Numerical investigation of noise generation and ... - PublicationsList.org

Numerical investigation of noise generation and

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radiation from modular bridge expansion joint

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(道路橋モジュラー型エクスパンションジョイントの

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騒音発生・放射に関する解析的研究)

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２００８年９月

埼玉大学大学院理工学研究科（博士後期課程） 生産科学専攻（主指導教員

山口

宏樹）

JHABINDRA PRASAD GHIMIRE

Numerical investigation of noise generation and

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radiation from modular bridge expansion joint

A thesis submitted in partial fulfillment of requirement for the degree of Doctor of

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Philosophy

by

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Jhabindra Prasad Ghimire

Department of Civil and Environmental Engineering Graduate School of Science and Engineering

Saitama University September, 2008

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My family

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Table of contents Abstract………………………………………………………………………….…… iv Acknowledgements.………………………………………………………….….…. vi List of tables and figures………………………………………………………………………. vii Nomenclature…………………………………………………………………………………... xv

Chapter 1: Introduction and state-of-the-art literature survey…………….… 1

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1.1 Background……………………………………………………………………………. 1 1.2 Noise from modular expansion joints……………………………………………….. 2

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1.3 Noise from highway and railway bridges…………………………………………… 5 1.4 Summary………………………………………………………………………………... 9 1.5 Research objectives and methodology………………………………………………11

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1.6 Thesis organization…………………………………………………………………….12

Chapter 2: Vibro-acoustic analysis of full scale model of modular expansion joint ………………………………………………………..…13 2.1 Background and objective……………………………………………………….……..13

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2.2 Description of full scale model of expansion joint…………………………………....13 2.3 Impact testing experiment of sound and vibration……………………………………15 2.4 Vibro-acoustic analysis of the joint-cavity system by FEM-BEM approach……….15

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2.4.1 Analytical procedure……………………………………………………………...15 2.4.2 Velocity response of the expansion joint……………………………………….16 2.4.2.1 Finite element modal analysis of the joint……………………………... 16 2.4.2.2 Velocity response estimation of the joint………………………………. 18

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2.4.3 Analysis of acoustic field………………………………………………………… 19 2.4.3.1 Modal acoustic transfer vector technique in indirect BEM…………… 19 2.4.3.2 Boundary element modeling of the joint-cavity system……………….. 21

2.5 Acoustic modal analysis of the cavity beneath the joint…………………………….. 23 2.5.1 Finite element method………………………………………………………….... 23 2.5.2 Indirect boundary element method (IBEM)...…………………….……………...24 2.6 Results and discussion……………………………………………….…………………..24 2.6.1 Acoustic modal parameters of the cavity………………………..……………... 24 2.6.2 Velocity response of the expansion joint………………………………….……..26 2.6.2.1 Velocity response in lateral and vertical direction……………………….26 2.6.2.2 Comparison of numerical and experimental results……………….……28 2.6.3 Sound pressure response……………………………………………………..…..30

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2.6.3.1 Sound pressure response inside the cavity…………………………..….30 2.6.3.2 Sound generation and radiation mechanism inside the cavity…….…..30 2.6.3.3 Comparison of numerical and experimental results……………….…….33 2.7 Reliability of discussion based on numerical results…………………………….….... 35 2.8 Conclusions……………………………………………………………………………...…37

Chapter 3: Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges……………………………..…38

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3.1 Background and objective………………………………………………………….….….38 3.2 Description of the modular joint between PC bridges……………………………..…...39 3.3 Noise and vibration measurement…………………………………………………..……41

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3.4 Vibro-acoustic analysis of the joint-cavity system by FEM-BEM approach….…..…..43 3.4.1 Analytical procedure………………………………………………………………...43 3.4.2 Velocity response estimation of the expansion joint…………………………..…44

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3.4.2.1 Finite element modal analysis of the joint ……………………………..….44 3.4.2.2 Velocity response estimation of the joint ……………………………….....45 3.4.3 Analysis of acoustic field…………………………………………………………….47 3.4.3.1 Indirect boundary element method………………………………………....47

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3.4.3.2 Boundary element modeling of the joint-cavity system…..………...…….49 3.4.3.3 Field point response and sound radiation characteristics………….…....50 3.5 Acoustic modal analysis of the cavity beneath the joint………………………….….....51

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3.6 Results and discussion………………………………………………………………...…..52 3.6.1 Velocity response of the expansion joint……………………………………...…..52 3.6.1.1 Velocity response on middle beams and support beams………….….…52 3.6.1.2 Effect of load positions on velocity response ………………………;….…55

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3.6.1.3 Experimental results………………………………………………...….…....57

3.6.1.4 Dynamic characteristics of the expansion joint……………………………60

3.6.2 Sound pressure response ……………………………………………………….....60 3.6.2.1 Sound pressure response at different field points……………………..….60 3.6.2.2 Sound generation and radiation mechanism inside the cavity………..…65

3.6.3 Sound radiation efficiency of the joint-cavity system…………………………..…68 3.6.4 Sound radiation characteristics of the joint-cavity system………………………..71 3.6.4.1 Effect of distance on directivity of sound radiation ………………………..71 3.6.4.2 Effect of acoustic characteristics of cavity on sound radiation …………..75 3.7 Analysis of joint vibration caused by transient loading…………………………………..81 3.7.1 Analysis method…………………………………………………………………..…...81 3.7.2 Vibration response of the expansion joint to transient loading………………..….83 3.8 Conclusions……………………………………………………………………………….…..87

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Chapter 4: Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge…....……89 4.1 Background and objective…………………………………………………………………..89 4.2 Description of two bridges with modular and finger joint………………………………...91 4.3 Experimental study of sound and vibration during vehicle pass-bys…………………...92 4.3.1 Sound and vibration measurement in two bridges ……….…………………….....92 4.3.2 Possible vibration power flow from the modular joint to the bridge…………….100 4.4 Vibration power flow estimation from the modular joint to the bridge………………….101

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4.4.1 Estimation procedure………………………………………………………………...101 4.4.2 Estimation of vertical mobility of the bridge………………………………………..103

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4.4.3 Estimation of velocity response of the modular joint ………….………………….112 4.4.4 Estimation of input vibration power to the bridge from the joint…………...…….118 4.5 Estimation of vibration response of the bridge using SEA…………………….…….....119 4.6 Estimation of sound radiation from the bridge………………………………………..….121

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4.7 Results and discussion……………………………………………………………...……..123 4.7.1 Velocity response of the bridge……………………………………………..….…..123 4.7.2 Sound pressure response at field points…………………………………….…….127

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4.8 Conclusions………………………………………………………………………………....130

Chapter 5: General discussion………….................................................................131 5.1 Sound radiation efficiency of joint-cavity system ………………………………………131

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5.2 Sound radiation characteristics of joint-cavity system to outside environment……..132 5.3 Limitations of the numerical and experimental studies in the thesis…………………133 5.4 Future applications of the thesis findings……………………………………………….135

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Chapter 6: General conclusions and recommendations for future work……..137 6.1 General conclusions from the thesis…………………..………………………………..137 6.2 Recommendations for future work………………………………………………………138

References……………………………………………………………………………….139

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Abstract Modular expansion joint is commonly used for large expansion joint movement. Because of the several advantages over other types of expansion joint, use of modular expansion joint has been increased recently. However, the noise generated and radiated from the modular joint has been a localized environmental problem in Japan and elsewhere. Understanding of noise generation and radiation is important from noise control point of view. Noise from modular expansion joint is mainly generated and radiated from the top of

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the joint on carriageway surface and from the bottom part of the joint which has a cavity beneath it for the maintenance purpose. From the previous studies, noise generation and

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radiation mechanism inside the cavity beneath the joint is not fully understood. There have been no reported studies on the noise radiation from the joint to the outside environment, which is necessary to be understood from noise control point of view. The

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vibration power of the joint during vehicle impact can be transmitted to the connected bridge. No reported studies were found on possible vibration power flow from the joint to the connected bridge and noise contribution from the bridge. The objectives of this study have been: (i) to understand the noise generation and radiation mechanism inside the

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cavity beneath the joint; (ii) to understand the characteristics and mechanism of noise radiation from the joint to the outside environment and (iii) to understand the possible

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vibration power flow from the joint to the bridge and noise radiation from the bridge.

A full scale model of modular expansion joint was considered to fulfill the first objective. Vibro-acoustic analysis was conducted by considering the joint and cavity which was

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underground as one system. Finite element method (FEM) - boundary element method (BEM) approach was used for the analysis. FEM was utilized for the dynamic response estimation of the joint structure and BEM was used to analyze the sound field inside the cavity of the joint-cavity system in the frequency range of 20-400 Hz. FE modal analysis was also conducted to obtain the acoustic modal parameters of the cavity beneath the joint. Impact testing experiment of noise and vibration were used to interpret the numerical results. It was concluded that dominant frequency components in the sound pressure response inside the cavity was due to vibration modes of the joint structure and/or acoustic modes of the cavity. In lower frequencies, peaks in the sound pressure response inside the cavity were mainly due to the vibration modes of the joint structure with possible interaction with acoustic modes of the cavity. At higher frequencies where

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the modal density of the acoustic modes was high, effect of acoustic resonance of cavity on sound pressure response was significant.

A modular expansion joint installed between prestressed concrete bridges was considered to fulfill the second objective. The cavity beneath the joint had openings at both ends along its length. Noise generated inside the cavity could radiate to the outside environment from these openings. FEM-BEM approach as in the previous study of full scale model joint was utilized to analyze the sound field inside as well as outside of the

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joint-cavity system. FE modal analysis was also conducted to obtain the acoustic modal properties of the cavity beneath the joint. Measurement of noise and vibration of

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expansion joint during vehicle pass-bys were also carried out. Sound field was analyzed in the frequency range of 50-400 Hz. It was concluded that the noise from the bottom side of the joint was caused by the excitation of structural modes of the expansion joint and/or

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acoustic modes of the cavity beneath the joint, which is consistent with a conclusion derived in a previous study with a full scale model joint. The sound radiation efficiency of the joint-cavity system appeared to be high at natural frequencies of vibration modes of the joint with significant vertical vibration of middle beams and support beams (coupled

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modes). Sound radiation efficiency of lateral modes of the joint appeared to be low. Noise from the joint–cavity system may be propagated most effectively at radiation angles of

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acoustic modes of the cavity, which can be predicted roughly from the fundamental theory of sound radiation from cavities and waveguides. The sound field around the joint-cavity system investigated in this study could be considered near field within 35 m from the joint-

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cavity center and far field at farther distances.

A modular expansion joint installed in a steel-concrete non-composite bridge was considered to fulfill the third objective. A simplified analytical approach using FEM and statistical energy analysis (SEA) was considered. FE model of the joint was used to estimate the dynamic response of the joint. SEA was used to estimate vibration response of the bridge. Measurement results of noise and vibration of expansion joint and bridge during vehicle pass-bys were used to interpret the analytical results. It was found that the simplified approach was able to predict the flow of vibration power from the expansion joint to the bridge and noise radiation from the bridge during vehicle pass-bys. This approach can be utilized to reduce the vibration power flow from the joint to the bridge.

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Acknowledgements First of all, I would like to express my sincere gratitude to my supervisor Assoc. Prof. Yasunao Matsumoto for his valuable guidance, help and encouragement throughout this study. Without his continuous support and sometimes critical comments this study would not have been at this stage.

I would like to thank Prof. Hiroki Yamaguchi for his suggestions and often critical

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questions that helped improve the thesis. I would also like to thank Prof. Hideji Kawakami

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and Assoc. Prof. Yoshiaki Okui for their comments to improve the thesis.

I am thankful to Mr. Itsumi Kurahashi for his help in experimental study. I would like to thank technical staff Mr. Yutaka Matsuhashi for his help to have the logistics I needed

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throughout this work.

I wish to thank Government of Japan for the financial support to do this research. I am also thankful to the staffs of Kawaguchi Metal Industries Co., Ltd., Saitama, Japan for

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their support in experimental study.

I am grateful to my parents and other family members who are in my home country for

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their advice and patience during this study period. My special thanks to my wife Sunita for her help and understanding throughout this study.

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At last but not the least, I would like to thank a number of people who have helped me in different ways in the last three years.

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List of tables and figures List of tables Table 2-1: Cross-section, size and materials for steel members Table 2-2 Spring constants of bearings and springs

List of figures joint on road surface (Ravshanovich et al., 2007)

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Fig. 1-1 Sound pressure spectra: (a) inside the cavity beneath the joint and (b) above the

Fig. 1-2 Acceleration spectra in one of the middle beam: (a) acceleration in lateral

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direction and (b) acceleration in vertical direction (Ravshanovich et al., 2007) Fig. 2-1 Plan and cross-section of full-scale test joint model

Fig. 2-2 Cross-sectional views of the cavity beneath the joint

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Fig. 2-3 Time series data of sound and vibration measurement during impact testing. (a) Lateral acceleration on middle beam, (b) vertical acceleration on middle beam and (c) Sound pressure inside the cavity beneath the joint Fig. 2-4 Flow chart of numerical analysis conducted in the study

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Fig. 2-5 FE model of the expansion joint: (a) 3-D view; (b) Cross-section view Fig. 2-6 Position and direction of point loads for velocity response evaluation

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Fig. 2-7 Boundary element model of the cavity for acoustic analysis: (a) Full 3-D model; (b) Sectional view showing measurement point

Fig. 2-8 FE model of the cavity for acoustic modal analysis

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Fig. 2-9 Frequency response of the sound pressure inside the cavity to acoustic point source. Key:

, at measurement point;

, acoustic mode identified by

FEM

Fig. 2-10 Some of the acoustic modes of the cavity identified from FEM Fig. 2-11 Lateral mobility of the expansion joint at measurement point on second middle beam Fig. 2-12 Some vibration modes of the joint with dominant vibration in lateral direction Fig. 2-13 Vertical mobility of the expansion joint at measurement point Fig. 2-14 Vertical mobility of the expansion joint at measurement point Fig. 2-15 Velocity responses at the measurement point in lateral direction. Key: Experimental-1;

, Experimental-2;

, Numerical

,

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Fig. 2-16 Velocity responses at the measurement point in vertical direction. Key: Experimental-1;

, Experimental-2;

, Numerical

Fig. 2-17 Sound pressure responses at measurement point (Numerical). Key: structural mode;

,

,

, acoustic mode

Fig. 2-18 Structural and acoustic modes generating sound around 160 Hz: (a) Structural mode at 160.35 Hz; (b) Acoustic mode at 161.10 Hz and (c) Acoustic mode at 161.70 Hz

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Fig. 2-19 Structural and acoustic modes generating sound around 268 Hz: (a) Structural mode at 267.6 Hz; (b) Acoustic mode at 267.91 Hz and (c) Acoustic mode at

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269.40 Hz

Fig. 2-20 Acoustic modes generating sound around 380 Hz: (a) Acoustic mode at 380.38 Hz; (b) Acoustic mode at 381.86 Hz and (c) Acoustic mode at 383.58 Hz

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Fig. 2-21 Sound pressure inside the cavity and natural frequencies of structural and acoustic modes. Key: , Numerical;

, Experimental-1;

, structural mode;

, Experimental-2;

, acoustic mode

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Fig. 2-22 Frequency response of the sound pressure inside the cavity at several points around the measurement point

Fig. 2-23 Frequency response of the sound pressure inside the cavity at measurement , 1 % modal damping;

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point. Key:

, 7 % modal damping

Fig. 2-24 Velocity responses in vertical direction at the measurement point during vehicle pass -bys over control beams. Key:

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ordinary car at 50 kmh-1;

, ordinary car at 40 kmh-1;

, wagon at 40 kmh-1;

,

, wagon at 50

kmh-1.

Fig. 3-1 Plan and cross-section of the expansion joint Fig. 3-2 Cross-sectional view of the cavity below the expansion joint Fig. 3-3 Positions of sound level meters inside the cavity and around the bridge during the experiment Fig. 3-4 Location of accelerometers on middle beams and support beams Fig. 3-5 Time series data of sound and vibration measurement during a truck pass-by. (a), (b) lateral and vertical acceleration respectively of third middle beam at the centre; (c) sound pressure inside the cavity and (d) sound pressure above the joint

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Fig. 3-6 Flow chart of numerical analysis conducted in the study Fig. 3-7 3-D finite element model of the joint Fig. 3-8 Lane width of the bridge, possible truck vehicle pass-by position on the joint Fig. 3-9 Position of four applied load cases Fig. 3-10 Boundary element model of the joint-cavity system Fig. 3-11 Directivity circles of different radius on a vertical plane Fig. 3-12 Finite element model of the cavity for acoustic modal analysis

Position-1;

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Fig. 3-13 Vertical velocity response at measurement points to Load-1 (Numerical). Key: Position-2;

Position-3;

Position-4

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Fig. 3-14 Vertical velocity response at measurement points to Load-3 (Numerical). Key: Position-1;

Position-2;

Position-3;

Position-4

Fig. 3-15 Some of the structural modes of the joint

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Fig. 3-16 Vertical velocity response at Position-1 to three different point loadings between second and third support beam around Load-1 position (Numerical) Fig. 3-17 Vertical velocity response at Position-1 to three different point loadings near the fifth support beam around Load-3 position (Numerical)

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Fig. 3-18 Vertical velocity response to the truck vehicle crossing the joint in Lane 1 (Experimental). Key:

Position-1;

Position-2;

Position-3;

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Position-4.

Fig. 3-19 Vertical velocity response to the truck vehicle crossing the joint in Lane 2 (Experimental). Key:

Position-1;

Position-2;

Position-3;

Position-4

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Fig. 3-20 Vertical velocity response on third middle beam and third support beam at measurement points.

Position-2 (middle beam, Load-3);

4 (support beam, Load-3);

Position-

Position-2 (Truck crossing in Lane 2);

Position-4 (Truck crossing in Lane 2).

Fig. 3-21: Numerically calculated sound pressure inside the cavity. Key: Load-1;

Load-2;

frequencies of structural mode;

Load-3;

Load-4.

natural

natural frequencies of acoustic mode

Fig. 3-22 Experimentally measured sound pressure at different field points to the truck vehicle crossing the joint in Lane-1. Key: m;

at 15 m

inside the cavity;

at 5

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Fig. 3-23 Experimentally measured sound pressure at different field points to the truck vehicle crossing the joint in Lane-2. Key:

inside the cavity;

at 5 m;

at 15 m Fig. 3-24 Numerically calculated sound pressure at different field points to Load-1. Key:

inside the cavity;

at 5 m;

at 15 m

Fig. 3-25 Numerically calculated sound pressure at different field points to Load-3. Key: inside the cavity;

at 5 m;

at 15 m

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Fig. 3-26 Sound pressure response inside the cavity. Load-3; Truck crossing in Lane 2; ¯ natural frequency of structural mode; U natural frequency of acoustic mode

5 m (Load-3);

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Fig. 3-27 Sound pressure at different field points outside the cavity. 15 m (Load-3);

5 m (Truck crossing in Lane 2);

(Truck crossing in Lane 2)

15 m

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Fig. 3-28 Structural mode of the joint in vertical direction at 150.17 Hz

Fig. 3-29 Acoustic modes of the cavity: (a) 154.06 Hz and (b) 155.82 Hz Fig. 3-30 Structural mode of the joint at 181.30 Hz

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Fig. 3-31 Mean quadratic velocity at the boundary of BE model. Key: Load-2;

Load-3;

Load-4

Fig. 3-32 Radiation efficiency of the joint-cavity system. Key: Load-3;

Load-1;

Load-4

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Load-2;

Load-1;

Fig. 3-33 Some of the vibration modes of the joint in lateral direction Fig. 3-34 Sound radiation pattern at 149 Hz

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Fig. 3-35 Sound radiation pattern at 392 Hz Fig. 3-36 Changes in the sound pressure with distance from the center of the joint at 300 degrees. Key: Hz;

332 Hz;

110 Hz;

122 Hz;

149 Hz;

282

392 Hz

Fig. 3-37 Some acoustic modes of the cavity beneath the joint Fig.3-38 Main lobe radiation angles of acoustic modes of the cavity.

, Natural frequency

of acoustic modes of different orders obtained from FEM Fig. 3-39 Time series of loading on the center beam at vehicle speed of 60 kmhr-1 to transient analysis

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Fig. 3-40 Fourier transforms of vertical velocity response of the joint at Position-2 on third middle beam to transient loading applied at Load-3 position (Fig.3-9) corresponding to different vehicle speeds. Key:

60 kmh-1;

80 kmh-1;

100 kmh-1 Fig. 3-41 Variation of multiples of tire pulse frequency, ft, with vehicle speed. ft ;

2ft ;

0.8ft ;

4ft; ¯ natural frequency of structural

3ft;

mode;U natural frequency of acoustic mode Fig. 3-42 Vertical velocity response on third middle beam at Position-2.

Load-3;

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single transient load applied at Load-3 position at 78 kmh-1;

experiment

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three consecutive transient loads applied at Load-3 position at 78 kmh-1;

Fig. 3-43 Beam pass frequency for different middle beam spacing. 50 mm;

60 mm;

70 mm;

40 mm;

80 mm; ¯ natural frequency of

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structural mode;U natural frequency of acoustic mode

Fig. 4-1 Cross-section of the bridges with finger type joint and modular type joint showing the position of accelerometers and sound level meters used in the experiment Fig. 4-2 Cross-section of cavity beneath the modular expansion joint

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Fig. 4-3 Top view of the modular joint installed in the bridge and position of the accelerometers in one of the middle beam during the measurement. Fig. 4-4 Time series of vibration and sound pressure response: (a) Acceleration response

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on concrete deck, (b) Sound pressure response under the bridge

Fig. 4-5 Sound pressure response under the bridge with modular joint Fig. 4-6 Sound pressure response at 5 m wayside of the bridge with modular joint

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Fig. 4-7 Sound pressure response under the bridge with finger joint Fig. 4-8 Sound pressure response at 5 m wayside of the bridge with finger joint Fig. 4-9 Comparison of sound pressure under the bridges. Key:

, Modular joint;

, Finger joint

Fig. 4-10 Comparison of sound pressure at 5 m wayside of the bridges. Key: Modular joint;

,

, Finger joint

Fig. 4-11 Vibration level at position (P3) of concrete deck of the bridge with modular joint Fig. 4-12 Vibration level at position (P1) of concrete deck of the bridge with finger joint Fig. 4-13 Comparison of vibration level on the deck of the bridges. Key: joint;

, Modular

, Finger joint

Fig. 4-14 Vibration level on the web of third girder (G3) of the bridge with modular joint Fig. 4-15 Vibration level on the web of third girder (G3) of the bridge with finger joint

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Fig. 4-16 Comparison of vibration level on the girder of bridges. Key: joint;

, Modular

, Finger joint

Fig. 4-17 Velocity level on middle beam of the joint at Position-1 Fig. 4-18 Velocity level on middle beam of the joint at Position-3 Fig. 4-19 Top view of the bridge and the modular joint Fig. 4-20 Cross-section view of the modular joint Fig. 4-21 Analytical procedure for the vibration power flow analysis from the expansion joint to the bridge and sound radiation from the bridge

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Fig. 4-22 Cross-section view of the bridge

Fig. 4-23 Cross-section detail of I-girders used in the bridge: (a) First girder (G1), (b)

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second (G2) and third girder (G3) and (c) fourth girder (G4)

Fig. 4-24 FE model of first span of first girder (G1): (a) full length view, (b) Detail view showing the stiffeners

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Fig. 4-25 Some of the vertical and lateral vibration modes of the I-girder from FE analysis Fig. 4-26 Application of dynamic point loadings on the girder Fig.4-27 Vertical mobility of I-girder to different loads by FEM. Average mobility (thick line) Fig.4-28 Comparison of vertical mobility of first I-girder. Key:

, FEM;

, infinite

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beam formula (Equation 4.3)

Fig.4-29 Vertical mobility of first I-girder. Key:

, middle

, high frequency

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frequency and

, low frequency;

Fig.4-30 Mobility of I-girders, concrete deck and combined mobility. Key: , 1st girder;

deck;

,4th girder;

, concrete

, combined (deck and 1st girder),

,combined (deck and 4th girder)

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Fig.4-31 Number of vibration modes of bridge components in 1/3 octave band frequency. Key:

, concrete deck;

, 1st girder web and

, 1st girder lower

flange

Fig.4-32 3-D FE model of the expansion joint Fig.4-33 Some of the vibration modes of the joint Fig.4-34 FE model of the expansion joint and applied load positions Fig.4-35 Velocity response on the first middle beam to Load-1. Key: , Position-2 and

, Position-1;

, Position-3

Fig.4-36 Velocity response on eight support beams to Load-1 Fig.4-37 Velocity response on the first middle beam to Load-2. Key: , Position-2 and

, Position-3

, Position-1;

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Fig.4-38 Velocity response on the support beams to Load-2 Fig.4-39 Velocity response at Position-1 on the first middle beam during vehicle pass-bys Fig.4-40 Velocity response at Position-2 on the first middle beam during vehicle passbys Fig.4-41 Velocity response at Position-3 on the first middle beam during vehicle pass-bys Fig.4-42 Frequency response of input vibration power to the bridge from the joint. Key:

, Load-1 and

,Load-2

Fig.4-43 Cross-section view of the bridge and different subsystems in SEA ,2nd and 3rd girder lower flange; , 1st and 4th girder web,

,1st and 4th girder lower flange;

,2nd and 3rd girder web

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deck;

,concrete

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Fig.4-44 Radiation efficiency of different subsystems. Key:

Fig.4-45 Square of the velocity response in different subsystems to Load-1. Key: concrete deck;

, 1st girder web (out of

, 1st girder lower flange

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plane);

,1st girder web (in plane);

Fig.4-46 Square of the velocity response in different subsystems to Load-2. Key: concrete deck; plane);

,

st

,1 girder web (in plane);

,

st

, 1 girder web (out of

, 1st girder lower flange

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Fig.4-47 Velocity response level on concrete deck at P2 during truck vehicle pass-bys Fig.4-48 Velocity response level on concrete deck at P3 during truck vehicle pass-bys

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Fig.4-49 Velocity response level on third steel girder (G3) web during truck vehicle passbys

Fig.4-50 Velocity response level on fourth steel girder (G4) web during truck vehicle passbys

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Fig.4-51 Comparison of velocity response level on concrete deck at P3 and on third girder (G3) web during truck vehicle pass-by. Key:

, deck and

,I-girder

web

Fig.4-52 Contribution from different subsystems to the total sound pressure response under the bridge to Load-1. Key: flange;

, girders web;

, concrete deck;

, girders lower

, Total bridge

Fig.4-53 Contribution from different subsystems to the total sound pressure response under the bridge to Load-2. Key: flange;

, girders web;

, concrete deck;

, girders lower

, Total bridge

Fig.4-54 Sound pressure level inside the cavity beneath the joint during vehicles pass-bys Fig.4-55 Sound pressure level under the bridge during vehicles pass-bys

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Fig.4-56 Sound pressure level at 5 m wayside of the bridge during vehicles pass-bys

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Fig.5-1 Cutoff frequency of the cavity for different depth

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Nomenclature Chapter 2

Acoustic transfer vector

ω = 2π f

Circular frequency (rad/second)

f

Cyclic frequency (Hz)

{v n (ω )}

Normal component of velocity vector

p(ω )

Sound pressure at field point

[ ATM (ω )]

Matrix of acoustic transfer vectors

{u(ω )}

Column vector of structural displacement components

[Φ]

Structural mode matrix

[Φn ]

Matrix of structural modal vectors normal to the boundary surface

{MPF (ω )}

Modal participation factor vector

{MATV (ω )}

Modal acoustic transfer vector

σY ΓY

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pY

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μY

Double layer potential

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Chapter 3

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{ATV (ω )}

Single layer potential

Boundary surface

Acoustic pressure at source point Y on the boundary surface

pY +

Acoustic pressure on positive side of the boundary surface

pY −

Acoustic pressure on negative side of the boundary surface

vnY

Normal acoustic velocity at a source point Y

∂p + ∂nY

Gradient of acoustic pressure at Y on the boundary on positive side

∂p − ∂nY

Gradient of acoustic pressure at Y on the boundary on negative side

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Unit outward normal vector

ρ

Mass density of the medium

G XY

Green function

∂GXY ∂nY

Gradient of green function

r

Distance between source and field point

k

Acoustic wave number

c

Velocity of sound in the medium

Wo,active

Active output power

Wi

Input power

v rms

Root-mean-square value of local normal velocity on the boundary

fc

Critical frequency (Hz)

k

Radius of gyration of second moment of area in meters

G H

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D

nY

Maximum source dimension

ld

Propagation angle of acoustic mode inside the cavity

Lx

Dimension of the cavity in X-direction

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θ

Ly

Dimension of the cavity in Y-direction

l m

Number of nodal planes of acoustic mode along Y-axis

ωlm

Cutoff angular frequency of (l , m ) acoustic mode

lπ Lx

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k xl =

PY

Number of nodal planes of acoustic mode along X-axis

k ym =

mπ Ly

Acoustic wave number component in X-direction

Acoustic wave number component in Y-direction

td

Duration of tire impact loading

Lc

Tire contact length

bf

Width of top flange of middle beam

v

Velocity of the vehicle

Tn

Natural period of the system

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Chapter 4

Input vibration power to the bridge from the expansion joint

Re(YBridge )

Real part of vertical point mobility of the bridge

F

Amplitude of dynamic force

vsup

Vertical velocity response of support beam of the expansion joint

K eq

Equivalent vertical spring constant

K bearing

Vertical spring constant of the bearing used in the joint

K spring

Vertical spring constant of the spring used in the joint

Ygirder

Vertical point mobility of I-girder

Zgirder

Vertical point impedance of I-girder

m'

Mass per unit length of I-girder

cB

Bending wave speed of I-girder

B

Bending stiffness of I-girder

E

Young’s Modulus of steel of I-girder

I

Second moment of area of I-girder about the strongest axis

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G H

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Cross-sectional area of I-girder

A

κ υ hw

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cR

Shear co-efficient Shear modulus

PY

G

D

Pin, Bridge

Poisson’s ratio

Thickness of I-girder web Rayleigh wave speed

hf

Thickness of flange of I-girder

If

Second moment per unit width of area of the flange

cshear

Shear wave speed

Ydeck

Vertical point mobility of concrete deck

B'

Bending stiffness of the deck as plate

m''

Mass per unit area of the deck

Econc

Young’s Modulus of concrete

tdeck

Thickness of the concrete deck

Page xviii

No of modes in 1/3 octave band

S

Surface area of plate

h

Thickness of plate

cL

Longitudinal wave speed of plate

Pdiss

Dissipated power from the bridge

Prad

Radiated power from the bridge

Pdissi

Power dissipated from ith subsystem of the bridge

ηi

Structural loss factor of ith subsystem of the bridge

D

n

vi 2

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Space averaged mean square velocity of ith subsystem of the bridge

Real part of point mobility of ith subsystem of the bridge

Re(Y j )

Real part of point mobility of jth subsystem of the bridge

Pradi

Acoustic power radiated from ith subsystem of the bridge

ρair

Mass density of air

cair

Velocity of sound in air

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σi

Sound radiation efficiency of ith subsystem of the bridge

P

PY

Surface area of ith subsystem of the bridge

Si

Perimeter of sound radiating surface

f fc

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α=

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Re(Yi )

Ratio of square root of frequency of interest to critical frequency

pfield

Sound pressure at field point

Lbridge

Length of bridge

Dfield

Distance of bridge from the bridge

Chapter 1 Introduction and state-of-the-art literature survey 1.1 Background Noise is generally considered to be unwanted sound. We hear sound when our ears are exposed to small pressure fluctuation in air. There are many ways in which pressure fluctuations are generated, but typically they are caused due to vibration of solid object.

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Noise caused by different means of traffic like road traffic, rail traffic and air traffic etc. have been an environmental problem today. In fact, among the environmental pollution

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factors that are affected by the use of transportation means, noise is perhaps the most commonly cited. This problem is exacerbated as the number of vehicles circulating in the urban network of roads and rails is steadily increasing while at the same time the number

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of quiet hours during the night has a tendency to reduce, although at a slower rate (Ouis, 2001).

In general every noise problem involves a system of three basic elements: noise source,

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transmission path and a receiver. When possible, the best way to remedy exposure to undesirable noise, both economically and aesthetically, is to control the noise emission at the source itself. But for most noise sources, the most corrective measure is making

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changes in the path. Moreover, different noise sources may have different acoustic characteristics. While some radiate a pure tone, others may radiate a random noise with more or less known frequency spectrum. So in this respect definition of noise problem is

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important.

For road traffic noise, an automobile has in general several noise generating sources. However, the dominant noise source of the automobile is due to the contact of rolling tire on the road. There is another source of road traffic noise when the vehicle passes through the bridges or viaducts and their expansions joints. The noise is generated and radiated due to vibration of the expansion joint, deck and girders of the bridge or viaduct. The mechanism of sound generation and radiation from the vibration of structures is known as vibro-acoustics. There are two components of sound in elevated structures like bridges and viaducts: (i) joint sound and (ii) span sound. The joint sound is radiated when vehicle passes over the expansion joint of the bridge and span sound is radiated when vehicle passes over bridge or viaduct span.

Chapter 1 Introduction and state-of-the-art literature survey

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Bridge expansion joints have a function to permit static and dynamic deformations of bridge deck due to various external effects, such as thermal changes, traffic loading, wind and earthquake etc. Requirements for expansion joints have been more sophisticated recently with the development of new bridge design and construction technology. Bridges with long spans or continuous-type spans require expansion joints bridging greater gaps than ordinary single-span bridges. Curved bridges and base-isolated bridges require expansion joints coping with multi-dimensional movements of bridge decks. There are several types of the expansion joints used in practice. Modular bridge expansion joints

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have capability of allowing greater movements in translation and rotation than ordinary expansion joints, such as finger type. Moreover, modular joint has a water and debris

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drainage facility so that superstructure and substructure beneath the expansion joint can be protected. Modular expansion joint are commonly used for expansion joint movement above 100 mm (Crocetti et al. 2003). Because of its several advantages over other types

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of the expansion joint, its use has been increased recently. However, it has been recognized that the modular bridge expansion joints may cause noise during vehicle passbys, which sometimes causes localized environmental problem.

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1.2 Noise from modular expansion joints

A typical modular expansion joint consists of a set of several parallel steel I-beams

PY

(referred to as middle beams in this thesis) aligned perpendicular to the bridge longitudinal axis, which is supported by a set of several parallel steel H-beams (referred to as support beams) in parallel with the bridge axis. In some other countries, middle beam has been referred as center beam or lamella. This joint structure allows dividing total movement

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required for the joint into small movements required to be accomplished by each gap between the middle beams. A cavity is constructed beneath the expansion joint for the maintenance purpose. The inspection of the expansion joint and replacement of damaged parts of the joint can be done from the cavity without obstructing the traffic on the road surface. Furthermore, there is a rubber sealing between adjacent middle beams so that water and debris can be drained out protecting superstructure and substructure beneath the joint. Because of these advantages, the application of modular bridge expansion joint to road bridges has been increased these days. However, it has been recognized that modular joints may generate and radiate loud noise during vehicle pass-bys, (Ramberger, 2002). It was reported in a study by Ancich et al. (2004) that noise generated by the

Chapter 1 Introduction and state-of-the-art literature survey

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impact of a motor vehicle tire on a modular expansion joint can propagate long distance and was audible up to 500 meters from a bridge in semi-rural environment.

Goroumaru et al. (1987) predicted the low frequency noise generated from vibrating highway bridge having rubber and finger type expansion joint by carrying out the sound and vibration measurements. They concluded the propagation characteristics of the joint sound as point sound source and that of span sound as line sound source. There have been some reported studies on noise and vibration of modular bridge expansion joints.

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Fobo (2004) have reported some control measures to reduce the noise from the expansion joint. These control measures were mainly adopted from the manufacturers of

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the modular expansion joint. A study by Barnard et al. (1997) suspected the noise caused from the modular expansion joints due to the acoustic resonance of the cavity beneath the joint. Ancich et al. (2004) discussed the possibility of interaction of structural modes of the

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joint and acoustic modes of the cavity located beneath the joint by comparing the natural vibration modes of the joint structure and acoustic modes of the cavity. They reported the effectiveness of Helmholtz absorber installation into the cavity walls beneath the joint in noise reduction. This study suggested that the noise generation and radiation mechanism

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around the joint involved possibly both parts of the bridge structure and joint itself as it is unlikely that there is sufficient acoustic power in the simple tire impact to have the noise

PY

level affecting the surrounding environment.

Matsumoto et al. (2007) investigated noise generation mechanism of modular joint by conducting the experimental study with full scale joint model. They also investigated the

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effectiveness of some control measures to reduce the noise. Another study by Ravshanovich et al. (2007) discussed the experimental results of full scale joint model used by Matsumoto et al. (2007) with the help of finite element modal analysis results of the joint structure. It was concluded from these studies that the noise generated from the top side of the joint mainly in the frequency range of 500-800 Hz may be related to resonances of the air gap formed by rubber sealing in between two adjacent middle beams and car tire. The noise generated from the bottom side of the joint mainly below 500 Hz was attributed to the vibration of the joint structure, such as the middle beams. As an example, the sound pressure inside the cavity beneath the expansion joint and on the road surface during car pass-bys experiment in Ravshanovich et al. (2007) is shown in Fig.

Chapter 1 Introduction and state-of-the-art literature survey

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1-1. The acceleration response of the middle beam in lateral and vertical direction is shown in Fig. 1-2.

(b)

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(a)

Fig. 1-1: Sound pressure spectra: (a) inside the cavity beneath the joint and (b) above the

PY

RI

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joint on road surface (Ravshanovich et al., 2007)

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Fig 1-2: Acceleration spectra in one of the middle beam: (a-1) acceleration in lateral direction and (a-2) acceleration in vertical direction (Ravshanovich et al., 2007)

Since vibrations of the joint structure appeared to be related with the noise generation from the bottom side of the joint, understanding of the dynamic characteristics of the joint structure is useful to develop effective measures for the reduction of the noise. There have been few studies on the dynamic response of the modular expansion joint to traffic loading, although the objectives of these studies were to investigate the durability of the joint structure exposed to repeated dynamic loadings due to vehicle pass-bys. In an analytical study by Steenbergen (2004), a mathematical model of an expansion joint was developed to identify the dynamic characteristics of the expansion joint including frequency response function and dynamic amplification factors. The importance of

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dynamic amplification factors in the design of the expansion joints was highlighted. Ancich et al. (2006) have reported the dynamic anomalies of modular expansion joint installed on a bridge. Vibration measurement in the expansion joint in this study showed that most of the measured vibration was in a narrow range of frequencies between 50 and 120 Hz. In addition, measurements of static and dynamic strain were carried out to assess the fatigue performance of the expansion joint. In addition to the noise problems fatigue cracking that has been observed on the modular joints is another problem of modular expansion joint. Fatigue cracking has significantly reduced the life of the expansion joint

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system. Fatigue performance and dynamic characteristics of the modular type expansion joint are reported in Roeder et al., (1994, 1998), Dexter et al., (2001) and Crocetti et al.,

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(2003). The overall durability of modular expansion joint has been studied in detail in Dexter et al. (2002).

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1.3 Noise from highway and railway bridges

Recently, the noise radiated from elevated structures has become one of the main sources of road traffic noise around elevated motorways. There are several types of bridges, viaducts for motorways. With the advancement in the design and construction

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technology, slender structures are chosen and, therefore, steel-concrete composite bridges and viaducts are common these days. Due to the slenderness of the structural components in steel-concrete composite bridges, the radiated noise level due to the

PY

vibration during vehicle pass-bys have been observed more as compared to the concrete only or masonry bridges (Walker et al.,1996). Similar noise problems are reported from the railway bridges and viaducts as well. The noise generated and radiated from these

CO

bridges and viaducts is called elevated structure noise. The embarrassment of the public due to traffic noise is an ever increasing problem for designers of roads and bridges. In case of bridges, noise emission is not only directed to above the carriageway, but also in direction below the carriageway. Noise is also intensified due to vibration of the bridge structural components. Particularly cumbersome are noise emissions due to impact, like they usually occur on a non-smooth carriageway and on expansion joints. As compared to the railway bridge structure borne noise prediction study, the wealth of research on highway elevated structure noise is less. In general, the traffic noise measured near a highway bridge depends on the following factors: (1) The vehicle noise including the engine and exhausts noise, (2) rolling noise emitted by the interaction between the road and tire, and (3) bridge vibration noise due to dynamic response of the

Chapter 1 Introduction and state-of-the-art literature survey

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bridge caused by moving vehicles. The bridge vibration noise is generated by the vibrating surface of bridge deck as well as the girders which compresses and expands the air alternatively and propagates the acoustic energy to the acoustic medium. Therefore, the acoustic energy radiated from the bridge deck and girders depends up on the road surface roughness, the vehicle speed and weight, shape and size of the deck and girders, materials of construction and stiffness degradation of the bridge etc.

The noise emitted by bridge or viaduct under traffic can, in principle, be calculated using

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finite element method or boundary element method that take into account of the geometry of the bridge. However, as the excitation frequency increases, the wavelength of vibrating

TE

modes in the structure decreases. To properly model high-frequency vibrations, either the order of shape functions in FEM or BEM must be increased or the size of mesh decreased. Therefore, finite element or boundary element models are disadvantageous for performing

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accurate high-frequency analysis because they become too large for efficient application. In addition, the traditional FEM and BEM are essentially deterministic analysis techniques. These methods require all the data for a problem to be known exactly. At low frequencies, data such as material properties and joint behavior are reasonably well known and the

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solution is not highly sensitive to their typical variations. However, at high frequencies, the required data for structural dynamic problems is uncertain, and thus the solution is highly

PY

sensitive to data variations. Therefore, at high frequencies, the statistical approach for analyzing structural and acoustic responses is more appropriate than the deterministic one. Statistical Energy Analysis (SEA) has become a widely accepted technique for modeling high frequency dynamic responses of vibro-acoustic systems. This technique

CO

has been extensively used for the noise and vibration analysis of engineering structures for medium to high frequency range.

Measurement data in the previous studies shows that steel plate girders may be the main source of elevated structure noise. Recently, problems with elevated structure noise in audio-frequency range around elevated motorways supposed to be caused by the vibration of structural elements of elevated motorways with steel plate girders are reported by Sakagami et al. (1998).They have carried out a study in which sound field radiated by a

baffled elastic plate of infinite length, which is a simple model of a web plate of steel plate girder, is analyzed theoretically. The results obtained from the study suggest that the steel plate girder can be the source of sound around bridges. The effect of plate parameters on

Chapter 1 Introduction and state-of-the-art literature survey

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noise radiation is also discussed in the study. A study by Michishita et al. (2000a) modelled the web of the steel plate as infinite unbaffled elastic strip and analyzed. The method of equivalent source was used in the analysis and it was confirmed that the calculated characteristics of radiated sound pressure from an infinitely long elastic plate show similar tendency to the main feature of measured elevated structure noise, and concluded that the steel plate girder can be the main source of elevated structure noise around the motorway. Michishita et al. (2000b) have carried a study to see the effect of deck of the bridge on sound radiation characteristics of the girders. Furthermore, the

D

variation in noise characteristics due to varying heights of a plate girder is also studied. The study showed that the concrete slab does not affect significantly the sound field

TE

radiated by a plate girder at most receiving points in the range of 500-1000 Hz which is the frequency range of main interest in the study on the elevated structure noise. There was another study from the same group of people, Michishita et al. (2003) on the effect of

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sound field of plate girders due to the parallel placing of plate girders in the bridge construction. Equivalent source method was employed during the analysis. It was found from the study that adjacent plate girder affects significantly the sound field radiated by

plate girder.

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the plate girder. The sound pressure is increased by the reflection due to adjacent steel

The noise radiation characteristics from a vibrating structure are important from noise

PY

control point of view. It has been shown that the use of vibration modes, which are generally used in vibration analysis and control, cause some difficulties in sound radiation analysis. For this reason, more appropriate concept called radiation modes, which can

CO

represent both vibration and radiation behavior has been utilized by Theerapong et al. (2004) for the identification of low frequency noise behavior in highway bridge. Imaichi et al. (1982), has analyzed the infra-sound radiation from a bridge by applying lifting surface technique. The relation between the bridge oscillation and corresponding infrasound radiation was made clear quantitatively. In the study by applying the method of estimation of infrasound radiation from Kazuno Bridge in Japan, it was shown analytically that the SPL around the bridge may reach about 90 dB which is near the measured result. Lee et al. (2005) have carried out the 3-D numerical analysis to predict the dynamic response and sound radiation characteristics of plate girder highway bridge considering the moving loads. They have also studied the coupling effect between the vibration noise and vehicle noise, in terms of the contribution of bridge vibration to the noise level. Augusztinovicz et al. (2006) used hybrid method consisting of both in situ experiments and numerical

Chapter 1 Introduction and state-of-the-art literature survey

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calculation to predict the noise radiation of a tram track to be installed onto an existing steel bridge. The vibro-acoustic response of the bridge was investigated by means of finite element method (FEM), boundary element method (BEM) statistical energy analysis (SEA) calculations. The limitation of each method for noise prediction was also discussed. Conventional FE and BE methods were able to characterize the whole system up to 100200 Hz only, while important noise components were to be expected between 500 and 2000 Hz.

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There have been several studies to understand the vibro-acoustic behavior of railway bridges. In some of the studies, low frequency noise generated and radiated has been

TE

analyzed by deterministic approaches like finite element method (FEM) or boundary element method (BEM). In other studies in which their interest was from low frequency to high frequency, statistical energy analysis (SEA) was utilized. Walker et al. (1996)

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investigated the noise contribution to total noise from the wheel/rail noise, and noise radiated from the bridge structure itself by finite element analysis, simple energy based prediction methods as well as from in-situ noise measurements and they found the good correlation between the predicted values and measured values. Janssens et al. (1996)

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have developed a simple but very effective analytical model based on noise path analysis for the estimation of noise levels due to train crossing the steel bridge. Comparisons with measurements on a number of typical bridges show very good agreement. They have

PY

concluded that the developed model can be used in design phase of bridges to reduce the overall noise level generation. Odebrant (1996) investigated the noise reduction methods to reduce the noise radiation from the railway bridges and were able to reduce the noise

CO

by 10 dB (A) by isolating the structural borne-sound and by using the sound absorbers on the lower side of the bridge deck. In relation to the noise prediction, the vertical mobility of steel girder of bridges was investigated in the study of Bewes et al. (2003). In this study, the driving point mobility of an infinite I-section beam within three different frequency ranges was discussed and its validity in application as an approximation to average point mobility of finite beam was carried out. Finite element method as well as boundary element method was used for this purpose. Wang et al. (2000) have used the resilient base plates on the railway bridge with the aim of reducing the noise level and they were able to reduce the noise level by 6 dB (A). Structure-borne noise and its dominant frequency range of a concrete viaduct were examined from experimental results as well as by finite element analysis in Ngai et al. (2002). It was found that peak level of structureborne sound was due to the vibration of concrete structure and dominant frequency range

Chapter 1 Introduction and state-of-the-art literature survey

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of noise radiation was between 20-120 Hz. Finite element method as well as simplified prediction model was used to minimize the direct and structure-borne train noise during the design phase of railway viaduct in Hong Kong in a study by Crockett et al. (2000).

As reported in Wittig (1983), the maximum A-weighted noise level during the passage of train on bridges in USA often exceeds 100 dB (A) at 10 m and at least for the older transit systems. He did the review of railway elevated structure noise problem in USA. The objective of the study was to develop the noise rating criteria, a noise impact survey of

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elevated structures in USA and to develop the analytical models to predict the noise from the elevated structures. Transmission of vibration from the rail to the various components

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of the bridge structure was calculated by using Statistical Energy Analysis (SEA) technique and noise radiation was estimated. In a study by Kurzweil, (1977), firstly, noise prediction and control techniques of railway bridges was reviewed. Then a new analytical

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model using SEA was introduced for the prediction of noise from railway bridges and viaducts. Remington et al. (1985) developed an analytical model based on SEA for the prediction of noise generation and radiation from a steel girders railway bridge. Also the effectiveness of some noise control measures was also investigated. Hardy et al. (1998)

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have used SEA for the noise prediction in railway bridges and have showed that reducing the vibration input or providing baffles around the structure is likely to be the most

PY

effective way of reducing the noise from a bridge. Harrison et al. (2000) have presented a rapid calculation method based on a particular form of statistical energy analysis (SEA). This method takes into account of the origin of the noise and provides a framework for the calculation of the propagation, dissipation and final radiation of vibration energy as noise.

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They have also been able to calculate the noise contribution of different structural components i.e., deck, girder web, girder flanges etc. of steel-concrete bridge to the total noise. Harren et al. (1999) have applied different numerical tools including AutoSEA

based on SEA to predict then noise radiation from railway bridges. A rapid calculation model for prediction of vibration flow from the rail to the bridge and noise radiation from the bridge has been presented by Bewes et al. (2006).

1.4 Summary To control the noise generated and radiated from the modular expansion joint, the mechanism of the noise generation and radiation need to be understood well. Although, the noise generation mechanism of the modular expansion joint have been studied

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previously as explained in the preceding section, noise generation and radiation mechanism inside the cavity beneath the joint has not been fully understood. Recent studies by Matsumoto et al. (2007) and Ravshanovich et al. (2007) attributed the peaks in noise response spectra inside the cavity to the peaks in the vibration response spectra of the modular expansion joint mainly the middle beams. However, there were some peaks on the noise response spectra which did not have corresponding peaks in vibration response spectra and vice-versa. The expansion joint is a vibrating structure and cavity beneath the joint has certain acoustic characteristics. To understand the noise generation

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and radiation mechanism inside the cavity, joint and cavity need to be considered as a system and possible interaction of vibration of the joint with the acoustic characteristics of

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the cavity need to be investigated. Vibro-acoustic analysis of the joint-cavity system is therefore indispensable.

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In actual situations, the cavity constructed beneath the modular joint has its openings. Noise generated from the bottom side of expansion joint will be radiated from the openings of the cavity to the outside environment. There have been no reported studies on the characteristics and mechanism of noise radiation from modular expansion joint-

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cavity system to the outside environment. Need of well understanding of the noise radiation characteristics of the expansion joint-cavity system to the outside environment,

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for noise control has motivated further research.

Modular expansion joint is loaded heavily by the wheels of passing vehicles and therefore subjected to heavy dynamic impact loading causing large vibration of structural

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components of expansion joints. Since the modular joint is connected with bridge span, vibration power transmission occurs from the modular joint through its support beams to the span of the bridge causing vibration in the bridge. This vibration power flow to bridge span ultimately causes noise radiation from the bridge. This problem will be exacerbated if the connected bridge is steel-concrete type which can radiate noise efficiently in broad frequency range. Therefore, understanding of possible vibration power flow from the joint to the bridge is necessary to reduce the vibration power flow and ultimately reduce the noise radiation. Though there have been several studies on the vibration power flow in railway bridges from the rail to the bridge and noise radiation from the bridge, no reported studies on the vibration power flow analysis from the modular joint to the bridge have

Chapter 1 Introduction and state-of-the-art literature survey

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been found except by Ravshanovich (2007) who suspected the possible vibration power flow from the modular joint to the steel-concrete bridge. 1.5 Research objectives and methodology As described in the previous section, the noise generation and radiation mechanism inside the cavity of the joint-cavity system is not fully understood. The noise radiation mechanism and characteristics of the joint-cavity system to the outside environment which is important from environmental noise control point of view have not been studied yet. The

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possible vibration power flow from the modular expansion joint to the connected bridge during vehicle pass-bys and noise radiation from the bridge in addition to the noise

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radiation from the joint-cavity system is not understood yet. Therefore, to solve these above mentioned problems the objectives of this study have been:

the joint-cavity system

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1. To understand the noise generation and radiation mechanism inside the cavity of

2. To understand the mechanism and characteristics of noise radiation from the joint-

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cavity system to the outside environment

3. To understand the possible vibration power flow from the modular expansion joint

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to the connected bridge and noise radiation from the bridge

To fulfill the above mentioned first two objectives, numerical study was conducted by

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using finite element method (FEM) and boundary element method (BEM). FEM was used for dynamic response analysis of the expansion joint and BEM was used to analyze the acoustic field inside as well as outside of the joint-cavity system by using the vibration response of the expansion joint as velocity boundary condition. This approach is known as FEM-BEM approach. To fulfill the third objective, a simplified approach using both FEM and statistical energy analysis (SEA) was utilized. FEM was used to estimate the dynamic response of the modular joint and SEA was used to estimate the vibration response of the bridge.

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1.6 Thesis organization This thesis is organized into six chapters. Chapter 1 provides introduction and state-ofthe-art literature survey to this effort. The objectives of this study are listed at the end of the chapter.

Chapter 2 investigates the noise generation and radiation mechanism inside the cavity of the full scale model of modular joint. FEM model of the expansion joint was developed

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and velocity response to the dynamic loadings was estimated. Boundary element model of the joint cavity system was developed and sound field inside the cavity was analyzed. The

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numerical results were discussed with impact testing experimental results of sound and vibration.

Chapter 3 is about the numerical investigation of noise radiation of the expansion joint-

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cavity system to the outside environment. A modular joint installed between prestressed concrete bridges is considered. Numerical analysis was conducted by FEM-BEM approach to analyze the sound field inside and outside of the joint-cavity system. Vehicle

results.

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pass-bys experiment data of noise and vibration were used to discuss the numerical

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Chapter 4 considers an analytical approach to understand the possible vibration power flow from modular expansion joint to the connected steel-concrete non-composite bridge. FEM was utilized for the dynamic response analysis of the expansion joint. Vibration

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power input to the bridge from the expansion joint was estimated. SEA was then utilized for estimation of average vibration response in the bridge. Noise radiation from the bridge was then estimated at field points around the bridge. Analytical results were discussed with the help of vehicle pass-bys experiment results.

Chapter 5 is about general discussion. Chapter 6 contains general conclusions drawn from the thesis. Some recommendations have been made for future studies at the end of the chapter.

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint 2.1 Background and objective Loud noise generated from the modular expansion joint was mainly due to two sources

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(Matsumoto et al.,2007 and Ravshanovich et al.,2007). First was the noise generated from the joint on the road surface due to the resonance of the air gap formed between the

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vehicle tire and rubber sealing between the middle beams of the joint. Second noise source was from the bottom part of the joint due to the possible interaction of vibration of joint structures with the acoustic characteristics of the cavity beneath the joint. The

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objective of this study was to understand the noise generation and radiation mechanism within the cavity of the joint-cavity system. A full scale model of expansion joint which was set up within the factory of the manufacturer was considered in this study. Vibro-acoustic analysis of the joint-cavity system was conducted by finite element method (FEM) -

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boundary element method (BEM) approach: dynamic response of the joint structure was analyzed by FEM and acoustic sound field inside the cavity was analyzed by BEM. Additionally, the acoustic modal parameters of the cavity were obtained from FEM and

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BEM. Sound and vibration results from impact testing experiment were compared with numerical results and noise generation and radiation mechanism inside the cavity was

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discussed.

2.2 Description of full scale model of expansion joint Figure 2-1 shows the full scale joint model used in the previous studies. This model was used in the previous studies Matsumoto et al. (2007) and Ravshanovich et al. (2007). The joint model was set up in the compound of the Kawaguchi Metal Industries Co., Ltd. in Saitama, Japan. A cavity was constructed underground beneath the joint to represent the space in real bridge expansion joint. The joint model consisted of three middle beams that were supported by four support beams. Control mechanisms consisting of a steel plank (referred to as control beam) and rubber springs installed between the top surface of a control beam and the bottom flange of a middle beam were designed to prevent excessive displacement between adjacent middle beams. There were steel frames placed

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vertically to accommodate a rubber springs between the frame and the bottom flange of a support beam and a polyamide bearing between the bottom flange of a middle beam and the top flange of a support beam. There were polyamide bearings beneath the ends of the support beams and rubber springs placed on top of the ends of support beams. Additionally, the modular bridge expansion joint has a drainage capability with the installation of a rubber sealing into each gap between two adjacent middle beams. The joint model was mounted on top of a test cavity constructed under the ground level to represent the cavity beneath the joint between adjacent bridge girders in real bridges (Fig.

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2-2). There was an opening at each end of the cavity. The wall of the cavity was covered

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by Styrofoam that provided sound absorption within the cavity to some extent.

Fig. 2-1 Plan and cross section of full scale test joint model

Fig. 2-2 Cross-sectional views of the cavity beneath the joint

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2.3 Impact testing experiment of sound and vibration In order to identify the dynamic characteristics of the joint, impact testing was carried out in the previous study by Ravshanovich et al. (2007). The vertical and lateral acceleration of the middle beams were measured at different locations and, additionally, the sound pressure was measured inside the cavity beneath the joint at 1.0 m below the top surface of the centre of the joint. Several measurements were carried out by impacting the middle beam from the top at different locations. The details of the experimental procedure and

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the results of experimental modal analysis were presented in Ravshanovich et al. (2007). A set of time histories of accelerations and sound pressure are shown in Fig.2-3. Transfer

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function was calculated by using the impact force measured by an impact hammer as input and the acceleration of the beam or the sound pressure inside the cavity as output. (b)

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G H

(a)

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(c)

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Fig. 2-3 Time series data of sound and vibration measurement during impact testing. (a)

Lateral acceleration on middle beam, (b) vertical acceleration on middle beam and (c) Sound pressure inside the cavity beneath the joint

2.4 Vibro-acoustic analysis of joint-cavity system by FEM-BEM approach 2.4.1 Analytical procedure In the vibro-acoustic analysis conducted in this study, firstly, a finite element model of the full scale joint model was developed so as to obtain the modal parameters of the joint. The velocity responses of the joint to different point loadings were then calculated by modal superposition so as to compare the numerical results with the experimental results from the impact testing. Finally, the velocity response obtained was used as the boundary

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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condition in the boundary element analysis of acoustic field in the cavity beneath the joint. The flow of the numerical analysis that was conducted is shown below in Fig.2-4.

FE modal analysis of the expansion joint

Velocity response of the joint to point loadings by FEM

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Vibro-acoustic analysis by FEM-BEM

Impact testing experiment results

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Sound field analysis of the joint-cavity system by BEM

Results interpretation

Acoustic modal analysis results of the cavity

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Fig. 2-4 Flow chart of numerical analysis conducted in the study

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2.4.2 Velocity response of the expansion joint 2.4.2.1 Finite element modal analysis of the joint Figure 2-5 shows the FE model of the joint developed in this study. The ANSYS 10.0 was used in the analysis described in this section. The middle beams, the support beams, the

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control beams and the frames were modeled with shell elements (SHELL63) so that the vibration response of the joint could be transferred directly to the boundary element model as velocity boundary condition. Bearings and rubber springs were simplified and modeled with spring elements (COMBIN14) as in the FE model developed in the previous study Ravshanovich et al. (2007). Geometric and material properties of different components in the FE joint model are shown in Tables 2-1 and 2-2. The size of the FE mesh in the analysis was 40 mm for shell elements so that the dynamic behavior of the joint could be investigated in the frequency range up to 1600 Hz according to the rule of thumb of 6 elements per wavelength.

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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Middle beam

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Spring

Support beam

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Bearing

Frame

(b)

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(a)

Fig. 2-5 FE model of the expansion joint: (a) 3-D view; (b) Cross-section view

Cross-section (mm) 2

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Table 2-1 Cross-section, size and materials for steel members

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Young’s Modulus (N/m ) Poisson ratio

Middle beam I-shape, 140 x 80 x 21 x 12.5 2.06 x 1011 0.3

Support beam H-shape, 120 x 90 x 20 x 12 2.06 x 1011 0.3

Control beam Rectangular, 80 x 25 2.06 x 1011 0.3

Table 2-2 Spring constants of bearings and springs

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Middle beams Polyamide Rubber spring bearing 8 5.67 x 10 1.50 x 106 8 2.02 x 10 5.07 x 105 6 3 3.94 x 10 9.76 x 10 5 2.15 x 10 5.56 x 102

Vertical (N/m) Lateral (N/m) Torsion (Nm) Rotation (Nm)

Support beams Polyamide Rubber spring bearing 8 5.97 x 10 5.83 x 106 0 1.97 x 106 4 0 3.89 x 10 0 2.27 x 103

Control beams Control spring 3.52 x 105 1.19 x 105 3 2.32 x 10 1.34 x 102

Modal analysis was conducted to identify the dynamic characteristics of the joint i.e. natural frequencies and mode shapes of the joint. Mode shapes were mass normalized during the analysis. There were a total of 211 structural vibration modes identified from the FE analysis in the frequency range between 0-800 Hz. The modal parameters of those structural modes were compared with the modal parameters identified in the impact testing in Ravshanovich et al. (2007). There was a reasonable agreement between the

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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numerical analysis and experiment in vibration modes dominated by vertical bending of the beams, while there were some discrepancies in vibration modes dominated by lateral motions of the joint, including rigid body mode and lateral bending. Some of the numerical mode shapes in lateral and vertical direction are shown later in results and discussion section.

2.4.2.2 Velocity response estimation of the joint

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Velocity response of the expansion joint was calculated with harmonic point loads of unit magnitude in the frequency range from 20 to 400 Hz with an interval of 1 Hz. The point

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loads were applied in the lateral and vertical directions simultaneously at the center of the top flange of the second middle beam, as shown in Fig. 2-6, to represent the loading on the joint during the impact testing experiment. The impact was inclined during the

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experiment. In Fig. 2-6, the circle mark at the centre of the third middle beam represents one of the measurement positions during the impact testing, and the data at this location was used for comparison between numerical and experimental results, as an example. Modal-based forced response analysis was carried out by using the structural modes

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identified as described in the preceding section. All the structural modes below 800 Hz were considered in the analysis, according to Citarella et al.(2007) stating that structural modes up to a frequency approximately double the maximum frequency of interest should

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be considered in the response calculation. In the previous studies, modular expansion joint was found lightly damped with modal damping ratio less than 3% for most of the lateral and vertical bending modes identified up to 500 Hz from the experiment

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(Ravshanovich et al.,2007) and less than 2% for all the first four vertical bending modes identified from the experiment (Ancich et al.,2006). A modal damping ratio of 1% was

assumed for all the modes considered in the analysis. The velocity response was calculated at all nodes of the FE model. The LMS Virtual.Lab Rev 6A was used in the analysis described in this section.

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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Point Loads

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Third middle beam

Y

X

Z

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Fig. 2-6 Position and direction of point loads for velocity response calculation

2.4.3 Analysis of acoustic field

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2.4.3.1 Modal acoustic transfer vector technique in indirect BEM In linear acoustics, the governing equation of time domain acoustics, the wave equations and governing equation of frequency domain acoustics can be derived from linearization of fundamental mass and momentum equations. Because of the linear system, linear

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output-input relationship can be established between the input of an acoustic system the mechanical vibration of structures and the output of an acoustic system, the sound pressure at field points in space. The acoustic transfer vector {ATV (ω )} is the term used

to represent the set of the acoustic transfer functions relating the normal component of velocities {v n (ω )} of the vibrating surfaces to the sound pressure p(ω ) at a single field

point. ATVs depend on the geometry of vibrating surfaces, acoustic treatment of the surfaces, field point position and frequency of interest. ATVs are independent of the structural response and therefore can be used to understand the acoustic performance in different loading conditions without changing the geometry of the structure. ATVs can be expressed by Citarella et al. (2007):

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

p(ω ) = {ATV (ω )}

T

{v n (ω )}

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(2.1)

where ω is the angular frequency. If there are more than one field points present then Equation (2.1) can be rewritten as:

{p(ω )} = [ ATM (ω )]{v n (ω )}

(2.2)

where [ ATM (ω )] is the matrix of ATVs.

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In ATV concept, structural response is to be known a priori. In many cases, structural responses of the mechanical systems are calculated using modal representation. Modal

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acoustic transfer vectors (MATVs) are the modal counterpart of acoustic transfer vectors (ATVs). MATVs express the acoustic transfer function in modal coordinates from a vibrating structure to a field point. In MATVs acoustic contribution from each structural

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mode is accounted. The acoustic response in the field point is then calculated by recombining the MATVs with the corresponding structural modal responses. In a modal response analysis, structural displacement at a given excitation frequency are calculated as a linear combination of structural modal vectors of the vibrating structure. As the

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structural pattern change with frequency, modal participation factors are also frequency dependent. The approach can be explained by: (2.3)

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{u(ω )} = [ Φ ].{MPF (ω )}

where {u(ω )} represents the column vector containing all the structural displacement components, [ Φ ] is the structural mode matrix whose columns are structural modal

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vectors and {MPF (ω )} is the column vector containing modal participation factors. The

Equation (2.3) can be projected to normal direction to the boundary surface and expressed in terms of velocity by:

{v n (ω )} = jω.[Φ n ].{MPF (ω )}

(2.4)

where [ Φ n ] represents the matrix whose columns are composed of structural modal vectors normal to the boundary surface. Now combining equations (2.1) and (2.4)

p(ω ) = {ATV (ω )} . j ω. [ Φ n ] .{MPF (ω )}

(2.5)

p(ω ) = {MATV (ω )} .{MPF (ω )}

(2.6)

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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where {MATV (ω )} = { ATV (ω )} . j ω. [ Φ n ] is the modal acoustic transfer vector and is the ensemble of acoustic transfer functions relating the contribution of individual structural modes to the sound pressure at single field point. Unlike in ATVs, MATVs cannot be computed only from acoustic parameters but in addition require the information of dynamic behavior of structure in terms of structural mode shapes. MATVs are also independent of structural loading. The detail of ATV and MATV can be found in literature Citarella et al. (2007) and Estorff et al. (2003).

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In conventional BEM analysis, the acoustic response is analyzed by solving the system of equations for each loading condition (Wu, 2000). This is time consuming in case of multi-

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load case condition and large and complex acoustic system because the BEM system of equations needs to be assembled and analyzed at each frequency and each load case. Use of ATV and MATV concept is different than that in conventional BEM analysis. As it is

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explained before, in these approaches first the acoustic transfer function is calculated from the vibrating surface to the sound pressure calculation point (field point) for the frequency range considered without taking into account the actual loading condition. Then the acoustic response is calculated for all loading conditions by combining the ATVs or

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MATVs with the normal structural velocity boundary condition vector. Since, acoustic transfer functions are calculations between the boundary elements of the boundary surface and field points. ATV-based analysis will be computationally faster than

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conventional BEM if the size of the boundary element model is large and numbers of field

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points are few.

2.4.3.2 Boundary element modeling of the joint-cavity system

The velocity response obtained in the FE analysis as described in section 2.4.2.2 was used as a boundary condition in BE analysis of the acoustic field in the cavity beneath the joint. The BE model of the cavity developed in this study is shown in Fig. 2-7. The boundary of the cavity was discretized into quadrilateral and triangular elements. The maximum size of acoustic boundary element was set as 100 mm so that the maximum frequency that could be analyzed was 566 Hz according to the rule of thumb of 6 elements per wavelength. The top boundary of the BE model was exactly the same as the FE model of the joint so that the velocity response of the joint could be transferred directly to the BE model. In this analysis, it was assumed that the vibration response calculated from the in vacuo structural modes of the joint calculated separately in the previous

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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section can be used to analyze the acoustic field of the joint-cavity system, i.e., a weak coupling was assumed between the expansion joint and cavity beneath it (vibrations of expansion joint were not influenced by the fluid), according to the criterion to assess the degree of coupling explained by Kim et al. (1999). According to the information received from the manufacturer of the Styrofoam, the sound absorption capacity of the Styrofoam was low in the frequency below 500 Hz. Therefore, the effect of the Styrofoam attached to the walls of the cavity beneath the joint was not considered in the analysis. The bare walls and floor of the cavity were considered rigid in the model. This assumption reduced

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computational cost in the numerical analysis. The sound speed and the density of the air

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Openings

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used in the BE analysis were 340 m/s and 1.225 kg/m3, respectively.

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Joint-cavity boundary

1m

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Measurement point

X

Z

(b)

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(a)

Y

Fig. 2-7 Boundary element model of the cavity: (a) Full 3-D model; (b) Sectional view showing measurement point Modal acoustic transfer vector (MATV) technique in indirect BEM (IBEM) was used to calculate the sound pressure inside the cavity. The field point used in this analysis was at the centre of the cavity and 1.0 meter below the top surface of the joint: this point was equivalent to the measurement location for sound pressure in the impact testing experiment. This method which uses modal acoustic transfer vectors (MATVs) was chosen here instead of conventional BEM analysis technique to save the computational time because there were large number of boundary elements in the BE model of the jointcavity system and we were interested to calculate the sound pressure at a single field

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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point which was equivalent to the measurement point inside the cavity. The frequency range considered for the acoustic analysis was 20-400 Hz. The LMS Virtual.Lab Rev 6A was used for the acoustic analysis.

2.5 Acoustic modal analysis of the cavity beneath the joint 2.5.1 Finite element method

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Finite element analysis of the cavity beneath the joint shown in Fig.2-2 was conducted to obtain the acoustic modal parameters of the cavity. The FE model of the cavity developed

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is shown in Fig.2-8.The acoustic space inside the cavity was modeled by cubical fluid element of mesh size 130 mm so that acoustic modes having natural frequencies up to 435 Hz could be calculated using the rule of thumb of 6 elements per wavelength. The openings on both sides of the cavity were modeled by applying impedance boundary

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condition: characteristic acoustic impedance of 416 kg/m2s was applied to model these openings (Fig.2-8). Some of the acoustic modes of the cavity identified from the modal

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analysis are shown later in Results and discussion section.

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Openings

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Expansion joint boundary

Y X

Fig. 2-8 FE model of the cavity for acoustic modal analysis

Z

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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2.5.2 Indirect boundary element method (IBEM)

Acoustic modal parameters of the cavity beneath the joint were also obtained by boundary element method. BE model of the cavity developed and shown in Fig.2-7 was used in the analysis. From the acoustic modal analysis results by FEM, though these are not shown here because of large numbers of modes were identified; it was observed that most of the acoustic modes had one of the antinodes at the bottom corners of the cavity. A acoustic point source of unit pressure amplitude was applied on the boundary element

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model of the joint-cavity system (Fig.2-7) at one of the bottom corner of cavity so that the maximum number of modes could be excited. Sound field of the cavity was analyzed by

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indirect boundary element method (IBEM) at the frequency step of 3 Hz in the frequency range of 20-400 Hz and sound pressure response was calculated inside the cavity at the

2.6. Results and discussion

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measurement point explained before (Fig. 2-7).

2.6.1 Acoustic modal parameters of the cavity

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Figure 2.9 shows the frequency responses of the sound pressure at measurement point to the point acoustic source applied at one of the bottom corner as explained in Section 2.5.2. Also shown in the figure are the natural frequencies of acoustic modes of the cavity

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identified from FE modal analysis for comparison with peak frequencies in the frequency response. In the figure, there are several dominant peaks of sound pressure responses at different frequencies. These peak frequencies are due to the excitation of the acoustic

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modes to the applied point source. Also, from the responses calculated inside the cavity, it can be seen that acoustic contribution from the particular acoustic mode at field point depends whether the field point is near to node or anti-node of the acoustic mode. That can be seen in the figure in which there is very small sound pressure at frequencies for example 160 Hz, 242 Hz and 278 Hz etc even though there were acoustic modes of the cavity. Some of the acoustic modes of the cavity and their natural frequencies identified by FEM are shown in Fig.2-10. The peak in the sound pressure response around these frequencies can also be observed in BEM results in Fig.2-9. These results also show the reliability of FE model of the cavity and BE model of the joint-cavity system.

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Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

Fig. 2-9 Frequency response of the sound pressure inside the cavity to acoustic point

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, acoustic mode obtained by FEM

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source. Key:

f=105.34 Hz

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f=65.65 Hz

f=124.21 Hz

f=172.41 Hz

Fig. 2-10 Some of the acoustic modes of the cavity obtained from FEM

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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2.6.2 Velocity response on the expansion joint 2.6.2.1 Velocity response in lateral and vertical direction

Figure 2-10 shows the mobility frequency responses functions calculated for lateral velocity at the centre of third middle beam (i.e. the location shown in Fig.2-6). The peaks in the velocity responses of the joint were attributed to vibration modes of the joint structure. For example, the peaks at 112 Hz and 161 Hz in the numerical results were interpreted as the excitation of lateral bending modes of the middle beam at 111.97 Hz

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and 161.49 Hz, respectively. As shown in Fig.2-12, these modes had one of the anti-

PY

RI

G H

112

TE

nodes at the loading point, i.e., the centre of the second middle beam.

Fig. 2-11 Calculated lateral mobility of the expansion joint at measurement point on

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second middle beam

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f= 111.97 Hz

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Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

f= 161.49 Hz

direction

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Fig. 2-12 Some of the vibration modes of the joint with dominant vibration in lateral

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Figure 2-13 shows the mobility calculated at the measurement point in the vertical direction in the numerical analysis. The peaks in the velocity responses of the joint in the vertical direction were attributed to the excitation of vertical modes of the joint structure.

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For example, the peaks in the numerical results at 160 Hz, 189 Hz, and 241 Hz were due to the excitation of structural modes at 160.35 Hz, 189.35 Hz and 241.59 Hz, respectively. These modes had dominant vertical bending of the middle beams with one of the anti-

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nodes at the loading point. The structural modes of the joint at 160.35 Hz and 189.35 Hz with dominant vertical bending of middle beams are shown in Fig.2-14.

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TE

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Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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RI

G H

Fig. 2-13 Calculated vertical mobility of the expansion joint at measurement point

f=160.35 Hz

f=189.35 Hz

Fig. 2-14 Some of the vibration modes of the joint with dominant vibration in vertical

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direction

2.6.2.2 Comparison of numerical and experimental results

Figures 2-15 and 2-16 compare the mobility frequency response functions calculated for the lateral and vertical velocity responses respectively at the centre of the third middle beam (i.e., the location shown in Fig. 2-6) with the corresponding experimental data (Ravshanovich et al. 2007). Two sets of experimental data are presented so as to show the repeatability of the measurement: the measurement appeared to be highly repeatable at frequencies below 300 Hz in lateral direction and for all frequencies considered in vertical direction. Although similar trend could be observed in the experimental and numerical results of the velocity responses, there were discrepancies in the magnitude

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

Page29

and the peak frequencies of the response. Possible reasons include errors in the modeling of the expansion joint, for example, errors in the spring constants used for rubber bearings and springs, and errors in the damping ratio chosen in the response calculation. Also, the loadings exerted by the impact hammer might not have been applied

RI

G H

TE

D

to the middle beam at 45 degrees from the road surface as in the numerical analysis.

Fig. 2-15 Velocity responses at the measurement point in lateral direction. Key: , Experimental-2;

, Numerical

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PY

Experimental-1;

,

Fig. 2-16 Velocity responses at the measurement point in vertical direction. Key: Experimental-1;

, Experimental-2;

, Numerical

,

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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2.6.3 Sound pressure response 2.6.3.1 Sound pressure response inside the cavity

The frequency response function calculated numerically between the sound pressure in the cavity and the input force is shown in Fig. 2-17. Additionally, the natural frequencies of the vibration modes of the joint and the acoustic modes of the cavity identified by FE analysis are indicated in the figure for comparison with the peak frequencies in the frequency response functions. There are several peaks observed in the frequency

PY

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G H

TE

of the joint and/or acoustic modes of the cavity.

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response of sound pressure which could be attributed to the excitation of structural modes

, structural

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Fig. 2-17 Sound pressure responses at measurement point (Numerical).Key: mode;

, acoustic mode

2.6.3.2 Sound generation and radiation mechanism inside the cavity

Sound generation from the interaction of vibrating structure with the enclosed acoustic cavity discussed in previous studies (Kim et al., 1999, Dowell et al., 1977, Cabeli, 1985 and Luo et al., 1978) and may be similar to sound generation from the joint model and the cavity beneath it investigated in this study. According to those previous studies, the peaks in the frequency response function calculated for the sound pressure in the cavity, as observed in Fig.2-17, may be attributed to three different mechanisms: (1) major contribution of a structural mode(s) of the joint with possible minor contribution of acoustic mode(s) of the cavity, (2) significant contribution from both structural mode(s) of the joint

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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and acoustic mode(s) of the cavity, and (3) major contribution of acoustic mode(s) of the cavity with possible minor contribution of structural mode(s). For example, the peak in the frequency response function for the sound pressure at around 160 Hz in Fig. 2-17 may be attributed to the resonance of the structural mode at 160.35 Hz (vertical bending mode) and the acoustic modes at 161.10 Hz and 161.71 Hz. The structural mode at 160.35 Hz and acoustic modes are shown in Fig.2-18. The excitation of this structural mode was observed in the velocity response in the vertical

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direction shown in Fig. 2-13. The acoustic mode at 161.10 Hz shown in Fig. 2-18 had low amplitude at the boundary with the joint so that this acoustic mode may have had little

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interaction with the structural mode. Although the acoustic mode at 161.71 Hz shown in Fig. 2-18 may have had higher possibility of interaction with the structural mode, the amplitude around the measurement point of this acoustic mode was less. Therefore, the

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peak in the frequency response function for the sound pressure at around 160 Hz may be caused mainly by the excitation of the structural modes with some minor effects from the acoustic modes. The less acoustic contribution from these acoustic modes around 160 Hz at the measurement point could be observed in Fig. 2-9 also. Similar discussion could be

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made on the peaks of the frequency response function for the sound pressure at around 112 Hz and 188 Hz where structural modes contribution may be dominant. The sound pressure peak at 268 Hz may be related to the structural mode of the joint at

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267.6 Hz (lateral bending mode) and the acoustic modes at 267.91 Hz and 269.40 Hz. These structural mode and the acoustic modes are shown in Fig. 2-19. Although the velocity response was small at around 267 Hz in Fig. 2-11 because of the small vibration

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of the third middle beam in the structural mode shown in Fig. 2-19, this vibration mode may be excited. The acoustic modes at 267.91 Hz and 269.40 Hz may be able to be excited by structural vibration of the joint because of high amplitude at the boundary with the joint. Therefore, the peak in the frequency response for the sound pressure at around 268 Hz may have been caused by the excitation of both the structural and acoustic modes. For some peaks in the frequency response function for the sound pressure, the contribution may be mainly from the resonance of acoustic modes. For example, Fig. 2-17 shows that there was no structural mode in the frequency range between 367 Hz and 397 Hz, while there were many acoustic modes in this frequency range. The sound pressure peak at around 380 Hz could be due to the resonance of the acoustic modes at 380.38 Hz,

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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381.86 Hz, and 383.58 Hz, along with many other acoustic modes around this frequency. Some of the acoustic modes at around 380 Hz are shown in Fig. 2-20. Significant response of sound pressure around 380 Hz can also be seen in Fig.2-9 which indicates the presence of acoustic modes and their significant acoustic contribution to the

(b)

(c)

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(a)

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measurement point.

Fig. 2-18 Structural and acoustic modes generating sound around 160 Hz: (a) Structural

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RI

mode at 160.35 Hz; (b) Acoustic mode at 161.10 Hz and (c) Acoustic mode at 161.70 Hz

(a)

(b)

(c)

Fig. 2-19 Structural and acoustic modes generating sound around 268 Hz: (a) Structural mode at 267.6 Hz; (b) Acoustic mode at 267.91 Hz and (c) Acoustic mode at 269.40 Hz

(a)

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Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

(b)

(c)

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Fig. 2-20 Acoustic modes generating sound around 380 Hz: (a) Acoustic mode at 380.38 Hz; (b) Acoustic mode at 381.86 Hz and (c) Acoustic mode at 383.58 Hz

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2.6.3.3 Comparison of numerical and experimental results

In Fig.2-21, the numerical result of sound pressure responses is compared with the experimental data measured in the impact testing (Ravshanovich et al. 2007). Two sets of experimental data in Fig. 2-21 show that the measurement appeared to be repeatable at

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frequencies higher than about 120 Hz. Although similar trend could be observed in the experimental and numerical results, there were discrepancies in the magnitude and the peak frequencies of the frequency response functions for several possible reasons. The

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discrepancies in the peak frequencies of the frequency response function calculated for the sound pressure, compared to the corresponding experimental data, may be partly due to differences in the natural frequencies of the vibration modes of the joint between the

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experimental and numerical results as discussed in the previous section. Additionally, there might be minor effects of the Styrofoam in the cavity beneath the joint in the experimental data, which was neglected in the numerical analysis. Spatial variation of the sound pressure around the measurement point appeared to be significant at higher frequencies, according to parametric investigation conducted. The Fig. 2-22 shows the sound pressure responses calculated on ten field points around the measurement point on vertical plane perpendicular to the cavity longitudinal axis and horizontal plane parallel to the cavity longitudinal axis. Eight field points considered on vertical plane were 10 cm away in all four directions from the measurement point. Two more points were on horizontal plane at centre line at 20 cm away on either direction of measurement point. The results show that some discrepancies could be expected mainly at higher frequencies if the field point for the numerical calculation of sound pressure was not identical to the

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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measurement point in the experiment. Error in damping ratio used in the velocity response analysis of the joint might have caused some difference between experimental and numerical results. Figure 2-23 shows the frequency responses of sound pressure at measurement point for two different modal damping ratios of 1% and 7%. The results show considerable effect of damping ratio in the frequency response of sound pressure. The difference between the real damping behavior of the expansion joint tested and value

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G H

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used in the analysis might have caused some errors.

Fig. 2-21 Sound pressure inside the cavity and natural frequencies of structural and acoustic modes. Key:

, structural mode;

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Numerical;

, Experimental-1; , acoustic mode

, Experimental-2;

,

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Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

Fig. 2-22 Frequency response of the sound pressure inside the cavity at several points

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RI

G H

around the measurement point

Fig. 2-23 Frequency response of the sound pressure inside the cavity at measurement point. Key:

, 1 % modal damping;

, 7 % modal damping

2.7 Reliability of discussion based on numerical results

There were differences in the frequency response observed between the numerical results obtained from the model developed in this study and the experimental results from the impact testing in the previous study (Ravshanovich et al. 2007), as shown in Figs 2-15, 2-16 and 2-21. The preceding sections show examples of those differences and discuss

Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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possible causes. Model updating was not attempted in this study because the experimental data available were limited while there were several sets of model properties that can be optimized. It was assumed that reasonable theoretical insights into the noise generation mechanism could be obtained from the model based on the reliable numerical technique with available mechanical properties and assumptions for model properties. The properties of structural vibration modes obtained from the numerical model were compared with the modal parameters identified in the impact testing (Ravshanovich et al. 2007). There was a reasonable agreement between the numerical and experimental

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modal properties in vibration modes dominated by vertical bending of the beams, such as the modes shown in Fig. 2-14. This may partially support the reliability of the discussion

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based on the numerical results obtained in this study. However, there were some discrepancies in vibration modes dominated by lateral motions of the joint, including rigid body mode and lateral bending. A similar conclusion was reported in the previous study

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(Ravshanovich et al. 2007) that compared modal properties identified in the experimental modal analysis with those in the analytical modal analysis with a 3-D frame model. Figure 2-24 shows the vertical velocity responses of the joint measured at the location

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shown in Fig. 2-6 during pass-bys of two types of vehicle (i.e., an ordinary car and a wagon) at two different speeds (i.e., 40 and 50 kmh-1) (Matsumoto et al. 2007 and Ravshanovich et al. 2007). The right wheels of those vehicles passed over the

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measurement and loading point shown in Fig. 2-6. Dominant frequency components in the vertical velocity responses were observed at frequencies below 220 Hz for all the experimental conditions, although there were variations caused by different conditions, as

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expected. Comparison of Fig. 2-24 with Fig. 2-16 shows similar characteristics in the results between the vehicle pass-by experiment and the impact testing, such as peaks at about 155 and 185 Hz and decreases at higher frequencies, although direct comparison cannot be made because Fig. 2-24 includes the effect of loading spectra while Fig. 2-16 shows the frequency responses. Those similarities observed in Figs 2-16 and 2-24 imply that the theoretical insights into the dynamic characteristics of the joint and their interaction with the acoustic characteristics of the cavity obtained from impact testing data could be useful in noise control in real situation.

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Chapter 2 Vibro-acoustic analysis of full scale model of modular expansion joint

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Fig. 2-24 Velocity responses in vertical direction at the measurement point during vehicle pass -bys over control beams. Key: car at 50 kmh-1;

, wagon at 40 kmh-1;

, ordinary

, wagon at 50 kmh-1.

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2.8 Conclusions

, ordinary car at 40 kmh-1;

Vibro-acoustic analysis of the full-scale model of the modular bridge expansion joint was carried out so as to obtain theoretical insights into the mechanism of noise generation

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from the bottom side of the joint. In the numerical analysis, sound field beneath the joint was investigated by FEM-BEM approach in the frequency range 20-400 Hz. Modal analysis was also conducted so as to understand the dynamic characteristics of the joint

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structure and the acoustic properties of the cavity beneath the joint. It was observed that dominant frequency components in the sound pressure generated inside the cavity was due to vibration modes of the joint structure and/or acoustic modes of the cavity. Some dominant frequency components in the sound pressure inside the cavity beneath the joint may be caused mainly by the vibration modes of the joint structure and others may be caused by the excitation of acoustic modes by the vibration modes of the structure. At higher frequencies where the modal density of the acoustic modes was high, there were dominant frequency components in the sound pressure that may be caused mainly by the excitation of acoustic modes of the cavity with minor contribution from vibration modes of the joint structure.

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges 3.1 Background and objective In chapter 2, vibro-acoustic analysis of the full scale model joint was conducted to

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understand the noise generation and radiation mechanism inside the cavity of the jointcavity system. In that case, the cavity beneath the joint was underground. In real bridge

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modular expansion joint the cavity beneath the joint is above the ground and its openings are exposed to the environment. Therefore noise generated from the bottom side of bridge expansion joint will be radiated from these openings to the outside environment.

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The characteristics and mechanism of noise radiation from modular expansion joint-cavity system to the outside environment has not been understood yet. Understanding of characteristics and mechanism of noise radiation from the expansion joint-cavity system to the outside environment which is important from noise control point of view motivated

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further research.

There have been several studies on the noise propagation inside the ducts and noise

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radiation outside from the openings of the ducts. These studies were mainly on cylindrical ducts to understand the noise radiation from the ducted turbo fans of aircraft engine and air conditioning duct etc. Some of the analytical and experimental studies assumed the

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infinite or semi-infinite length of the duct to understand the acoustic radiation from the duct end (Candel, 1973, Homicz et al., 1975, Rice, 1977, Rice et al., 1979, Chapman 1994 and

1996). Geometric theory of diffraction was used to investigate the noise radiation in thin walled axisymmetric cylinder (Keith et al., 2002a, Hocter, 2000) and scarfed cylinder (Keith et al., 2002b). An approximate formulation was derived and sound radiation pattern of cylindrical duct was investigated by Hocter (1999).

Effect of finite length of the duct on sound radiation was investigated by Wang et al. (1984), Hamdi et al. (1981, 1982, and 1986), Malbequi et al. (1996), Shao et al. (2005a, 2005b) and Choi et al. (2006). Boundary integral equation method was utilized to predict the ducted fan engine noise radiation in some studies (Dunn et al., 1999 and Carley, 2003). Morfey (1969) investigated the radiation efficiency of acoustic duct modes and

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 39

concluded that radiation efficiency of duct modes is close to unity above their cut off frequency. Cumings et al., (1980) derived theoretical models to predict the low frequency noise radiation from the vibrating walls of rectangular ducts and have been able to predict the far field directivity of noise radiation.

The objective of this study was to understand the mechanisms and characteristics of noise radiation to the outside environment from an existing modular bridge expansion joint installed in an expressway prestressed concrete (PC) bridge. In this study, the modular

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expansion joint, the cavity and the surrounding sound field were considered as a vibroacoustic system so as to understand the dynamic and acoustic characteristics of the

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system. Frequency responses of this system were calculated with harmonic dynamic loadings applied to the joint as an input and sound pressures in the sound field around the joint as an output. The input harmonic loadings were applied to the joint at locations where

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dynamic loads may be applied during vehicle pass-bys. The output sound pressure was calculated at an arbitrary location. Vibro-acoustic analysis was conducted by using FEMBEM approach similar to the used in Chapter 2 so as to understand the relation between structural vibration and noise generation from the modular expansion joint. FE analysis of

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the joint was conducted to understand the dynamic characteristics and the evaluation of vibration response. The sound field inside the cavity and around the joint was calculated

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from the indirect boundary element method (IBEM) with the vibration response of the joint estimated from the FE analysis as boundary condition. Additionally, the acoustic modal parameters of the cavity were identified from the finite element analysis. The numerical results of both noise and vibration analysis were compared with measurements of noise

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and vibration of the expansion joint during vehicle pass-bys.

3.2 Description of the modular joint between PC bridges Figure 3-1 shows the plan and cross-section of the expansion joint installed between PC girder bridges. The width of the two lane bridge was 12.16 m. The expansion joint length was the same as the width of the bridge, i.e., 12.16 m. The joint was installed in the bridges in a skewed position such that the angle between the bridge axis and expansion joint axis was 760.

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 40

Fig. 3-1 Plan and cross-section of the expansion joint

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The expansion joint consisted of five middle beams that were supported by eight support beams. Control mechanisms consisting of a steel plank (referred to as control beam) and

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rubber springs installed between a control beam and the bottom flange of a middle beam were designed to prevent excessive displacement between two middle beams. There were steel frames placed vertically to accommodate a rubber springs and a polyamide bearing between a middle beam and a support beam. There were polyamide bearings

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beneath the ends of the support beams and between the support and middle beams as shown in Fig. 3-1 to allow relative movements between the support beams and the bridge girder and relative movements between support beams and middle beams, respectively. Similarly, rubber springs were placed on top of the ends of support beams and between the support beams and the frames. Additionally, the joint had a water and debris drainage capability with the installation of rubber sealing into the gap between two adjacent middle beams. There was a cavity beneath the joint between adjacent bridge girders as shown in Fig. 3-2. The cavity was surrounded by the joint at the top, the bridge girders at sides, and the concrete pier at the bottom. The two sides of the cavity along its length were open.

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 41

Fig. 3-2 Cross-sectional view of the cavity below the expansion joint

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3.3 Noise and vibration measurement

Noise generated when vehicles passed over the expansion joint was measured with sound level meters, RION NL-21 and NL-32, as shown in Fig.3-3. The sound pressure

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inside the cavity beneath the joint was measured at the center of the cavity at 1.5 meter above the floor of the cavity. Sound pressure above the joint was measured with a sound level meter mounted on the parapet wall beside the joint. A sound level meter was placed below the joint on the ground near the pier at 1.5 meters above the ground which was grassy land. To measure the sound radiated outside from the joint-cavity system, two sound level meters were placed on the ground at 1.5 meters above the ground at 5 meters and 15 meters from the edge of the bridge slab. These two sound level meters were placed transverse to the bridge axis and along the longitudinal axis of the cavity beneath the joint.

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 42

Fig. 3-3 Positions of sound level meters inside the cavity and around the bridge during

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the experiment

Additionally, vibration responses of the joint during vehicle pass-bys were measured with piezoelectric accelerometers, RION PV-85, at several locations simultaneously, as shown in Fig. 3-4. Lateral and vertical accelerations were measured at the locations in the first,

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third and fifth middle beams, and only vertical acceleration was measured on second and third support beams. The lateral direction was defined as the direction perpendicular to

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the longitudinal axis of the middle beam.

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Position-1

Position-3

Position-2

Position-4

Fig. 3-4 Location of accelerometers on middle beams and support beams Signals from the sound level meters and accelerometers were digitized at 10,000 samples per second after anti-aliasing filtering. The measurement was carried out for all types of vehicles ranging from heavy trucks to small cars that crossed the joint from both traffic

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 43

directions. A set of time histories of sound pressures and accelerations are shown in Fig.3-5, as an example. The figure shows that there were transient responses when the front and rear wheels of the vehicle excited the joint. The low frequency components observed in the sound pressure response above the joint were the aerodynamic noises as the vehicle approached the joint. Spectral analysis was conducted for the time histories of sound pressures and accelerations. Time history data of 0.7 seconds that include the transient response signals as shown in Fig.3-5 were used in Fourier analysis by using the scanning averaging method (Harris, 1995). Hanning window having a length of 0.4096 s

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was used in the analysis with an overlap of 75 %. The frequency resolution of spectra was

(c)

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(a)

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therefore 2.44 Hz.

(d)

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(b)

Fig. 3-5 Time series data of sound and vibration measurement during a truck pass-by. (a), (b) lateral and vertical acceleration respectively of third middle beam at the centre; (c) sound pressure inside the cavity and (d) sound pressure above the joint 3.4 Vibro-acoustic analysis of the joint-cavity system by FEM-BEM approach 3.4.1 Analytical procedure The vibro-acoustic analysis conducted in this study is similar to that was done in case of full-scale mode joint in Chapter 2. Firstly, a finite element model of the expansion joint installed between the bridges was developed to identify the modal parameters of the joint. The velocity responses of the joint to different point loadings applied to represent the

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 44

dynamic loading that might have been applied during the vehicle pass-bys were then calculated by modal superposition technique. Finally, the velocity response thus obtained was used as boundary condition in the boundary element analysis of acoustic field inside and outside of the joint-cavity system. The flow of the numerical analysis that was conducted is shown below in Fig.3-6.

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Velocity response of the joint to point loadings by FEM

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FE modal analysis of the expansion joint

Vibro-acoustic analysis by FEM-BEM

Results interpretation

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Vehicle passbys experiment

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Sound field analysis of the joint-cavity system by BEM

Acoustic modal analysis results of the cavity

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Fig. 3-6 Flow chart of numerical analysis conducted in the study 3.4.2 Velocity response estimation of the expansion joint 3.4.2.1 Finite element modal analysis of the joint

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A three dimensional FE model of the expansion joint was developed in a similar manner to that of full scale model joint model developed in Chapter 2 by using ANSYS 10.0. The middle beams, support beams, control beams and frames were modeled with shell elements (SHELL63) so that velocity response of the joint could be transferred directly into the boundary element analysis of the sound field as velocity boundary condition. The polyamide bearings and rubber springs were simplified and modeled by using spring elements (COMBIN14). The size and material properties of all the components of the joint were same as in full-scale joint model described in Chapter 2. The maximum size of the FE meshes in the analysis was 60 mm so that the dynamic behavior of the joint could be investigated up to 900 Hz. The 3-D FE model of the joint is shown in Fig. 3-7.The modal analysis was carried out to obtain the natural frequencies and mode shapes of the joint.

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 45

The mode shapes were mass normalized as in case of full-scale model joint. There were a total of 663 vibration modes identified at frequencies below 800 Hz by modal analysis.

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Control beam

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Some of the vibration modes of the joint are shown later in results and discussion section.

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Middle beam

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Support beam

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Fig. 3-7 3-D finite element model of the joint 3.4.2.2 Velocity response estimation of the joint The velocity response of the joint was calculated by applying harmonic point loadings to the joint. Harmonic loadings were used so that the frequency response of a vibro-acoustic system consisting of the joint and the cavity would be identified. Although there should be differences between the vibration responses to transient dynamic loadings caused by different vehicles and the responses to stationary harmonic loadings, the harmonic loadings were used because of the numerical tools used for acoustic analysis described later in the Chapter. A combination of vertical and lateral harmonic point loads was exerted on the middle beams of the joint so as to consider a possible inclined impact of a vehicle tire to the joint. The point loadings were applied to the joint at locations where

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 46

loadings might be applied during a vehicle pass-by. Fig. 3-8 shows the position of the traffic lanes on the expansion joint. The width of each lane was 3.5 meters. Fig. 3-8 also shows the names of the structural components of the expansion joint used in this study. Since the joint was not placed perpendicular to the bridge axis, a vehicle crosses the expansion joint in an inclined orientation. A lane separator between the lanes was located at a certain distance from the center line (CL) of the bridge width as shown in Fig. 3-8. Therefore, the positions of loading on the two traffic lanes were not considered symmetric about the center of the joint, so that vibration responses of the expansion joint might be

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different depending on traffic direction. By assuming that a heavy truck with its axle width of 1.8 m passes at the center of a traffic lane, possible positions of external loading due to

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a truck pass-by were determined, as shown in Fig. 3-8. A pass-by of a heavy truck was assumed because, it was found from the experimental results that heavy trucks caused greater vibration responses of the expansion joint. It can be seen in Fig. 3-8 that a truck

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impacts the joint approximately at the middle of adjacent two support beams in the left lane (referred to as Lane 1 in this study) and near the support beams in the right lane (referred to as Lane 2). Therefore, the loadings were applied at the middle of adjacent two

CL

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support beams in Lane 1 and near the support beam in Lane 2.

Fig.3-8 Lane width of the bridge, possible truck vehicle pass-by position on the joint

Four different load cases were considered in the numerical analysis to calculate velocity responses of the expansion joint as shown in Fig. 3.9. The point loads were exerted on the central middle beam (referred to as the third middle beam) at four different positions: (i) between the second and third support beams (Load-1), (ii) between the third and fourth support beam (Load-2), (iii) near the fifth support beam (Load-3), and (iv) near the sixth support beam (Load-4). Only one loading position was considered for each load case upon considering the fact that, when one tire of an axle is on the third middle beam,

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 47

another tire of the same axle is off the joint because of the skewed position of the joint. Additionally, velocity responses of the joint were calculated with harmonic loadings applied at several different positions so as to investigate the effect of loading position on the vibration responses, because the exact position of the loading during a truck pass-by was not known. Also, it is to be noted that vehicle-structure interaction was not considered

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in the study.

First support beam

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Fig. 3-9 Position of four applied load cases

Harmonic point loads with unit amplitude were applied in the frequency range between 50 and 400 Hz at 1 Hz interval. This frequency range in the analysis was determined based

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on the frequency range of the noise generated from the joint-cavity system observed in the measurement. Modal-based forced response analysis was carried out by using the structural modes identified by the modal analysis in the frequency range below 800 Hz by using LMS Virtual.Lab Rev 6A. A modal damping ratio of 1% was assumed for all the modes considered in the analysis. 3.4.3 Analysis of acoustic field 3.4.3.1 Indirect boundary element method The sound field around the expansion joint was computed by the indirect boundary element method (IBEM). The IBEM involves non-physical double layer potential μY (the difference in the acoustic pressure across the boundary element model) and single layer

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 48

potential σ Y (the difference in the normal gradient of the pressure) (Raveendra et al., 1998, Vlahopoulos et al., 1998 and Zhang et al., 2003):

μY = pY + − pY − ,

σY =

(3.1)

∂p + ∂p − − = − j ρω ⎡⎣vnY + − vnY − ⎤⎦ ∂nY ∂nY

(3.2)

where pY and vnY are the acoustic pressure and the normal acoustic velocity at a source

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point Y on the surface of the boundary element model ΓY . ρ is the mass density of the medium and ω is circular frequency. The superscripts + and - indicate values on the

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positive side and the negative side of the surface ΓY , respectively, which are defined by

nY , the unit outward normal vector to the acoustic domain at the source point Y on the

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surface.

The acoustic pressure at a field point X in the acoustic domain is expressed by the Helmholtz-Kirchoff integral equation in terms of the single and double layer potentials (Raveendra et al., 1998):

⎡

∫ ⎢⎣GXY σY −

ΓY

⎤ ∂G XY μY ⎥ d ΓY , ∂nY ⎦

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pX = −

(3.3)

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Here, GXY is the free-space Green’s function for three-dimensional problem, which is expressed in terms of wave number k =

ω c

and the distance between the source and field

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locations r G XY =

1

4π r

e − jkr

where c in k =

ω c

(3.4)

is the velocity of sound in the medium. The acoustic velocity at the field

point X can be expressed by an integral equation in terms of the single and double layer potentials by taking into account the relationship between acoustic velocity and acoustic pressure:

⎡ ∂G ⎤ ∂px ∂ 2GxY = − j ρω vxi = − ∫ ⎢ xY σ Y − μY ⎥ d ΓY , ∂xi ∂xi ∂xi ∂nY ⎦ ΓY ⎣

(3.5)

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 49

where xi is a principal direction and vxi is the component of the acoustic velocity along direction xi. Equations (3.3) and (3.5) can be extended to the acoustic pressure and acoustic velocity on the surface of the model and these equations can be related with the boundary conditions so as to solve the problem. Detailed explanations about the boundary element method can be found elsewhere (Raveendra et al., 1998; Vlahopoulos et al., 1998 and

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Zhang et al., 2003).

In the direct boundary element method, there is a distinction between interior and exterior

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analysis depending on whether the primary variables are defined on the interior or exterior side of a model. In the indirect method, however, the primary variable contains information about both sides of a boundary surface so that acoustic medium on both sides of the

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boundary surface is considered simultaneously. This enables openings and multiple connections to be included in an indirect boundary element model (Vlahopoulos et al., 1998, Kopuz et al, 1996).

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3.4.3.2 Boundary element modeling of the joint-cavity system

The boundary element model of the joint-cavity system developed is shown in Fig. 3-10.

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The boundary of the cavity was discretized by using quadrilateral and triangular elements. The maximum size of boundary element was 120 mm so that the highest frequency that could be analyzed was 472 Hz, according to the thumb rule of 6 elements per wavelength. The two openings at the end along the length of the cavity and other small openings

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between the rubber bearings of the bridge girders were included in the model. The length of the cavity beneath the joint was shorter than the length of the joint as seen in Fig. 3-10. The top boundary of the BE model was made exactly the same as the FE model of the joint so that the velocity response of the joint could be transferred directly to the BE model. The sound speed and the density of the air used in the analysis were 340 ms-1 and 1.225 kgm-3 respectively.

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 50

Joint-cavity boundary

1.5 m

Z

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Field Point

Y

X

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Fig. 3-10 Boundary element model of the joint-cavity system

As in the vibro-acoustic analysis conducted in Chapter 2, in this study, it was assumed that the vibration response calculated from the in vacuo structural modes of the joint calculated separately in the previous section can be used to analyze the acoustic field of

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the joint-cavity system, i.e., a weak coupling was assumed between the expansion joint and cavity beneath it (vibrations of expansion joint were not influenced by the fluid). Layer

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potentials of the boundary surface were calculated by the indirect boundary element method with the velocity responses of the joint described in Section 3.4.2 as a boundary condition. The analysis of the acoustic field was conducted with LMS Virtual.Lab Rev 6A.

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3.4.3.3 Field point response and sound radiation characteristics

The layer potentials obtained in the previous section can be used to calculate the sound pressure at any field point inside the acoustic domain by using Equation (3.3). In this study, field points to calculate sound pressure were chosen to be identical to the measurement location in the experiment described in Section 3.3 (Fig. 3-3). One of the field points was inside the cavity at horizontally center and at 1.5 meter above the bottom floor of the cavity. Also, the sound pressure was calculated at two field points outside of the cavity which were at 5 meters and 15 meters from the edge of the joint.

It would be important to understand the sound radiation characteristics of the joint-cavity system with respect to environmental problem. The cavity beneath the joint was surrounded by concrete of the PC girder of the bridge and the concrete pier as shown in

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 51

Fig.3-2. The sound generated inside the cavity propagates outside mainly from the two open ends of the cavity along its length.

The analysis of the directivity of sound radiation of the joint-cavity system was conducted so as to identify the radiation characteristics of the sound in the vertical plane passing through the middle of the cavity along its length. Fig.3-11 shows the concentric directivity circles having their center at the joint-cavity center. The circles radii varied from 8 meters to 100 meters at an interval of 4 meters. The sound pressure was calculated at points on

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each circle circumference at an angular interval of 3 degrees. The angles were measured in anti-clockwise direction from the positive Z-axis which was parallel to the cavity length

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as shown in Fig.3-11. The effects of obstacles such as the bridge pier beneath the joint, the bridge girder and the slab were not included in the analysis assuming that the effect of these obstacles would not be significant for the sound propagation in the plane used in the

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analysis. Also, the effect of ground was not included in the analysis.

0 degree

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Expansion joint

Directivity circle

Fig.3-11 Directivity circles of different radius on a vertical plane 3.5 Acoustic modal analysis of the cavity beneath the joint

In addition to the analyses described above, the cavity beneath the joint was modeled by finite elements to identify the acoustic modal parameters (i.e., acoustic modes and their corresponding natural frequencies). The FE model of the cavity developed is shown in Fig. 3-12. The acoustic field inside the cavity was modeled by cubical fluid element of mesh size 130 mm so that acoustic modes having natural frequencies up to 435 Hz could be

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 52

calculated based on the rule of thumb of 6 elements per wavelength. The openings on both ends of the cavity length and other small openings between the bearings of bridge girder were modeled by applying impedance boundary condition: characteristic acoustic impedance of 416 kgm-2s-1, corresponding to the sound speed of 340 ms-1 and the air density of 1.225 kgm-3, was applied to model these openings. Some of the acoustic

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modes of the cavity are shown later in the Results and discussion section.

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Expansion joint boundary

Y

X

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Openings

Z

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Fig. 3-12 Finite element model of the cavity for acoustic modal analysis

3.6 Results and discussion 3.6.1 Velocity response of the expansion joint 3.6.1.1 Velocity response on middle beams and support beams

Figure 3-13 shows examples of vertical velocity responses calculated in the numerical analysis to Load-1 in Lane-1 at two positions in the third middle beam, between the second and third support beams (Position-1 in Fig. 3.4) and between the third and fourth support beams (Position-2). Also shown in Fig. 3-13 are the vertical velocity responses calculated at the center of the second and third support beams (Positions-3 and 4, respectively, in Fig. 3-4). These four positions correspond to the measurement locations in the experiment. Figure 3-14 shows the vertical velocity responses calculated in the numerical analysis to Load-3 in Lane 2 calculated at same positions to Load-1 explained

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 53

above. Here Load-1 and Load-3 were chosen so as to show the examples of results from both lanes.

There were dominant peaks in the vertical velocity responses of the middle beams and support beams in Fig.3-13 and 3-14 at frequencies about 111, 122, 135, 150 Hz and from 228 to 238 Hz. These peaks may be caused by the excitation of a vertical vibration mode of the joint at a natural frequency of about 113.33, 121.27, 134.70, 150.17 Hz, few modes from 229 to 240 Hz, respectively. In these vibration modes, motions of the middle beams

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and the support beams were coupled each other. For example, some structural modes of the joint in vertical vibration at 110.72 Hz, 113.33 Hz, 121.27 Hz and 134.70 Hz are shown

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in Fig. 3-15 which show coupled vibration of middle beams and support beams. In the structural vibration mode at 110.72 Hz shown in Fig. 3-15,,almost all parts of the joint move in phase in the vertical direction, while the center beams vibrate out-of-phase in the

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lateral direction. The maximum amplitude in the vertical direction was observed near the center of the joint. In the vibration mode at 184.87 Hz shown in Fig. 3-15, bending/lateral/torsional vibration of middle beam with out-of-phase motions within the joint was observed. This may reduce the efficiency of the sound generation.

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It can be seen also that the vibration response magnitude of the third middle beam was significantly greater than the support beams in the frequency range between 160 and 220

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Hz and between 240 and 270 Hz in Fig. 3-13 and between 160 to 190 Hz and 240-270 Hz in Fig. 3-14. This was due to the excitation of structural modes of the joint with dominant vertical vibration in middle beams but less vibration in support beams. For example, structural modes of the joint at 174.38 Hz and 184.87 Hz are shown in Fig. 3-15 Hz. Also

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it can be seen in these modes that middle beam were vibrating out of phase with each other. The vibration response pattern of the expansion joint was different to Load-1 and Load-3 due to different contribution of the structural modes of the joint to these loadings.

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 54

Position-2;

Position-3;

Position-4

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Position-1;

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Fig. 3-13 Vertical velocity response at measurement points to Load-1 (Numerical). Key:

Fig. 3-14 Vertical velocity response at measurement points to Load-3 (Numerical). Key: Position-1;

Position-2;

Position-3;

Position-4

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 55

f=113.33 Hz

f=134.70 Hz

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f=121.27 Hz

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f=110.72 Hz

f=184.87 Hz

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f=174.38 Hz

Fig. 3-15 Some of the structural modes of the joint 3.6.1.2 Effect of load positions on velocity response

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Figures 3-16 and 3-17 show the examples of the velocity responses calculated in the numerical analysis so as to investigate the effect of load position on the vibration responses of the expansion joint. The Fig.3-16 shows the velocity responses at Position-1 to three different loadings on the third middle beam in Lane 1 around Load-1 position. One of the loads was at mid span between second and third support beam and one each at either side of mid span between support beams. The Fig.3-17 shows the velocity responses at Position-1 to three different loadings on the third middle beam in Lane 2 around Load-3 position. One of the loads was above the fifth support beam and one each at either side of the support beam. Although general trend of the vibration response remains similar, there were variations in the vibration responses to those different loadings at some frequencies. The variation in the vibration responses observed at some

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 56

frequencies could be due to different contribution of structural modes of the joint to these

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loadings.

Fig. 3-16 Vertical velocity response at Position-1 to three different point loadings between

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second and third support beam around Load-1 position (Numerical)

Fig. 3-17 Vertical velocity response at Position-1 to three different point loadings near the fifth support beam around Load-3 position (Numerical)

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 57

3.6.1.3 Experimental results

Figure 3-18 shows the vertical velocity responses of the middle beam and the support beams, as an example of the measurement result while a heavy truck crossed the joint in Lane 1. Similarly Fig. 3-19 shows the vertical response of the middle beam and support beams while a heavy truck crossed the joint in Lane 2. The measurement locations of the data shown in Fig.3-18 and 3-19 correspond to the locations of the numerical results shown in Fig. 3-13 or 3-14.

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The vibration responses measured in the experiment showed similar trends to those calculated numerically, although the loadings applied to the joint in the experiment were

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not known and different from those in the numerical analysis. In Fig.3-18, dominant peaks in the middle beam response appeared at frequencies of about 110, 122, 131, 144, 183, 202 Hz and from 222 to 235 Hz. Significant support beam vibration was observed at

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frequencies of about 110,122, 131, 144 Hz and from 222 to 235 Hz. The middle beam response was much greater than the support beam response in the frequency range between 160 and 220 Hz. In Fig.3-19, dominant peaks in the middle beam response appeared at frequencies of about 120, 134, 151, 180, 200 Hz and from 225 to 232 Hz.

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Significant support beam vibration was observed at frequencies of about 120, 134, 150 Hz and from 225 to 231 Hz. The middle beam response was greater than the support beam

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response in the frequency range between 160 to 210 Hz. The data recorded with other vehicles that crossed the joint in Lane 2 also showed similar trends to those shown in Fig. 3-19, although variations were observed. It can also be observed that the velocity response of the joint to vehicles crossing in Lane 1 and Lane 2 is different and this was

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observed in the numerical results as well. There was a significant difference between middle beam and support beam response below 100 Hz and 240 to 275 Hz for the vehicle crossing in Lane 1 where comparable response was observed in these frequencies for the vehicle crossing in Lane 2.

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 58

Fig. 3-18 Vertical velocity response to the truck vehicle crossing the joint in Lane 1. Position-2;

Position-3;

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Position-1;

Position-4

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. Key:

Fig. 3-19 Vertical velocity response to the truck vehicle crossing the joint from Lane 2. .Key:

Position-1;

Position-2;

Position-3;

Position-4

Figure 3-20 shows the comparison of numerically obtained and experimentally measured vertical velocity response at one location (Position-2) on third middle beam and on third support beam (Position-4). Numerical results are shown to Load-3 and experimental results are shown to the heavy truck vehicle crossing in Lane 2. Truck vehicle considered here to show the results is the same that is shown in Fig. 3-19. Figure 3-20 implies that

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 59

the motion of the middle beams and motion of the support beams were coupled together in some of the peaks observed: for example, at about 110 Hz and 356 Hz, both the velocity response of the middle beam and that of the support beam showed a peak. Although the loadings applied to the joint in the experiment were not known, they were clearly different from the harmonic loading in the numerical analysis. Thus, although the numerical results cannot be compared directly with the experimental results, some similar trends may be observed between the vibration responses measured experimentally and

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those calculated numerically. At frequencies around 120 Hz and around 355 Hz, the vertical response of the middle beam was comparable with that of the support beam in the

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numerical analysis and the experiment. In the frequency range 140–180 Hz, the middle beam response was greater than the support beam response. The data recorded with other vehicles also showed similar trends to those shown in Fig. 3-20, although variations

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were observed possibly because different vehicles had different dynamic loadings and impacted the joint at different locations, depending on the type of vehicle, vehicle speed,

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and so on.

Fig. 3-20 Vertical velocity response on third middle beam and third support beam at measurement points. (support beam, Load-3); (Truck crossing in Lane 2).

Position-2 (middle beam, Load-3); Position-2 (Truck crossing in Lane 2);

Position-4 Position-4

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 60

3.6.1.4 Dynamic characteristics of the expansion joint

The modal analysis of the joint yielded a total of 663 vibration modes at frequencies below 800 Hz, as mentioned above. Numerical modeling of the full-scale model joint explained in Chapter 2 have shown reasonable agreements in the modal properties between the numerical and experimental modal analyses (Ravshanovich et al, 2007). Roeder (1998) has conducted a numerical modal analysis of a modular expansion joint in the "single support bar swivel joist design", and reported that "the modes of vibration for this modular expansion joint were closely spaced with hundreds needed to include the predominate

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portion of the mass in three-dimensional vibration". This may be consistent with the many

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closely spaced vibration modes identified for the joint in this study.

At frequencies around 110 Hz, a vibration mode in which almost all parts of the joint move in phase in the vertical direction was identified for the joint in this study, as shown in Fig.

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3-15. A "translational (bounce/bending) mode" in the vertical direction, in which all parts of the joint vibrated in phase, was identified for a modular expansion joint in a "hybrid design" at 71 Hz by Ancich et al. (2006). In the frequency range around 70 Hz, vibration modes in which the whole body of the joint moved in phase in the lateral direction were

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identified in this study as the lowest order modes of the joint. Roeder (1998) has stated that the vibration modes of the joint with the lowest natural frequencies "were associated with horizontal movement", which suggests similar characteristics to those of the joint

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investigated in this study. The differences in modal properties reported in different studies may be due to the different designs of the modular expansion joint. 3.6.2 Sound pressure response

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3.6.2.1 Sound pressure response at different field points

The sound pressures in the cavity beneath the joint calculated numerically in this study are shown in Fig. 3-21 for all four load cases considered. The natural frequencies of structural modes of the joint and acoustic modes of the cavity are also shown in the figure. The peaks in the sound pressure appeared for all load cases at frequencies around 110, 122, 150, 227, 332 and 392 Hz. These peaks may be caused by the excitation of structural modes of the joint and/or acoustic modes of the cavity.

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 61

Load-1;

Load-2;

structural mode;

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Fig. 3-21 Numerically calculated sound pressure inside the cavity. Key: Load-3;

Load-4.

natural frequencies of

natural frequencies of acoustic mode

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Figure 3-22 shows the sound pressure recorded in the measurement during a heavy truck pass-by in Lane 1. The heavy vehicle considered here is the same as the one used to

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show the velocity responses in Fig. 3-18. The figure shows the sound pressure measured inside the cavity as well as at locations outside of the cavity: at 5 m and 15 m from the edge of the slab as shown in Fig. 3-3. There were dominant peaks in the sound pressure inside the cavity at around 70, 110,124,131,153 Hz and from 225 to 232 Hz. Figure 3-23

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shows the sound pressure recorded in the measurement during a heavy truck pass-by in Lane 2. The heavy vehicle considered here is the same as the one used to show the velocity responses in Fig. 3-19. There were dominant peaks in the sound pressure inside the cavity at around 70, 122, 150 and 178 Hz and from 225 to 250 Hz. Fig.3-24 shows the sound pressure calculated numerically with Load-1 at the locations where experimental results are shown in Fig.3-22. In the figure, dominant sound pressure peaks inside the cavity appeared at frequencies of about 110, 122, 149, 230 to 236, 332 and 392 Hz. Fig.325 shows the sound pressure calculated numerically with Load-3 loading at the corresponding locations where experimental results are shown in Fig.3-23. In the figure, dominant sound pressure peaks inside the cavity appeared at frequencies of about 110, 122, 149, 227, 281, 332 and 392 Hz.

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 62

Fig. 3-22 Experimentally measured sound pressure at different field points to the truck at 5 m;

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at 15 m

inside the cavity;

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vehicle crossing the joint in Lane 1. Key:

Fig. 3-23 Experimentally measured sound pressure at different field points to the truck vehicle crossing the joint in Lane 2. Key: at 15 m

inside the cavity;

at 5 m;

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 63

inside the cavity;

at 5 m;

at 15 m

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Key:

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Fig. 3-24 Numerically calculated sound pressure at different field points to Load-1.

Fig. 3-25 Numerically calculated sound pressure at different field points to Load-3. Key: inside the cavity;

at 5 m;

at 15 m

Figure 3-26 shows the sound pressure at the measurement point inside the cavity, as shown in Fig. 3-3, calculated numerically with Load-3. Figure 3-26 also shows the sound pressure recorded at the corresponding point in the measurement during a heavy truck pass-by in Lane 2 that is the sound pressure response shown in Fig. 3-23. Although the

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 64

magnitudes of the sound pressure response obtained from the numerical analysis cannot be compared directly with those obtained in the measurements, some similarities were observed in the numerical analysis and the measurement, such as dominant frequency components.

Figure 3-27 shows the sound pressure calculated with Load-3 at locations outside of the cavity: at 5 m and 15 m from the edge of the joint as shown in Fig. 3-3. The sound pressure spectra measured at the corresponding locations outside the cavity during a

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truck vehicle pass-by in Lane 2 in the experiment are also shown in Fig. 3-27. The heavy vehicle considered here is the same as the one used to show the sound pressure

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response inside the cavity in Fig. 3-23 or 3-26.

Fig. 3-26 Sound pressure response inside the cavity.

Load-3;

Truck

crossing in Lane 2; ¯ natural frequency of structural mode; U natural frequency of acoustic mode.

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 65

Fig. 3-27 Sound pressure at different field points outside the cavity. crossing in Lane 2).

5 m (Truck crossing in Lane 2);

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15 m (Load-3);

5 m (Load-3); 15 m (Truck

3.6.2.2 Sound generation and radiation mechanism inside the cavity

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The peaks in the frequency response function calculated for the sound pressure inside the cavity may be attributed to the excitation of the structural modes of the joint and/or the

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acoustic modes of the cavity. Sound generation due to the interaction between a vibrating structure and an enclosed acoustic cavity has been discussed in previous studies (Kim et al., 1999, Dowell et al., 1977, Cabeli et al., 1985 and Luo et al., 1978). These studies

showed that dominant peaks in the sound pressure inside the cavity could be due to: (1)

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major contribution of a structural mode of the joint with possible minor contribution of acoustic modes of the cavity, (2) significant contribution from both structural mode of the joint and acoustic modes of the cavity, and (3) major contribution of acoustic modes of the cavity with possible minor contribution of structural modes. These mechanisms were considered as possible causes of peak frequency components in the sound pressure inside the cavity beneath the full-scale joint model of modular expansion joint in the previous study in Chapter 2. Similar discussion can be made in this study so as to understand possible causes of the peaks in the sound pressure observed with the jointcavity system investigated.

For example, the sound pressure peak at 150 Hz observed in Fig. 3-21 may be attributed to the excitation of a vibration mode of the joint at 150.17 Hz with significant vertical

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 66

vibration. This hypothesis may be supported by significant vibration response of the joint at about 150 Hz as seen in Fig. 3-13 and 3-14. The acoustic modal analysis of the cavity beneath the joint showed that there were acoustic modes at 154.06 Hz and 155.82 Hz. The structural vibration mode of the joint with dominant vertical vibration at 150.17 Hz is shown in Fig.3-28. Also, the acoustic modes of the cavity at 154.06 Hz and 155.82 Hz are shown in Fig. 3-29. The comparison between the mode shape of the structural mode shown in Fig. 3-28 and the pressure distribution in the top plane of the cavity in the acoustic modes shown in Fig. 3-29 implies that there can be significant interaction

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between the structural mode and the acoustic modes. However, the contribution from the acoustic modes to the sound pressure peak at 150 Hz shown in Fig. 3-21 would be minor

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because there was a nodal plane in both acoustic modes that include the location for the pressure calculation. The sound pressures at around 110, 122, 230, 250 and 392 Hz may also be caused by the structural modes and/or the acoustic modes with varying

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contributions from those modes depending on frequency.

Fig. 3-28 Structural mode of the joint in vertical direction at 150.17 Hz

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 67

(b)

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(a)

Calculation point

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Calculation point

Fig. 3-29 Acoustic modes of the cavity: (a) 154.06 Hz and (b) 155.82 Hz

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In Fig. 3-21, there were some other peaks in the sound pressure response that may be associated with no structural modes but acoustic modes. This implies that those sound pressure peaks were caused by the acoustic modes only. Those peaks were observed

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especially in the higher frequency range where the acoustic modal density was high. For example, the sound pressure peak at around 332 Hz may be due to the dominant contribution from acoustic modes only, since there were no structural modes of the joint at

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around this frequency while there were many numbers of acoustic modes as shown in Fig.3-21.

In Fig.3-21, there were also some frequencies in the sound pressure frequency responses

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with less sound pressure. For example, sound pressure from 175 to 220 Hz was small for all load cases. This could be possibly because there were vibration modes of the joint with dominant vibration of the middle beams in the vertical and torsional directions and less vibration of the support beams in this frequency range. Also, in these modes, the middle beams moved out of phase with each other so that the acoustic pressure generated from different middle beams may have been cancelled each other out. As an example, a structural mode of the joint at 181.30 Hz is shown in Fig. 3-30 in which the middle beams moved out of phase with each other.

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Fig. 3-30 Structural mode of the joint at 181.30 Hz

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 68

3.6.3 Sound radiation efficiency of the joint-cavity system

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The sound radiation efficiency is an index which expresses how effectively a vibrating structure converts mechanical energy into sound energy. The radiation efficiency is defined as the ratio of the active output power to the input power (SYSNOISE Manual, 1999):

Wo,active

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Radiation efficiency =

Wi

(3.6)

where Wo,active is the active output power that is the real part of the output power and

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corresponds to the average power radiated by the vibrating structure during one vibration cycle. In the indirect boundary element method, the active output power is given by: Wo,active =

1 ρ c Re( μ.v n ∗ )dS 2

∫

(3.7)

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S

Here, ρ is the density of the acoustic medium, c the velocity of sound in the medium, μ

the double layer potential (or jump of pressure) and vn∗ the complex conjugate of the normal velocity. Wi in Equation (3.6) is the input power that is the power associated with

mechanical vibrations. It is obtained by integrating the squared normal velocities of the boundary surface as:

∫

2

Wi = ρ c v rms dS = S

1 ρ c v n 2 dS 2

∫

(3.8)

S

where vrms is the root-mean-square value of the local normal velocity boundary conditions

vn . The input power depends only on the prescribed velocity boundary conditions.

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 69

The mean quadratic velocities calculated at the boundary of the boundary element model for all load cases are shown in Fig. 3-31. The sound radiation efficiencies calculated for the joint-cavity system with all load cases used in this study are shown in Fig. 3-32. The figures show that the mechanical power at frequencies below 100 Hz was not converted efficiently into acoustic power, even though there was significant input power in this frequency range, as indicated by significant velocity seen in Fig. 3-31, mainly due to the vibration modes of the joint with significant lateral vibration of middle beams. This implies that these vibration modes of the joint were not efficient in sound radiation. For example

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some of the vibration modes of the joint in lateral direction in this frequency range are

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shown in Fig.3-33. Vibration modes of any structure below the critical frequency cannot radiate sound efficiently. The critical frequency f c of the middle beam of the joint in lateral and vertical directions was calculated by using the simplified expression (Jeyapalan et al.,

fc =

3.64 k

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1979):

(3.9)

where k is the radius of gyration of second moment of area in meters. The critical

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frequency of the lateral vibration of the middle beam calculated using Equation (3.9) was 180 Hz. The critical frequency of the vertical vibration of the middle beam was 67 Hz indicating that all vertical vibration modes above this frequency could radiate sound

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efficiently.

Fig. 3-31 Mean quadratic velocity at the boundary of BE model. Key: Load-2;

Load-3;

Load-4

Load-1;

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Fig. 3-32 Radiation efficiency of the joint-cavity system. Key: Load-4

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Load-3;

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Load2;

Load-1;

f=182 Hz f=82 Hz f=75 Hz Fig. 3-33 Some of the vibration modes of the joint in lateral direction

High radiation efficiency was observed in the frequency range from 100 to 160 Hz. In this frequency range, there were several numbers of vertical vibration modes of the joint with significant vibration in both middle and support beams (coupled modes). In these modes, all middle beams vibrated in phase with the support beams (for example first four modes in Fig.3-15 and Fig 3-28). In the frequency range from 160 to 225 Hz, the radiation efficiency was low, possibly because there were vibration modes of the joint with dominant coupled vertical and torsional vibrations of the middle beams and less vibration of the support beams (for example last two modes in Fig. 3-15 and Fig.3-30). Also, as explained

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 71

in the previous section, in these vibration modes, the middle beams moved out of phase with each other and the acoustic power radiated from different middle beams may be cancelled each other out. Furthermore, there were vibration modes of the joint with significant vibration in lateral direction in this frequency range (Fig.3-33). The radiation efficiency was high at frequencies above 275 Hz. At higher frequencies above 275 Hz, the modal density of the acoustic modes of the cavity was high and the interaction between vibration modes of the joint and many numbers of acoustic modes might have caused

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high sound radiation efficiency. 3.6.4 Sound radiation characteristics of the joint-cavity system

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3.6.4.1 Effect of distance on directivity of sound radiation

The directivity patterns of sound radiation of the joint-cavity system calculated in a vertical plane explained in Section 3.4.3.3 is shown for example with Load-3 at 149 Hz (lower

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frequency with dominant sound pressure peak inside the cavity) and 392 Hz (higher frequency with dominant sound pressure peak inside the cavity) are presented in Figs. 334 and 3-35, respectively, at 12, 24, 48 and 96 m from the center of the expansion jointcavity system. The figures show that the directivity pattern changes with distance near the

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source but remains more or less the same at far distances from the source. Fig. 3-36 shows the changes in the sound pressure with distance from the center of the expansion joint for some frequencies at which dominant sound pressure peaks inside the cavity were

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observed. The sound pressure was obtained at an angle of 3000 that corresponds to the direction of one of the far field main lobes in the directivity patterns for all frequencies shown in Fig. 3-36. In the figure, it is seen that the change in the sound pressure was not

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uniform at distances closer than 35 m possibly because of near field effect of the sound source. At distances farther than 35 m, the sound pressure decreases by approximately 6 dB with doubling the distance from the source. According to the definition of far field (Junger et al., 1986), the sound field at 35 m and further may be considered as far field for the joint-cavity system.

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 72

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Fig. 3-34 Sound radiation pattern at 149 Hz

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 73

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Fig. 3-35 Sound radiation pattern at 392 Hz

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 74

40 m

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-6 dB

80 m

Fig. 3-36 Changes in the sound pressure with distance from the center of the joint at 3000. Key:

392 Hz

122 Hz;

149 Hz;

282 Hz;

332

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Hz;

110 Hz;

The near field and far field of the joint-cavity system is governed by the sound radiation from the bottom part of the joint structure and cavity beneath the joint through the

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openings of the cavity on both side along the cavity length as well as sound radiation from the expansion joint structure from the top and overhanging portion of the joint as shown in Fig. 3-10. Noise from the bottom part of the joint is radiated outside through the

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propagation of acoustic modes of the cavity. Bies and Hansen (1988) have presented three criteria to be satisfied by far field of a sound source in a free field: r >> λ (2π ) , r >> ld , r >> π ld 2 (2λ )

(3.10)

where r is the distance from the source to a field point, λ is the wavelength of radiated sound, and ld is the maximum source dimension. They have stated that "the "much

greater than" criteria in the above three expressions refer to a factor of three or more". The third expression in Equation (3.10) is also defined as Rayleigh distance in Carley (2003) and Keith et al. (2002a and 2000b). In this study, the greater dimension of the cavity cross-section, i.e., the height of the cavity (2.62 m, as shown in Fig. 3-2), can be considered to be the maximum source dimension l, because the sound radiation from the joint–cavity system is dominated possibly by the sound radiation of acoustic modes of the

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 75

cavity, as discussed later in this chapter. For this expansion joint-cavity system, the governing criterion for far field is the third expression in Equation (3.10). For the sound speed of 340 ms-1, the minimum distance of far field from the source may then be approximately obtained as 10.5 m at 110 Hz, 26.8 m at 282 Hz, and 37.3 m at 392 Hz. In order to compare those distances with the results shown in Fig. 3-36, the distance between the center and edge of the cavity (3.7 m) has to be added to the above distances, which yields 14.2, 30.5 and 41.0 m at 110, 282 and 392 Hz, respectively. It seems that those distances are consistent with the trend of the change in sound pressure with the

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distance from the source shown in Fig. 3-36.

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3.6.4.2 Effect of acoustic characteristics of the cavity on sound radiation

The sound from the joint during vehicle pass-bys may be affected partly by the propagation of acoustic modes of the cavity beneath the joint. The characteristics of the

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sound radiated from the openings of the cavity along its longitudinal axis may depend upon the radiation characteristics of the structural components of the expansion joint, mainly the middle beams, and the acoustic characteristics of the cavity. The contribution of the acoustic characteristics of the cavity to sound propagation may be understood

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based on the propagation of sound from an ideal rectangular cavity. The cutoff frequency of a rectangular cavity below which only plane wave modes could propagate inside the

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cavity is obtained by (Kinsler et al., 2000): Cutoff frequency =

c 2L

(3.11)

where c is sound velocity and L is the greater dimension of cavity cross-section, i.e., the

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height of the cavity. Higher order modes (i.e., modes other than plane wave mode) are formed inside a rectangular cavity by the constructive and destructive interference of outgoing and incoming plane waves inside the cavity (Kinsler et al., 2000). Each higher order acoustic mode of the cavity with an opening at both ends along its longitudinal axis had a certain cutoff frequency, above which the mode could propagate inside the cavity. Therefore, the propagation angle of a particular acoustic mode of the cavity from its longitudinal axis depended on the cutoff frequency of the acoustic mode and frequency of interest. The propagation angle θ that the propagating vector of a particular acoustic

mode makes with longitudinal axis of the cavity can be expressed as: 1/ 2

ω ⎡ ⎤ cos θ = ⎢1 − ( lm )2 ⎥ ω ⎦ ⎣

(3.12)

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 76

where ω is the angular frequency of interest and ωlm is the cutoff angular frequency of the ( l , m ) mode which can be calculated as: 2 ωlm = c k xl2 + k ym

where k xl =

(3.13)

lπ mπ and k ym = are the wave number components in X-axis and Y-axis, Lx Ly

respectively. Here, l and m represent the number of nodal planes along the horizontal

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and vertical axes (X-axis and Y-axis in Fig.3-12, respectively). Lx and Ly are the dimensions of the cavity in X-axis and Y-axis, respectively. From Equation (3.12), it can

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be seen that the propagation angle for an acoustic mode is close to 90 degrees (i.e., normal to the longitudinal axis of the cavity) at a frequency close to its cutoff frequency and decreases with increases in frequency above its cutoff frequency. Additionally, Rice et

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al. (1979) concluded that the propagation angle inside a cylindrical duct was coincident with the principal lobe of far field radiation.

With respect to the effect of the acoustic characteristics of the cavity beneath the joint on

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sound propagation, the cutoff frequency of the cavity cross-section was 56 Hz from Equation (3.11). The cutoff frequencies of the acoustic modes of the cavity were calculated from Equation (3.13) and used to obtain the radiation angle of the main lobe of

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acoustic modes in far field using Equation (3.12). Some of the acoustic modes of the cavity that were obtained from the FE modal analysis as explained in Section 3.5 are shown in Fig. 3-37. The radiation angle of the main lobe of acoustic modes of the cavity of

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orders (0, 1) to (1, 4) which were present in the frequency range of 50-400 Hz are shown in Fig. 3-38. Also shown in the figure are the natural frequencies of acoustic modes of different orders of the cavity identified from FEM.

From figure 3-38 it can be seen that most of the identified acoustic modes of the cavity can radiate sound at angles larger than 300 from the longitudinal axis of the cavity. Figures 3-34 and 3-35 show that sound radiation pattern of the joint-cavity system depended on frequency. At 149 Hz shown in Fig. 3-34, one of the main radiation lobes in far field was at approximately 51 degrees. In Fig. 3-38, the radiation angle of a main lobe in far field at 149 Hz was 48 degrees for the (0, 2) mode and 20 degrees for the (0, 1) mode. Therefore, the radiation from the (0, 2) mode may be dominant in the radiation at this frequency. At 392 Hz shown in Fig. 3-35, there were many numbers of radiation lobes in far field at

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 77

different angles from the cavity axis. There was significant sound radiation along the cavity axis (i.e., horizontal direction in the figure) at 392 Hz, unlike at 149 Hz. At frequencies significantly higher than the cutoff frequency of an acoustic mode, the radiation angle of the mode from the cavity axis decreases. Therefore, at high frequencies, the sound could radiate in directions close to the cavity axis by lower order acoustic modes and at greater angles from the cavity axis by acoustic modes having its cutoff frequency close to the frequency of interest. The radiation characteristics at 392 Hz that show many lobes at different angles may be caused by different acoustic modes radiated

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at different angles as shown in Fig. 3-38. Radiation from the lower part of the joint within the cavity length may occur through the propagation of acoustic modes of the cavity.

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Sound radiation from the top and overhanging portion of the joint might have some effect

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on the radiation patterns of the joint-cavity system.

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 78

Fig. 3-37 Some acoustic modes of the cavity beneath the joint

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 79

Fig. 3-37 Some acoustic modes of the cavity beneath the joint (contd.)

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 80

90

70 60 50

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40 30 20 10 0 75

100

125

150

175

200 225 250 Frequency (Hz)

275

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50

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Main lobe radiation angle (degrees)

80

300

325

Fig.3-38 Main lobe radiation angles of acoustic modes of the cavity.

350

375

400

, Natural frequency

of acoustic modes of different orders obtained from FEM

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The sound pressure calculated at 5 and 15 meters from the edge of the slab presented in the Fig. 3-24 to Load-1 showed that the sound pressure at 5 meters tended to be higher

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than that at 15 meters as expected (e.g., at 150 Hz, 182 Hz, 323), while, at some frequencies (e.g., 110 Hz, 134 Hz), the sound pressures at these two field points were comparable. Similarly, the sound pressure calculated at 5 and 15 meters from the edge of the slab presented in Fig.3-25 to Load-3 showed that the sound pressure at 5 meters

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tended to be higher than that at 15 meters (e.g., at 150 Hz, 225 Hz, 281 Hz), while, at some frequencies (e.g., 110 Hz, 122 Hz, 134 Hz), the sound pressures at these two field points were comparable. These apparent effects of distance on the sound pressure may be caused by the directivity pattern of sound radiation. The sound pressure obtained in the measurement at the corresponding locations, i.e., at 5 and 15 meters, shown in Fig. 3.22 to the vehicle crossing in Lane 1 also showed that the sound pressure was higher at 5 meters than at 15 meters (e.g., at 110 Hz, 122 Hz, 134 Hz, 150 Hz), comparable at some frequencies (e.g., 230 Hz, 270 Hz), and higher at 15 meters than at 5 meters (e.g., at 161 Hz, 297 Hz), Although the measurement results might include some contribution from vehicle and tire noises. Similarly, the sound pressure measured at the same locations, i.e., at 5 and 15 meters, shown in Fig. 3-23 to the vehicle crossing the joint in

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 81

Lane 2 also showed that the sound pressure was higher at 5 meters than at 15 meters (e.g., at 122 Hz), comparable at some frequencies (e.g., 146 Hz, 250 Hz), and higher at 15 meters than at 5 meters (e.g., at 205 Hz).

3.7 Analysis of joint vibration caused by transient loading

Although harmonic loadings applied at a single point were used to calculate the frequency response of the joint–cavity system because of the limitation of numerical tools used in this study, the dynamic loadings on the joint during vehicle pass-bys have temporal and

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spatial variations in practice. The characteristics of the response of the joint to dynamic vehicle loadings will be different from those to stationary harmonic loadings to some

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extent. Therefore, in this section, a preliminary investigation of the vibration response of the joint to transient loadings, partly analogous to vehicle loadings, is described. The objective

of

the

preliminary

investigation

was

to

investigate

the

fundamental

3.7.1 Analysis method

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characteristics of the dynamic response of the joint to transient loadings.

The loading on the joint from a vehicle is due to the impacts between tires and the joint

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structure. The characteristics of tire impact loading on the middle beam of the joint will depend on various factors such as: type of vehicle, vehicle speed, tire contact length, and

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the width of the top flange of the middle beam. In previous numerical studies of fatigue of the modular expansion joint, tire impact loading on a middle beam was modeled by a symmetrical triangular pulse (Roeder,1998) and a half-cycle sine pulse (Steenbergen, 2004 and Crocetti and Edlund, 2003), by assuming a constant vehicle speed. In this study,

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a symmetrical triangular pulse force was used for simplicity. The effect of a possible dynamic interaction between vehicle and joint structure was not considered, as in previous studies (Roeder, 1998 and Crocetti and Edlund, 2003). This may be a reasonable assumption because the frequency range of interest in noise generation is much higher than the natural frequencies of vibration modes of the vehicle with a significant modal mass, such as bounce mode and axle-hop mode (Kim et al., 2005), which will have an effect on the vehicle loading.

The duration of the tire impact loading on a middle beam of the joint, td , was calculated by dividing the sum of the tire contact length, Lc , and the width of the top flange of the middle

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 82

beam, b f , by the vehicle speed, v (Steenbergen, 2004, Roeder, 1998 and Crocetti and Edlund, 2003): Lc + b f

td =

(3.14)

v

The reciprocal of the duration of the tire impact loading, td , was referred to as the "tire pulse frequency" by Ancich et al. (2004). In this study, while the tire contact length was assumed to be 0.20 m and the width of the top flange of the middle beam was fixed at

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0.08 m, the vehicle speed was in the range 50–100 kmh-1. A unit maximum magnitude was used for the symmetrical triangular pulse forces in the analysis. For example, the

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transient loading model used in the analysis for standard three-axle truck vehicle running at 60 kmh-1 is shown in Fig. 3-39.

1N

0.0084 0.01680

0.2514

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0

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F (t)

0.2598 0.2682 0.3474 0.3558

0.3642

t (s)

Fig. 3-39 Time series of loading on the center beam at vehicle speed of 60 kmhr-1 to

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transient analysis

The above defined loading in the vertical direction was applied on the third middle beam at the same locations as in the analysis with harmonic loadings described in the preceding

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section. Firstly, a single pulse force was applied at a location to obtain the vibration response of the joint so as to understand the fundamental characteristics of the joint response to transient loading. Changes in the space distribution of the force in the top flange of the middle beam during vehicle pass-by were not considered in the analysis for simplicity. It was assumed that the spatial variation of the force had a minor effect on the excitation of vertical vibration modes of the joint, which appeared to be more efficient in sound radiation, as described above. The FE model of the joint was exactly the same as the model used in the analysis with harmonic loadings. Transient analysis was carried out using the modal superposition technique in ANSYS 10.0. The structural modes of the joint that were identified in the preceding section were used in the analysis. The integration time step was 0.0002 s so that the frequency range in the analysis was limited to 250 Hz and below to reduce computational load.

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 83

3.7.2 Vibration response of the expansion joint to transient loading

Figure 3-40 shows the spectra of the velocity responses of the joint at Position-2 calculated in the transient analysis with the loadings corresponding to vehicle speeds of 60, 80 and 100 kmh-1 applied at Load-3 position. The figure clearly shows that the dynamic response of the joint was dependent on the vehicle speed. The differences in the response caused by the loadings corresponding to different vehicle speeds may be

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attributed to the effect of the duration of loading.

Fig. 3-40 Fourier transforms of vertical velocity response of the joint at Position-2 on third middle beam to transient loading applied at Load-3 position (Fig.3-9) corresponding to 60 kmh-1;

80 kmh-1;

100 kmh-1

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different vehicle speeds. Key:

The relationship between the duration of loading and the response of a single-degree-offreedom (SDOF) system for a symmetrical triangular pulse loading has been discussed in some textbooks (Haris, 1995 and Chopra, 2001). It is understood that a vibration mode will be excited efficiently when the ratio of the duration of loading, td, to the natural period of the system, Tn, and, in other words, the ratio of the natural frequency of the system, Fn,

to the tire pulse frequency, ft, are close to odd numbers. When a vibration mode satisfies the ratios td/Tn and Fn/ft by being close to even numbers, the vibration mode will not be excited efficiently. These characteristics were observed in the velocity responses of the joint calculated shown in Fig. 3-40. For example, for a vehicle speed of 60 kmh-1, the ratios td/Tn and Fn/ft are about 3 for the natural frequency, Fn, of 178 Hz, and about 2 and 4 for Fn of 119 and 238 Hz, respectively. It seems that these frequencies corresponds to a

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 84

peak frequency of about 173 Hz and trough frequencies of about 118 and 238 Hz in the velocity response calculated for a vehicle speed of 60 kmh-1 shown in Fig. 3-40. A similar argument can be made for the results with 80 and 100 kmh-1 shown in the figure. Figure 3-41 shows the relation between the vehicle speed and the tire pulse frequency multiplied by integers from 1 to 4. It also shows the natural frequencies of the structural modes of the joint and the acoustic modes of the cavity. The natural frequency Fn that satisfies the ratios td/Tn and Fn/ft being integers from 1 to 4 can be found for different

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vehicle speeds. Figure 3-41 also shows the relationship between the vehicle speed and the tire pulse frequency multiplied by 0.8, because the maximum response of the SDOF

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system will be induced when the ratios td/Tn and Fn/ft are about 0.8. The figure indicates that, within the range of vehicle speed practically expected, the tire pulse frequency can be a factor affecting the dynamic response of the joint, as discussed elsewhere Ancich et

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al. (2004).

Fig. 3-41 Variation of multiples of tire pulse frequency, ft, with vehicle speed.

ft;

2 ft ;

3 ft ;

U natural frequency of acoustic mode

0.8ft ;

4ft; ¯ natural frequency of structural mode;

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 85

Figure 3-42 compares the vertical vibration responses of the middle beam at Position-2 calculated numerically with the harmonic loading Load-3 and the transient loading applied at Load-3 position calculated for a vehicle speed of 78 kmh-1. The figure also shows the experimental data measured at Position-2 during a truck pass-by at 78 kmh-1, which are the same data shown in Fig. 3-19. Although the magnitude of the spectra cannot be compared each other in the figure because each spectrum has a different meaning, the effect of the tire pulse frequency observed in the response to the transient loading, i.e., a trough at around 160 Hz, was not found in the response to the harmonic loading, as

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expected. The effect of the tire pulse frequency was not clear also in the experimental data, possibly because the analysis did not account for multiple loadings caused by the

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multiple tires of vehicle. Additionally, the effect of the tire pulse frequency may appear in more complicated manner for a joint in a skewed position than a joint aligned perpendicular to the longitudinal axis of the bridge, because the tires on two sides of

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multiple axles of vehicle will impact the middle beams individually. Figure 3-42 shows the vertical vibration response of the middle beam at Position-2 calculated numerically with a series of three symmetrical triangular pulse forces, each of which was the same as the pulse force described in the previous section. That series of pulse forces was applied at

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the corresponding location to Load-3. The time intervals between three pulse forces were determined based on a standard three-axle truck running at a speed of 78 kmh-1 as shown in Fig. 3-39. The result presented in Fig. 3-42 shows that the effect of the tire pulse

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frequency observed at around 160 Hz was less clear in the response to the three pulse

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forces compared to the response to the single pulse force.

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Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 86

Fig. 3-42 Vertical velocity response on third middle beam at Position-2.

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single transient load applied at Load-3 position at 78 kmh-1; consecutive transient loads applied at Load-3 position at 78 kmh-1;

Load-3; three

experiment

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In addition to the effect of tire pulse frequency, Ancich et al. (2004) have also investigated the effect of the "beam pass frequency" that is given by the vehicle speed divided by the sum of the middle beam spacing and the width of the middle beam top flange. A vibration

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mode of the joint induced by vehicle tire impacts will be enhanced when tire impacts to adjacent middle beams occur in phase with the vibration response. Figure 3-43 shows the relationship between the beam pass frequency and the vehicle speed for different spacings between the middle beams of the joint from 40 mm to 80 mm. 80 mm is the maximum allowable spacing between the middle beams of the joint studied. Figure 3-43 also shows the natural frequencies of the structural modes of the joint and the acoustic modes of the cavity. Although no numerical analysis to investigate the effect of the beam pass frequency was undertaken in this study, it is expected that a greater vibration response will be observed at frequencies corresponding to the beam pass frequency, for example, in the range around 150 Hz for the legal speed limit of 80 kmh-1, and at odd number multiples of the beam pass frequency.

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D

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 87

Fig. 3-43 Beam pass frequency for different middle beam spacing. 60 mm;

70 mm;

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50 mm;

40 mm;

80 mm; ¯ natural frequency of structural

mode;U natural frequency of acoustic mode

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Additionally, for the expansion joint investigated in this study, the effects of the tire pulse frequency and the beam pass frequency on the response of the joint–cavity system may be more complex than the discussion above allows because of the skewed position of the

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joint, as mentioned briefly in the discussion on the effect of the tire pulse frequency. Although the effect of the skewed position may be minor compared to the accuracy of the above quantitative discussion, further study may be required to understand the effect of the alignment of the joint. 3.8 Conclusions

Vibro-acoustic analysis of an existing modular expansion joint installed between PC highway bridges was carried out so as to understand the noise generation and radiation mechanism. Measurements of noise and vibration of the expansion joint during vehicle pass-bys were also carried out. The sound field was analyzed in the frequency range of 50-400 Hz. It was concluded that the noise from the bottom side of the joint was caused by the excitation of structural modes of the expansion joint and/or acoustic modes of the

Chapter 3 Vibro-acoustic analysis of modular expansion joint installed between prestressed concrete bridges Page 88

cavity beneath the joint, which is consistent with a conclusion derived in a previous study with a full-scale model of modular expansion joint in Chapter 2. The sound radiation efficiency of the joint-cavity system appeared to be high at natural frequencies of vibration modes of the joint with significant vertical vibration of middle beams and support beams. The radiation efficiency of lateral vibration modes of the joint appeared to be low. Noise from the joint–cavity system may be propagated most effectively at radiation angles of acoustic modes of the cavity, which can be predicted roughly from the fundamental theory of sound radiation from cavities and waveguides. The sound field around the joint-cavity

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system investigated in this study could be considered near field within 35 meters from the joint-cavity center and far field at farther distances. The boundary between near field and

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far field in the sound field around the joint–cavity system may be predicted approximately by the previous findings of the characteristics of radiation field of a sound source by

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dimension.

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considering the greater dimension of the cavity cross-section as the maximum source

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge 4.1 Background and objective

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The noise level observed around bridges with modular expansion joint has been much greater than that around the bridge with other types of expansion joint like finger type joint.

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As explained in previous studies in Chapter 2 and 3, the loud noise can be generated from the modular expansion joint-cavity system involving different mechanisms. In addition, there is a possibility that vibration power of expansion joint due to vehicle impact while

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crossing the joint can be transmitted to the connected bridge and bridge structure vibrates and radiates noise to the environment. The noise radiation from the bridge due to vibration transmission from the joint could be significant in case of steel-concrete bridge which can radiate significant noise in broad frequency range. In this case the noise problem from the

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modular expansion joint would be more serious if connection points of modular expansion joints with the bridge are not properly isolated. It will be always desirable to understand

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the vibration power flow from the modular expansion joint to the connected bridge and estimate the noise contribution from the bridge to the overall noise around the modular expansion joint. This information could help to reduce the vibration power transmission to

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the bridge from the joint by adopting some control measures.

As stated in section 1.3 there have been several studies in railway bridges on the vibration power flow from the rails to the bridge and noise radiation from the bridge (Bewes et al., 2006, Harrison et al., 2000, Janssens et al., 1996 and Wang et al., 2004). All of these studies were based on the statistical energy analysis (SEA) which is the energy based approach for the vibration and noise response estimation in high frequency. In SEA, whole structure is subdivided into different vibrating subsystems. Energy flowing coefficients from one subsystem to other is calculated along with the power dissipated in each subsystems. Then, power balance equations are used to calculate the vibration power distribution in each subsystem. From the calculated vibration power, velocity response of each subsystem is estimated and used to estimate the radiated sound power. Also, it is to

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 90

be noted while dividing into subsystems that each subsystem can have only one type of wave propagation.

There are several literatures that can be referred about SEA and its application in noise and vibration analysis. A book by Lyon (1975) is the first attempt on general applicability of SEA to dynamical systems. A brief introduction of SEA has been presented by Woodhouse (1981) exploring the ideas of the approach to vibration analysis. A critical review of SEA has been presented by Fahy (1994). SEA has been used for study of

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structure-borne sound transmission in large welded ship structures in Hynna et al. (1995). FEM was utilized in the study to reduce the calculation time. Study about the lower

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frequency limit in the SEA analysis has been investigated by Craik et al, (1991). There have been some studies where SEA along with FEM was utilized for structure-borne

Fredo et al., 1997).

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sound transmission in complex structures (Steel et al., 1994, Shankar et al., 1997 and

There have been no previous reported studies on the vibration power flow from the modular joint to the bridge and noise radiation from the bridge. The objective of this study

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was to understand the possible vibration power flow from the modular joint to the connected steel-concrete non-composite bridge and estimate the noise radiation from the

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bridge. Firstly, the possibility of the vibration power flow from the modular joint to the connected bridge was hypothesized by comparing the vibration and noise response of two similar steel girder bridges: one with modular joint and another with finger joint. A simplified approach which uses the finite element method (FEM) for vibration response

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estimation of the joint and statistical energy analysis (SEA) for vibration response estimation of the bridge and noise radiation from the bridge was utilized. Here, deterministic methods like FEM and BEM were not feasible for analysis of the whole joint and bridge system considering the size of the structure and frequency range to be analyzed. Due to the large size of the bridge and with rule of thumb of 6 elements per wavelength, deterministic approaches like FEM and BEM were not feasible for the higher frequency range say up to 1000 Hz, which is the frequency of interest in noise problems from bridges. A large bridge can have many structural vibration modes even below the lowest audible frequency. Therefore a simplified model based on SEA was considered in this study. Though, this simplified approach cannot give the detailed information about the noise prediction like the directivity of the noise radiation from the source, still this kind of

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 91

formulation can be useful to predict the problem at first hand and sometimes during the design stage and reduce the noise from the expansion joint and bridge system. The vibration response on the expansion joint, bridge and noise radiation from the bridge was estimated and compared with experimental results of vehicle pass-bys experiment. 4.2 Description of two bridges with modular and finger joint Figure 4-1 shows the cross-section of the two bridges constructed in an expressway. These two lane bridges were located across the same river and direction of traffic was

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opposite in two bridges. The length of the bridges was 184 meters with five continuous spans. Both bridges were steel-concrete non-composite type. Bridge in the left side in the

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Fig. 4-1 had 5 steel I-girders and concrete deck. The finger type expansion joint was installed in this bridge. The bridge at the right hand side of the figure had 4 steel girders and concrete deck. The modular type expansion joint was installed in this bridge. The

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thickness of the concrete deck and height of the steel-girders is shown in the figure. The cavity beneath the modular joint is shown in Fig.4-2. Expansion joint in both bridges was

P2

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installed near one of the abutments of the bridge.

P1

P1

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1m

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G5

G4

G3

G2

G1

P3

P2 1m

G1

G2

G3

G4

5m

5m 1.5 m

: Sound level meter;

: Accelerometer

Fig. 4-1 Cross-section of the bridges with finger type joint and modular type joint showing the position of accelerometers and sound level meters used in the experiment

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 92

Concrete deck I-girder

Modular joint

Cavity

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Abutment

Fig. 4-2 Cross-section of the cavity beneath the modular expansion joint

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4.3 Experimental study of sound and vibration during vehicle pass-bys 4.3.1 Sound and vibration measurement in two bridges Measurement of sound and vibration during vehicle pass-bys was carried out in a separate study (Ravshanovich, 2007). During the vehicle pass-bys experiment, sound

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pressure was measured at different locations around the modular expansion joint and bridge by using the sound level meters RION NL-21 and NL-32. Sound pressure was measured inside the cavity beneath joint at 1 meter below the road surface. Also sound

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pressure was measured on the road surface beside the joint, under the bridge 1.5 m above the ground surface and at 5 m wayside of the bridge along the expansion joint length. Locations of the sound level meters are shown in Fig.4-1. Additionally vibration

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response of the bridge was measured at different locations on the concrete deck and web of steel girders using the accelerometers RION PV-85. Vibration was measured on the

concrete deck at three locations between the girders. Similarly, bending vibration of the all four steel girders web was measured. The positions of the accelerometers are shown in Fig.4-1. These measurement points on the deck and girders web were around 5 m away from the expansion joint. Measurement was carried out for several numbers and types of vehicles including the heavy trucks, buses and light vehicles like normal cars. The direction of vehicle traffic impacting the modular joint was from the bridge towards the joint. Since direction of traffic in both bridges was opposite, the vehicle was moving from the road towards bridge while impacting the finger joint.

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 93

Figure 4-3 shows the modular expansion joint installed in the bridge. The expansion joint consisted of two middle beams and these were supported by eight support beams. Accelerometers were placed on the first middle beam at three different locations (Fig.4-3) to measure the lateral and vertical response during vehicle pass-bys.

1.0 m

1.5 m

Position-1

1.0 m 9.42 m

Position-2

1.5 m

Position-3

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1.05 m

1st middle beam

1.0 m

1.05 m

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1st support beam

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Fig.4-3 Top view of the modular joint installed in the bridge and position of the accelerometers in one of the middle beam during the experiment

Similar to the bridge with modular expansion joint, sound and vibration measurement was

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carried at different locations around the bridge with finger joint during vehicle pass-bys. Sound pressure was measured on the road surface beside the joint, under the joint, under the bridge and at wayside of the bridge as in the bridge with modular joint. Vibration

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response of the bridge deck at two different locations and on the web of five girders was measured. Positions of the sound level meters and accelerometers are shown in Fig. 4-1. The positions of the accelerometers on the bridge deck and web of girders were at 5 m

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from the expansion joint. The accelerometers and sound level meters types used in the measurement were same as that were used in the bridge with modular joint. As in the bridge with modular joint, measurement was carried out for several numbers and types of vehicles including the heavy trucks, buses and light vehicles like normal cars.

Signals from the sound level meters and accelerometers were digitized at 10,000 samples per second after anti-aliasing filtering. A set of time histories of acceleration response on bridge deck and sound pressure under the bridge with modular joint for one heavy truck vehicle crossing the modular joint are shown in Fig.4-4, as an example. The figure shows that there were transient responses when the front and rear wheels of the vehicle excited the joint. In this study sound and vibration responses were analyzed in one third octave band for the time histories of sound pressures and accelerations. Time history data of 0.4

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 94

seconds in Fig.4-4 after vehicle impacts on the joint were used in the analysis. The sound pressure and vibration response in terms of velocity were expressed in decibel (dB). The reference values used in the sound pressure and velocity response to express in dB were

2 × 10 −5 Pa and 1× 10−9 m / s respectively. It was observed from the experimental results that heavy truck vehicles caused the greater level of vibration on the expansion joint and bridge. Therefore, measurement data for heavy truck vehicles were considered in the

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analysis.

(b)

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(a)

Fig. 4-4 Time series of vibration and sound pressure response: (a) Acceleration response on concrete deck, (b) Sound pressure response under the bridge

Figures 4-5 and 4-6 show the sound pressure responses measured under the bridge and at 5 m wayside of the bridge (Fig.4-1, right) respectively in the bridge with modular expansion joint. Results for 7 different heavy truck vehicles are shown in the figures. Similarly, Figs. 4-7 and 4-8 show the sound pressure responses at the corresponding locations in the bridge with finger type joint (Fig.4-1, left). Figures 4-9 and 4.10 show the comparison of sound pressure under the bridge and at 5 m wayside of the bridge respectively in two bridges while truck vehicle crossed the bridge.

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Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 95

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Fig. 4-5 Sound pressure response under the bridge with modular joint

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Fig. 4-6 Sound pressure response at 5 m wayside of the bridge with modular joint

Fig. 4-7 Sound pressure response under the bridge with finger joint

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Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 96

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Fig. 4-8 Sound pressure response at 5 m wayside of the bridge with finger joint

Fig. 4-9 Comparison of sound pressure under the bridges. Key:

, Modular joint;

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, Finger joint

Fig. 4-10 Comparison of sound pressure at 5 m wayside of the bridges. Key: Modular joint;

, Finger joint

,

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 97

Figure 4-11 shows vibration level measured at position P3 as shown in Fig. 4-1 on concrete deck of the bridge with modular joint. Figure 4-12 shows the vibration level measured at position P1 on concrete deck of the bridge with finger joint and Fig. 4-13

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shows the comparison of vibration level on these positions for one truck vehicle crossing.

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Fig. 4-11 Vibration level at position (P3) of concrete deck of the bridge with modular joint

Fig. 4-12 Vibration level at position (P1) of concrete deck of the bridge with finger joint

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Fig. 4-13 Comparison of vibration level on the deck of the bridges. Key: , Finger joint

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joint;

, Modular

Figure 4-14 shows the vibration level on the web of third girder (G3) of the bridge with modular joint. Figure 4-15 shows the vibration level on the web of third girder (G3) with finger joint and Fig. 4-16 shows the comparison of vibration level on these two locations

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for one heavy truck vehicle crossing. Truck vehicle considered for sound pressure and vibration level comparison between two bridges was same. Figures 4-17 and 4-18 show the vibration level in the vertical direction of a middle beam of the modular joint at

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measurement positions Position-1 and Position-3 respectively (Fig. 4-3).

Fig. 4-14 Vibration level on the web of third girder (G3) of the bridge with modular joint

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Fig. 4-15 Vibration level on the web of third girder (G3) of the bridge with finger joint

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Fig. 4-16 Comparison of vibration level on the girder of bridges. Key: joint; , Finger joint

Fig. 4-17 Velocity level on middle beam of the joint at Position-1

, Modular

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Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 100

Fig. 4-18 Velocity level on middle beam of the joint at Position-3

4.3.2 Possible vibration flow from the modular joint to the bridge

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The magnitude of the sound pressure at different locations in two bridges cannot be compared directly because the size of the bridge is not exactly same though deck thickness and depth and thickness of plate girders were similar. The width of bridge with finger joint was wider and numbers of I-girders in that bridge was more. Also, the vehicle

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loadings on both bridges could be different. However, from the above results on sound pressure at two locations around the two bridges, it can be seen that the noise response

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around the bridge with modular expansion joint is much greater than that around the bridge with finger type joint. The difference between the noise level is significant above 160 Hz centre frequency of 1/3 octave band under the bridge and below 50 Hz and above

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160 Hz at 5 m wayside of the bridge.

From Fig 4-11 to 4-13, it can be seen that vibration level on concrete deck during vehicle crossing is much greater on the bridge with modular joint than that on the bridge with finger joint. Greater difference can be observed in the frequencies above 160 Hz centre frequency. Similar pattern can be observed on the vibration in steel girders web though results for one girder are shown here from Fig.4-14 to 4-16. The vibration level on the bridge girder with modular joint is greater than that in the bridge girder with finger joint. The difference is much greater in the frequencies above 160 Hz. From Figs. 4-17 and 418, it can be seen that vertical vibration of the middle beam has dominant peak around 160 Hz and significant vibration in lower frequencies and higher frequencies as well. Also can be noted is that vibration response at Position-1 is much smaller than that in Position-

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 101

3. We can see that there is a dominant peak in vibration response around 160 Hz on middle beam, concrete deck and steel girder in the bridge with modular joint. There is dominant vibration peak on the concrete deck and steel girders in the higher frequencies at 250 Hz and 400 Hz also. There was no vibration measurement on the modular joint’s support beams which were directly connected to the bridge. From the vibration response results of middle beam of the modular joint, concrete deck and steel girders of the bridge with modular joint, it can be said that there is a possibility of vibration power transmission from the modular joint to the bridge. Because of this vibration transmission, the noise

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could have been radiated from the bridge vibration and higher sound pressure level observed on the bridge with modular joint could possibly be due to the noise contribution

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from the modular joint-cavity system itself and noise contribution from the bridge. 4.4 Vibration power flow estimation from the modular joint to the bridge

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4.4.1 Estimation procedure

The vibration power of the modular joint is transmitted to the bridge through the support beam connections with the bridge. The support beams of the joint are supported on the bridge deck. Figure 4-19 shows the top view of the bridge and modular joint. There are

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eight connection points through the end of eight support beams of the joint. There is a polyamide bearing at the bottom of support beam end and rubber spring at the top as shown in Fig.4-20. The arrows in the figure show the connection of support beam with the

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concrete deck.

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Girder web

Girder flange edge

Bridge deck Connection point

Fig. 4-19 Top view of the bridge and the modular joint

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Fig. 4-20 Cross-section view of the modular joint

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Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 102

The vibration power flow from any vibrating system through the point connection to the

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bridge can be estimated by (Janssens et al., 1996 and Cremer et al., 2005):

Pin, Bridge = Re(YBridge ) × F 2

(4.1)

where Pin, Bridge is the input vibration power to the bridge, Re(YBridge ) is the real part of

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point mechanical mobility of the bridge and F is the amplitude of the dynamic force at the connection point to the bridge. Here, only the real part of the point mobility of the bridge is considered because it is responsible for the vibration power flow in long distances. In case

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of modular joint, joint is connected to the bridge through the bearings and springs at support beams end (Fig.4-20). For the support beam vibrating with velocity vsup , equivalent spring constant of the bearing and spring at support beam end connected to

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the bridge as K eq and vibration frequency f in Hz, Equation (4.1) can further be expressed as:

{

}

Pin,Bridge = Re YBridge ×

2 K eq

4π f

2 2

v s2up

(4.2)

Each parameter in Equation (4.2) is important to estimate the vibration power input to the bridge. In following sections estimation of each parameter will be explained in detail. Velocity response on support beams of the expansion joint to the applied dynamic loadings will be estimated by FEM and point mobility of the bridge will be estimated by using the simplified formulation. Here, the vibration power flow is considered in the vertical direction only. This is because, the bearings at the bottom of support beam (connection point with bridge) are free to move except in vertical direction. Though, there was a rubber

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 103

spring at the upper side of the support beam connected to the deck, the spring constant of the springs in vertical direction was quite small as compared to that of bearing. Distribution of input vibration power in the bridge structural components will be estimated by using SEA and vibration response thus obtained will be used to estimate the radiated noise from the bridge. The vehicle pass-by experiment results of sound and vibration will be used to interpret the analytical results. Figure 4-21 shows the analytical procedure for the input vibration power estimation to the bridge, vibration distribution in the bridge and estimation of noise radiation from the bridge. Each of the steps will be explained in detail

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in following sections.

Velocity response of the joint to point loadings by FEM

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Point mobility of the bridge

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Input vibration power estimation to the bridge

Vehicle pass-bys experiment results

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Average vibration response estimation in the bridge structural components by SEA

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Acoustic power radiation from the bridge structural components

Sound pressure at field points

Vehicle pass-bys experiment results

Fig. 4-21 Analytical procedure for the vibration power flow analysis from the expansion joint to the bridge and sound radiation from the bridge

4.4.2 Estimation of vertical mobility of the bridge

Mechanical point mobility of the bridge is the ratio of vibration velocity to the applied force. The response measurement point and direction should be same as loading point and direction. The point mobility of the bridge depends on the loading point on the bridge,

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 104

structural configuration of the bridge etc. The bridge considered in the study consisted of concrete deck of 0.23 m thickness and four steel I-girders. The upper flanges of steel Igirders were connected to the concrete deck. The cross-section of the bridge is shown in Fig.4-22 which was also shown in Fig.4-1. The cross-section detail of the I-girders used in the bridge is shown in Fig.4-23. The vertical point mobility of the bridge would be different at different locations. For example, mobility of bridge in between the steel I-girders would be different than the mobility of bridge just above the web of the steel I-girders. To calculate the effective mobility of the bridge, mobility of steel girders and concrete deck

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need to be considered separately first and then combined mobility can be calculated by

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considering both.

G1

G2

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2.5 m

(a)

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Fig. 4-22 Cross-section view of the bridge

2.5 m

G4

G3

2.5 m

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 105

0.40 x 0.018 m

0.45 x 0.021 m

0.55 x 0.024 m

1.85 x 0.011 m

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1.85 x 0.009 m

1.85 x 0.011 m 0.55 x 0.02 m

0.50 x 0.016 m

(a)

(b)

0.60 x 0.02 m

(c)

G H

Fig. 4-23 Cross-section detail of I-girders used in the bridge: (a) First girder (G1), (b) second (G2) and third girder (G3) and (c) fourth girder (G4)

Vertical point mobility of I-girder:

RI

The vertical point mobility of I-girders depends on frequency. In low frequencies, mobility can be calculated by considering the I-girders as Euler beam; in middle frequencies, it can

PY

be calculated by Timoshenko beam and in higher frequencies the mobility can be calculated by assuming as plates.

Low frequency:

CO

In low frequencies, the real part of the mobility of I-girders can be calculated as infinite Euler beam by (Janssens et al., 1996 and Cremer et al., 2005): ⎧⎪ 1 Ygirder = Re ⎨ ⎩⎪ Zgirder

⎫⎪ 1 ⎬= ⎭⎪ m ' cB

(4.3)

where Ygirder is the real part of the point mobility of the I-girder, Z girder is the point impedance of I-girder, m ' is the mass per unit length and cB is the bending wave speed of the I-girder. The bending stiffness cB can be given by: 1/ 4

⎛ B ⎞ cB = ⎜ ⎟ ⎝ m'⎠

ω

(4.4)

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 106

where B = EI is the bending stiffness of the I-girder about the strongest axis. E is the Young’s Modulus of steel , I is the second moment of area of the girder about the strongest axis and ω is the circular frequency of vibration.

Middle frequency:

For the middle frequencies, shear effect must be taken into account and impedance can be approximated according to Timoshenko’s beam theory by (Janssens et al., 1996): 1

(4.5)

2 A G ρκ

D

Ygirder ≈

TE

where A is the cross-sectional area of the I-girder, ρ is the mass density , G is the shear modulus and κ is the shear co-efficient. Shear modulus is calculated by, G =

E . 2(1 + υ )

κ=

5 + 5ν 6 + 5ν

High frequency:

(4.6)

RI

where ν is the Poisson’s ratio.

G H

Shear coefficient κ for rectangular cross-section is calculated as (Hutchinson, 2001):

At high frequencies, longitudinal waves occurring in both length and height directions in

PY

the girders lead to a reduction in impedance (increase in mobility) relative to that of beam formula. The mobility in the high frequency region can be approximated by (Bewes et al.,

⎡ ⎢ 1 ⎢ ≈⎢ 2 1 ⎢ (ω / 4) ⎛⎜ 1 − υ + ⎜ ⎢ ρ cR 2 hw ⎝ Ehw ⎣

CO

2006):

Ygirder

⎤ ⎥ If E ρ hf ⎥ +8 ⎥ ⎞ 1− υ2 ⎥ ⎟⎟ ⎥ ⎠ ⎦

−1

(4.7)

where cR is the Rayleigh wave speed in the beam, hw and h f are the thickness of the girder web and flange respectively and I f is the second moment per unit width of area of the flange . The Rayleigh wave speed in the girder cR is calculated by (Rahman et al., 2006):

cR = C ∗cshear

(4.8)

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 107

where C ∗ = 0.874 + 0.196υ − 0.043υ 2 − 0.0553 and cshear is the shear wave speed and is given by cshear =

G

ρ

.

Vertical mobility in low frequency by FEM:

To confirm the reliability of use of the above formulation, detail numerical analysis of Igirder was conducted. Because of the large number of finite elements needed, mobility estimation from FEM in high frequency analysis was not feasible. Vertical mobility of first I-

D

girder was calculated by FEM analysis in low frequency range. The first girder (G1) on the first span having 31.5 m length was considered in the analysis. The girder was modeled

TE

by shell elements (SHELL63) in ANSYS 10.0. The horizontal and vertical stiffeners that were in the girder were also modeled by shell elements (SHELL63). The maximum size of the mesh was 10 cm. The FE model of I-girder is shown in Fig.4-24. Modal analysis was

G H

carried out to identify the natural frequencies and mode shapes of the girder. Some of the vibration modes of the girder are shown in Fig.4-25. To calculate the vertical point mobility of the bridge 10 different harmonic point loads were applied on the top flange of the girder above the web in half span as shown in Fig. 4-26 and velocity response was calculated in

RI

vertical direction at each loading point and average mobility was estimated. Due to the

CO

PY

large size of the girder, response calculation was limited up to 400 Hz.

31.5 m

(a)

(b)

Fig. 4-24 FE model of first span of first girder (G1): (a) full length view, (b) Detail view showing the stiffeners

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 108

f=30.99 Hz

TE

D

f=21.44 Hz

f=60.35 Hz

G H RI

f=99 Hz

f=69 Hz

f=127 Hz

CO

PY

Fig. 4-25 Some of the vertical and lateral vibration modes of the I-girder from FE analysis

Vertical point load

Fig. 4-26 Application of dynamic point loadings on the girder

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 109

Figure 4-27 shows the vertical mobility of the girder at different loading points and average mobility is shown by thick line. It can be seen from the figure that the response in lower frequencies has distinct peaks and is due to the excitation of structural modes of the joint. In the higher frequencies due to the high modal density, the response has no significant peaks. The vertical mobility of the I-girder calculated from FEM and estimated from infinite beam formula using Equation (4.3) is compared in Fig. 4-28. It can be seen in the figure that infinite beam representation in low frequencies has error in the response due to the limited number of structural modes of the girder. In higher frequencies, infinite beam

RI

G H

TE

D

formulation approximately represents the response of the girder.

line)

CO

PY

Fig.4-27 Vertical mobility of first I-girder to different loads by FEM. Average mobility (thick

Fig.4-28 Comparison of vertical mobility of first I-girder. Key: beam formula (Equation 4.3)

, FEM;

, infinite

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 110

Vertical mobility of first I-girder in the frequency up to 800 Hz obtained from Equation (4.3) to (4.5) is shown in Fig.4-29. The mobility from the low frequency formula crosses the value from high frequency around 390 Hz. The mobility of I-girder in further analysis is calculated below 390 Hz from low frequency formula at and above 390 Hz by high

G H

TE

D

frequency formula.

Fig. 4-29 Vertical mobility of first I-girder. Key:

, middle

, high frequency

RI

frequency and

, low frequency;

Vertical mobility of concrete deck:

PY

In the bridge, steel girders are connected with concrete deck. The mobility of the concrete deck also in low frequency has modal resonance peaks and in the higher frequencies due to the modal overlap, distinct peaks in the response disappear. The response of the

CO

concrete deck can be estimated by assuming the plate. In higher frequency, real part of point mobility can be calculated assuming infinite plate by (Cremer et al., 2005):

Ydeck =

1

'

3.5 B m

''

(4.9)

Econc tdeck 3 where B = is the bending stiffness of the plate, Econc is the Young’s Modulus 12(1 − υ 2 ) '

of concrete, tdeck is the thickness of concrete deck and m'' is the mass per unit area of the deck.

The vertical point mobility of bridge just above the I-girder is the combined mobility of Igirder and concrete deck. The combined mobility of built-up structures consisting of

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 111

different beams and plates and vibration power flow between them have been considered in detail in Grice et al. (2000a,2000b and 2000c), Grice et al.(2002) and Ji et al.(2003). The combined mobility of the bridge can be calculated by assuming the parallel connection:

1 YBridge

=

1 Ydeck

+

1

(4.10)

Ygirder

In Fig. 4-19, first and eighth support beam of the expansion joint are near the first girder and fourth girder respectively. Other support beams are in between the girders. Therefore,

D

to calculate the input vibration power from first and eighth support beam, combined

TE

mobility of first girder and deck and fourth girder and deck respectively was used. The vertical point mobility of steel girders, concrete deck and combined mobility calculated for first and fourth girder is shown in Fig. 4-30. To estimate the vibration power from other

CO

PY

RI

G H

support beams mobility of concrete deck was used.

Fig.4-30 Mobility of I-girders, concrete deck and combined mobility. Key: deck;

st

, 1

girder;

,4

th

girder;

, concrete

, combined (deck and 1st girder);

,combined (deck and 4th girder)

Lower frequency limit in the analysis:

As it was observed in the Fig. 4-28 that infinite beam representation gives error in the lower frequencies because of the resonance characteristics of the response caused by few modes presence in low frequency. Use of SEA in frequencies where number of structural modes is not sufficient will not give reliable results. To have reliable estimate from SEA, there should be at least 5 vibration modes in the frequency band considered

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 112

(Cremer et al., 2005). The approximate formula to calculate the modal density of a plate in flexural vibration is given by (Cremer et al., 2005): n=

S 3.6 × h × cL

(4.11)

Where S is the surface area of the plate, h is the thickness of the plate and cL is the longitudinal wave speed on the plate which is calculated by cL =

E . The number ρ (1 −ν 2 )

D

of modes in 1/3 octave band frequency was calculated on the concrete deck, girders web and flange. Figure 4-31 shows the number modes in 1/3 octave band for concrete deck,

TE

1st girder web and lower flange. It can be seen that above 100 Hz number of modes in all structural components is greater than 5. This range would be applicable for other girders web and flanges because number of modes will either be equal or greater than that in first

G H

girder web or flange because of the sizes of other girders. The lower frequency limit of the

CO

PY

RI

analysis using SEA was therefore decided as 100 Hz centre frequency of 1/3 octave band.

Fig.4-31 Number of vibration modes of bridge components in 1/3 octave band frequency. Key:

, concrete deck;

, 1st girder web and

, 1st girder lower flange

4.4.3 Estimation of velocity response of the modular joint

Figures 4-3 and 4-20 show the plan and cross-section respectively of the modular joint installed in the bridge. The bridge had two lanes and traffic direction was same in both lanes. Traffic direction was such that vehicle moved from the span of the bridge towards the expansion joint. 3-D finite element model of the expansion joint was developed as in Chapters 2 and 3. The modular joint consisted two middle beams and eight support

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 113

beams. Also there were control beams, frames and polyamide bearings and rubber springs at different locations. The size of the all components was exactly same as in the expansion joint that were explained in Chapters 2 and 3. Here also, middle beams, support beams, control beams and frames were modeled by shell elements (SHELL63) and rubber springs and polyamide bearings were modeled by spring elements (COMBIN14). The size of the FE mesh was 40 mm. The FE model of the expansion joint is shown in Fig. 4-32. The FE modal analysis was carried out so as to understand the dynamic characteristics: natural frequencies and mode shapes and modes were obtained

D

up to 1600 Hz. Some of the vertical vibration modes of the joint with significant support beam and middle beam vibration are shown in Fig.4-33. These modes with significant

TE

support beam vibration in vertical direction may be responsible to transmit the vibration

CO

PY

RI

G H

power to the bridge.

Middle beam

Fig. 4-32 3-D FE model of the expansion joint

Control beam Support beam

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 114

f=180 Hz

f=250 Hz

RI

G H

f=247 Hz

TE

D

f=176 Hz

f=279 Hz

PY

f=273 Hz

Fig.4-33 Some of the vibration modes of the joint

To estimate the velocity response of the expansion joint to the dynamic loadings that

CO

might have been applied during the vehicle crossing, dynamic point loadings were considered as in Chapter 3. Figure 4-34 shows the possible vehicle impact position on the expansion joint for the truck vehicle of axle width about 1.8 m if the truck passes from the centre of the lane. During the experimental study, it was observed that most of the vehicle passed from the right lane of the bridge. Two load cases were considered for two tire impacts. In each load case, one vertical and one lateral load were applied on the first middle beam as shown in Fig. 4-34. Harmonic point loading of unit magnitude was applied at 1 Hz interval in the frequency range of 100-800 Hz. Velocity response was calculated for each load cases at the measurement points (Fig. 4-3) on the first middle beam and at support beams end at the bearing connection point (Fig. 4-19). Velocity response was then estimated in 1/3 octave band centre frequencies. Here one 1/3 band is considered

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 115

because the SEA which will be explained and used later requires broad-band analysis due to the sufficient number of vibration modes of the structure needed to each frequency band. Velocity response in one third octave band was calculated by calculating the average spectral energy and divided by the bandwidth. The velocity responses of the middle beam at measurement points and at support beams connection points to the bridge to Load-1 are shown in Figs. 4-35 and 4-36 respectively. Similarly, velocity responses at corresponding points to Load-2 are shown in Figs. 4-37 and 4-38

Load-1

1.8 m

PY

RI

G H

CL

TE

Load-2

D

respectively.

CO

Fig. 4-34 FE model of the expansion joint and applied load positions

Fig. 4-35 Velocity response on the first middle beam to Load-1. Key: , Position-2 and

, Position-3

, Position-1;

TE

D

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 116

PY

RI

G H

Fig. 4-36 Velocity response on the 8 support beams to Load-1

Fig. 4-37 Velocity response on the first middle beam to Load-2. Key: , Position-3

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, Position-2 and

Fig. 4-38 Velocity response on the 8 support beams to Load-2

, Position-1;

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 117

As explained before in Section 4.3, vibration response measured during vehicle pass-bys at different locations were analyzed in one third octave band. Figs. 4-39, 4-40 and 4-41 show the velocity responses at three measurement points on the middle beam during

G H

TE

D

seven different truck vehicle pass-bys.

CO

PY

RI

Fig.4-39 Velocity response at Position-1 on the first middle beam during vehicle pass-bys

Fig.4-40 Velocity response at Position-2 on the first middle beam during vehicle pass-bys

TE

D

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 118

Fig.4-41 Velocity response at Position-3 on the first middle beam during vehicle pass-bys

Though experimental and numerical results cannot be compared directly, some similarity

G H

can be observed in the results. For example, significant velocity response can be observed from 160 to 250 Hz at Position-2 in numerical analysis to Load-1. In Load-2, significant response can be observed at 160 and 500 Hz at Position-2 and 250 Hz at

RI

Position-3. The response at Position-1 is smaller in both load cases. This could be due to the excitation of vibration modes of the joint having nodes or little vibration at the loading point (Fig. 4-33). In the support beams, dominant response can be observed at 160 Hz to

PY

Load-1 and at 250 Hz to Load-2. Dominant vibration response peak can be observed around 160 Hz in experimental results at Position-2 and around 160, 250 to 315 Hz at Position-3. The response at Position-1 is smaller as observed in the numerical analysis.

CO

4.4.4 Estimation of input vibration power to the bridge from the joint

After estimating the vertical mobility of the bridge and velocity response on the support beams the input vibration power to the bridge from the joint to the applied loadings on the joint was calculated from Equation (4.2). The input power was also calculated in one third octave band frequency. The equivalent spring constant in vertical direction K eq in Equation (4.2) was calculated by considering the parallel connection of polyamide bearing and rubber spring at the bottom and top of the support beam respectively. K eq = K bearing + K spring

(4.12)

where K bearing is the spring constant of bearing in vertical direction and K spring is the spring constant of rubber spring in vertical direction. The spring constants of bearing and spring used in the expansion joint were 5.97E+08 N/m and 5.83E+06 N/m respectively.

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 119

We can see that spring constant of bearing is about 100 times greater than that of rubber spring. Therefore, the vibration power input to the bridge from the joint was mainly dependent on the spring constant of the bearing used. Moreover, it can be seen in Equation (4.2) that input power is proportional to the square of the spring constant. The reduction of spring constant of the bearing reduces the input power to the bridge. Figure 4-42 shows frequency response of input vibration power to the bridge from the joint to the applied two load cases. In the figure, it can be seen that there is dominant response peak at 160 Hz centre frequency of 1/3 octave band to Load-1 and 250 Hz to Load-2. This

D

could be due to the vibration velocity peak observed on support beams response at 160 Hz to Load-1 (Fig. 4-36) and 250 Hz to Load-2 (Fig. 4-38). Vibration power input in the

TE

higher frequencies is less even though there was significant vibration on the support

PY

RI

G H

beams because of the frequency term in the denominator in Equation (4.2).

CO

Fig. 4-42 Frequency response of input vibration power to the bridge from the joint.

Key:

, Load-1 and

,Load-2

4.5 Estimation of vibration response of the bridge using SEA

The estimated vibration power in the previous section to the bridge deck from the modular joint is ultimately distributed to the different structural components of the bridge and dissipated through damping and possibly from radiation. The vibration power that is flowing to the deck from the joint through its support beams is transmitted to the I-girders web and lower flanges and is ultimately dissipated. The vibration power distribution in each structural component of the bridge was estimated by using Statistical Energy Analysis (SEA).

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 120

Use of SEA in this study is to estimate the average vibration response in the bridge and then use the estimated vibration response to predict the sound radiation from the bridge. The bridge structure is subdivided into four different subsystems as below. These subsystems are also shown in Fig. 4-43. (1) Concrete deck in bending vibration (2) I-girders web in plane vibration

(1)

(3)

(4)

G H

(2)

TE

(4) I-girders lower flange in bending vibration

D

(3) I-girders web out of plane vibration

RI

Fig.4-43 Cross-section view of the bridge and different subsystems in SEA

In SEA, different subsystems are assumed to be resonant, linear and undergoing

PY

harmonic motion so that the vibration energies can be represented by either maximum potential or kinetic energies. In this study, it was assumed that there was no damping loss at the junction of the concrete deck and I-girders top flanges and I-girders web and lower flanges. This is known as conservatively coupled system (Norton et al., 2003). In SEA,

CO

different subsystems are strongly coupled if the coupling loss factor between connected subsystems is more than the structural loss factor (damping) of each subsystem. Strong coupling generally occurs in lightly damped systems. Here strong coupling means that vibration energy can flow well between subsystems. In the previous literature Janssens et

al. (1996) and Harrison et al. (2000), strong coupling was assumed between different

subsystems of the bridge. The concrete deck and steel girders can be considered as lightly damped systems. The structural loss factor of concrete and steel is approximately 0.015 and 0.0006 respectively (Norton et al., 2003). In this study too, strong coupling between different subsystems was assumed.

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 121

Velocity response in the different subsystems can be calculated by using the power balance equation: Pin,Bridge = Pdiss + Prad ≈ Pdiss

(4.13)

where Pdiss is the dissipated power from the bridge due to damping, Prad is the radiated power from the bridge which is very small as compared to dissipated power in bridge and hence can be neglected (Janssens et al., 1996). The power dissipated from subsystem i can be calculated by:

D

Pdissi = ωηi mi v i 2

(4.14)

TE

where ηi is the structural loss factor of the subsystem i, mi is the mass per unit area and

v i 2 is the mean square vibration velocity response on the subsystem.

is the space

averaged velocity of the subsystem. According to SEA, for strongly coupled subsystems,

G H

ratio of square of their velocity response is equal to the ratio of their real part of point mobility (Cremer et al., 2005), i.e, vi 2 vj

2

≈

Re(Yi ) Re(Y j )

(4.15)

RI

The mobilities of different subsystems were calculated by assuming infinite size as explained in Section 4.4.2. The point mobility of concrete deck, girder web (out of plane

PY

bending), and lower flange was calculated from Equation (4.9). The point mobility of steel web (in plane motion) was calculated in a similar way to the steel girder mobility as explained before in section 4.4.2 excluding the flanges in the calculation. After knowing the mobility of each system, structural loss factors of concrete and steel and input

CO

vibration power to the bridge, space averaged velocity response of each subsystem was calculated by using Equations (4.13), (4.14) and (4.15). To calculate the velocity response, unit length of the bridge was considered i.e. surface area and mass of each subsystems were calculated per unit length of the bridge.

4.6 Estimation of sound radiation from the bridge

The vibrating subsystems of the bridge radiate sound to the surrounding environment. The radiated sound power from each subsystem can be calculated by: Pradi = ρair cair σ i Si v i 2

(4.16)

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 122

where ρair is the density of the air and cair is the velocity of sound in air. σ i is the radiation efficiency of the structural subsystem and Si is the surface area of the subsystem. Here, ρair and cair are taken as 1.225 kg / m3 and 340 m / s respectively. Average mean square velocity of each subsystem is already estimated in the previous section. The in plane vibration of web does not radiate sound. The radiation efficiency of concrete deck, steel girders web and lower flanges in bending was calculated in different frequencies assuming them as baffled strips. The radiation efficiency of each subsystem

D

at different frequencies was calculated from following expressions given by Xie at al.

for kc a ≤ 3

1/ 4

( kc a )

⎛

f ⎞

σ = ⎜1 − c ⎟ f ⎠ ⎝

−1/ 2

for kc a > 3

f fc

(4.17)

for f = fc

RI

σ = 1.2 − 1.3

for f < fc , α =

G H

⎛ 1+ α ⎞ (1 − α 2 )ln ⎜ ⎟ + 2α 4 f Pc ⎝ 1− α ⎠ + σ = 2air η π fc 4π Sfc (1 − α )3 / 2

TE

(2005):

for f > f c

PY

where P is perimeter, S is the surface area of the subsystem. fc is the critical frequency and η is the structural loss factor and a is the shorter edge of the subsystem. Critical 2 cair where cL is the longitudinal wave 1.8cL h

CO

frequency of a subsystem was calculated by f c =

speed and h is the thickness of the subsystems. The radiation efficiencies of different subsystems that were calculated at different frequencies are shown in Fig. 4-44. It can be seen that radiation efficiency of concrete deck is much greater than that of girder webs and lower flanges in lower frequencies indicating that concrete deck radiates sound efficiently in lower frequencies as well. Radiation efficiencies of webs were smaller than that of lower flanges and decks in the frequency range considered. The radiated sound power was calculated from each subsystem using Equation (4.16).

Fig. deck;

4-44

Radiation

efficiency

of

different

,2nd and 3rd girder lower flange;;

subsystems.

Key:

,concrete

,1st and 4th girder lower flange;

,

,2nd and 3rd girder web

G H

1st and 4th girder web,

TE

D

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 123

The estimated radiated sound power from the bridge subsystems were then used to calculate the sound pressure at any field points around the bridge. To calculate the sound

RI

pressure at any field point around the bridge, vibrating bridge subsystems were considered as line sources. The directivity of sound radiation was not considered and all the sources were assumed to be incoherent. The sound pressure at any field point can be

PY

estimated by (Janssens et al., 1996):

p 2field = ρair cair

1

π Lbridge Dfield

⎛ 0.5Lbridge arctan ⎜ ⎝ Dfield

⎞ ⎟ Prad ⎠

(4.18)

CO

where Lbridge is the length of the bridge (here 1 meter length was considered) and D field

is the distance of field point from the bridge. Sound pressure radiated from each subsystem was calculated and added to get the total sound pressure at any field point. The sound pressure was calculated at two field points under the bridge and at 5 m wayside of the bridge which were equivalent to measurement points as shown in Fig. 4-1.

4.7 Results and discussion 4.7.1 Velocity response of the bridge

Figures 4-45 and 4-46 show the frequency response of square of the velocity in different subsystems to Load-1 and Load-2 respectively. It can be observed from the figures that dominant velocity response in each subsystem was at 160 Hz to Load-1 and at 250 Hz to

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 124

Load-2. Also, it can be observed in both figures that velocity response of the concrete deck was much smaller than that in the I-girders web (out of plane bending) and lower flanges as expected.

Figures 4-47 and 4-48 show measured vibration velocity level at two locations P2 and P3 respectively on concrete deck (Fig. 5-1). Figures 4-49 and 4-50 show the measured velocity level in the third girder (G3) web and fourth girder (G4) web respectively. It can be seen that the velocity level was dominant at 160 Hz at P2 and P3 positions on the deck for

D

almost all truck vehicles crossing. Significant response can be observed from 250 to 400 Hz at P3 for some of the vehicles crossing although response varied for different vehicles.

CO

PY

RI

G H

at 160 to 250, 400 Hz on fourth girder (G4).

TE

Similarly, significant vibration can be seen at 160, 250 and 400 Hz on third girder (G3) and

Fig. 4-45 Square of the velocity response in different subsystems to Load-1. Key: concrete deck;

,1st girder web (in plane);

, 1st girder lower flange

,

, 1st girder web (out of plane);

TE

D

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 125

Fig. 4-46 Square of the velocity response in different subsystems to Load-2. Key: st

,1

girder web (in plane);

PY CO

,

girder web (out of plane);

RI

, 1 girder lower flange

, 1

st

G H

concrete deck;

st

Fig. 4-47 Velocity response level on concrete deck at P2 during truck vehicle pass-bys

TE

D

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 126

PY

RI

G H

Fig. 4-48 Velocity response level on concrete deck at P3 during truck vehicle pass-bys

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Fig. 4-49 Velocity response level on third steel girder (G3) web during truck vehicle passbys

Fig. 4-50 Velocity response level on fourth steel girder (G4) web during truck vehicle passbys

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 127

Though experimental and analytical results cannot be compared directly, there were some similarities that could be observed. There was a dominant vibration response peak at 160 Hz on all subsystems in both analytical results (Load-1) and experimental results on concrete deck (P2 and P3) and girders web (G3). In analytical results to Load-2, there was dominant vibration response at 250 Hz. In the experimental results significant response at 250 Hz can be observed for one vehicle on concrete deck (P3) and on girder web (G3 and G4). Figure 4.51 shows the velocity response on concrete deck (P3) and third girder (G3) during one truck vehicle pass-by. It can be seen that vibration level on

D

concrete deck was much smaller than that in I-girder web. Dominant peaks in both

PY

RI

G H

coupling between deck and I-girder web.

TE

concrete deck and I-girder can be observed at same frequencies. This shows the vibration

Fig. 4-51 Comparison of velocity response level on concrete deck at P3 and on third , deck and

,I-girder web

CO

girder (G3) web during truck vehicle pass-by. Key:

4.7.2 Sound pressure response at field points

The radiated sound pressure from the vibrating surface of the bridge calculated using Equation (4-18) is shown in Figs. 4-52 and 4-53 to Load-1 and Load-2 respectively. Both figures show the sound pressure contribution from the concrete deck, girders web, girders lower flange and total sound pressure from all subsystems. From the figures it can be seen that dominant peak of frequency response of sound pressure at 160 Hz to Load-1 and 250 Hz to Load-2. Different sound pressure contribution from different subsystems in different frequencies can clearly be observed. It can be observed that sound pressure contribution was dominated by concrete deck below 250 Hz, was comparable with steel girders between 250 to 315 Hz and in higher frequencies domination was from steel

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 128

girders (webs and lower flanges). Concrete deck contribution was dominant in low frequencies due to the high radiation efficiency of the concrete deck in lower frequencies (Fig. 4-44) and large radiating area even though vibration level of concrete deck was much lower than in steel girders web and lower flanges (Figs. 4-45 and 4-46). In higher

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frequencies steel girders dominated due to their increased radiation efficiencies.

Fig. 4-52 Contribution from different subsystems to the total sound pressure response , concrete deck;

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under the bridge to Load-1. Key:

, Total bridge

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, girders web;

, girders lower flange;

Fig. 4-53 Contribution from different subsystems to the total sound pressure response under the bridge to Load-2. Key: , girders web;

, concrete deck;

, Total bridge

, girders lower flange;

Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 129

Figures 4-54, 4-55 and 4-56 show the sound pressure level measured during vehicle pass-bys inside the cavity beneath the joint, under the bridge and at 5 m wayside of the bridge respectively. There is dominant sound pressure peak under the joint, under the bridge at 160 Hz and from 160 to 250 Hz at wayside of the bridge. The sound pressure under the joint could be due to the vibro-acoustics of the joint-cavity system. There can be some similarity between the analytical results and experimental results on sound pressure under the bridge and at 5 m wayside of the bridge mainly on frequencies of dominant peak of sound pressure. There could be some contribution on the sound pressure

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measured at field points from the noise generated on the road surface.

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Fig. 4-54 Sound pressure level inside the cavity beneath the joint during vehicle pass-bys

Fig. 4-55 Sound pressure level under the bridge during vehicle pass-bys

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Chapter 4 Vibration power flow from modular expansion joint to steel-concrete non-composite bridge and noise radiation from the bridge Page 130

Fig. 4-56 Sound pressure level at 5 meters wayside of the bridge during vehicle pass-bys

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4.8 Conclusions

A simplified analytical approach using finite element method (FEM) and statistical energy analysis (SEA) was considered for the vibro-acoustic analysis of steel-concrete noncomposite bridge with modular expansion joint at its one end. FE model of the expansion

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joint was developed and velocity response was estimated to the applied dynamic loadings to represent the loadings that might have been applied during the vehicle pass-bys. The velocity response of the expansion joint was then used to predict the input vibration power

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to the bridge from the joint. SEA was used to estimate the vibration response on the structural components of the bridge and radiated sound pressure at field points around the bridge was calculated. Sound and vibration measurement of the expansion joint and

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bridge during vehicle pass-bys was also conducted in a separate study. Sound and vibration response analysis was carried out in the frequency range of 100-800 Hz. It was found that the simplified approach considered in this study was able to predict the flow of vibration power to the bridge from the expansion joint and noise radiation from the bridge during vehicle impact. However, the simplified approach based on SEA fails to predict the response in the lower frequency where the response is dominated by the resonance of few structural modes. This simplified method can be utilized to reduce the vibration power flow from the modular joint to the bridge.

Chapter 5 General discussion 5.1 Sound radiation efficiency of joint-cavity system The sound radiation efficiency of the joint-cavity system was discussed in section 3.6.3 of Chapter 3. It was shown that sound radiation efficiency at frequencies below 100 Hz was

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low even though there was significant vibration response mainly due to the vibration modes of the joint with significant lateral vibration of middle beams (Fig. 3-33). This

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implies that these vibration modes of the joint were not efficient sound radiators. Figure 332 also showed that there was high radiation efficiency from 100 to 160 Hz due to the vertical vibration modes of the joint with significant vibration in middle beams and support

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beams (coupled modes). Significant sound pressure response inside the cavity was observed at natural frequencies of these vibration modes.

From 160 to 225 Hz the radiation efficiency was low even though there was significant

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vibration response in both lateral and vertical direction. In this frequency range, there were vertical vibration modes with significant vibration in middle beams which were vibrating out of phase to each other and less vibration in support beams. Also, there were several

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modes with significant middle beams vibration in lateral direction in this frequency range. This showed that both vertical vibration modes in which middle beams were vibrating out

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of phase to each other and lateral vibration modes did not radiate sound efficiently.

In consistent with this finding, from the full scale model of the modular joint studied in Chapter 2, it was found that significant sound pressure inside the cavity was due to the vertical vibration modes of the joint with significant vibration in middle beams and support beams (Fig. 2-18 and 2-21).The sizes of different components of modular joint used in full scale model and in real bridges investigated in this thesis are the sizes that may be commonly used in practice. Therefore in modular expansion joint-cavity system, vertical vibration modes of the joint with significant vibration in middle beams and support beams (coupled modes) can radiate sound efficiently. Sound radiation efficiency of lateral vibration modes of the joint may be low.

Chapter 5 General discussion

Page 132

5.2 Sound radiation characteristics of joint-cavity system to outside environment As discussed in section 3.6.4 of Chapter 3, noise from the joint-cavity system may be radiated at propagation angles of acoustic modes of the cavity. Plane wave modes of the cavity can radiate sound at all direction whereas each higher order mode (other than plane wave mode) can radiate sound outside at radiation angle (equivalent to propagation angle of acoustic mode inside the cavity). From the directivity patterns of sound radiation from the joint-cavity system on a vertical plane discussed in section 3.6.4 (Fig. 3-34 and 335), it was observed that significant sound pressure was radiated at certain angles from

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the longitudinal axis of the cavity.

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Higher order modes are formed inside the cavity above the cutoff frequency of cavity which is calculated by Equation (3.11). Generally, the cavity beneath the modular type joint may have depth (size of the cavity in vertical direction) larger than its width (size

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along bridge axis). Therefore, the cutoff frequency of the cavity is governed by its depth size. Cutoff frequencies of the cavity for different possible depths of the cavity in real bridge modular joint are shown in Fig. 5-1. The figure shows that for possible depths of the cavity say above 1 meter, higher orders modes of the cavity, which can radiate sound

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at propagation angles which are away from the longitudinal axis of the cavity, are formed in the frequency range in which significant noise generation and radiation inside the cavity

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of joint-cavity system is observed from the experiment. This shows that sound radiation from modular expansion joint-cavity system to outside environment can be away from the longitudinal axis of the cavity. The number of acoustic modes of certain order depends on the length of the cavity as well: the longer the length of the cavity, higher the number of

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acoustic modes of certain order. For the cavity beneath the joint in large bridge width like of four lanes, six lanes etc., there may be significant sound radiation near the longitudinal axis of the cavity axis as well in higher frequencies from lower order acoustic modes.

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Chapter 5 General discussion

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Fig.5-1 Cutoff frequency of the cavity for different depth

5.3 Limitations of the numerical and experimental studies in the thesis The numerical investigations carried out in this thesis to understand the noise generation and radiation of the modular expansion joint-cavity system using FEM-BEM approach

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have provided deeper insight into the problem. The noise is generated and radiated inside the cavity of the joint-cavity system from the excitation of structural modes of the joint

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and/or acoustic modes of the cavity. The generated and radiated noise inside the cavity of the joint-cavity system is ultimately radiated to the outside environment from the openings of the cavity. Analysis of vibration power flow from the modular joint to the connected bridge and noise radiation from the bridge was investigated with simplified approach using

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the FEM and SEA. Numerical results in the above studies were interpreted with experimental results as discussed in Chapters 2, 3 and 4. However, there were some limitations of the numerical and experimental studies carried out in this thesis.

In the study in Chapter 2, numerical results were compared with experimental results from impact testing experiment. There were discrepancies between the experimental and numerical results. This could be due to several possible reasons as explained in Chapter 2. There may be some errors in experimental results as well. There were limited measurement data of sound and vibration in the experiment. Sound pressure was measured inside the cavity at a single point which may not be enough to understand the

Chapter 5 General discussion

Page 134

noise characteristics inside the cavity. There were limited impact positions and vibration response measurement positions on the joint.

In the numerical study in Chapter 3, the dynamic loadings on the joint during the vehicle pass-by were not known. The harmonic point loadings were considered in the numerical study to represent the dynamic loadings during vehicle pass-bys. However, the loading on the joint from the vehicle impact has temporal and spatial variations. The characteristics of the response of the joint to dynamic vehicle loadings will be different from those to

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stationary harmonic loadings to some extent. Harmonic loadings were chosen because of the limitations of numerical tools used in the study. In the numerical investigation carried

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out in this study to understand the noise radiation from the joint-cavity system to the outside environment only geometrical spreading was considered. Effect of ground and other obstacles such as the pier, bridge deck and girders which may have some effect on

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the noise radiation pattern of the joint-cavity system were not considered. Rigid ground surface increases the sound pressure at receiver point by reflection. Impedance ground surface causes the interference of direct and reflected sound radiation at the receiver point. Also the atmospheric effects like air absorption, atmospheric turbulence,

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temperature gradient etc. which may have certain effects on sound radiation and propagation were not considered in the study. To consider these above mentioned effects

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on sound propagation from joint-cavity system to the surrounding, simplified propagation models can be utilized. The height of source and receiver, impedance of ground surface, atmospheric variations etc. may be the input parameters of the models.

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Numerical results in Chapter 3 were interpreted with experimental results of vehicle passbys. Although the magnitude of the experimental and numerical results could not be compared directly because the loading in the experiment was not known and was certainly different than in numerical analysis, there were similarities in the experimental and numerical results such as dominant frequency components as discussed in Chapter 3. There may be some errors the experimental study as well. For example, in the sound pressure measured inside the cavity and at field points on the ground, there could be some effect of wind which was present during the experiment. Also, there may be noise contribution at field points from the top surface of the joint such as tire noise, air gap resonance noise etc. The noise generation and radiation from the top surface of the joint was out of the scope of this study.

Chapter 5 General discussion

Page 135

In the numerical study in Chapter 4 as in Chapter 3, the dynamic loadings considered in the numerical analysis were to represent vehicle impact was harmonic point loadings. Also in this study, magnitude of experimental and numerical results could not be compared directly. However, there were some similarities like dominant frequency components in the response of sound and vibration. SEA applied in this study assumes excitation on the structure as steady state. However, the excitation on the bridge during vehicle pass-by is rather transient like. The numerical study based on SEA estimates vibration response in each structural components of the bridge as space averaged

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response. However, there were limited measurements points in concrete deck and as well as on steel girder webs during the experiment. To have reliable space average data, more

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measurement points may be needed in different structural components of the bridge. Also, in the numerical analysis strong vibration coupling was assumed between the different structural components of the bridge as this assumption was reasonable in the lightly

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damped systems. This important assumption which decides the vibration response distribution in each structural component could not be confirmed because of the lack of experimental data such as coupling loss factors measurement. The sound pressure response estimation using this approach does not consider the directivity of sound

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radiation. The sound radiation from the bridge structural components may have certain directivity pattern. Therefore, the results obtained from this approach may have some

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discrepancies with experimental results.

5.4 Future applications of the thesis findings The objective of the studies in this thesis was to understand the noise generation and

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radiation mechanism of the modular expansion joint. The findings from this study helped to understand the mechanism of noise generation and radiation inside the cavity and noise radiation to the outside environment from the openings of the cavity. After clear understanding of mechanism and characteristics of noise generation and radiation, next step is to reduce the noise by using the suitable techniques. Even though there were some limitations of numerical as well as experimental studies, it is believed that findings from this thesis could be helpful to control the noise caused by modular expansion joint installed in road bridges.

At first, any noise control engineer thinks to control the noise at the source itself. Noise which is generated and radiated inside the cavity beneath the joint is ultimately radiated to

Chapter 5 General discussion

Page 136

the outside environment from the openings of the cavity. The studies in Chapters 2 and 3 investigated the noise generation and radiation mechanism inside the cavity of the jointcavity system. Excitation of structural vibration modes of the joint and/or acoustic modes of the cavity caused the high sound pressure inside the cavity. Noise mitigation measures which involve the frequency change in structural and acoustic modes such as use of stiffer structural component of the joint may not be effective because of the modal densities of structural and acoustic modes and coupling between them observed in Chapters 2 and 3. Some mitigation measures considering the propagation of acoustic modes of the cavity

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may be effective to reduce the noise from the bottom part of the joint.

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The investigation on noise radiation characteristics of the joint-cavity system to the outside environment enabled to understand the noise radiation pattern which was found directional and frequency dependent. The noise is radiated outside at propagation angle

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of acoustic modes of the cavity. As discussed in section 5.2, the main lobes of sound radiation may be at certain angle from the longitudinal axis of the cavity. Far field radiation angle can be approximated from the size of the cavity and frequency of sound radiation. Affected regions around the joint from the noise radiation from the bottom side of the joint

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may therefore be identified from the height of bridge or viaduct and receiver points such as different storey level of the buildings. The near field and far field of sound radiation

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around the joint-cavity system can be approximated from the size of the cavity. This information could be helpful to locate sound measurement points for environmental noise assessment studies.

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The study carried out in Chapter 4 to understand the vibration power flow from the joint to the bridge and noise radiation from the bridge showed the possibility of vibration power flow prediction from the joint to the bridge noise radiation from the bridge around the joint in addition to the noise from the joint-cavity system. The simplified approach chosen in the study makes possible the prediction of the noise radiation from the bridge, which cannot be solved by deterministic approaches like FEM or BEM because of the large size of the structure and high frequency of interest. This approach can be utilized at the design stage to reduce the vibration power flow from the joint to the bridge by considering vibration isolation and hence reduce the noise radiation.

Chapter 6 General conclusions and recommendations for future work The noise generated and radiated from modular expansion joint has been a localized environmental problem in Japan and elsewhere. Understanding of noise generation and

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radiation is important from noise control point of view. From the previous studies noise generation and radiation mechanism inside the cavity beneath the joint is not fully

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understood. Studies on noise radiation from the modular joint to the outside environment have not been reported yet. Possible vibration power flow from the modular joint to the connected bridge and noise radiation from the bridge in addition to the noise from the joint

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itself could exacerbate the noise problem around the joint. The objectives of this study have been to consider these issues.

A full scale model of modular joint was considered to investigate the noise generation and

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radiation mechanism inside the cavity. A modular joint installed between the prestressed concrete bridges was considered to understand the characteristics and mechanism of noise radiation from the joint and cavity beneath it to the outside environment. Vibro-

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acoustic analysis was carried out in these both studies in the frequencies below 400 Hz which is the frequency range in which noise problem from bottom part of the modular joint has been observed in experimental studies. A modular joint installed in a steel-concrete

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non-composite bridge was considered to understand the possible vibration power flow from the joint to the bridge and noise radiation from the bridge.

6.1 General conclusions from the thesis The following are the main conclusions drawn from this thesis. ¾ Noise from the bottom side of the modular expansion joint was caused by the excitation of structural modes of the joint and/or acoustic modes of the cavity beneath the joint. Dominant noise peaks inside the cavity in lower frequencies were from the structural modes of the joint with possible interaction with acoustic modes of the cavity. In higher frequencies acoustic resonance of the cavity showed significant effect.

Chapter 6 General conclusions and recommendations for future work

Page 138

¾ The sound radiation efficiency of the joint-cavity system appeared to be high at natural frequencies of vibration modes of the joint with significant vertical vibration of middle beams and support beams (coupled modes).The radiation efficiency of lateral vibration modes of the joint appeared to be low. ¾ Noise from the joint-cavity system may be propagated most effectively at radiation angles of acoustic modes of the cavity, which can be predicted approximately from

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the fundamental theory of sound radiation from cavities and waveguides. ¾ The boundary between near field and far field in the sound field around the joint-

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cavity system may be predicted approximately by the previous findings of the characteristics of radiation field of sound source by considering the greater

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dimension of the cavity cross-section as the maximum source dimension. ¾ A simplified approach based on statistical energy analysis presented in this study for the vibration power flow analysis can be helpful to predict the vibration power flow from the modular expansion joint to the connected bridge and estimate the

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noise contribution from the bridge. This approach can be utilized to reduce the vibration flow from the joint to the bridge.

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6.2 Recommendations for future work

¾ After understanding the noise generation and radiation mechanism of modular joint, next step is to investigate the control of noise. The most effective way would be to

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reduce the noise at the source itself. Investigation of noise control measures which reduces the noise inside the cavity of the joint-cavity system merits further research.

¾ For the reliable application of simplified approach that was utilized in the vibration power flow analysis from the expansion joint to the bridge, more experimental vibration data are needed at sufficient numbers of locations in the bridge structural components to have an average estimate of the response of the bridge. Vibration measurement data on expansion joint including the support beams as well as on bridge could be helpful for reliable estimation the coupling loss factors that are needed for the application of detail SEA model.

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