - ' - N i - v i 5 2 o > - ' S 5 W C D - N | - v l G 0 - » O , a > I O - & . i i o M - ' - ' - ' O S O - ' ( » N o i 4 k - ' - » f t S - ' Q ^ - » - » - k - > . - » 0 0 0 - ' co ro CO CO - » - » - » - » •_. o o o Q - t » . c o - j . i » . c o r j c j i c o r o - v i o - J - o o o c n ^ c n g n c D f v j c o - ' - ' - c o ^ i t J i . & - ' C o - N i - ' U i - » I I I CD co(DuiSia->ocs(0 o o o o o o o o o o 2 2 S 2 I 8 2 B 2 'H 00 o s CO (0 3" 0) CO ro cn 3" CO V) CD co TJ 3" 0) •n 3" 0) cn TJ IT a> s CA CD 13 3" C0 8 CO o S i C Q> CD 3 a co 2.1 D> Q . <?• Si 3 a . 8-i Bi 6 § S 00 RECOMMENDATIONS FOR FUTURE RESEARCH While this work explored many aspects of the dynamics of the system logging bridge—logging truck, there are several additional aspects worth studying. First, a model of a logging truck with a total weight comparable to that of the design truck from the 1996 C H B D C code should be developed for numerical simulations. The weight of the frilly loaded truck used for numerical analysis in this work was 441 k N while the design truck weights 740 kN. Lack of data on such a truck did not permit its inclusion in the present work. Second, a three-dimensional numerical model of the bridge and the vehicle could be developed. This would enable to perform a more detailed dynamic analysis in order to verify code values of dynamic load allowance for deck joints, load transfer, etc. Third, experimental modal characteristics of the bridge could be obtained using another method, e.g. forced vibration testing using the hammer or the shaker. Although the hammer testing was also conducted on the Lillooet River Bridge, it was very limited and did not offer enough data to allow reliable determination of the modal characteristics. Fourth, displacement transducers should be used for measuring the bridge response during the forced vibration testing. This would permit to calculate the experimental dynamic amplification factor directly from Equation 1.1 without integration of acceleration records. Fifth, the roughness of the bridge deck could be measured precisely and the measured values could be used as input data in numerical simulations. Characteristics of the truck used for dynamic testing could be independently measured, because the data supplied by the manufacturer of the truck may not be valid for a truck that has been in service for several years. 117 CONCLUSIONS A number of conclusions can be drawn from the results of this investigation. These have been grouped into four sections; namely Ambient Vibrations Testing, Forced Vibrations Testing, Numerical Analysis, and Comparison of Numerical Analysis with Code Proposals. AMBIENT VIBRATIONS TESTING Determining natural frequencies and natural modes of a bridge using ambient vibrations and signal processing functions is a very practical and useful method. Modal characteristics from experiments can be used for verification of a numerical model. FORCED VIBRATIONS TESTING Determining dynamic amplification factors of a bridge using forced vibrations testing is a useful method. Measurements with the same truck speed should be repeated several times in order to eliminate distorted signals from results, i f they occur. Damping of a bridge can be easily obtained from the free vibration response, during the forced vibration testing, after the passage of the testing truck. NUMERICAL ANALYSIS Direct time integration of equations of the motion is a very useful method for solving forced vibration problems if the mass, stiffness or damping matrices are not constant during the analysis. Modal characteristics of the system logging bridge ~ logging truck change significantly with position of the truck on the bridge. The roughness of the bridge deck has a very significant influence on the response of the logging wooden bridge under the moving truck. It is possible to successfully simulate the response of the bridge due to the moving truck 118 by the linear model of the bridge implemented in the program B V I . The program yields good results of the numerical simulations if the characteristics of the roughness profile are known. COMPARISON OF NUMERICAL ANALYSIS RESULTS WITH PROPOSED RECOMMENDATIONS OF 1996 CHBDC The numerical analysis yielded dynamic amplification factors (DAF) for both the short and the long spans of the Lillooet River Bridge. The maximum average values, over all speeds considered, are about the same as those proposed in the 1996 issue of the C H B D C code. The code proposals imply that the dynamic load allowance for the design truck should be 0.175 if three or more axles are present and if the bridge is made from wood. The values of D A F from the numerical analysis relate to the logging truck of total weight of 441 kN. The code values for D A F (in terms of dynamic load allowance) relate to a design truck of total weight of 740 kN. Although the weights of the trucks are different, the values of D A F were used for the comparison, because the characteristics of the 740 k N truck were not available. The conclusion from the comparison is that the value of the dynamic load allowance for the global effect of the live load, on logging bridges with glulam girders, is appropriate. If a heavier truck is used in the numerical analysis the D A F values would be smaller than those of the 441 k N truck. This indicates, that the code D A F is not overestimated. 119 NOMENCLATURE a,a vector of nodal displacements or accelerations, respectively « 0 ' a l constants A area of the cross section, point on the beam amplitude of a component of the roughness function c constant of the dashpot C damping matrix ce damping matrix of the beam element d,d,d global vectors of the displacement, velocity and acceleration e eccentricity of the axle of the vehicle from the C.G. of the body of the vehicle E Young's modulus of elasticity G shear modulus h(x) roughness function I moment of inertia IT moment of inertia of the body of the vehicle hJi inertia forces k shear constant of cross section, spring constant K stiffness matrix Ke stiffness matrix of the beam element I length of the beam element L vector of shape functions, kinetic energy mT mass of the body of the vehicle M mass matrix, vector of shape functions Me mass matrix of the beam element P(R) probability of exceeding a given value R Q vector with zero or unit values R right hand side, given value of a barrier power spectral density at circular frequency co 120 T natural period u,uA axial displacement U strain energy w vertical displacement Wi virtual work x, y, z coordinates of position on the bridge s axial strain y shear strain c; nondimensional coordinate, modal damping ratio 4) phase shift p volume density a standard deviation, axial stress co circular frequency 0 shear rotation y/ rotation of the cross-section 121 ABBREVIATIONS A A S H O American Association of State Highway Officials; today: A A S H T O B V I Bridge-Vehicle Interaction (phenomenon or name of program) BVTDAF Bridge-Vehicle Interaction-Dynamic Amplification Factors (name of program) B V I E Bridge-Vehicle Interaction - Evaluation of results (name of program) BVTNV Bridge-Vehicle Interaction - Natural Vibrations (name of program) B V I R Bridge-Vehicle Interaction Reduced (name of program) B V I R E Bridge-Vehicle Interaction Reduced - Evaluation of results (name of program) C . G Center of Gravity CHBDC Canada Highway Bridge Design Code CSMIP California Strong Motion Instrumentation Program D A F Dynamic Amplification Factor D L A Dynamic Load Allowance d.o.f. Degree Of Freedom E M P A Swiss Federal Laboratories for Materials Testing and Research E W East-West (direction) F B A Force-B alance-Accelerometer HBES Hybrid Bridge Evaluation System HRB Highway Research Board (USA); today: TRB km/h kilometers per hour L5441 Truck - 5 axles, weight 441 k N (a fully loaded logging truck) NRC National Research Council of Canada NS North-South (direction) OHBDC Ontario Highway Bridge Design Code T3162 Truck - 3 axles, weight 162 k N (the empty logging truck used for the testing) U B C University of British Columbia UD Up-Down (direction) 122 REFERENCES A A S H O Road Test, History and Description of Project (1961). H R B Special Report 61A, NAS-NRC Publication 816. Bata, M . , Plachy, V . (1977) Some Notes on the Dynamic Tests of Prestressed Concrete Bridges in situ. Proc. R I L E M Symp. "Testing in situ", Vol . 1. Budapest, Hungary. Bata, M . , Plachy, V . and Travnicek, F. (1987) Dynamics of Structures (in Czech). SNTL Prague, Czech Republic. Billing, J .R, Agarwal, A .C . (1990) The Art and Science of Dynamic Testing of Highway Bridges. Developments in Short and Medium Span Bridge Engineering 1 (1990), CSCE, ed. by B. Bakht, R. Dorton, L . Jaeger. Bily, V . (1991) Analysis of System Consisting of Bridge Structure and Moving Vehicle by the Component Element Method. Proc. 2nd Int. Conf. on Traffic Effects on Structures and Environment, Nizke Tatry, Slovakia. Bracewell, R .N. (1986) The Fourier Transform and Its Applications (2nd ed. rev.) New York, McGraw - Hill . CSMIP (1985) Standard Tape Format for CSMD? Strong-Motion Data Tapes, Report OSMS 85-03. California Department of Conservation Division of Mines and Geology, Office of Strong Motion Studies, Sacramento, California. Canadian Standard Association (1988) Design of Highway Bridges CAN\CSA-S6-88. Ottawa, Canada. Canadian Standard Association (1995, February) Draft of the Canadian Highway Bridge Design Code. Ottawa, Canada. Cantieni, R. (1992) Dynamic Behavior of Highway Bridges Under the Passage of Heavy Vehicles. E M P A - Swiss Federal Laboratories for Materials Testing and Research, Report No. 220, Dubendorf, Switzerland. Celesco Industries Inc. (1988) Specifications for the Model PT101 Position/Displacement Transducer. Canoga Park, California. Clough, R.W. and Penzien, J. (1993) Dynamics of Structures, 2nd Edition. McGraw - Hill , New York, USA. Cook, R.D., Malkus, D.S. and Plesha, M E . (1989) Concepts and Applications of Finite Element Analysis. 3rd Edition. John Wiley & Sons, New York, USA. Csagoly, P.F., Campbell, T.I., Agarwal, A .C . (1972) Bridge Vibration Study. Ontario M T C Report R R 181. 123 Drosner, S., (1989) Beitrag zur Berechnung der dynamischen Beanspruchungen von Briicken unter Verkehrslast. Diss., R W T H Aachen, Fakultat fur Bauingenieur und Vermessungswesen. Eberhardt, A.C. , Walker, W.H. (1972) A Finite Element Approach to the Dynamic Analysis of Continuous Highway Bridges. University of Illinois, Civ. Eng. Studies, Struct. Research Series 394. Felber, A.J . (1993) Development Of A Hybrid Bridge Evaluation System. Ph.D. thesis, University of British Columbia, Canada. Flugge, W. (1962) Handbook of Engineering Mechanics. McGraw - Hill , New York, USA. Fryba, L . (1972) Vibration of Solids and Structures Under Moving Loads. Noordhoff International Publishing, Groningen. Green, M.F. (1990) The Dynamic Response of Short-Span Highway Bridges to Heavy Loads. Ph.D. Thesis, University of Cambridge, United Kingdom. Green, M.F. , Cebon, D. and Cole D.J. (1995) Effects of Vehicle Suspension Design on Dynamics of Highway Bridges. Journal of Structural Engineering, Vol . 121, No. 2, Feb. 1995, p. 272-282. Horyna, T. (1989) Dynamic Analysis of Highway Bridge in Teplice (in Czech). Dipl. Engineer thesis. Czech Technical University in Prague, Czech republic. Humar, J.L. (1990) Dynamics of Structures. Prentice Hall, New Jersey, USA. Humar, J.L. and Kashif, A . H . (1995) Dynamic Response Analysis of Slab-Type Bridges. Journal of Structural Engineering, Vol . 121, No. 1, Jan. 1995, p. 48-62. Hwang, E.S., Nowak, A.S. (1989) Dynamic Analysis of Girder Bridges. TRR 1223. Kajikawa, Y . , Honda, H . (1987) Studies of Bridge Vibration Over Twenty Years. Japan Research Committee on Bridge Vibration. Kinemetrics (1991) Operation Instructions For FBA-11 Force Balanced Accelerometer. Pasadena, California. Microsoft Corporation. (1991) Microsoft MS-DOS User's Guide and Reference - Operating System 5.0. Microsoft Corporation (1993) Microsoft Fortran Power Station Optimizing Compiler, version l.Of. User's Guide and Reference. Ministry of Transportation of Ontario, Quality and Standard Division (1991) Ontario Highway Bridge Design Code, Ontario. 124 Mitchel, C.G. (1987) The Effect of the Design of Goods Vehicle Suspensions on Loads on Roads and Bridges. TRRL Dept. of Transport, Res. Report 115. Mulcahy, N X . , Pulmano, V . A . , Traill-Nash, R.W. (1983) Vehicle Properties for Bridge Loading Studies. IABSEProc. P-65/1983. Newland, D.E. (1975) An Introduction to Random Vibrations and Spectral Analysis. Longman Group Ltd., New York, USA. Nicol-Smith, C A . (1988) Why Glulam Isn't Used in Bridges Anymore - British Columbian Experience with Logging Bridges. International Conference on Timber Engineering, Seattle, WA, 1988. Pirner, M . et al (1989) Dynamics of Structures - Technical Guide No. 33 (in Czech). SNTL, Prague, Czech Republic. Pkware, Inc. (1983) Eigensystem Subroutine Package (Eispack). Glendale, Wisconsin. Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (1992) Numerical Recipes in Fortran. Cambridge University Press, New York, USA. Rice, S.O. (1944) Mathematical Analysis of Random Noise. Bell System Technical Journal, Vol . 23, 1944, pp. 282-332 and Vol . 24, 1944, pp. 46-156. Reprinted in Selected Papers in Noise and Stochastic Processes, ed. N . Wax, Dover, New York, 1954. 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. Weaver, W., Jr. and Johnston, P.R. (1987) Structural Dynamics by Finite Elements. Prentice Hall, New Jersey, USA. 125 A P P E N D I X A BRIDGE-VEHICLE INTERACTION SOFTWARE Appendix A gives a summary of the programs developed for purposes of this project. Four main codes have been written: Bridge-Vehicle Interaction (BVI), Bridge-Vehicle Interaction Evaluation (BVIE), Bridge-Vehicle Interaction - Dynamic Amplification Factors (BVTDAF) and Bridge-Vehicle Interaction - Natural Vibrations (BVTNV). This part of the thesis gives operating instructions for these programs and shows examples of the input files. Structures of the output files are described if necessary. A.1 OVERVIEW OF THE PROGRAMS Performing simulations of bridge-vehicle interaction is a computationally demanding task. Four main computer programs have been developed for purposes of this project: • Bridge-Vehicle Interaction (BVI). This program, given a bridge, a vehicle and its speed, performs the simulation of the passing of the vehicle over the bridge. It calculates the response of the bridge due to the vehicle moving with a constant speed. The time histories of the static component of the displacement, the total displacement, the velocity and the acceleration of selected degrees of freedom are the output of the program. This program has modification called Bridge-Vehicle Interaction Reduced (BVIR). The program BVTR performs the same analysis as BVI . The only difference between these programs is that the output file of BVTR contains only the displacement time histories at selected degrees of 126 freedom. The size of the BV1K. output file is about 25% of the size of the B V I output file from the same analysis. • Bridge-Vehicle Interaction - Evaluation (BVJE). This program permits to evaluate results of simulations obtained by B V I . The user can check the results on the screen and then generate the time histories of the output quantities. The program has a modification called Bridge-Vehicle Interaction Reduced - Evaluation (BVIRE). The program B V I R E is used for evaluation of the results from the program B VLR. • Bridge-Vehicle Interaction - Dynamic Amplification Factors (BVTDAF). The programs B V I and BVTR can work in batch mode and perform a series of the simulations. It is a common task because the input data for B V I or BVLR can vary in some parameters. It can happen if the user wants to study the influence of the changing parameters on the response of the bridge. It brings a problem of solving a series of simulations. Then, the user needs to evaluate the results of this series as a single task. The program B VTDAF works with output files of several simulations performed by B V I or BVTR. The output of BVLDAF is a table of the dynamic amplification factors, defined in Chapter 1, for the series of simulations. • Bridge-Vehicle Interaction - Natural Vibrations (BVTNV). This program solves the natural vibrations of the bridge itself or the vehicle on rigid foundation or the coupled system bridge-vehicle with the vehicle at a given position on the bridge. The program also permits the user to watch selected natural mode and the corresponding frequency of the coupled system bridge-vehicle when the vehicle is moving over the bridge. The above introduced codes are written in Fortran-77 and compiled with Microsoft Fortran Power Station Compiler - Version 1.0F. This is a 32 bit compiler and it works in protected 127 mode of the C P U of an I B M - PC compatible computer. Therefore, a processor 80386 or higher is necessary to run the programs. These programs also need the file D O S X M S F . E X E (Microsoft Fortran Power Station memory manager) be copied on the hard drive in the directory with a path to it. Finally, the programs B V I E and B V I R E require at least V G A graphics and the font files COURB.FON and TMSRB.FON from Microsoft Fortran Power Station library be copied on the hard drive. The required amount of R A M for running the programs is 4 M B . The recommended C P U for running the simulations (BVI or BVTR) is 80486 - 66 M H z processor or higher, based on the author's experience with the software. The executable and the source codes of the programs together with examples of the input and output files are available at the Computing Lab, Civil Engineering, U B C . A.2 PROGRAM BVI - OPERATING INSTRUCTIONS A.2.1 R U N N I N G T H E P R O G R A M The B V I program can be run by typing the following at the DOS prompt: bvi filename where filename is the name of the input file without extension. The extension has to be '.dat'. and the input file has to be in the current directory. The program B V I has to be in the directory with a path to it or in current directory. If the input file is found and successfully read, the following message appears on the screen: BridgeA/ehicle Interaction, reading data from: filename.dat time step: istep from: nsteps time: time distance: dis 128 where: istep is the number of the current time step in the simulation (changing during the simulation); nsteps is the total number of the time steps in the simulation; time is the real time since the beginning of the simulation; dis is the distance of the center of gravity of the vehicle from the beginning of the bridge. This message shows the user how the simulation is proceeding. The simulation is successfully completed i f istep reached the value of nsteps and no other messages appeared on the screen. The program permits the user to model the bridge from up to 20 spans, 20 types of the cross-section and 10 types of material. The vehicle can have up to 10 axles. The maximum number of degrees in the current version is 500. This includes the degrees of freedom of the bridge (three per node) and the degrees of freedom of the vehicle (two plus the number of the axles). A.2.2 D E S C R I P T I O N A N D E X A M P L E O F T H E INPUT F I L E The input file for B V I program consists of three parts: information about the bridge, information about the vehicle and information about the simulation. The input data consists of both real and integer valued numbers and, the units of the input data are also important. Therefore, the required format ('R' for real and T for integer) and the required unit in parentheses follow the name of the input quantity in the following. If the input value is dimensionless, that is, e.g., the number of the spans or the scaling factor, only the format is given in the parentheses. Bridge part: 129 one line with N M A T , the number of materials (I) used to model the bridge; N M A T lines, i.e., one line for each material with the Young's modulus of elasticity (R, Pa), the shear modulus (R, Pa) and the mass per unit volume (R, kg/m3); one line with N X S E C , the number of the cross-sections (I); N X S E C lines, i.e., one line for each cross-section with: the moment of inertia of the cross-section for evaluation of the stiffness matrix (R, m4), the area of the cross-section for evaluation of the stiffness matrix (R, m2), the shear constant of the cross-section (R), the moment of inertia of the cross-section for evaluation of the mass matrix (R, m4) and the area of the cross-section for evaluation of the mass matrix (R, m2); one line with N S P A N , the number of the spans (I) of the bridge; N S P A N times a set of two or more lines (one sequence for each span): The first line of this set with the length of the span (R, m) and the number of the elements (I) in the span. The next one or more lines contain information about different parts with the same characteristics of the span. Each of these lines begins with the numbers of the first and the last element of the part of the span (I); the numbering of the elements is local for each span, starting from 1. The next three values in the same line are: the length of the element (R, m), the number of material (I) and the number of the cross-section (I), one line for input of boundary conditions. The number of the values in this line must be equal to the number of the spans increased by 1, that is the number of the supports. The acceptable codes (I) for identification of the boundary conditions are numbers 1 and 2. The first and the last supports have to have code 1, which indicates a hinge support. The intermediate supports can have codes 1 (for connected spans at the support - continuous 130 beam) or 2 (for independent rotations of the neighboring spans at the support - simply supported beam); one line with the modal damping ratio (R) of the bridge; one line with two values of the circular frequencies (rad/s) for evaluation of the damping matrix (see explanation in section 2.3 for details); one line with two values: the maximum value of the bridge deck roughness function (R, m) and the probability of its exceeding (R); one line with NFREQ, the number of the frequency intervals (I), where the roughness of the bridge is generated; NFREQ lines, i.e., one line for each interval with two frequency values (R, Hz): the beginning and the end of the frequency interval. Vehicle part: one line with the values of the mass of the body of the vehicle (R, kg) and the moment of inertia (R, m4) of the vehicle's body about the horizontal axis of the body, perpendicular to the direction of moving of the vehicle; one line with N A X L E S , the number of the axles of the vehicle (I); N A X L E S lines, i.e., one line for each axle (first for the front axle) with: the mass of the axle (R, kg), the distance (R, m) to the axle before (or, in the case of the front axle, the distance from the center of the gravity of the body to the first axle), the spring constant (R, N/m) of the "upper spring", the spring constant (R, N/m) of the "lower spring" and the dashpot constant (R, Ns/m). The notation, used here, corresponds to Figure 2.4. 131 Simulation part: • one line with one value (I) to select the method of the numerical integration; the code is: 1 for the linear acceleration method and 2 for the constant average acceleration method; • one line with the speed of the vehicle (R, km/h); • one line with the value (R, s) of the time from the beginning of the simulation to the time, when the front axle of the vehicle is entering the bridge; • one line with the value (R, s) the time from the time, when the rear axle is leaving the bridge to the end of the simulation; • one line with the time step (R, s) for the numerical integration; • one line with the scaling factor (R) for values of the roughness function (if set to zero, the subroutines associated with the roughness are not executed); • one line with the scaling factor (R) for the weight of the vehicle (the gravity acceleration considered in the program is g=9.81 m/s2); • one line with the number (I) of the time steps, after which the output information is stored; • one line with N D E G , the number of the degrees of freedom (I) for the output, that is, for how many degrees of freedom the user wants to get results; • one line with N D E G numbers, the numbers of the degrees of freedom for the output (the degrees of freedom are numbered according to Equation 2.39). The example of the input file, which follows, is the input file for the simulation of the response of the Lillooet River Bridge under the vehicle T5441 moving by the speed of 30 km/h. The complementary explanation of the input values is preceded by a '! ' character in each line. 132 A.2.3 STRUCTURE OF THE OUTPUT FILE The output file contains the information about the simulation in its beginning (lines 1 - 11). This is followed by three lines with the values used by program BVTE (lines 12 - 14, see the explanations after exclamation marks). Finally, the output file contains for each time step the line with the number of the degree of freedom, the static displacement, the total displacement, the velocity and the acceleration. The example of the part of the output file follows: Bridge - Vehicle Simulation Input file: 10130 .dat Constant acceleration method Roughness generated Weight of the truck considered Velocity of the truck: 30.00 km/h Time step: .005000 s Number of time steps: 996 Time for initial conditions: 1.00 s Time for free vibration: .50 s Printout after each 10 steps 996 .50000E-01 47 ! No. of time steps, time step for output total No. of variables 2 ! No. of variables on output 7 28 ! numbers of variables available for output 7 .0000D+00 .0000D+00 .0000D+00 .0000D+00 28 .0000D+00 .0000D+00 .0000D+00 .0000D+00 7 .2084D-04 -.1305D-06 -.5219D-04 -.2088D-01 28 .0000D+00 .0000D+00 .0000D+00 .0000D+00 7 .4170D-04 -.6995D-06 -.1754D-03 -.2842D-01 28 .0000D+00 .0000D+00 0000D+00 .0000D+00 7 .6256D-04 -.1263D-05 -.4976D-04 .7869D-01 28 .0000D+00 .0000D+00 .0000D+00 .0000D+00 7 .8344D-04 -.8368D-07 .5213D-03 .1497D+00 28 .0000D+00 .0000D+00 .0000D+00 .0000D+00 134 A.3 PROGRAM BVIE - OPERATING INSTRUCTIONS A.3.1 RUNNING THE PROGRAM The program B V I E can be run by typing the following at the DOS prompt: bvie filename numvar where filename is the name of input file (which is the output file of the program BVI) without extension that has to be '.out'. The input file has to be in the current directory. The program B V I E has to be in the directory with a path to it or in the current directory. The argument numvar is the number of the variable, the results of which are to be evaluated. If the input file is found and successfully read, the following message appears on the screen: BVI - Evaluation of Results, reading data from: filename.oui If the argument numvar starts with an '* ' which has to be followed by one character V (for the static displacement), 'd ' (for the total displacement), V (for the velocity) or 'a' (for the acceleration), an output file with extension '.aaa' (Felber, 1993, p.247-248) is generated after graphics of the program BVJJE is terminated. This file contains the time history of the desired quantity for variable numvar. This file can be used as an input file for the programs U L T R A (Felber, 1993), T H V (Appendix E) and FRF (Appendix D). A.3.2 DESCRIPTION OF THE GRAPHIC REPRESENTATION OF THE RESULTS A graphic display appears after the input data have been successfully read. The screen shows time histories of the static displacement, the total displacement, the velocity and the acceleration at the point of interest. The dynamic amplification factor from the simulation is 135 given in INFO selection of the menu. The graphic part of the program is provided with the detailed on-screen help information and, therefore, no further explanation will be given here. A.4 PROGRAM BVIDAF - OPERATING INSTRUCTIONS A.4.1 RUNNING THE PROGRAM The program B V I D A F can be run by typing the following at the DOS prompt: bvidaf filename ngroup nveloc dvelo velmin Only the argument filename is compulsory. The other arguments are optional but order sensitive. Their default values are: ngroup=6; nveloc=6; dvelo=10; velmin=10. The explanation of these arguments is following: • ngroup is the number of the groups of the simulations; • nveloc is the number of the velocities of the vehicle used for the simulations; • dvelo is the increment of the velocity of the vehicle (in km/h); velmin is the minimum velocity of the vehicle used in the simulations. A.4.2 DESCRIPTION OF THE OUTPUT FILE Output file of the program B V I D A F is a table of the dynamic amplification factors from the series of the simulations done by the programs B V I or BVIR. Each group of simulations is represented by one column of the factors and each velocity of the vehicle is represented by one line of the factors. The example of the output file of the program B V I D A F follows (default values of the optional parameters were used): 136 speed gro.1 gro.2 gro. 3 gro.4 gro. 5 gro. 6 10 ,1.204 1.373 1.244 1.311 1.273 1.244 20 1.189 1.359 1.194 1.356 1.241 1.222 30 1.106 1.165 1.116 1.288 1.152 1.187 40 1.020 1.060 1.031 1.054 1.033 1.031 50 1.027 1.049 1.038 1.029 1.033 1.017 60 1.099 1.059 1.068 1.103 1.097 1.066 A.4 PROGRAM BVINV - OPERATING INSTRUCTIONS The program B V I N V can be run by typing the following at the DOS prompt: bvinv filename where filename is the name of the input file without extension that has to be '.dat'. The input file has to be in the current directory and it has the same structure as the input file for the program B V I . The program B V I N V has to be in the directory with a path to it or in the current directory. If the input file is found and successfully read, the following menu appears on the screen: Bridge - Vehicle Interaction: Natural vibration Input file: filename.dat Main menu, enter your selection: 1...solve eigenproblem of the bridge 2...solve eigenproblem of the vehicle on rigid foundation 3...solve eigenproblem of coupled system bridge-vehicle 4...watch chosen frequency with moving vehicle 5...terminate program The user can obtain natural frequencies and modes for the selected structures. Results are stored in the output file that has the same name as input file and extension '.eig'. The results are stored together with explanatory comments and, therefore, no further explanation of the structure of the output file will be given here. 137 t • . r-A P P E N D I X C DETAILS OF THE DYNAMIC TESTS This appendix provides information about dynamic tests conducted on the Lillooet River Bridge. Section §C. 1 contains the details of the ambient vibration testing. This is followed by details of the forced vibration testing (§C2). C.l DETAILS OF THE AMBIENT VIBRATION TESTING The bridge was instrumented with FBA-11 sensors during ambient vibration testing. The locations of sensor and reference points are shown in Figure C . l . Since the Lillooet River Bridge has three independent simply supported spans, three reference points were used during the ambient vibration testing. Eighteen measurement setups were done during the testing. Table C. 1 lists each setup along with the filename containing the ambient vibration data (in '.bbb' format), conditioning details (attenuation and filtering) and the comments on measured signals. The characteristics that were constant for all the setups are given at the bottom of Table C . l . Channels 1 to 6 were used for the measurements as the moving channels while channels 7 and 8 were used as the reference channels. Channel 7 was used as the reference channel in the vertical direction while channel 8 was the reference in the horizontal direction. Examples of the measured signals are shown in Figures C.2 and C.3. Figure C.2 shows one segment of data obtained from testing the long span of the bridge (setup 2). Figure C.3 shows one segment of data obtained from testing the short span (setup 15). 140 8 •c I o o 3 O &0 § I I o o 3 |+8 CO .1 0 OH 1 co c3 CD o S "S o o I 3 I I I 3 4-S I L a I 4 I U I U I *+- oo ~ -o fc. . 1. 4> •-I E2 2 s $ + « + 5 + 535 + 8 $Q t ^ 1 " a 8 § 8 2 -a J © > § i .1 I * u a •I-I so «a O Figure C. 1. Instrumentation of the Lillooet River bridge for the ambient vibration testing. 141 O o ST 3. CA a . o CA CA CA ? l 3*2; o It t/i CD o 2" o 2 o i f o o CA O © f a , IP °> 8 <§ CA P rj- CA ft • • I CO ] CD 1 CO 00 CD 1 CO 0) CD I CD I Nil CO CO co 00 Cn CD < O l < en co < cn co —i cn cn < cn cn CD |0> cn cn to < CO < CO H cn < NJ NJ 00 NJ =5 cn cn CD 1 cn CO < CO < cn < cn —i -vi < H CO CD CD < < CO < CO HCO < CO CD 1 iw -fc. o < NJ < CO CD < <*>] CD NJ cn NJ NJ CO cn CD I o Z3 o I1 C D 0) 3 C D 3 O 3 CD 1 NJ < NJ NJ NJ CD CO CO < CO 00 < CO cn < CO cn —i co -v l < CO 1^ H 3 NJ < < CD S CO < cn < cn H CD I O CD O < NJ < CD < CD I 001 CD O 00 CO cn CO CD o CD < CP < CO < CD o CO 00 < NJ O < < CO H CO < CP ~vl < cn < cn CD < CD CD O cn co co < CO 00 < CO cn < CO cn —I co < CO 01 < NJ O CD I CO I 00 col d5 CD $ O CO NJ < CO < CO < CO CO CO < l CO CO CD o CO NJ 00 < CO o < '3 H NJ NJ CD H cnl CO NJl CD O NJ NJ < NJ CO < NJ CO < NJ CO NJ cn < NJ cn NJ I col CO *-»* f—t-C D I1 C O Q -O 0) 5' " D O O I * 1 CA w CA o l 0) C D CA D) D . Q.I NJ| T 3 O O CO| ^1 col CO c T 3 CD O NJ NJ < NJ NJ < NJ < NJ H NJ < NJ C D 3 C D l o l NJ IO CO IO N> < NS H s o' i — CD CA O O C D 5' CO l o l cn '§ CO O rr co C D C o' CD > C O C D CA O — l CA 3 3 C D 3 I : O a* CB' a ^. I. o CA B' C T Q O o o 3 p . a * CTQ CD 0 CD CA CD 1 CA Figure C.2. Ambient Vibration Testing of the Lillooet River Bridge - an example of the acceleration records measured on the long span (setup 2). 143 Figure C.3. Ambient Vibration Testing of the Lillooet River Bridge - an example of the acceleration records measured on the short span (setup 15). 144 C.2 DETAILS OF THE FORCED VIBRATION TESTING The response of the Lillooet River Bridge due to the moving truck T3162 was measured at six points on the long span and at two points on the short span. The locations of these points are shown in Figure C.4. The response was measured by the FBA-11 sensor at each point. These sensors occupied channels 1 to 8 of the data acquisition system. Channels 9 and 10 were used by trigger sensors to measure the times when the testing truck first entered the bridge and then left the bridge. These sensors were developed for this testing in the Earthquake Engineering Research Laboratory of the Department of Civil Engineering, U B C . The information obtained from these sensors was also used to determine the actual speeds of the testing vehicle during each passage. During the forced vibration testing, an attempt to measure the total vertical displacements of the bridge was made. Four displacement transducers Celesco model PT101 were mounted at points 19, 20, 45 and 46 in order to measure relative displacements of these points with respect to ground. These sensors occupied channels 11 to 14 of the data acquisition system. The measurement range of these transducers was 25.4 cm and sensitivity 39.37 mV/V/cm (Celesco Industries Inc., 1988). The displacement transducers were added to the data acquisition system before this test and tested in laboratory conditions with very good results. However, the proper installation of these sensors on the site was impossible due to the strong current and the high water level of the Lillooet river during the time of the testing. The strings connecting the sensors with the ground could not be installed vertically. They had to be anchored near the shore and, therefore, the resolution of the obtained displacement records was poor. The portion of the data measured by the displacement transducers was not used for further analysis in this thesis. 145 Seventeen setups were measured during the forced vibration testing. The details about these setups are given in Table C.2. The locations of all used sensors together with the numbers of the corresponding channels and the types of the sensor are summarized in Table C.3. Table C.3 Locations and types of the sensors for the forced vibration testing. Channel No. Sensor Location Type of the Sensor 1 point 19 FBA-11 2 point 20 FBA-11 3 point 23 FBA-11 4 point 24 FBA-11 5 point 27 FBA-11 6 point 28 FBA-11 7 point 45 FBA-11 8 point 46 FBA-11 9 between points 1 and 3 trigger time sensor 10 between points 53 and 55 trigger time sensor 11 point 19 displacement transducer 12 point 20 displacement transducer 13 point 45 displacement transducer 14 point 46 displacement transducer Examples of the measured acceleration records from channels 1 to 8 are shown in Figures C.5. and C.6. Figure C.5. shows one segment of data obtained during the measurement when 146 the speed of the testing truck was 13.36 km/h (setup 1). Figure C.6 shows one segment of data obtained during setup 16 when the speed of the testing truck was 58.39 km/h. The impact hammer testing was also conducted on the Lillooet River Bridge. The response of the bridge due to the hammer impact at point 20 was measured at eight points FBA-11 sensors located at the same positions as it was during the forced vibration testing. The data obtained from the hammer testing was not analyzed in this thesis, but it is available for future further analysis. 147 Figure C.4. Instrumentation of the Lillooet River bridge for the forced vibration testing. 148 6H a & £ % i-» O 00 S" <: 2 . w o ' P Co O co 3 i 8.1: CO CD % CO CO CD X t—»• CO CO cn b cn 1 b 0 1 cn cn cn O) ro cn b 0 1 CD cn cn ro CO c n 1 cn cn O ro cn c o CO cn o o o o CO cn cn 1 CO CO cn 1 ro oo co CO 0 1 ro o o cn ro o o ro o o CO ro o o ro ro ro co ro o o ro cn cn 1 ro o o CO ro 0 1 ro o o o o CO ro CO cn 1 ro o o ro o o co ro cn oo cn ro co co cn 1 00 CO CO cn o o ro o o CD o 3 5L CD CD Q . 3 CO 00 C O CD c J O . H 3 CD CO O g O 0) CO CD t o CD Z N »» 2 . c 2. 3 </> O" CD O C O Z 3 c r CD CD 5T o & CD* O to 0 C? CD 1 o 00 «-•-crq CD O 0 CD 1 CD >-l w ere CD i a. CD 00 O sr CD on CD •I oo Figure C.5. Forced vibration testing of the Lillooet River Bridge - an example of the accelerations measured during the passage of the vehicle T3162 (setup 1) 150 Figure C.6. Forced vibration testing of the Lillooet River Bridge - an example of the accelerations measured during the passage of the vehicle T3162 (setup 16) 151 A P P E N D I X D FREQUENCY RESPONSE FUNCTION (FRF) PROGRAM - OPERATING INSTRUCTIONS The program Frequency Response Function (FRF) is a tool to calculate the frequency response function, the phase and the coherence function of two real valued time histories x(t) and v(r). Assuming that the signal x(t) is the input and the signal y(t) is the output of a single degree of freedom viscously damped system, the complex valued frequency response function H„,(/) can be obtained from (Bracewell, 1986): where C^(f) is the real component of the one-sided cross spectral density function; Qxy(f) is the imaginary component of the one-sided cross spectral density function; Gxx(f) 1S t n e one-sided auto spectral density function of the signal x( t). The phase of the frequency response function <I> (f) is obtained using: D.l DESCRIPTION (D.l) 0„(J) = tan {CM)) (D.2) The phase angle displayed in degrees is ranging between 0° < <D < 360°. The ordinary coherence function y^ , ( / ) between the two signals is computed using: 152 The coherence yly(f) is displayed in scalar values ranging from 0 to 1. The program FRF was written by Dr. Carlos E . Ventura, Civil Engineering, U B C , together with the author of this thesis. The latest version in the time of writing this thesis was version 2.0a from January 1995. D.l INSTALLATION, SYSTEM REQUIREMENTS AND PROGRAM EXECUTION The program FRF should run on any I B M - PC compatible computer with a 80386 processor or higher and 4 M B of R A M or more. Since the program has a possibility for graphic representation of the results, the graphic part of the hardware is important for successful use of the FRF program. The program was developed and used on PC computers with V G A graphic cards and with monitors supporting V G A graphics. The previous graphic hardware versions (CGA, E G A , etc.) were not tested with the program. Therefore, it is recommended to use a computer with V G A graphics or higher. The program FRF can be copied into any directory and executed from any other directory provided there is a path to it. The program requires the font files COURB.FON and TMSRB.FON from the Microsoft Fortran Power Station library be copied on the hard drive. To run the program, the user should type at the DOS prompt: FKFfilel file2 result t1 start Hend t2start t2end scalef smoothf decimf where: filel is the name of the input file (the input signal); file2 is the name of the input file (the output signal); 153 result is the name of the file for storing the results; t1 start is the start time of the signal in the filel (optional parameter); tlend is the end time of the signal in the filel (optional parameter); t2start is the start time of the signal in the file2 (optional parameter); t2end is the end time of the signal in the file2 (optional parameter); scalef is the scaling factor for computed FRF (optional parameter, default=l); smoothf is the factor for smoothing FRF values (optional parameter); decimf is the decimation factor for storing results (optional parameter, default=l). The FRF program is equipped with a detailed help on the input parameters. This help can be evinced by calling the program without input parameters, that is entering command 'FRF' at the DOS prompt. This help provides the user with sufficient information on successful selection of the input parameters and, therefore these parameters will not be dealt with here. However, it is desirable to note that the program allows the user: • work with input files of formats *.bbb (Schuster, 1994, p. 143-144), *.aaa (Felber, 1993, p. 247-248) and *.v2 (CSMIP, 1985) - entering the extension of the input files is optional; • add or subtract the signals in the input files; • select a method for smoothing the results. Two smoothing methods are implemented in the program: the Moving Average Method (Press at al., 1992) - evoked it smoothf is positive and the Savitzky-Golay Smoothing Filter (Press at al., 1992) - evoked if smoothf is negative. 154 After the program was successfully called, the results are calculated. Then the graphic part of the program is initialized if the first character of the output file in the command line is an '*' , otherwise the graphic option is not called. The graphic functions of the program allow the user to check visually the results using functions: Zoom, Scale, Redraw, Bitmap, Cursor and Info. The on-line help on these functions is available to the user when using the graphics and, therefore these graphics functions will not be explained here. The final job of the program is to store the results into the specified filename. The user can approve or cancel storing the results. D.3 EXAMPLE Let us assume that the frequency response functions of the files Ibfv0503.bbb (the input signal) and Ibfv0505.bbb (the output signal) is to be calculated using the program FRF. It can be done by entering the command: frf lbfv0503 lbfv0305 *frf0305 0 40 0 40 1 5 5 The used options indicate that the starting time is 0 s for both signals, the end time is 40 s for both signals, the scaling factor for results is 1, the smoothing factor is 5 and the decimation factor is 5. The results will be stored to the file frf0305. The '* ' character in the beginning of this filename indicates that the graphics will be initialized. This command was used for calculating the frequency response function of the files that are shown in Figures D. la. and D. lb. The results are shown in Figures D. lc . to D. le. 155 a) Signal lbfv0503: signal! -2.0 - | I | I | I | I | TT 15 20 25 30 35 time (s) c) Amplitude of the Frequency Response Function: |FRF| 15 frequency (Hz) 30 360 d) Phase angle: <|> 180 H T 15 frequency (Hz) e) Coherence function: y T 10 15 20 frequency (Hz) 30 Figure D. 1. Program FRF - an example of input data and results; a) and b) - the input signals; c), d) and e) - results of the analysis on the input signals. A P P E N D I X E TIME HISTORY VIEWER (THV) PROGRAM - OPERATING INSTRUCTIONS E.l DESCRIPTION The program Time History Viewer (THV) is a tool to view several time history records. Its function is to display all the records on a computer's screen in order to allow the user efficient visual observation and comparison of these signals. Thus the user is able to say quickly if any anomalies or signal contamination has occurred in the signals. The main purpose of developing the program was increasing effectiveness of the field dynamic tests when the measured signals from one setup have to be visually checked. The program FRF was written by the author of this thesis together with Dr. Carlos E. Ventura, Civil Engineering, U B C . The first version of the program, version 1.0, was written in December 1994. The enhanced version of T H V is being prepared for the beginning of 1996. E.2 INSTALLATION, SYSTEM REQUIREMENTS AND PROGRAM EXECUTION The program T H V should run on any I B M - PC compatible computer with a 80386 processor or higher and 4 M B of R A M or more. Since the program has a possibility of graphic representation of the results, the graphic part of the hardware is important for successful using of the T H V program. The program was developed and used on PC computers with V G A 157 graphic cards and with monitors supporting V G A graphics. The previous graphic hardware versions (CGA, E G A , etc.) were not tested with the program. Therefore, it is recommended to use a computer with V G A graphics or higher. The program T H V can be copied into any directory and executed from any other directory provided there is a path to it. The program requires the font files COTJRB.FON and TMSRB.FON from the Microsoft Fortran Power Station library be copied on the hard drive. To run the program, the user should type at the DOS prompt: thv filel file2 file3 file4 file5 file6 file! file8 file9 where: filel to file9 are the names of the files to be viewed. By doing so the time histories stored in the files filel to file9 are plotted on the screen. The current version (version 1.0) of the program is capable to read files of the formats *.bbb (Schuster, 1994, p. 143-144), *.aaa (Felber, 1993, p. 247-248) and *.v2 (CSMTP, 1985) -using the extensions of the input files is optional. The maximum number of viewed files is sixteen and each file can have up to 64 kB points. It is a common practice during the dynamic testing that one setup of the recorded files contains files with very similar names differing only in one character. This is also the case of the files recorded by the data acquisition program A V D A (Schuster, 1994), which is used at TJBC, Civil Engineering. The program A V D A names the files recorded in one setup so, that the last character of the file name is the number of the channel. Then, the following files can be stored by the program A V D A : 158 lbfvOlOl.bbb, lbfv0102.bbb, lbfv0103.bbb, lbfv0104.bbb, lbfv0105.bbb, lbfv0106.bbb, lbfV0107.bbb, lbfv0108.bbb, lbfv0109.bbb, lbfvOlOa.bbb, lbfvOlOb.bbb, lbfvOlOc.bbb, lbfvOlOd.bbb, lbfvOlOe.bbb, lbfvOlOf.bbb, lbfvOlOg.bbb. A l l these files can be viewed by a simple command: thv lbfvOlO? 16 where '?' is a wild character used for the changing last character of the filenames. The file name with the wild character should be followed by the number of the files to be viewed, which is 16 in this case. If the wild character is used and the number of the files is not entered, the default value 16 is considered. It is assumed that the number substituted for the wild character in the first file name is 1. After the program was successfully called the graphic part is initialized. The graphic functions of the program allow the user to check visually the signals using functions: Zoom, Select, Cursor, Vertical, Redraw, Bitmap, DosShell and Info. The on-line help on these functions is available to the user when using the graphics and, therefore these graphics functions will not be explained here. 159
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Dynamic analysis of bridges with laminated wood girders Horyna, Tomás 1995
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Title | Dynamic analysis of bridges with laminated wood girders |
Creator |
Horyna, Tomás |
Date Issued | 1995 |
Description | The response of bridges due to loading by heavy vehicles is one of the important aspects of bridge design. The effects of traffic on the response are significant if the bridge is a short and light-weight structure and the vehicle is very heavy. This is the case of a loaded logging truck passing over a logging bridge. The passage of a vehicle over a bridge is a complex dynamic problem and, for simplicity, bridge design codes treat it in a pseudostatic manner. Generally, the codes require a static analysis of the bridge using a specified design truck. The static response of the bridge is then multiplied by a dynamic load allowance factor in order to account for the dynamic response. The timber industry is one of the most important in the province of British Columbia and transportation of its products is a significant part of this industry. Logging bridges made from wood form an important element of this transportation network and their optimal design is, therefore, of importance to the province. However, provisions of bridge design codes are mostly supported by investigations conducted on highway bridges. Such is the case of the new Canadian Highway Bridge Design Code (CHBDC), which is now being prepared in Canada for release in 1996. It is desirable, therefore, to investigate if the recommendations of CHBDC for wooden bridges correspond with the actual response of a logging bridge loaded with a passing logging truck. The objective of this study was to develop a tool to perform dynamic analysis of the system logging bridge - logging truck and to verify, that the dynamic load allowance factor from the CHBDC represents adequately the dynamic amplification factor obtained from dynamic analysis of the system logging bridge - logging truck. A numerical model to simulate the passage of a vehicle over a bridge was developed. The model was calibrated with results of ambient and forced vibration testing of an existing bridge. Finally, dynamic amplification factors were obtained from numerical simulations and compared with CHBDC provisions. The results of this study showed that these provisions are generally adequate for wooden logging bridges. |
Extent | 9717683 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-01-30 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0050320 |
URI | http://hdl.handle.net/2429/3996 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1995-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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