Statistical Models of the Lambert Road Bridge: Changes in Natural Frequencies Due to Temperature

Transcription

1 Utah State University All Graduate Theses and Dissertations Graduate Studies Statistical Models of the Lambert Road Bridge: Changes in Natural Frequencies Due to Temperature Nickolas Ryan Foust Utah State University Follow this and additional works at: Part of the Civil and Environmental Engineering Commons Recommended Citation Foust, Nickolas Ryan, "Statistical Models of the Lambert Road Bridge: Changes in Natural Frequencies Due to Temperature" (2014). All Graduate Theses and Dissertations This Thesis is brought to you for free and open access by the Graduate Studies at It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of For more information, please contact

2 STATISTICAL MODELS OF THE LAMBERT ROAD BRIDGE: CHANGES IN NATURAL FREQUENCIES DUE TO TEMPERATURE by Nickolas Ryan Foust A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Civil and Environmental Engineering (Structural Engineering and Mechanics) Approved: Marvin W. Halling Major Professor Paul J. Barr Committee Member Gilberto E. Urroz Committee Member Mark R. McLellan Vice President for Research and Dean of the School of Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 2014

3 ii COPYRIGHT NOTICE Copyright Nickolas Ryan Foust 2014 All Rights Reserved

4 iii ABSTRACT Statistical Models of the Lambert Road Bridge: Changes in Natural Frequencies due to Temperature by Nickolas Ryan Foust, Master of Science Utah State University, 2014 Major Professor: Dr. Marvin W. Halling Department: Civil and Environmental Engineering Structural health monitoring (SHM) using ambient vibration has become a tool in evaluating and assessing the condition of civil structures. For bridge structures, a vibration-based SHM system uses the dynamic response of a bridge to measure modal parameters. A change in a structure s modal parameters can indicate a physical change in the system, such as damage or a boundary condition change. These same modal parameters are sensitive to environmental factors, mainly temperature. Statistical models have been utilized to filter out modal parameter changes influenced by temperature and those caused by physical changes. Statistical models also help describe the relationship between modal parameters and environmental conditions. The Lambert Road Bridge is a concrete integral abutment bridge located south of Sacramento, California, and is studied through this paper. A SHM system has been installed and has been recorded for 3 years. Three months of SHM records will be used to understand how the bridge s natural frequencies typically change due to temperature.

5 iv First, temperature was observed to be the driving force behind many of the SHM records. A linear relationship was found between the structure s natural frequency and temperature. Collinearities between potential predictor variables were noticed. Initial linear regression analyses were completed with a bridge average temperature. Certain strain gauge regression models were used as base models to eliminate other regression models that potentially were altered by aliasing. These base models, and the other seven corresponding models, showed a direct linear relationship between temperature and natural frequency. It was concluded that changes in boundary conditions due to bridge expansion have a greater effect on global dynamic properties than material property changes due to temperature. Stepwise linear regression followed the initial regression modeling. Eight thermocouple readings were consistently being selected in the stepwise process and were chosen to be the main predictor variables. Due to collinearities among the predictor variables, ridge regression was completed to eliminate any unstable variables. The final six sensors locations indicate that longitudinal, transverse, and depth gradients are all important factors in the linear regression models of this relationship. Comparing the multiple linear regression models to single-variable regression models with the highest averaged adjusted R 2 values, a minimum percent difference of 21% and 19% was seen for the first and second natural frequencies, respectively. It was also concluded that these multiple linear regression models explained more of the variability in the natural frequencies and would be a better model to use to filter out temperature effects. (94 pages)

6 v PUBLIC ABSTRACT Statistical Models of Two Concrete Bridges: Changes in Natural Frequencies Due to Temperature Nickolas Ryan Foust Structural health monitoring (SHM) systems have become tools for understanding how a civil structure truly behaves. In bridge structures, vibration-based SHM systems can be used to monitor the bridge s modal parameters. Changes in modal parameters may indicate a change in the structure itself, boundary conditions, or damage. These modal parameters also vary with environmental factors, mainly temperature. Statistical models help show the correlation between modal parameters and temperature and can be used to filter out unimportant temperature effects. If done successfully, physical changes can be indicated and further inspection can be accomplished to minimize further damage. The bridge structure utilized in this study is located in Sacramento, California. It is a two-span, integral abutment, cast-in-place concrete, box-girder bridge. A SHM system consisting of velocity transducers, thermocouples, tilt-meters, and strain gauges has been monitoring the structure s behaviors and environmental conditions. A long-term view of the bridge behavior was used in this study. Temperatures through the bridge deck, along the length, and transversely are used in many different statistical models. Final regression models that include the most important temperature readings are chosen.

7 vi ACKNOWLEDGMENTS I would like to thank Dr. Marvin Halling for his years of teaching and making this research opportunity available to further my education in civil engineering. I would also include those professors that I have worked with or have taught me through the past years at Utah State University: Paul Barr, James Bay, Bruce Bishop, Joe Caliendo, William Rahmeyer, and Gilberto Urroz. All of their contribution and interest in each student s education, time in class preparation, love for engineering, and friendships have helped and inspired me through the past years. Their lessons will be added to those that have motivated me to improve and will be remembered throughout my career and life. I would also acknowledge the countless efforts, love, and encouragements of my parents, R. Michael and Leasa Foust. The values they have taught me through the years have guided me to where and who I am today. Through the good times in life, you were there to celebrate; through the hard times, you were there for understanding and encouragement. Your love and examples will never be forgotten and I will always be indebted to you. I also appreciate the many others who have helped me in various ways in life: my sisters and their families, grandparents and extended family, friends, colleagues, coaches, and co-workers. Thanks for the good experiences we have had together, the moral support, and encouragement to accomplish my goals. Nickolas Ryan Foust

8 vii CONTENTS Page ABSTRACT... iii PUBLIC ABSTRACT... v ACKNOWLEDGMENTS... vi LIST OF TABLES... viii LIST OF FIGURES... ix CHAPTER 1 TOPIC INTRODUCTION AND LITERATURE REVIEW... 1 Introduction... 1 Literature Review... 2 Literature Review Summary CALIFORNIA LAMBERT ROAD BRIDGE California Lambert Road Bridge Description Structural Health Monitoring System Fast Sampling Rate Record Processing STATISTICAL MODELING PROCESS Predictor Variable Exploration and Selection Initial Linear Regression Models Initial Stepwise Regression Model Multiple Linear Regression with Ridge Regression Final Regression Models SUMMARY AND CONCLUSION REFERENCES... 93

9 viii LIST OF TABLES Table Page 1 Sensor Names, Descriptions, & Locations I Sensor Names, Descriptions, & Locations II Potential Natural Frequencies Average Bridge Temperature Linear Regression Coefficients I Average Bridge Temperature Linear Regression Coefficients II Stepwise Regression Variables I Stepwise Regression Variables II Correlation Coefficients Regression Coefficients & Values for SG-G4-BF-CC with Time Lags Adjusted R 2 Values with Time Lags (1 st Natural Frequency) Adjusted R 2 Values with Time Lags (2 nd Natural Frequency) Linear Regression Model Statistical Values Single-Variable Regression Models (First & Second Natural Frequency) Percent Differences for First Natural Frequency Percent Differences for Second Natural Frequency... 90

10 ix LIST OF FIGURES Figure Page 1 Bridge Location from Sacramento, California (Google Maps, 2014) Satellite Plan View (Google Maps, 2014) Elevation Looking East Typical Cross-Section Bent Cap & Column Elevation (Hodson, 2010) Column Section (Hodson, 2010) Abutment Detail (Hodson, 2010) Plan View with Sections Tiltmeters at Sections A-A, D-D, & H-H Sensor Locations at Section B-B & F-F Sensor Locations at Sections C-C & G-G Deck Thermocouple (DTC) Locations Sensor Locations at Section E-E System Layout from Below Vibrating Wire & Hitec Strain Gauges Tiltmeter Uninstalled Velocity Transducer & Deck Thermocouples Web Thermocouples Data Acquisition Systems Tiltmeter Time Domain Record Example... 37

11 x 21 Processed Tiltmeter Example Probable Natural Frequency Example from Tiltmeter Over Smoothing Example on a Tiltmeter Record Noise Cancelation Example on a Tiltmeter Record Folding Example Typical FFT of a Strain Gauge Consecutive FFT of Strain Gauges Initial Processing Run & Potential Peaks of Six Sensors Graphical User Interface (GUI) Example & Options West Temperature Gradient March 2 nd -6 th West Temperature Gradient March 2 nd -6 th (Depth View) West Temperature Gradient March 2 nd -6 th (Time View) Temperatures, Natural Frequencies, Tiltmeter, & Strain Gauges Collinearity between Sensors Temperatures and Natural Frequencies I Scatterplot of VT-UD-BB & DTC-West Scatterplot of VT-UD-CC & DTC-West Temperatures, Natural Frequencies, & Strain Gauge II West Temperature Gradient & Bridge Average Temperature Average Bridge temperature & Natural Frequencies Linear Regression for Average Bridge Temperature & SG-G3-TF-CC Linear Regression for VT-UD-CC & DTC-West Linear Regression with 15 Minute Time Lag Increments... 72

12 x 44 Ridge Regression, 1 st Iteration Ridge Regression, 2 nd Iteration Ridge Regression, 3 rd Iteration... 82

13 CHAPTER 1 TOPIC INTRODUCTION AND LITERATURE REVIEW In the first chapter, an introduction to structural health monitoring (SHM) will be given. Then, a literature review will be given. Finally a summary of the literature and some important topics that will be used in this thesis will be given. Introduction The ability to assess and evaluate a structure s health by an efficient and cost effect method is growing in importance. This is due to the increasing number of structures being constructed to support demand, many of which are more complex. The structural integrity and health of older structures are unknown. Structural health monitoring (SHM) has become a trending research topic due to the potential to help evaluate the civil infrastructure. Many topics within SHM are still being researched in order to simplify the general application to the circumstances of large, unique structures. The application of SHM within the civil infrastructure is more complex due to the uniqueness of each structure. SHM uses a structure s modal parameters for evaluation. In order to start evaluating a structure s condition, environmental effects, mainly temperature, must be taken into consideration. Temperature causes the material properties to slightly change, which can affect the modal properties. In order to further describe the study topic, a literature review is presented first. This is to give a broader view of the direction of this type of research and its application in the civil engineering field. Each paper will be individually summarized, with certain aspects that pertain to the California Lambert Road

14 2 Bridge being considered. After the literature review, Chapter 2 will give descriptions of the bridge, the SHM system in place, and the record processing steps taken to account for temperature effects on the bridge. Chapter 3 will give the statistical modeling steps used to create a base regression model to describe the temperature effects. Chapter 4 will summarize the study, its findings, and give the conclusions. Literature Review Structural Health Monitoring (SHM) within the field of civil engineering has been defined as the implemented process of a damage detection strategy of civil structures. This detection strategy uses observations of modal parameters of the structure s normal performance, which must be established as a standard, to compare periodic observations. When compared to the established standard, these periodic observations can indicate a change in the system s performance. Within other engineering fields, such as aerospace and mechanical engineering, SHM has been a successful tool in assessing a structure s condition and detecting damage. Application of this assessment tool within civil engineering has not been utilized as frequently. This is due to the unique modal parameter variations that come with each structure s behavior. The variations that are present in each structure make the application of SHM more specialized, hence more expensive, within the civil discipline. SHM has been successful and economical in these other fields due to the general application. The hopes to bring the same success and implementation into the civil field are some of the driving factors in this research area.

15 3 Research has been rising in the field of SHM with the increasing need of a more effective, economical, accurate, and efficient way to assess the condition of civil structures. The understanding of how a civil structure s performance changes over its lifetime is also not well understood or, currently, taken into consideration at the design stage. This is another reason research in SHM is increasing. The literature that is reviewed in this section deals specifically with the processes or analyses in SHM or the application of these to bridge structures. The research papers that give analytical direction, a similar experimental setup to that presented in this thesis, or a history of where SHM is in the civil field are summarized in this section. Temperature Effect on Vibration Properties of Civil Structures: a Literature Review and Case Studies (Xia et al., 2012) This paper introduces SHM and how it can be used in condition assessment, specifically for civil structures. It also, describes many studies that give examples of existing problems within SHM, mainly variations in vibration properties due to temperature effects. These effects are shown to be in most civil structures including bridges, buildings, and laboratory experiments. The paper describes different statistical models used to relate these effects. After, it gives a quantitative analysis, laboratory comparative study, and two case studies. It was concluded that variations in frequencies are mainly cause by material changes in the modulus under different temperatures. This paper summarizes many of the results that studies have given that attempt to explain the temperature effect on vibrational properties. The paper introduces the topics of vibration-based structural condition assessment and SHM by describing some of the

16 4 problems that structural temperature and environmental variations create. Results from multiple studies are referenced to show that temperature variations are shown to be as or more significant than structural damage. It is stated that if temperature effects are not taken into account, vibration-based condition assessment can be impacted. Other variables that play important roles in making assessments more reliable are mentioned to be boundary condition changes, non-uniform temperature distributions, and thermal inertia effects. This paper explains and refers to the findings from many papers that show how a structure s vibrational properties and temperature relate. Variations in natural frequencies, mode shape, and damping are how the vibrational properties are divided. The vibrational properties, in this same order, describe the extent to which the relationship to temperature is understood. The percentage change, if available, in each of these vibrational properties is given. After the relationship between temperature and vibrational properties are described, the statistical models on modal frequencies are described. The models described include: regression models, autoregressive models, and principle component analysis. Within all three categories of statistical models and tools listed in this paper, the referenced papers were categorized into groups. These groups consisted of the following categories: linear regression and multiple linear regression models, control chart analysis, support vector machine technique, autoregressive (AR) and autoregressive with an exogenous (ARX) model, and principle component analysis. A quantitative analysis was performed for this study. The analysis found that the percent change of the natural frequency of steel, aluminum, and reinforced concrete (RC)

17 5 beams with temperature are 0.018, 0.028, and 0.15% per degree Celsius, respectively. These values are then used to compare with the results from the laboratory comparative study, case studies, and in the conclusion and discussion sections. In the laboratory comparative study, the percent change found for steel, aluminum, and RC beams were 0.036, 0.056, and 0.30%, respectively. Close to half of the variations in natural bending frequencies are shown to be from modulus changes due to temperature. No correlation between damping and temperature was found. The effects that temperature had on mode shapes were not covered by this paper. Two case studies of the Tsing Ma Suspension Bridge and the Guangzhou New TV Tower were described in this paper. The suspension bridge had an average percent frequency change of 0.018%, which is close to that found for the steel beam. The TV Tower percent change was not listed, although the slope of the linear fitted curve was noted to be 1.5 x 10-3 / C, which was close to half the concrete laboratory slope of -3.0 x 10-3 / C. In conclusion of these case studies, the paper implied that even in a large-scale structure, variations in bending frequencies are mainly due to effects of modulus changes due to temperature. The applicable conclusion of this paper that relates to the topic of interest is the following: the changes of natural frequencies seen in civil structure are mainly produced by material changes (modulus) under different temperatures.

18 6 A Review of Structural Health Monitoring Literature (Sohn et al., 2004) An overall view of the topics within SHM is given by this paper. This paper focuses on SHM within structural engineering in the years between 1996 and The review first, describes the purpose and definition of SHM. It then, divides SHM into four categories with specific topics within these categories. These topics are listed and discussed giving specific literature examples. Strong and weak points within these topics are listed. The review also gives specific direction that could bring progress to SHM in some of these topics. Structural Health Monitoring is defined by: the process of implementing a damage detection strategy for aerospace, civil and mechanical engineering infrastructure (Sohn et al., 2004). The authors explain that SHM monitoring is a statistical pattern recognition problem. Statistical pattern recognition models can be described by four categories including: first, Operational Evaluation; second, Data Acquisition, Fusion and Cleansing; third, Feature Extraction and Information Condensation; and fourth, Statistical Model Development for Feature Discrimination. These four categories and their respective topics are discussed independently below. The paper lists four questions to explain the topics within the Operational Evaluation category. First, how damage is defined for the civil structure? Second, what operational and environmental conditions are present? Third, what limitations are caused by these conditions? Fourth, what are the economic or life safety reasons behind performing SHM? The studies within the selected year range were mostly laboratory tests. In these tests, operational or environmental variability is small. Economic or life

19 7 safety considerations are not shown within these laboratory tests. The majority of these laboratory tests prescribe specific damage versus quantifying damage that is inflicted by the constant operational and environmental conditions. The category of Data Acquisition, Fusion and Cleansing has the following topics: first, selecting the type, number, and locations of sensors; second, the data acquisition, storage, and transmittal hardware; third, how long and often should measurements be taken; and forth, what techniques are used in filtering, cleansing, and fusion. Feature extraction is noted to receive the most attention in literature. Finding structure properties used to distinguish between an undamaged and damaged structure is the process of feature extraction. Information condensation is the process of compressing the data collected into smaller amounts of data that are still usable in detecting damage. Data fusion is somewhat related and can be seen as a form of information condensation. The category of Feature Extraction and Information Condensation describe the main damage-sensitive properties as resonant frequencies, mode shapes, or properties derived from mode shapes. The paper states that investigators from the chosen time period are using system features transitioning from a time-invariant, linear system to a time varying, non-linear system as a result of damage. The last category described in the literature review is Statistical Model Development. This category is noted to have received the least attention in the current and past reviews. Supervised learning is a type of algorithm that includes readings from the undamaged and damaged structure. Unsupervised learning algorithms include readings from the undamaged structure only. Unsupervised learning is said to be effective

20 8 for identifying the onset of damage. Supervised learning is used more to find the location of damage. It is concluded that some of the main problems that need to be addressed are: first, the performance and validation of statistical techniques in real operational environments; and second, the system s sensitivity to environmental and operational conditions that may be present. Structural Health Monitoring of Civil Infrastructure (Brownjohn, 2007) The topic of SHM is introduced as a developing but useful tool for inspection and assessment of structures. The paper was written to describe the motivations for SHM more in certain areas of civil infrastructure. SHM receives less attention within residential and commercial structures due to potential obligations and consequences if an owner were to know the poor health or performance of their structure. Industrial or structures that support a business or public safety are considered higher risks and more research has been done in these areas due to the need and funding. Every structure within civil engineering is unique and could require special training, which can be expensive, to establish the normal or baseline performance of a structure. In other SHM fields, common rules can be prescribed, which is less expensive and tends to be more utilized. In order to focus on applicable topics this paper covers for the thesis topic, the following sections will be described: first, the objectives of monitoring; second, the history and motives of SHM in bridges and buildings; third, the case study of the Tuas Second Link bridge; and finally, the present directions in civil SHM.

21 9 The main research objective for SHM within the civil field is the optimization process. With every structure being unique, the requirement of special and expensive training to apply SHM principles to view the structural health exists. The main aim of research is to develop effective and reliable ways to acquire, manage, integrate, and interpret structural performance data in order to get the maximum amount of information for the lowest cost. Removing the subjective human element from getting reliable health performance information would lower the cost. The history and motives of SHM in bridges and buildings described by this paper include some of the earliest instances of monitoring. These earlier instances, Carder (Carder, 1937) and the University of Washington (University of Washington, 1954), were mainly motivated to document dynamic behaviors. These monitoring programs or studies took record of the Golden Gate and Bay bridges along with the Tacoma Narrows Bridge, respectively. Studies have mainly focused on larger, more important lifeline structures. This paper gives SHM as a more beneficial tool for short-span highway bridges since global response changes can be seen without visual inspection. For buildings, SHM has served more purposes than structural performance. SHM systems have been used for understanding earthquake and wind loadings more accurately. In both bridges and buildings, SHM data can be used to update, verify, and validate finite element models and their readings. This paper gives a case study of a large span suspension bridge and a concrete box girder bridge. In order to summarize the applicable topics to the thesis topic, only the case study concerning the concrete box girder bridge is described. The Tuas Second Link Bridge was opened in During the construction of this bridge, a SHM system was

22 10 installed. Strain gauges, accelerometers, and thermocouples were installed. This case study was chosen to be described since the bridge was a concrete box girder bridge along with the location of the thermocouples. The thermocouples could be used to find the temperature gradient. Although the bridge type and sensor locations were similar to the thesis project, the case study mainly focused on the strain readings and structural changes that occurred through the bridge s construction phases. Temperature was also found to be an important factor through different processes. The case study was shown mainly to show an example of the usefulness of SHM systems to give a motivation for SHM systems. Present directions for SHM include the following: sensors, data storage, data transmission, database management leading to feature extraction, data mining, load/effect model development from study of data, learning from past experience, and decision making from SHM data and models. The present thesis topic deals with many of the described directions this paper presents. Environmental Variability of Modal Properties (Cornwell et al., 1999) This study covers SHM and in particular a certain variable that affects at least one known modal property: temperature. Within civil engineering, SHM can be used to assess or evaluate the performance or health of a civil structure. SHM assessment varies from visual or experimental. Civil structures are unique and research is aiming to optimize SHM systems, both the equipment and procedures used throughout the SHM process. SHM depends on measuring and evaluating the modal parameters. Three common modal parameters are described to be: resonant frequencies, mode shapes, and modal damping.

23 11 Changes in these dynamic parameters could indicate a change in stiffness, damping, mass, or loading. This research shows that there are other important variables that must be taken into consideration in order to assess a structure s performance. The research focuses in on the environmental effects of temperature. The experiment that was conducted to show the effects of temperature was performed on the Alamosa Canyon Bridge. This bridge is a seven-span, simplysupported, reinforced concrete beam bridge. The experiment was conducted on the first span of the bridge. A total of 30 accelerometers were placed in a grid-like fashion over the span, 58 inches apart in the direction of the bridge width and 76 to 98 inches apart in the direction of the bridge length. All thermometers were located at the midspan at the following locations: one on the bottom of the concrete deck at center width, two on the exterior web of the outside girder, and two on the top of the concrete deck above the exterior girder. Expansion of the bridge was limited by dirt inside the expansion joints. The experiment took a total of 24 hours. A total of 30 averages were used to take the frequency measurements every two hours, which took about 30 to 45 minutes to complete. The average temperature readings were also taken for these time periods. The correlations between the following temperature readings and the resonant frequencies were compared: individual temperature measurements, average temperature at the top and bottom of the deck, and temperature differences from the east and west sides of the bridge. The highest correlation factor came to be 0.94 and was obtained by the temperature differences from each side of the bridge. The first three resonant frequencies varied by 4.7%, 6.6%, and 5.0%, respectively, over this day. This experiment indicates that another important variable that affects modal parameters is temperature or

24 12 temperature gradients. It was noted that a similar correlation obtained could also be obtained by using a time-shifted analysis. This is due to frequency changes lagging behind temperature changes. The findings of this study indicate that the temperature differential across the deck correlated with the resonant frequencies the best. The need to understand the principles behind the effects of temperature in modal parameters is explained. The development for procedures and processes to filter out temperature effects in structural assessment is necessary. Comparative Study of Damage Identification Algorithms Applied to a Bridge: I. Experiment (Farrar and Jauregui, 1998) This study covers the specific area within SHM of damage identification within civil structures. The paper presents the SHM goal of accessing civil structures, specifically detecting damage at the earliest stage possible. The authors list many methods that have already been developed, which do not fit within the SHM realm. These methods include: Ultrasonic, Acoustic, Magnetic Field, Radiography, Eddy-Current, and Thermal Field. The authors also give the limits of these methods. These methods require a knowledge that damage exists, an approximate location of the damage, the damaged area must be accessible and near the structure s surface. SHM does not have these requirements, although it does have other demands that have to be considered. The goal to mitigate bridge failures is mentioned. The current process listed by the paper to monitor bridges and mitigate bridge failures is visual inspection techniques. It is stated that damage can occur between inspection intervals and that damage can go

25 13 undetected during an inspection. The goal to mitigate bridge failures could fail because of the occurrences listed. The idea that the goal of bridge failure mitigation could be accomplished through constantly monitoring bridges with SHM systems might be appropriate since a major change in structural performance might be detected. This is an option as long as the methods and applications of SHM are fully understood and accepted. Within the focused area of damage detection, the research paper gives the four levels of damage detection. These include the following: the first level, identifying that damage has occurred; the second level, identifying that damage has occurred and its location(s); the third level, identifying that damage has occurred with locating and estimating the severity; the forth level, identifying that damage has occurred with locating and estimating the severity and determining the remaining useful life of the structure. As of the time of this study, the limitations to which SHM can be used in these levels were not fully known. The variables that are used in SHM principles to fulfill these damage detection levels are the structure s natural frequencies, the mode shapes, and modal damping. These properties are functions of mass, damping, stiffness and boundary conditions. In order to further understand SHM, an experiment was performed on the Interstate-40 (I-40) bridge in New Mexico. The intention of the experiment was to show the changes to the structure s natural frequencies, mode shapes, and modal damping due to damage that simulated fatigue cracking, which is common to plate-girder bridges. The steel plate bridge consisted of a three-span, continuous, steel girder configuration that supports a concrete deck. A thermal expansion joint was located at both ends of the three-

26 14 span bridge, one at an abutment and the other at a pier. Four levels of damage were induced to the bridge incrementally from light to heavy damage. The damage was in the form of different size cuts to the web of the north plate girder. The excitation, data acquisition system, and the parameters used in each are described. The damage influences to the natural frequencies were not noticeable until the worst case scenario. The structure s first three natural frequencies changed from 2.48, 2.96, and 3.50 hertz to 2.30, 2.84, 3.49 hertz, respectively, when bottom flange was finally cut in the worst case scenario. In the previous scenario, the natural frequencies were measured to be 2.46, 2.95, and 3.48 hertz. The only other modal variable reported, damping percentages, did not show any connection to the scenarios. This showed that the changes in the natural frequencies due to the prescribed damage was either being masked by other variables or not noticeable until the damage was more significant. The authors concluded that the boundary conditions must be taken into account in the study and that linearity of the structure should be checked to some degree. The following needs are also mentioned in the paper: more sophisticated methods to examine modal data for indications of damage and the quantification of environmental effects on the measured modal properties through statistical analysis. Damage Detection Accommodating Varying Environmental Conditions (Giraldo et al., 2006) This study covers the specific area within SHM of identifying and locating damage or a loss of stiffness in a lab created building structure. This is accomplished with temperature effects being considered. The evidence of the increasing need to effectively,

27 15 economically, and efficiently assess many civil structures with the increase of quantity and complexity is described. An example of the city of Seattle, that has over 150 bridges, needing an overall infrastructure assessment after a major event, such as an earthquake, is given. It mentions that SHM is of major interest for assessment purposes and that many studies are being conducted in order to simplify the process for the application in the civil engineering field. Within this field, each structure is unique and the process to apply SHM is very specialized and expensive. The results of a study are presented in this paper. The proposed technique to use SHM as an assessment tool utilized the following analyses to identify the existence and location of damage: Natural Excitation Technique (NExT), Eigen system Realization Algorithm (ERA), and a nonlinear principal component analysis (NL-PCA). The NExT and ERA analyses are used for modal identification. Stiffness identification is then used by selecting an appropriate identification model (ID-model), which limits the capabilities of SHM techniques applications. These models typically simplify the structural dynamic response. PCA is then completed and the residual error as a percentile reduction in element stiffness rather than the absolute value is used to indicate damage. An advantage of this proposed technique is that the residual error change is independent of the environmental factors. The limitation that this method will not give the extent of damage is mentioned. These analyses are further described in the paper. A brief example of a two element structure is given. The PCA of this structure is shown graphically and an example of how the residual error is used to determine stiffness losses in one of the elements. Once this is described, an example of an outlier analysis is shown for this specific example to set the criteria for damage alert.

28 16 Finally, a numerical example on a four-story experimental building is given. A finite element model of the actual structure was used to produce ambient vibration responses. Ambient vibration, sensor noise, and varying environmental factors were simulated. An appropriate ID-model was developed and the analyses were performed. Eight damage cases were utilized to simulate the response of the system and compare to the undamaged case. This technique resulted in indicating the correct elements had lost stiffness in five of the eight cases. The other three cases, false negatives were indicated; although in one of these cases, a false positive was accompanied by correctly identifying one element that had lost stiffness. The paper concluded that the residual error was successfully used as a damage indicator. It mentions that the approach used is limited to cases where stiffness losses effect the dynamic properties. It is also mentioned that this approach could be limited to structures that do not experience stress stiffening of members due to thermal expansion since the test specimen was free-standing. Influence of In-Service Environment on Modal Parameters (Alampalli, 1998) This research is a study on SHM, which was referred to as remote bridge monitoring systems (RBMS) through this research paper. It brings to light that the bridge inspection and assessment process relies largely on visual inspection to detect a bridges health or condition. It states that bridge owners are looking for more reliable tools to assist in inspections and assessment. The paper references other studies that have been successful in indicating damage through SHM. The paper also points out that, along with any other experimental techniques, SHM produces variable results when repeated. Causes

29 17 for variability are listed as: test environment, electrical disturbance, in-service environment, and variation among operators. These causes for variability are also stated to cause as much variability in modal parameters as damage, as shown in referenced papers. The research and experiment of this study was focused on in-service environment and testing methods. An experiment, intended to capture the effects of in-service environment and testing methods, was established and conducted on a single-span bridge in New York. The bridge spans Mud Creek and consisted of two W18x64 beams which supported a reinforced concrete deck. The two W18x64 beams that spanned the creek were embedded in concrete abutments at both ends. Damage was to be induced after taking ten measurements of the modal parameters that would serve as the baseline or normal bridge behavior. After damage was to be induced, the modal parameters were measured again to compare to the baseline measurements. This study used only the natural frequencies of the bridge, the first three were selected. The air temperature was also measured at the same times the bridge frequencies were measured. The baseline measurements were taken at the average temperature of 46.6 F, with the minimum temperature being 42 F. After the initial baseline measurements were taken, damage was induced and twenty other measurements were taken. These twenty measurements were divided into two groups. A group with temperatures well above freezing (32 F) and another group with temperatures below or close to freezing. The results showed that the group well above freezing indicated a change in structural behavior since the modes all slightly decreased. The group with temperatures near freezing actually showed an increase of stiffness from the undamaged structure.

30 18 It was concluded that above freezing temperatures had no significant effect on the natural frequencies. The temperatures below freezing suggested a stiffening of the bridge which leads to the belief that the supports were frozen and acted more fixed. It was concluded that the baseline should be established on at least one complete cycle on inservice conditions. These in-service environmental conditions should be well understood before establishing criteria to detect damage. Literature Review Summary Much of the literature that was reviewed defined SHM and listed the advantages of using it as an assessment tool. The literature has also described the difficulty and limits it currently has. Much of the research has been dedicated to understand and quantify the relationship of temperature to modal properties. This confirms that the relationship must be considered and cannot be neglected. Most of the literature has considered each study structure to be unique, which shows to be a good consideration since many of the results or findings have varied. Although some of the literature has given suggestions that describe the generally accepted trends in this relationship, there are exceptions to these suggestions. Boundary conditions or physical changes to the system, such as a frozen bearing or frozen soil, can make these exceptions. The literature does show a common procedure or steps to measure and quantify their findings. Statistical models of the relationship have been created and used as a baseline to filter out temperature effects. The same steps of using a statistical model to describe the temperature to frequency relationship of the bridge used in this study will be utilized.

31 19 CHAPTER 2 CALIFORNIA LAMBERT ROAD BRIDGE In this chapter, three things will be discussed. First, a detailed description of the studied bridge is presented. Second, a description of the Structural Health Monitoring (SHM) system that is installed on the bridge will be given. Finally, the records extracted, the data management, and the processing techniques utilized in this study will be described. California Lambert Road Bridge Description The Lambert Road Bridge was built in 1975 and is South of Sacramento. The bridge takes Interstate-5 (I-5) southbound traffic over Lambert Road. Two traffic lanes are carried by this cast-in-place, four-cell, box-girder concrete bridge. This double-span bridge measures a total length of m (258 ft). It is supported by a bent cap and column, such that each span measures a length of m (129 ft). The bride spans in the North-South direction and has an 8 skew with each supporting abutment and the centered bent cap. Figure 1 shows the location of the Lambert Road Bridge relative to Sacramento, taken using Google Maps (Google Maps, 2014). Figure 2 shows a plan view taken using Google Maps (Google Maps, 2014) while Figure 8 shows a plan view with the dimensions labeled. Figure 3 shows an elevation looking east of the lambert road bridge.

32 20 Figure 1: Bridge Location from Sacramento, California (Google Maps, 2014) The Lambert Road Bridge is a box-girder design that measures a total width of 12.8 m (42 ft) and a road width of 12.2 m (40 ft). This road width carries two 3.66 m (12 ft) traffic lanes and the east and west shoulders that measure 3.05 (10 ft) and 1.83 m (6 ft), respectively. The two surrounding barriers measuring 0.3 m (1 ft) make the difference between the total width and the roadway width.

33 21 North Figure 2: Satellite Plan View (Google Maps, 2014) Figure 3: Elevation Looking East Structurally, this bridge is a four-cell, box-girder design. The average deck thickness measures 200 mm (8 in.). The deck overhangs the box-girder cells a distance of 0.92 m (3 ft). The bottom of the box-girder, or the bottom flange, is on average 150 mm

34 22 (6 in.) thick. The five girders, or webs, connecting the deck and the bottom flange are each 0.30 m (1 ft) thick. The outermost girders are inclined from vertical by 30 and make the two outermost cells trapezoidal. The bottommost part of the trapezoidal cells measure 1.58 m (5 ft-2 in.) from wall-to-wall. The two most inner cells measure 2.44 m (8 ft) in wall-to-wall width. All four cells measure 1.32 m (4 ft-4 in.) in wall-to-wall height, which makes the total height of the box-girder 1.68 m (5 ft-6 in.). Figure 4 shows a typical cross-section view of the box-girder bridge. Figure 4: Typical Cross-Section There are three diaphragms located on the deck; a 1.83 m (6ft) thick intermediate diaphragm at the bent cap and two 203 mm (8 in.) thick diaphragms at each midspan. These reinforced concrete diaphragms have a similar 8 skew as the abutments. The bent cap is supported by a bent column that measures 1.07 m (3 ft-6 in.) wide. The supporting column tapers at a 14-to-1 slope outwards (east and west) when moving upward from the ground to the bent cap. The column foundation measures 5.48 m by 3.66 m (18 ft by 12 ft) and 1.07 m (3.5 ft) thick. Each girder was prestressed strands that followed a parabolic

35 23 path. Figure 5 shows an elevation view of the bent cap and column (Hodson, 2010). Figure 6 shows a section cut of the column (Hodson, 2010). Figure 5: Bent Cap & Column Elevation (Hodson, 2010) Figure 6: Column Section (Hodson, 2010)

36 24 The ends of the box-girder bridge are supported by integral abutments with wing walls attached. The abutments are 0.46 m (1.5 ft) thick and are supported by a reinforced pile cap, between which is a neoprene bearing pad. The pile caps measure 0.46 m (1 ft-6 in.) thick by 1.22 m (4 ft) wide by m (42.5 ft) long. Each pile cap is supported by seven mm (16 in) diameter, cast-in-drilled-hole, concrete piles. The design loading of the piles is 623 kn (70 tons). Figure 7 shows the abutment detail (Hodson, 2010). Figure 7: Abutment Detail (Hodson, 2010) Structural Health Monitoring System A structural health monitoring (SHM) system was installed on the Lambert Road Bridge. Two Campbell Scientific Dataloggers, a CR1000 and a CR5000, were utilized as the data acquisition systems. The SHM system has been recording the bridge response and certain environmental conditions since May Seventy-one sensors were

37 25 installed and include the following: 16 Hitec strain gauges, 4 Geokon vibrating-wire strain gauges, 4 Mark Sercel velocity transducers (geophones), 3 Geokon tiltmeters, and 44 Omega thermocouples. Since the SHM had started recording, the sample rates and recording times have changed numerous times. The following recording descriptions are for the data used through this study and neglect records sampled under other conditions. The records can be categorized into two groups based on their sample rates. The velocity transducers, tiltmeters, and 12 of the Hitec strain gauges had a sample rate of 50 hertz, which is grouped into the fast sampling rate category, which takes a 5 minute data burst of records. Every sensor, except the velocity transducers, also took a measurement every 15 minutes, which is grouped into the slow sampling rate category. The tiltmeters also measured its own temperature and records this value every 15 minutes. The recordings in both the fast and slow sampling rate categories are recorded for further understanding of the bridge behavior throughout the structure s lifespan. In order to give the locations of the sensors, named section cuts are made through the bridge. These sections follow the 8 skew that the abutment and bent cap established. A plan view with the section cuts is shown in Figure 8. Section A-A and Section H-H correspond respectively to the north and south abutment. Section D-D corresponds to the longitudinal center of the bridge. A length of L = m (129 ft) is noted for the two spans. This same L is used to describe the spacing between the section cuts. The distance measured from Section A-A to Section B-B is 0.30L. From Section B-B to Section C-C measures 0.30L. From Section C-C to Section D-D measures 0.40L. This same spacing is used for the south span when respectively measuring to Sections F-F, G-G, and H-H from

38 26 the Section D-D. Section E-E is 2.41 m (7 ft-11 in.) south of Section D-D. All sensors are located on, or close to, these sections. Table 1 and Table 2 summarize the sensor location and names. Figure 8: Plan View with Sections The longitudinal locations are given by the sections. The transverse and depth locations are now given. The transverse dimensions are always given from the west side of the bridge and the depth locations are measured from the top-of-deck surface. Sections A-A, D-D, and H-H have tiltmeters located at the transverse location of 5.03 m (16 ft-6 in.) and at a depth of 508 mm (1 ft-8 in.). The tiltmeter names and locations in Table 1 and Table 2 correspond to Figure 9. Sections B-B and F-F have a velocity transducer and two Hitech strain gauges paired at the same location as two thermocouples. At these two sections, the velocity transducers are located at the transverse direction of 5.03 m (16 ft-6 in.) right below the deck, a depth of 152 mm (6 in.). Both sensor pairs of the strain gauge and thermocouple are located transversely at 6.4 m (21 ft). The depth locations of these pairs are 460 mm (1

39 27 ft-6 in.) and 1680 mm (5 ft-6 in). These sections and the sensor names corresponding to Table 1 and Table 2 are shown in Figure 10. Figure 9: Tiltmeters at Sections A-A, D-D, & H-H Figure 10: Sensor Locations at Section B-B & F-F Sections C-C and G-G have a velocity transducer and six groupings of other sensors. These sensor groupings contain a mixture of Hitech strain gauges, vibrating wire

40 28 strain gauges, and thermocouples. The transverse and depth locations of these pairings are explained solely in Table 1 and Table 2 due to their various locations. The velocity transducers at these two sections are located at the same transverse and depth locations as in Sections B-B and F-F, 5.03 m (16 ft-6 in.) and 152 mm (6 in.), respectively. These sensor names and some locations are also labeled in Figure 11. Figure 11: Sensor Locations at Sections C-C & G-G Section E-E has the other 30 thermocouples. These thermocouples can be divided into two groups intended to measure the temperatures through the depth of the bridge. They measure the temperatures at different depths of the bridge to understand the bridge temperature gradient. The west group is situated transversely at 1.57 m (5 ft-2 in.), while the east group is at 3.10 m (10 ft-2 in.). The five, lower, web thermocouples of the west group are not exactly at this transverse location, due to the angle of the outermost boxgirder wall, but Table 1 and Table 2 gives these exact locations. The web thermocouples

41 29 of the east group are at a transverse location of 3.51 m (11 ft- 6 in.). The depths and configuration of the ten deck thermocouples of each group is given in Figure 12. Figure 12: Deck Thermocouple (DTC) Locations The deck thermocouples are actually located inside the deck. When the SHM system was installed, a hole was bored most the way through the deck from within the box-girder cells. The deck thermocouples were distributed through a plastic pipe which was placed within this hole through the deck. The plastic pipe was then epoxied in place and filled with more epoxy. The lower web thermocouples were epoxied to the boxgirder bridge as closely to the transverse location of the deck thermocouples. The depths

42 30 of the web thermocouples of both the east and west groups are also in Table 1 and Table 2. Section EE is shown in Figure 13. Figure 13: Sensor Locations at Section E-E The sensor names used in Table 1 and Table 2 will be used throughout the rest of this study on the Lambert Road Bridge. Figure 14 through Figure 19 show some of the installed sensors and data acquisition system for the SHM system. Figure 14 shows the installed system layout from below with protective boxes. Figure 15 shows a vibrating-wire strain gauge next to a Hitec strain gauge and Figure 16 shows an installed tiltmeter. Figure 17 shows an uninstalled velocity transducer and the deck thermocouples inside the pipe configuration that was inserted through the bridge deck. Figure 18 shows the thermocouples on the web or vertical girder. Figure 19 shows the data acquisition systems.

43 31 Table 1: Sensor Names, Descriptions, & Locations I Section Name Description Longitudinal Location (m) Transverse Location (m) Depth Location (mm) AA TM_NAWall_AA Tilt Meter, Notth Abutment Wall BB CC VT_UD_BB Velocity Transducer, Under Deck SG_G3_TF_BB Strain Gauge, Girder 3, Top Flange TC_G3_TF_BB Thermocouple, Girder 3, Top Flange SG_G3_BF_BB Strain Gauge, Girder 3, Bottom Flange TC_G3_BF_BB Thermocouple, Girder 3, Bottom Flange VT_UD_CC Velocity Transducer, Under Deck SG_G1_BF_CC Strain Gauge, Girder 1, Bottom Flange TC_G1_BF_CC Thermocouple, Girder 1, Bottom Flange SG_G2_BF_CC Strain Gauge, Girder 2, Bottom Flange VWS_G2_BF_CC Vibrating Wire, Girder 2, Bottom Flange TC_G2_BF_CC Thermocouple, Girder 2, Bottom Flange SG_G3_TF_CC Strain Gauge, Girder 3, Top Flange TC_G3_TF_CC Thermocouple, Girder 3, Top Flange SG_G3_BF_CC Strain Gauge, Girder 3, Bottom Flange TC_G3_BF_CC Thermocouple, Girder 3, Bottom Flange SG_G4_BF_CC Strain Gauge, Girder 4, Bottom Flange VWS_G4_BF_CC Vibrating Wire, Girder 4, Bottom Flange SG_G5_BF_CC Strain Gauge, Girder 5, Bottom Flange TC_G5_BF_CC Thermocouple, Girder 5, Bottom Flange DD TM_Pier_DD Tiltmeter, Central Pier EE DTC_WEST_01 Deck Thermocouple, West Device, DTC_WEST_02 Deck Thermocouple, West Device, DTC_WEST_03 Deck Thermocouple, West Device, DTC_WEST_04 Deck Thermocouple, West Device, DTC_WEST_05 Deck Thermocouple, West Device, DTC_WEST_06 Deck Thermocouple, West Device, DTC_WEST_07 Deck Thermocouple, West Device, DTC_WEST_08 Deck Thermocouple, West Device, DTC_WEST_09 Deck Thermocouple, West Device, DTC_WEST_10 Deck Thermocouple, West Device, WTC_G1_01 Web Thermocouple, Girder 1, WTC_G1_02 Web Thermocouple, Girder 1, WTC_G1_03 Web Thermocouple, Girder 1, WTC_G1_04 Web Thermocouple, Girder 1, WTC_G1_05 Web Thermocouple, Girder 1, DTC_EAST_01 Deck Thermocouple, East Device, DTC_EAST_02 Deck Thermocouple, East Device, DTC_EAST_03 Deck Thermocouple, East Device, DTC_EAST_04 Deck Thermocouple, East Device, DTC_EAST_05 Deck Thermocouple, East Device, DTC_EAST_06 Deck Thermocouple, East Device, DTC_EAST_07 Deck Thermocouple, East Device, DTC_EAST_08 Deck Thermocouple, East Device, DTC_EAST_09 Deck Thermocouple, East Device, DTC_EAST_10 Deck Thermocouple, East Device, WTC_G2_01 Web Thermocouple, Girder 2, WTC_G2_02 Web Thermocouple, Girder 2, WTC_G2_03 Web Thermocouple, Girder 2, WTC_G2_04 Web Thermocouple, Girder 2, WTC_G2_05 Web Thermocouple, Girder 2,

44 32 Table 2: Sensor Names, Descriptions, & Locations II Section Name Description Longitudinal Transverse Location (m) Location (m) Depth Location (mm) VT_UD_FF Velocity Transducer, Under Deck SG_G3_TF_FF Strain Gauge, Girder 3, Top Flange TC_G3_TF_FF Thermocouple, Girder 3, Top Flange SG_G3_BF_FF Strain Gauge, Girder 3, Bottom Flange TC_G3_BF_FF Thermocouple, Girder 3, Bottom Flange VT_UD_GG Velocity Transducer, Under Deck SG_G1_BF_GG Strain Gauge, Girder 1, Bottom Flange TC_G1_BF_GG Thermocouple, Girder 1, Bottom Flange SG_G2_BF_GG Strain Gauge, Girder 2, Bottom Flange VWS_G2_BF_GG Vibrating Wire, Girder 2, Bottom Flange TC_G2_BF_GG Thermocouple, Girder 2, Bottom Flange SG_G3_TF_GG Strain Gauge, Girder 3, Top Flange TC_G3_TF_GG Thermocouple, Girder 3, Top Flange SG_G3_BF_GG Strain Gauge, Girder 3, Bottom Flange TC_G3_BF_GG Thermocouple, Girder 3, Bottom Flange SG_G4_BF_GG Strain Gauge, Girder 4, Bottom Flange VWS_G4_BF_GG Vibrating Wire, Girder 4, Bottom Flange SG_G5_BF_GG Strain Gauge, Girder 5, Bottom Flange TC_G5_BF_GG Thermocouple, Girder 5, Bottom Flange HH TM_SAWall_HH Tiltmeter South, Abutment Wall FF GG Figure 14: System Layout from Below

45 33 Figure 15: Vibrating Wire & Hitec Strain Gauges Figure 16: Tiltmeter

46 34 Figure 17: Uninstalled Velocity Transducer & Deck Thermocouples Figure 18: Web Thermocouples Once the SHM system was installed, the data acquisition was programmed to measure the slow sampling rate records every 15 minutes, as explained before. From May 2011 to November 2012, the data acquisition system was programmed to record the fast sampling rate recordings twice daily, for 12 minute intervals staring at 05:01 and 17:01.

47 35 Figure 19: Data Acquisition Systems In November 2012, the data acquisition system was reprogrammed to record the fast sampling rate records for 5 minutes, starting at the following times: 00:00, 03:00, 06:00, 09:00, 12:00, 15:00, 18:00, and 21:00. As mentioned before, a sampling rate of 50 hertz was utilized through all these recordings. It has been recording at these times ever since. In both the fast and slow sampling rate records, daily records have been lost or overwritten due to various circumstances. This has not yet been an issue in statistical modelling since the number of records has grown tremendously large with the ongoing recording procedure. Only the chosen three month time period, January to March 2013, is used in this study. The dates used through each process are described before the results are given. One important note on this particular SHM system and its components needs to be mentioned. The Campbell Scientific Dataloggers and the velocity transducers do not have

48 36 any anti-aliasing filters installed. A low-pass filter would be ideal for most SHM systems utilized for civil structures. Since a low-pass filter was not installed on the system, folding of higher frequencies onto lower frequencies in Fourier Transformations could be problematic in determining where the structure s natural frequencies are located. This may also reverse the direction the natural frequencies would move due to temperature (material properties) or boundary condition changes (system changes). The measures taken in this study to ensure folding would not cause major effects on the results or conclusions are described in the next section. This is discussed after the signal processing technique options are discussed and chosen. Fast Sampling Rate Record Processing In order to understand the environmental effects on this bridge, the fast sampling rate records were processed. Fast Fourier Transformation (FFT) techniques from signal processing were utilized to find the structure s natural frequencies. An FFT will take the time domain record and transform it into the frequency domain. An example of a time domain record, in this case the tiltmeter (TM-SAWall-HH) record from March 2 nd, is given in Figure 20. The x-axis of any sensors time domain record will be in some form of seconds. The y-axis units, however, would be the actual units the sensor measures. For example, strain gauges measure units of micro-strain and the velocity transducer measure units of inches-per-second. The y-axis units, and the tiltmeter units, of Figure 20 are angular degrees ( ).

49 37 Figure 20: Tiltmeter Time Domain Record Example The look of the time domain records are similar, although the magnitudes of the record depends on the units being measured and the ambient vibration caused by traffic or wind. In Figure 20, medium sized amplitudes are seen around 20, 36, and 48 seconds which might indicate smaller vehicles passing over the bridge. Larger amplitudes at 80 and 96 seconds might indicate semi-trucks crossing the bridge. The important information extracted from the time domain records is not the amplitudes. The frequencies are the important part extracted by the FFT. The frequencies depend on the forcing frequency of the ambient vibration, which is only applied for the time the traffic is on the bridge, and the structure s response frequency. This response frequency can be investigated using frequency domain analysis such as an FFT plot. The structure s frequencies are more noticeable in the processed record. Figure 21 shows the same tiltmeter fast sampling rate record that has been processed using an FFT.

50 38 Figure 21: Processed Tiltmeter Example The FFT plots give critical information about the system. As shown on the previous figure, the lower x-axis range from 0 to 1.5 hertz has large y-values. This lower range is typical in FFT plots and can be ignored. In the case of this study, the magnitudes (y-values) of the peaks are not required; however, the relative magnitude of a peak compared to the neighboring points does matter. A large difference in magnitude is important and might indicate a natural frequency. Noise will create smaller, but consistent peaks. Figure 22 show the large difference in magnitude of a probable natural frequency, the value of this frequency, and the consistent peaks that noise creates.

51 39 Figure 22: Probable Natural Frequency Example from Tiltmeter There are many processing options in signal processing. In this study, Matlab s Pwelch function with a 30 second window, 50% overlap, and the number of points in the discrete Fourier transform (NFFT) is set to 2 12 (4096 points). The 50% overlap, which helps with leakage, is common in digital processing and was chosen to be used. An initial run through of March s records was performed with different overlaps and NFFTs. Over smoothing occurred when smaller windows were selected. Figure 23 shows a 10 second window on the top graph and a 5 second window on the graph below. Figure 23 is the same tiltmeter record in Figure 21 and Figure 22. This smoothing can affect the location of the natural frequency peaks. A 30 second window was chosen for this study.

52 40 Figure 23: Over Smoothing Example on a Tiltmeter Record The NFFT chosen also affected the frequency domain records. Choosing a lower NFFT value speeds up the calculation time, but alters the frequency domain records since the number of points for the curve is related to the NFFT number. Figure 24 shows the jagged curve of FFT records with lower NFFT values. This figure is the same processed tiltmeter record as before. The upper graph shows the records produced with the NFFT value set to 2 10 (1024 point). This graph has 513 points between 0 and 25 hertz. Between neighboring points, there is a hertz gap, which cancels out some noise and makes a less jagged curve. This might not yield accurate results since there is noise. The lower graph with the NFFT value set to 2 8 (256 points) is given to show more drastic results.

53 41 There are 129 points between 0 and 25 hertz, with a hertz gap. This yields worse results. A balance of calculating time and accuracy was found with the NFFT value set to 2 12 (4096), which gives 2049 points between 0 and 25 hertz and a hertz gap between neighboring points. To see the same frequency domain record with the chosen NFFT of 4096, see Figure 24. Figure 24: Noise Cancelation Example on a Tiltmeter Record As mentioned earlier, the data acquisition system used to record the information does not have the ideal low-pass filters needed to eliminate any problems with folding. Folding is a source of aliasing. Aliasing is an effect that causes one frequency to become indistinguishable from another, or a false frequency. In folding, higher frequencies are

54 42 imprinted on lower frequencies. Since a 50 hertz sampling rate was used in recording, the Nyquist frequency of 25 hertz is the x-axis limit in the frequency domain. The Nyquist frequency is always half of the sample rate. Increasing the sample rate would increase the x-axis limit and higher frequencies could be seen. If different sample rates were used on different recordings and the processed records were compared, a better understanding of the effects of folding would be more evident. The data acquisition system s sample rates are limited with the amount of sensors that are connected. This was not an option to narrow out any results. An example of higher frequencies imprinting on lower frequencies is found in Figure 25. If the highest part of Figure 25 shows the true frequencies being recorded at 50 hertz without a low-pass filter, folding would occur in the FFT processing. The Nyquist frequency would be 25 hertz and folding would occur at this frequency. The frequency magnitudes in all even ranges (Range 2 in Figure 25) would be imprinted, or added to the frequency magnitudes of Range 1, in the reverse direction. The frequencies in all odd ranges (Range 3 in Figure 25) are imprinted on Range 1 in the same direction. The bottom left part of the figure shows why this is named folding. The bottom right of this figure show the observed frequency domain recurred with folding. A low-pass filter would eliminate any true frequencies that are above the desired range. Using a low-pass filter would give observed frequencies very similar to the true frequencies.

55 43 True Frequencies hertz Range 1 Range 2 Range3 Range 2 Range 3 Observed Frequencies Range hertz Figure 25: Folding Example Since a low-pass filter was not installed on the data acquisition system, extra steps were added to potentially eliminate biased or wrong results. The principles of structural dynamics show that all modes of the same natural frequency must shift in the same direction if either boundary conditions or material properties change. If the frequencies in the odd ranges shift in one direction, then any frequencies in the even ranges of the same modes would shift in the opposite direction. These could be eliminated or the shifts could be measured in the opposing direction. This requires that the shift direction of each mode is known from an unbiased sensor or sensors. This also requires that additional peaks be

56 44 known modes of each other, recorded and compared. It requires additional regression models to be made for these peaks and additional comparison between these regression models. Although these requirements were time consuming, they were necessary to validate the records and findings. The Hitech strain gauge records and the regression models they produced were used as a basis. These strain gauges were on the bottom of the bridge girders. In this position, they are less sensitive to localized frequencies the deck might be experiencing due to the forcing frequencies of the traffic vibrations. The actual structure s frequencies are more noticeable. None of the strain gauges indicated any frequency peaks above 8 hertz. Some strain gauges only indicated one peak consistently in the range of 2 and 9 hertz. Other strain gauges indicated 2 to 3 consistent peaks in this same range. The natural frequencies that these gauges indicated were also further validated by an earlier study on the bridge. In 2010, Thurgood performed dynamic testing on this bridge (Thurgood, 2010). The Data Physics acquisition system used to perform this dynamic test utilized low-pass filters during testing. With the ideal low-pass filters used during this test, the observed natural frequencies by the SHM system that correspond to the previous test would be further validated. The bridges natural frequencies were determined to be 3.30, 4.45, 8.25, 10.49, and hertz (Thurgood, 2010). The 3.30 hertz mode shape showed the two spans alternating vertically (one span deflects upward while the other deflects downward) from each other. The 4.45 hertz mode shape showed the two spans both deflecting upward or downward at the same time (Thurgood, 2010). The third mode shape of 8.25 hertz showed a torsional mode (Thurgood, 2010). Figure 26 shows a typical

57 45 FFT of SG-G4-BF-CC. The next figure, Figure 27, shows three strain gauge records for two consecutive times. The sensor names and recording times are labeled on the figure. Figure 26: Typical FFT of a Strain Gauge Figure 27: Consecutive FFT of Strain Gauges

58 46 In order to process thousands of these records, many Matlab scripts were written. The text files from the acquisition system had to be organized and imported into Matlab. Once imported, appropriately sized Matlab variable files were created. These are easier and faster to import than text files. These files were stored and used as a database for programed files that were written to process the records and make the regression models. As mentioned before, any peaks that indicate a shift in natural frequency had to be compared to the base, strain gauge models. This required that any peaks that did not correspond to the peaks in the strain gauges be recorded. The additional peaks that acted consistent with the base sensors could be further studied, while those that did not act consistent were eliminated. All peaks that might have indicated a natural frequency were considered, but first, values for these peaks must be discovered. A preliminary processing run of the records in January to March, 2013, was completed to find all peaks to be considered. To cancel out any unwanted peaks or troughs created by noise, the FFT values for each recording were added together. When the noise peaks or troughs from many different records are added together, they tend to cancel each other out. Figure 28 shows the three month initial run for the following sensors: SG-G1-BF-GG, SG-G2-BF- GG, VT-UD-CC, VT-UD-GG, TM-NAWall-AA, and TM-Pier-DD. Any peaks that showed up on one sensor were also considered on any similar sensor of the same make and model. For example, the strain gauge named SG-G3-TF-CC usually only had one peak near 3.15 hertz. Another strain gauge, made by the same company and in a similar position, named SG-G3-BF-CC had two consistent peaks with an occasional third. The first, second, and third peaks were close to 1.55, 3.15, and 4.25 hertz, respectively. Noticeable peaks near these same frequencies were recorded for both strain gauges. If

59 47 SG-G3-TF-CC, or other strain gauges, did not show a peak, a frequency was not recorded. Figure 28: Initial Processing Run & Potential Peaks of Six Sensors The peaks considered for each type of sensor or sensor groups are given in Table 3. Two of the Hitech strain gauges, SG-G3-TF-FF and SG-G3-BF-FF, will not be listed since only slow sampling rate records were recorded for these sensors. All others had fast sampling rate records. The natural frequency from the previous study mentioned before are also listed (Thurgood, 2010). Asterisks are frequencies that were not present. As shown in Table 3, many of the observed natural frequencies by the SHM system are similar to those observed in Thurgood s study. As seen in Figure 28, each sensor group also shows different natural frequencies better than the other groups. Many of these frequencies were expected show similar results through this study.

60 48 Sensor Groups: Table 3: Potential Natural Frequencies Number of Peaks Considered: Natural Frequency Peaks (hertz) Hitech Strain Gauges * * * * * Velocity Transducers Tiltmeters 7 * Thurgood Study (2010) 5 * * * The environmental factors that affect the natural frequency ranges are the focus. The number of peaks or frequencies listed for each sensor type in Table 3 will be the extent of environmental effect on this bridge. Mode shapes and damping ratios will be left out. Even without a low-pass filter installed on the SHM system, the peaks locations are consistent with Thurgood s study (Thurgood, 2010) and confidence can be put into these records. As mentioned before though, if the shift direction of one natural frequency is known, the other natural frequencies will shift in the same direction. The shift direction of the strain gauge s observed natural frequencies were still used to eliminate any conflicting results. Once the data was organized and the approximate peak locations were known, processing the individual records from the same months followed. With the natural frequencies able to shift due to environmental conditions and error, a graphical user interface (GUI) was programmed. The GUI would complete the FFT calculation and display the output graphically. Default range values for the natural frequencies were also programmed to select the highest peak within the approximate frequency range. This was programed to save time and minimized manual range selection. The GUI allowed the user to choose one of the three following options: first, to neglect the natural frequency due to record quality; second, chose the peak within the default range if the location was correct;

61 49 or third, choose to set another range for the peak if the record quality was good but the peak was outside the default range. A picture of the GUI and options is in Figure 29. This is a record for a tiltmeter, TM-Pier-DD, and shows the second natural frequency being chosen automatically. Figure 29: Graphical User Interface (GUI) Example & Options The GUI saved the following information: first, a true or false value if the record was good; second, the frequency of the peak; third, the peak s set frequency range; and forth, the amplitude. For this study, the frequency range was important to save in order to try different processing techniques. Although all the saved values are not used in this study, these could be used in future research.

62 50 No processing steps were needed for the slow sampling rate recordings, although judgment and statistical testing were used to find better correlations in the regression models. This will be described in subsequent sections.

63 51 CHAPTER 3 STATISTICAL MODELING PROCESS In Chapter 3, the following processes will be described. First, the initial exploration of the relationship between temperature and natural frequency is described. This is to show some of the considerations that were made in the statistical models. This also helped to see the nature of the relationship. Second, initial linear regression is completed to verify and further describe this relationship. The temporal aspect of aligning peak-to-peaks is also initially tested here. Third, stepwise regression was utilized to find common variables that should be included on the final models. These common variables correlated with most fast sampling rate sensors in the stepwise regression models. This was an approach to consider the spatial aspect in the statistical models. Fourth, multiple linear regression was completed. Ridge regression was also utilized in this section to eliminate any unstable variables. The temporal peak-to-peak alignment was also verified to be consistent in this section. Fifth and finally, a final regression model was created and chosen to condense and describe the findings for the three month period. Predictor Variable Exploration and Selection In this section, an understanding of the different temperatures that this bridge undergoes is observed. The main temperature observations will be through the depth of the bridge, although the same observations may be evident in transverse or longitudinal directions. The temporal and spatial considerations for a regression model are described and shown with the temperature observations.

64 52 In order to better understand the relationship between natural frequency and environmental effects, mainly temperature, an understanding of the independent variable is needed. The relationship between the independent and dependent variable can be more evident if the independent pattern over time is understood. Considerations for regression models can also more evident with this understanding. Figure 30 shows temperatures measured through the depth of the bridge (west deck and web thermocouples) for the time period of March 2 nd to the 6 th. Figure 30: West Temperature Gradient March 2 nd -6 th Some important observations can be seen in Figure 30. The thermocouples through the bridge deck surface, the first 8 in., record much higher and lower temperatures than the lower thermocouples. There is also a daily cycle of the bridge being heated and then cooled as shown by the figure. One important note is that all of

65 53 these readings have correlations between each other. Looking at the depth versus temperature plane of this same figure, a higher order polynomial could be used to fit the data. Hence, these variables can be spatially correlated. See Figure 31 below. Figure 31: West Temperature Gradient March 2 nd -6 th (Depth View) Looking at the time versus temperature plane, a temporal correlation between these predictor variables can also be seen (see Figure 32). ARIMA regression models can be implemented with time series regression models and can be an option for taking into account some of the temporal collinearities.

66 54 Figure 32: West Temperature Gradient March 2 nd -6 th (Time View) Both the temporal and spatial collinearities between predictor variables may cause problems with interpreting the regression models. Collinearity exists when the predictor variables contain parts of the same information. In the case of multilinear regression, each variable can be considered a direction. Each direction is hopefully orthogonal to each other. If variables are collinear, then each variable or direction is not necessarily orthogonal and certain procedures must be taken to better understand the relationship between the predictor and dependent variables. There are types of regression models that help deal with variables that are collinear, such as Principle Component Analysis or Partial Least Squares regression. A simple model that will give insight to the nature of the relationship will first be chosen and variables may be added with testing and judgment. This will minimize error and the time required to create and interpret a complex model.

67 55 In order to make better assumptions to start a regression model, a better understanding of the relationships between all variables is investigated. In order to get any variables that might be useful to start modeling, other sensors records were organized. Each sensor reads a different measurement and hence a different unit; for example, the thermocouples read in units of degrees Fahrenheit ( F), tiltmeters in angular degrees ( ), and strain in micro-strain (μ-strain). If these measurements are related, which is known through principles of mechanics and heat transfer, these measurements would show similarities and relationships. If these measurements were plotted on the same graph, the plot would be difficult to understand because of scale. In order to compensate for this scaling problem, the following was calculated. First, the standard deviation from each sensor s readings was found for the recording time. Secondly, the sensor s measurements (for the same recording time) are divided by the standard deviation from the same measurements. It should be noted that in the first step, the standard deviation is in the same units as the measurement. For example, a sensor might have measured for a time period that produced 20 F for 1 standard deviation. If the original measurements were divided by this 20 F/1 standard deviation, the new unit of measure would be standard deviations. This will also get the curves on a similar scaling and give a better visual for interpreting the relationships between variables. It should also be noted that the fast sampling rate sensors took fewer measurements, so the standard deviation of these sensors are smaller values. In order to compare the change in temperatures and frequency over time, the standard deviations are shifted up or down. In Figure 33, two strain gauges and tiltmeters were chosen to represent the other similar sensors located on the bridge. All values in the figure were calculated by the

68 56 sensors original measurements being divided by the standard deviation of the five day period (March 2 nd through 6 th ). This was done in order to eliminate units and compare the measurements at a common scale. Only two days, March 3 rd and 4 th, are included to see the similarities between the measurements. All z-units were standard deviations that were shifted up and down to show common relationships or collinearities between possible predictor variables. Only four of the thermocouples are shown to be concise: two located through the deck and two on the web. Figure 33 shows that all of these are also collinear. In order to see potential relationships, the natural frequencies from a velocity transducer (VT-UD-BB), were also plotted. Figure 33: Temperatures, Natural Frequencies, Tiltmeter, & Strain Gauges As seen in the previous figure, many of the sensor s readings are showing the same trend with time. These trends all follow the trend that the bridge temperature takes,

69 57 which indicates that temperature is the driving cause of the other measurement fluctuations. As temperature increases, the bridge expands and causes the tiltmeters measurements to increase. This expansion can also cause an internal stress if it is resisted. The tiltmeter readings show a resistance of expansion; therefore, the stresses should fluctuate with a uniform temperature increase or decrease. This is seen with the tiltmeters and the strain gauges. If the temperature gradient is non-linear through the bridge, internal resistance also creates internal stresses. The high order polynomial curve seen in Figure 31 indicates the internal stress should fluctuate with temperature gradient. This is less evident by the plots. There does seem to be a correlation between the natural frequencies and the other parameters. In order to minimize the procedures to correct for collinearity in the future, certain redundant variables were eliminated. Figure 34 isolates the records from DTC-West-07, the tiltmeters, and the strain gauges. Figure 34: Collinearity between Sensors

70 58 It is seen that both SG-G3-TF-EE and TM-NAWall-AA contain nearly the same data as the thermal couples through the higher part of the deck. This is relatively consistent through the day and night. It is also seen that TM-SAWall-HH is nearly a mirrored image of the TM-NAWall-AA and contains nearly the same information, just in an opposite direction. This is also expected, when a structure overall temperature increases, an expansion occurs. This expansion should cause the bridge to extend further into the abutments and to some degree tilt the abutments. The slope of the abutments and pier should mimic the expansion and contract effects and hence the bridge temperature. The collinearity of these records is seen and using all of these variables in a regression model would be redundant, since the same information is contained in each recording. Since there is redundant information and temperature is the driving cause of these other factors, only the thermocouple readings will be used in statistical regression models. In order to see the possible relationship between temperature and natural frequency, these are isolated in the following figure. The top and bottom temperatures from the previous figure, DTC-West-07 andwtc-g1-05, are plotted with the natural frequencies. The natural frequencies and temperatures have been vertically shifted to keep the spaces between the readings consistent. A better relationship can be seen in Figure 35. As shown, the peaks of the natural frequency happen sometime between the peak of the top deck thermocouples and the lower deck thermocouples. The slopes before and after the temperature peaks and the natural frequency peaks are also similar, which is good for aligning a time lag if necessary. Statistical regression models can be made with the temperatures and natural frequencies with any or a group of temperature sensors.

71 59 Figure 35: Temperatures and Natural Frequencies I The SHM records show a correlation between the bridge temperature and natural frequencies. To better understand the relationship between a structure s natural frequency and temperature, scatterplots were created for March s records. The scatterplots are the 15 minute average temperatures of the listed sensors that correlate with the five minute time the fast sampling rate sensors were recording. Figure 36 shows the natural frequencies measured by VT-UD-BB versus the bridge temperatures measured by DTC- West-04. Figure 37 shows the natural frequencies measured by VT-UD-CC versus the bridge temperatures measured by DTC-West-07.

72 60 Figure 36: Scatterplot of VT-UD-BB & DTC-West-04 Figure 36 and Figure 37 show a linear relationship between the independent and dependent variables. The relationship looks to be a direct (positive) and linear for all frequencies, maybe greater in magnitude for the lower frequencies, and there looks like there could be less error in the lower frequencies. The nature of the relationship is assumed linear for this study.

73 61 Figure 37: Scatterplot of VT-UD-CC & DTC-West-07 From this section, the SHM records indicate the following: a linear relationship between temperature and frequency and collinearity between potential predictor variables (both spatial and temporal). The spatial collinearities are viewed through the depth of the bridge and the temporal collinearities are seen with the gradient over time. These spatial and temporal collinearities are present longitudinally and transversely along the bridge as well.

74 62 Initial Linear Regression Models As seen in the previous section, a linear relationship between the temperature and natural frequencies were observed. In this section, linear regression models are made to better understand this relationship. Both temporal and spatial considerations are made in these models. These models are initial models to be improved upon. In order to create linear regression models, certain considerations must be taken into account with the collinearity observed in the predictor variables; these were explained in the previous section. The predictor variables showed to be temporally and spatially collinear. In order to simplify these and create useful models, engineering principles and statistical results will be used to pick variables that will help describe this relationship. In this section, the following steps will be completed: first, an average bridge temperature is calculated from the weighted average of the deck and web temperatures to their cross-sectional areas; second, temporal considerations are considered for the natural frequencies and the calculated bridge temperature; third, a linear regression model is created; and finally, the results are discussed. In order to show how the average bridge temperature might be a good predictor variable, Figure 38 is shown again.

75 63 Figure 38: Temperatures, Natural Frequencies, & Strain Gauge II In Figure 38, DTC-West-07 (76.2 mm (3 in.) below the deck surface) records a peak temperature right before the natural frequency peaks. WTC-G1-05, which is on the bottom of the bridge, records a maximum temperature after the natural frequency peaks. Since the correlation is linear and positive, the natural frequency peak should align with a temperature peal. The average bridge temperature condenses 15 thermocouple readings into one and its peak will happen close to the natural frequency. This would create a peak somewhere between the top bridge temperatures and the lower bridge temperatures. This somewhat compensates for both temporal as spatial collinearities. The Kuppa Method (Moorty, 1991) is a simplified method to calculate the average bridge temperature. This method reduces the number of temperature values considerably. The Kuppa Method is given in Equation 1. Equation 1

76 64 In this equation, i represent the number of cross-sectional elements through the depth of a bridge. A i, E i, α i, and T i are the area, modulus of elasticity, coefficient of thermal expansion, and the temperature, respectively, of each i th element. It is a weighted average of the temperatures to the cross-sectional area if E and α are constants. For this section, only the west deck grouping will be used through the subsequent steps. Figure 39 shows the average bridge temperature calculated for March 1 st through the 15 th, along with the bridge gradient. Figure 39: West Temperature Gradient & Bridge Average Temperature As seen in Figure 39, the peaks of the average bridge temperature align in between the peaks of the top and bottom bridge thermocouples as desired. To see where this temperature temporally lines up with the natural frequencies, the average bridge

77 65 temperature is plotted with DTC-West-07, WTC-G1-05, and the natural frequencies of VT-UD-BB in Figure 40. Figure 40: Average Bridge temperature & Natural Frequencies The units of Figure 40 are units of standard deviation as in previous figures. A smaller time range was chosen to show that the bridge average temperature simplifies some of the temporal and spatial correlations. This time period also shows that the time period from January 22 nd to the 24 th has smaller amplitudes and fluctuations; this pattern is also evident in the natural frequencies. This again shows that the signals are more aligned with each other. In order to study the relationship of temperature and natural frequencies further, the bridge average temperature was used to create regression models for every sensor s potential natural frequencies. The information from the base, strain gauge regression models is used to eliminate other sensor s natural frequencies that show the opposite

78 66 correlation. Figure 41 shows a model that was chosen to represent the majority of the models. Figure 41: Linear Regression for Average Bridge Temperature & SG-G3-TF-CC The top plot in Figure 41 shows the linear regression model fitted for sensor SG- G3-TF-CC and the average bridge temperature. The 95% confidence interval is shown and a residual in Figure 41 by the dashed line. The beta coefficients (β # ) in this figure are given with the variable names p1 and p2 in the bottom right table of Figure 41. The error plot is shown below the correlation plot. The majority of the errors are within the 95% confidence interval. Later, better correlations are found and the error terms are better, but the possibility of outlier is seen here. This is only one of all the models created. Table 4 and Table 5 show the beta coefficients (β # ), their standard error (Std. Error), the t-test

79 67 statistic (tstatistic), and the probability value (pvalue). The Mean Square Error (MSE) and adjusted R 2 values for all the regression models are also shown. As seen in Table 4 and Table 5 many of the frequencies that were processed can be eliminated. As a note, any R 2 values reported in this report are adjusted R 2 values for convenience of comparing various models. When less than 5% of the 720 fast sampling rate records did not show a natural frequency peak, asterisks were inserted across that row and the linear regression models were neglected. When the adjusted R 2 values were negative, lower than 0.400, or the slope (β 1 ) was not consistent with the base strain gauges, the models were eliminated. Higher frequencies cannot be validated without knowing it is a mode of one of the lower frequencies. The values of these eliminated models are listed, but the rows start and end with an asterisk and are not highlighted. The base strain gauges show a positively sloped correlation and any natural frequency indicating a negatively sloped correlation is eliminated. These natural frequencies also show an asterisk at the beginning and end of their row, but the slopes are darkly highlighted to show the negative β 1 coefficient. All natural frequencies lightly shaded had adjusted R 2 values that were close to or above and consistent to those around it. These also indicated positively sloped correlations. Some of the negatively sloped models actually showed a higher adjusted R 2 value, hence less error in the model, than the others. Since they cannot be validated at this time, they will be neglected from this study. The sensors and natural frequency modes used further on in this study correspond to those lightly highlighted. The two verified natural frequencies were close to 3.10 and 4.20 hertz. All sensors showed the 3.10 hertz natural frequency and all but one showed the 4.20 hertz frequency. These are now the only frequencies that are studied.

80 68 Table 4: Average Bridge Temperature Linear Regression Coefficients I Sensor Groups & Names: Sensor Groups & Names: Hitech Strain Gauge Velocity Transducer Tiltmeter Tiltmeter Hitech Strain Gauge Velocity Transducer Natural Frequency Peaks 1.50 Hertz Group 3.10 Hertz Group Statistical Values β 0 Std. Error β 0 tstatistic β 0 pvalue β 0 β 1 Std. Error β 1 tstatistic β 1 pvalue β 1 MSE R 2 β 0 Std. Error β 0 tstatistic β 0 pvalue β 0 β 1 Std. Error β 1 tstatistic β 1 pvalue β 1 MSE R 2 SG-G3-TF-CC * * * * * * * * * * E E SG-G1-BF-CC * * * * * * * * * * E E SG-G2-BF-CC E E E E E E SG-G3-BF-CC E E E E E E SG-G4-BF-CC * * * * * * * * * * E E SG-G5-BF-CC * * * * * * * * * * E E SG-G3-TF-FF * * * * * * * * * * E E SG-G1-BF-FF * * * * * * * * * * E E SG-G2-BF-FF * * * * * * * * * * E E SG-G3-BF-FF * * * * * * * * * * E E SG-G4-BF-FF * * * * * * * * * * E E SG-G5-BF-FF * * * * * * * * * * E E VT-UD-BB * * * * * * * * * * E E VT-UD-CC * * * * * * * * * * E E VT-UD-FF E E E E E E VT-UD-GG E E E E E E TM-NAWall-AA * * * * * * * * * * E E TM-Pier-DD * * * * * * * * * * E E TM-SAWall-HH * * * * * * * * * * E E Natural Frequency Peaks Hertz Group Hertz Group Statistical Values β 0 Std. Error β 0 tstatistic β 0 pvalue β 0 β 1 Std. Error β 1 tstatistic β 1 pvalue β 1 MSE R 2 β 0 Std. Error β 0 tstatistic β 0 pvalue β 0 β 1 Std. Error β 1 tstatistic β 1 pvalue β 1 MSE R 2 SG-G3-TF-CC E E * * * * * * * * * * SG-G1-BF-CC E E * * * * * * * * * * SG-G2-BF-CC E E * * * * * * * * * * SG-G3-BF-CC E E * * * * * * * * * * SG-G4-BF-CC E E * * * * * * * * * * SG-G5-BF-CC E E * * * * * * * * * * SG-G3-TF-FF E E * * * * * * * * * * SG-G1-BF-FF E E * * * * * * * * * * SG-G2-BF-FF E E * * * * * * * * * * SG-G3-BF-FF E E * * * * * * * * * * SG-G4-BF-FF E E * * * * * * * * * * SG-G5-BF-FF E E * * * * * * * * * * VT-UD-BB E E E E E E VT-UD-CC E E * * * * * * * * * * VT-UD-FF E E E E E E VT-UD-GG E E E E E E TM-NAWall-AA E E * * * * * * * * * * TM-Pier-DD * * * * * * * * * * * * * * * * * * * * TM-SAWall-HH E E E E E E

81 69 Table 5: Average Bridge Temperature Linear Regression Coefficients II Sensor Groups & Names: Sensor Groups & Names: Velocity Transducer Velocity Transducer Tiltmeter Tiltmeter Natural Frequency Peaks (htz) Hertz Group Hertz Group Statistical Values β 0 Std. Error β 0 tstatistic β 0 pvalue β 0 β 1 Std. Error β 1 tstatistic β 1 pvalue β 1 MSE R 2 β 0 Std. Error β 0 tstatistic β 0 pvalue β 0 β 1 Std. Error β 1 tstatistic β 1 pvalue β 1 MSE R 2 VT-UD-BB E E E E VT-UD-CC E E * * * * * * * * * * VT-UD-FF E E E E VT-UD-GG E E E E TM-NAWall-AA E E E E TM-Pier-DD E E * * * * * * * * * * TM-SAWall-HH * * * * * * * * * * E E Natural Frequency Peaks (htz) 20.0 Hertz Group Hertz Group Statistical Values β 0 Std. Error β 0 tstatistic β 0 pvalue β 0 β 1 Std. Error β 1 tstatistic β 1 pvalue β 1 MSE R 2 β 0 Std. Error β 0 tstatistic β 0 pvalue β 0 β 1 Std. Error β 1 tstatistic β 1 pvalue β 1 MSE R 2 VT-UD-BB E E E E VT-UD-CC E E E E E VT-UD-FF E E E E VT-UD-GG E E E E E TM-NAWall-AA E E * * * * * * * * * * TM-Pier-DD E E * * * * * * * * * * TM-SAWall-HH E E * * * * * * * * * *

82 70 Although the bridge average temperature was able to eliminate some natural frequencies and confirm others, the adjusted R 2 values averaged On average, the models were able to account for 50% of the variation. There might have been some loss of information by using the bridge temperature average to describe the relationship. To see if any quality was lost, a quick regression model was built with the records from VT- UD-CC and DTC-West-07 just to compare statistical values. This will be for the natural frequency around 3.10 hertz. Figure 42 shows the regression model plot. Figure 42: Linear Regression for VT-UD-CC & DTC-West-07 As seen in Figure 42, the adjusted R 2 value has increased from to 0.603, which means the actual temperature at one spot of the bridge describes more variability in the regression model than the average bridge temperature. Also, the standardized residual (error) plot at the bottom of the figure verifies the linear relationship and

83 71 homoscedasticity; although, there might be a couple outliers in the measured data. In order to see if this model is affected by some of the spatial considerations, 12 more regression models were built by using the same dependent variable paired with temperatures from earlier times. Creating models with these earlier times would line up the peaks and compensate for the time lag or temporal collinearities. In each of the 12 model iterations, the temperatures from the previous 15 minute iteration were correlated with natural frequency. A total range of the lagged variables studied was three hour. Figure 43 shows the 3 of the 12 models graphically and a table of the statistical values calculated for each. Each fit number in the table corresponds to the time lag iteration, the first with a 15 minute lag. The three plots in Figure 43 show similar regression models. The y-values (ordinates) in this figure do not change, but the x-values (coordinates) do slightly. The table in Figure 43 does not give the intercept or slope of the linear regression models. The table does give the adjusted R 2 values to the right of their respective plots. The three models with the highest adjusted R 2 are shown with their intercepts and slopes. Fit 4, corresponding to a 60 minute time lag, had the highest adjusted R 2 value (0.612), as seen in Figure 43 s table.

84 72 Figure 43: Linear Regression with 15 Minute Time Lag Increments In this section, linear regression models were made for the bridge average temperature and every possible natural frequency. Certain natural frequencies were selected due to consistency between all models and the adjusted R 2 values. The natural frequencies that lie close to 3.10 and 4.20 hertz have a positive linear relationship with bridge temperature. Although the tiltmeter records show a better correlation with higher natural frequencies, these were eliminated due to their inconsistency of slope with the base strain gauge models which may be caused by folding. A quick regression model was made for a single thermocouple and natural frequency, rather than the model created with

85 73 the bridge average temperature. This model explained more of the variability than the bridge average temperature did. The standardized residual plots of this model also supported the linear model assumptions. Finally, the temporal issue of aligning the natural frequency and temperature peaks was addressed by creating regression models with lagged temperature readings. Similar adjusted R 2 values for models created with no lag to an hour lag were found, but the best corresponded to a 45 minute lag. Initial Stepwise Regression Model In order to build a model that is useful for the majority of the natural frequencies, stepwise linear collinear regression was completed in this section. First, however, an explanation of why the regression coefficients can be misleading is given. Then, stepwise linear regression was completed to find the common variables that best describe the variation in natural frequencies. This reduces some spatial redundancies. Stepwise regression was utilized to create models utilizing the previous section s chosen natural frequencies, the 3.10 and 4.20 hertz. Collinearity still exists between predictor variables; which means, the coefficients estimated using regression models cannot be used to interpret importance. Even with temperatures being the same units, or the scenario of creating regression models by transforming temperatures into units of standard deviations, collinearity makes an infinite amount of possibilities for the coefficients that correspond to these correlated variables. A quick example of this is given with two independent, linearly correlated variables, T 1 and T 2, and a linearly, dependent variable F 1. A multiple linear regression model is assumed and given by Equation 2.

86 74 Equation 2 If two consecutive observations were made with this equation, then subtracted one from the other, the β 0 term cancels out and Equation 2 can be written as Equation 3. Equation 3 If the variable T 2 can be described as a linear function of a constant, C 1, and T 1, the changes in temperatures can also be written as Equation 4 by subtracting subsequent observations. Equation 4 Substituting Equation 4 into Equation 3 and simplifying, one would get the following equations: Equation 5 through Equation 7. Equation 5 Equation 6 Equation 7 Equation 7 is a line with intercept β 1 and slope β 2 that runs through the point F 1 / T 1. Constant C 1 does not vary. There are an infinite number of lines that run through one specific point. For this reason, the coefficient values may not give specific importance for each variable. Considerations and steps can help eliminate possible unstable collinearity variables and will be used in a later section of this paper. For now, linear stepwise regression is used to find fewer variables that account for the most variation in the frequencies. Linear stepwise regression was used for all natural frequencies and had all thermocouples as possible predictor variables. Any added predictor variables cannot

87 75 decrease the R 2 value from the previous linear regression models since the added coefficients may be set to zero and the previous linear regression model could be obtained. The adjusted R 2 value, however, can decrease since this also depends on the number of predictor variables. In order to compare models, an adjusted R 2 value is reported for every model. The stepwise regression models all reported adjusted R 2 values higher than previous models. The variables used in the final stepwise linear regression models are reported in Table 6 and Table 7. The adjusted R 2 values ranged from to for the first natural frequency, while it ranged from to for the second. From the tables, the important variables stepwise regression chose are consistent for each natural frequency. Even between the first and second natural frequency, the chosen variables do not differ greatly. DTC-West-06 and DTC-West-07 were both important for the second natural frequency, but due to the fact that they are located close to each other and the first frequency depends solely on DTC-West-07, this was selected. The temperature sensors chosen to be used as predictor variables through the rest of the study are the following: TC-G3-TF-BB, TC-G3-BF-CC, TC-G3-TF-FF, TC-G3-TF-GG, DTC-West-07, WTC-G1-01, DTC-East-07, and WTC-G2-05. Since these still may be fairly collinear, the coefficients cannot show importance as shown earlier. The next section will further verify the use of these variables.

88 76 Table 6: Stepwise Regression Variables I Natural Frequency Sensors (3.10 Hertz) Section SG-G3-TF-CC SG-G1-BF-CC SG-G2-BF-CC SG-G3-BF-CC SG-G4-BF-CC SG-G5-BF-CC SG-G3-TF-FF SG-G1-BF-FF SG-G2-BF-FF SG-G3-BF-FF SG-G4-BF-FF SG-G5-BF-FF VT-UD-BB VT-UD-CC VT-UD-FF VT-UD-GG TM-NAWall-AA Thermocouples: TM-Pier-DD TM-SAWall-DD TC-G3-TF-BB x x x x x x x x x TC-G3-BF-BB TC-G1-BF-CC TC-G2-BF-CC TC-G3-TF-CC x TC-G3-BF-CC x x x x x x x x x x x x x x x x x x x TC-G5-BF-CC x DTC-WEST-01 DTC-WEST-02 DTC-WEST-03 DTC-WEST-04 DTC-WEST-05 DTC-WEST-06 DTC-WEST-07 x x x x x x x x x x x x x x x x x x DTC-WEST-08 DTC-WEST-09 DTC-WEST-10 WTC-G1-01 WTC-G1-02 WTC-G1-03 WTC-G1-04 WTC-G1-05 DTC-EAST-01 DTC-EAST-02 DTC-EAST-03 DTC-EAST-04 DTC-EAST-05 DTC-EAST-06 DTC-EAST-07 x x x x x x x x x x x x x x DTC-EAST-08 DTC-EAST-09 DTC-EAST-10 WTC-G2-01 WTC-G2-02 WTC-G2-03 WTC-G2-04 x x WTC-G2-05 x x x x x x x x x x x x x x x x x x x TC-G3-TF-FF x x x x x x x x x x TC-G3-BF-FF x TC-G1-BF-GG TC-G2-BF-GG x x TC-G3-TF-GG x x x x x x x TC-G3-BF-GG x x x TC-G5-BF-GG BB CC EE FF GG

89 77 Table 7: Stepwise Regression Variables II Natural Frequency Sensors ( Hertz) Thermocouples: TC-G3-TF-BB TC-G3-BF-BB TC-G1-BF-CC Section SG-G3-TF-CC SG-G1-BF-CC SG-G2-BF-CC SG-G3-BF-CC SG-G4-BF-CC SG-G5-BF-CC SG-G3-TF-FF SG-G1-BF-FF SG-G2-BF-FF SG-G3-BF-FF SG-G4-BF-FF SG-G5-BF-FF VT-UD-BB VT-UD-CC VT-UD-FF VT-UD-GG TM-NAWall-AA TM-Pier-DD* TM-SAWall-DD TC-G2-BF-CC TC-G3-TF-CC TC-G3-BF-CC x x TC-G5-BF-CC DTC-WEST-01 DTC-WEST-02 DTC-WEST-03 DTC-WEST-04 DTC-WEST-05 DTC-WEST-06 x x x x x x x x x x x DTC-WEST-07 x x x x x x DTC-WEST-08 DTC-WEST-09 DTC-WEST-10 WTC-G1-01 x x x x x x x WTC-G1-02 WTC-G1-03 WTC-G1-04 WTC-G1-05 x x DTC-EAST-01 DTC-EAST-02 DTC-EAST-03 DTC-EAST-04 DTC-EAST-05 x x DTC-EAST-06 DTC-EAST-07 DTC-EAST-08 DTC-EAST-09 DTC-EAST-10 x WTC-G2-01 x x x WTC-G2-02 WTC-G2-03 x x WTC-G2-04 WTC-G2-05 x x x x x x x x x x x x x x x x x TC-G3-TF-FF x x x x x x x x x x x x TC-G3-BF-FF TC-G1-BF-GG TC-G2-BF-GG TC-G3-TF-GG x x x TC-G3-BF-GG BB CC EE FF GG TC-G5-BF-GG x x x x x

90 78 These variables give insight on common scenarios that might affect the natural frequencies of the bridge. Certain environmental scenarios can be captured by these chosen variables. The upper bridge deck temperatures for example, might relate information about the intensity and direction of solar radiation. The magnitude of the deck thermocouples could indicate the intensity of the radiation. The difference between the east and west deck thermocouples could represent how solar radiation hits one part of the bridge earlier or later in the day. The difference between the top and bottom temperatures could indicate faster changes in weather, such as rain, wind, or precipitation initially freezing on the bridge. The thermocouples below the bridge or on the interior cells might have more information about the ambient air temperature, since these are less exposed to weather. The spacing of the variables indicates that the temperature gradients in all directions are important variables in this problem. In this section, stepwise linear regression was utilized to determine the predictor variables that will be used through this study. A total of eight thermocouple records were chosen. These chosen variables were consistently present in most the stepwise regression models. These variables were also mostly consistent when comparing the first and second natural frequencies models. The positions of the thermocouples help explain the importance of thermal gradient in all directions: transversely, longitudinally, and through the depth.

91 79 Multiple Linear Regression with Ridge Regression In this section, multiple linear regression models were made. Only the chosen variables from the previous section are used. The location of these sensors helps with the spatial collinearities of the predictor variables. First, ridge regression is used to establish any previously chosen variables that are unstable. These variables were taken out of the regression model. The temporal collinearities were then addressed through a strain gauge s natural frequencies. The final regression model is chosen in a later section. Ridge regression is useful when collinear predictor variables are present. Ridge regression is a way to verify the stability of variables. Ridge regression inflates the variance values (the diagonal) of the variance/covariance matrix used in multiple linear regression models. The variance/covariance matrix is a matrix that shows the variance between all predictor variables. The ideal situation is to have the variances large and covariance small. With collinearity, this is not the case. Ridge regression inflates the variance values incrementally; making regression models along the way and keeps track of ridge parameters. With each incremental increase, the differences of these parameters determine if the variables are stable or unstable. If the values are considered stable, the variables can be kept, even though their coefficients in the un-inflated model might be small. First, the correlation coefficients are calculated for the predictor variables. Table 8 gives the values calculated for the predictor variables. This shows the collinearity between predictor variables.

92 80 Table 8: Correlation Coefficients Variables: TC-G3-TF-BB TC-G3-BF-CC TC-G3-TF-FF TC-G3-TF-BB TC-G3-BF-CC TC-G3-TF-FF TC-G3-TF-GG DTC-West DTC-East WTC-G WTC-G TC-G3-TF-GG DTC-West-07 DTC-East-07 WTC-G1-01 WTC-G2-05 An ideal ridge regression plot would have horizontal lines for each ridge parameter corresponding to a predictor variable. Most, however, do not have ideal characteristics but show which variables level out after inflation and which do not. The variables that level out are stable. The unstable variables may be eliminated. If a ridge parameter changes from a positive to a negative on the graph, or vice versa, these are a good indicator of an unstable variable that can affect the model. Another unstable indicator is the initial slope change when the inflation is added. The more drastic the slope change, the more likely the corresponding predictor variable is affecting the model. Variable elimination is an iterative process since the ridge parameters are dependent on all the predictor variables. If an unstable variable is taken out of the model, the ridge parameters may behave differently. The plots are less definite, but a good visual representation of the ridge parameters. The values obtained in ridge regression on the

93 81 following figures were used in determining the stability of each variable. Figure 44 shows the first iteration of ridge regression. Figure 44: Ridge Regression, 1 st Iteration As seen in the figure, both TC-G3-TF-EE and WTC-G1-01 cross the x-axis and switch signs. In the first iteration, TC-G3-TF-EE was taken out due to the larger change in slope in the 0 to 0.1 range. Figure 45 shows the second iteration. As shown in Figure 45, the nature of some of the lines changed a little. WTC-G1-01, however, still changes sign after a small amount of inflation is applied. This variable is also eliminated. The change between the first iteration and second iteration is also evident in the other variables from the locations they cross one another or the slopes. The slopes in these iterations remain constant to show the drastic slope change in the eliminated variables and the change in the other ridge parameter values. Figure 46 show the third iteration.

94 82 Figure 45: Ridge Regression, 2 nd Iteration Figure 46: Ridge Regression, 3 rd Iteration