Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Probabilistic assessment of damage states using dynamic response parameters Quiroz, Laura Maria 2011

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2011_fall_quiroz_laura.pdf [ 1.23MB ]
Metadata
JSON: 24-1.0063203.json
JSON-LD: 24-1.0063203-ld.json
RDF/XML (Pretty): 24-1.0063203-rdf.xml
RDF/JSON: 24-1.0063203-rdf.json
Turtle: 24-1.0063203-turtle.txt
N-Triples: 24-1.0063203-rdf-ntriples.txt
Original Record: 24-1.0063203-source.json
Full Text
24-1.0063203-fulltext.txt
Citation
24-1.0063203.ris

Full Text

PROBABILISTIC ASSESSMENT OF DAMAGE STATES USING DYNAMIC RESPONSE PARAMETERS by  Laura Maria Quiroz  Civil Engineer, Universidad Nacional de Rosario, 2009  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES  (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)  August 2011  © Laura Maria Quiroz, 2011  Abstract To acknowledge and account for the uncertainties present in civil engineering applications is an area of major importance and of continuing research interest. This Thesis focuses on an application of Bayes' inference rule to evaluate the probability of damage in structures, using measured modal parameters and a set of possible damage states. The hypothesis is that observed changes in dynamic characteristics are due to damage accumulation over time. The main objective is to identify the most likely damage scenario from a set of previously defined damage states. These are characterized in terms of vectors, i, the components of which are the parameters, ij, that are associated with the stiffness contribution, Kj, from each substructure undergoing damage. These stiffness matrices are uncertain as a result of random geometric and material properties. For different combinations of the damage parameters and realizations of the random variables, the modal parameters are calculated solving the basic eigenvalue problem. The results are used to calculate the statistics of the parameters given a specific damage state, the likelihood functions, as these are needed to calculate the probability of a given a set of measurements given a damage state. Each damage state Di is associated with a prior probability P(Di). In order to calculate its posterior probability, given a set of measurements, a Bayesian updating is implemented, in which the prior probability is updated by means of the likelihood functions, f(r|Di), which represent the probability density function of the modal parameter, r, given the damage state, Di. This Thesis discusses the effectiveness of the approach in identifying a particular damage state referred to as damage scenario. It is shown that measurement of multiple modal parameters is required to identify, quickly and with confidence, a given damage state. The discussion also considers the effect ii  of error in the measurements, and the number of repeated measurements that are required to achieve a substantial confidence as to the presence of a particular damage state. Ranking of the estimated probabilities, after a set of measurements, offers guidance to the engineer as when and where to conduct a direct inspection of the structure.  iii  Table of Contents Abstract .................................................................................................................................... ii  Table of Contents ................................................................................................................... iv  List of Tables .......................................................................................................................... vi  List of Figures ........................................................................................................................ vii  Acknowledgements ................................................................................................................. x  Dedication ............................................................................................................................... xi  1 INTRODUCTION............................................................................................................ 1  1.1   MOTIVATION.......................................................................................................... 1   1.2   OBJECTIVES ........................................................................................................... 1   1.3   SCOPE .................................................................................................................... 2   1.4   LITERATURE REVIEW ............................................................................................. 3   1.5   THESIS ORGANIZATION .......................................................................................... 8   2 PROBABILISTIC FORMULATION .......................................................................... 10  2.1   BAYES' RULE ....................................................................................................... 10   2.2   APPLICATION OF BAYES' RULE TO DAMAGE DETECTION ..................................... 12   2.3   LIKELIHOOD FUNCTIONS AND THEIR CALCULATION ............................................ 15   2.3.1  UNIVARIATE LIKELIHOOD FUNCTION BY SAMPLING ......................................... 17  2.3.2  MULTIVARIATE LIKELIHOOD FUNCTION BY SAMPLING ..................................... 19  2.3.3  UNIVARIATE LIKELIHOOD FUNCTION BY FORM .............................................. 22  3 METHODOLOGY: STRUCTURAL MODEL AND DAMAGE STATES ............. 24  3.1   ESTABLISHING THE STRUCTURAL MODEL............................................................ 24   3.2   DEFINING POTENTIAL DAMAGE STATES .............................................................. 26   4 CASE STUDY ................................................................................................................ 28  4.1   STRUCTURAL MODEL........................................................................................... 28   4.2   DAMAGE STATES ................................................................................................. 34   4.3   LIKELIHOOD FUNCTIONS ...................................................................................... 39  iv  4.3.1  UNIVARIATE LIKELIHOOD FUNCTIONS FOR THE CALCULATED RESPONSES ....... 39  4.3.2  MULTIVARIATE LIKELIHOOD FUNCTIONS FOR THE CALCULATED RESPONSES ... 48  4.4   BAYESIAN UPDATING .......................................................................................... 55   5 PARAMETRIC STUDIES ............................................................................................ 67  6 CONCLUSIONS AND FURTHER WORK ................................................................ 73  References .............................................................................................................................. 76  Appendices ............................................................................................................................. 79  Appendix A ......................................................................................................................... 79  Appendix B ......................................................................................................................... 81   v  List of Tables Table 4.1: Definition for the 64 damage states in terms of the damage indicators................. 35  Table 4.2: Parameters of fitted distributions for 1 with RELAN.......................................... 42  Table 4.3: Parameters of fitted distributions for 2 with RELAN.......................................... 42  Table 4.4: Parameters of fitted distributions for 3 with RELAN.......................................... 42  Table 4.5: Parameters of fitted distributions for 21 with RELAN ......................................... 46  Table 4.6: Parameters of fitted distributions for 31 with RELAN ......................................... 46  Table 4.7: Realizations of calculated  .................................................................................. 48  Table 4.8: Realizations of calculated 1 ................................................................................. 51  Table 4.9: Realizations of calculated 2 ................................................................................. 51  Table 4.10: Realizations of calculated 3 ............................................................................... 51  Table 4.11: Measured data sets for  and D7 as damage scenario ......................................... 57  Table 4.12: Measured data sets for 1 and D7 as damage scenario ........................................ 57  Table 4.13: Measured data sets for 2 and D7 as damage scenario ........................................ 57  Table 4.14: Measured data sets for 3 and D7 as damage scenario ........................................ 57   vi  List of Figures Figure 2.1: Bayes' rule for events ........................................................................................... 11  Figure 2.2: Bayesian updating formulation scheme ............................................................... 14  Figure 2.3: Generic sample cumulative distribution function ................................................ 18  Figure 2.4: Sample and different types of CDF ...................................................................... 19  Figure 4.1: Three storey frame................................................................................................ 29  Figure 4.2: Lognormal CDF for the basic stiffness random variables.................................... 31  Figure 4.3: Mode shapes for the three storey frame ............................................................... 33  Figure 4.4: First frequency variation with the damage state ................................................... 37  Figure 4.5: Second frequency variation with the damage state .............................................. 38  Figure 4.6: Third frequency variation with the damage state ................................................. 38  Figure 4.7: Sample CDFs for the calculated frequencies ....................................................... 40  Figure 4.8: Fitted distributions for 1 ..................................................................................... 40  Figure 4.9: Fitted distributions for 2 ..................................................................................... 41  Figure 4.10: Fitted distributions for 3 ................................................................................... 41  Figure 4.11: CDFs obtained using FORM for the calculated natural frequencies.................. 43  Figure 4.12: Comparison for the third natural frequency ....................................................... 44  Figure 4.13: Sample CDFs for 1 components ....................................................................... 44  Figure 4.14: Fitted distributions for 21 .................................................................................. 45  Figure 4.15: Fitted distributions for 31 .................................................................................. 46  Figure 4.16: CDFs obtained using FORM for the calculated first mode shape components . 47  Figure 4.17: Comparison for 21 ............................................................................................. 47  vii  Figure 4.18: Correlation between first and second natural frequencies.................................. 50  Figure 4.19: Correlation between first and third natural frequencies ..................................... 50  Figure 4.20: Correlation between second and third natural frequencies ................................ 50  Figure 4.21: Correlation between first mode shape components ............................................ 53  Figure 4.22: Correlation between second mode shape components ....................................... 53  Figure 4.23: Correlation between third mode shape components........................................... 53  Figure 4.24: Bi-variate Normal distribution for the first mode shape .................................... 54  Figure 4.25: Bi-variate Normal distribution for the second mode shape ................................ 54  Figure 4.26: Bi-variate Normal distribution for the third mode shape ................................... 54  Figure 4.27: Initial prior probabilities ..................................................................................... 55  Figure 4.28: Bayesian updating results, posterior probabilities of each damage state using individual natural frequencies ................................................................................................. 59  Figure 4.29: Bayesian updating, posterior probabilities using individual first mode components ............................................................................................................................. 61  Figure 4.30: Bayesian updating, posterior probabilities using individual second mode components ............................................................................................................................. 62  Figure 4.31: Bayesian updating, posterior probabilities using individual third mode components ............................................................................................................................. 63  Figure 4.32: Bayesian updating, posterior probabilities of each damage state using the vector of natural frequencies .............................................................................................................. 65  Figure 4.33: Bayesian updating, posterior probabilities of each damage state using the first mode shape vector................................................................................................................... 65   viii  Figure 4.34: Bayesian updating, posterior probabilities of each damage state using the second mode shape vector................................................................................................................... 66  Figure 4.35: Bayesian updating, posterior probabilities of each damage state using the third mode shape vector................................................................................................................... 66  Figure 5.1: Bayesian updating, posterior probabilities for different damage scenarios after the first evaluation, using  as input ............................................................................................ 70  Figure 5.2: Bayesian updating, posterior probabilities for different damage scenarios at the first evaluation, using 3 as input ........................................................................................... 70  Figure 5.3: Bayesian updating, posterior probabilities for different errors at the first evaluation, using  as input .................................................................................................... 72   ix  Acknowledgements I would like to express my sincere gratitude and appreciation to my research advisors Dr. Terje Haukaas for his endless encouragement, supervision and support, and Dr. Ricardo Foschi for his guidance and advice throughout my thesis. It is a truly honor for me to have had the opportunity to share your knowledge in the structural reliability field. I wish to acknowledge all of my professors for the excellent education they provided me during the course of the Master program. I would also like to thank my fellow students, members of the Reliability Group at UBC, for their insightful comments and discussion. I would like to express my deepest gratitude to my friends and family for the encouragement and understanding they provided me through all the challenges I have faced. Last but certainly not least, I would like to thank my husband, his love and support are the key to my success.  Funding for this research was provided by Natural Sciences and Engineering Research Council of Canada through the Strategic Project Grant (STPGP 336498-06) Infrastructure Risk; and the Research Grant (RGPIN 5882-04) Neural networks for reliability and performance-based design in earthquake engineering, University of British Columbia.  x  Dedication  To my parents  xi  1 INTRODUCTION  1.1  MOTIVATION  Knowledge about the current state of a structure is required when evaluating the remaining service life and possible repair strategies, not only for normal use over time but after major events like earthquakes and other hazards. How to make a determination of the state of a structure is not a simple problem. Beyond visual inspection by experienced engineers, a formal procedure for quantitative evaluation is needed. These procedures are generally known as structural health monitoring (SHM). SHM is the process of continuously monitoring structures for damages that may occur over time, to ensure safety and adequate performance. The approach involves the measurement of structural responses (dynamic or static) and a correlation between these responses and different states of damage or deterioration. This Thesis is motivated by the need to develop a probabilistic assessment procedure that accounts for the uncertainty associated with the prediction of different damage states.  1.2  OBJECTIVES  Motivated by the need to identify potential damage during the service life of a structure, including after extreme loading events, this Thesis addresses several specific objectives. The overarching goal is to detect the location and severity of potential damage states. It is also an objective to assess and communicate the uncertainty associated with the damage predictions. This is addressed by adopting a Bayesian approach. The methodology thus includes prior 1  probabilities for each damage state, input dynamic measurements and, through the introduction of the likelihood of observing those measurements given a pre-defined damage state, update its probability of occurrence. The observations that constitute the input to the Bayesian procedure are envisioned to come from ambient vibration testing. It is not an objective of this Thesis to discuss in detail the process of data gathering. The measurements are assumed to come from an instrumented structure, nearly continuously monitored. Raw instrument data, e.g. accelerograms, are streamed into a data processing framework. In turn, estimates of the structure’s natural vibration frequencies and associated mode shapes are calculated. This processing of raw data is also outside the scope of this Thesis: it is assumed that the input already consists of calculated frequencies and mode shapes. These estimates are also associated with uncertainty, and one of the objectives is to evaluate the effect of measurement errors in the probabilities associated with the different damage states. Ultimately, the engineer in charge of the structure is responsible for making decisions about inspection and possible repairs. The objective of this Thesis is to develop a probabilistic framework in support of this decisionmaking process. Another central objective of the Thesis is to study the robustness of the proposed methodology, assessing its precision and effectiveness in identifying damage states.  1.3  SCOPE  The applicability of the proposed methodology is wide-ranging. It can be utilized to assess damage in a variety of structures that have experienced continuous or sudden deterioration. The computations are based on measured structural responses, which in this Thesis are  2  assumed to be dynamic characteristics obtained from measurements. In particular, the vibration frequencies and vibration modes are employed. Other structural response parameters could have been used, such as deflections from static load tests. Estimation of the dynamic response characteristics may be carried out by techniques such as “operational modal analysis” when ambient vibration data is available, and by “experimental modal analysis” when forced vibrations are generated. However, although the approach makes use of structural characteristics that are measured in-situ, the different methods by which these characteristics are obtained are not discussed in this Thesis.  1.4  LITERATURE REVIEW  The body of literature in the field of damage assessment is particularly extensive. As a critical component of the SHM process, damage detection methodologies have been subject of research for several decades both for mechanical and structural systems. In the case of civil engineering, the process initially involves the observation of a structure over time using periodically spaced measurements. Subsequently, it requires the extraction of damage sensitive features from these measurements, and finally, the statistical analysis of these features to determine the current state of the structural system (Farrar and Worden 2007). During this process, global and local structural properties are assessed on the basis of measured variables. Then, the structures are periodically supervised with the aim of minimizing the safety risk and of keeping the maintenance cost as low as possible. Farrar and Worden (2007) presented a general discussion on the challenges related to damage identification and provided a historical overview for the SHM technology development including a variety of disciplines. The authors referred to SHM as the ‘grand challenge’ for 3  the engineering community since significant developments implies multi disciplinary research efforts amongst fields such as structural dynamics, signal processing, computational hardware and “statistical pattern recognition”. Given the vast amount of literature, the purpose of this review is mainly to establish the methodology presented in this Thesis in the context of damage detection schemes. This review mainly focuses in damage detection methods that employ probabilistic techniques, to acknowledge/account for the uncertainties involved in such problem. A detailed review of the general SHM literature was presented in a series of reports by Los Alamos National Laboratory (Doebling et al. 1996; Sohn et al. 2004; Figueiredo et al. 2009). The first report includes an exhaustive survey of technical literature until the late 1990's, classifying the methods according to required measured data and analysis technique. It also categorizes the applications according to the type of structure analyzed (beams, trusses, plates, bridges etc.). The second report was presented as an update of the previous version and reviews the publications appearing in the technical literature between 1996 and 2001. It is organized following the definition of the “statistical pattern recognition paradigm”, namely as a four part process, which includes: Operational evaluation, data acquisition, feature extraction and statistical model development. The latest report contains a comparison of SHM algorithms applied to standard data sets obtained for an aluminum frame structure. It focuses mostly in the feature extraction/system identification techniques, namely the process of finding or identifying the modal parameters from vibration data. A more recent state of the art review on techniques regarding damage detection algorithms for civil structures was produced by Turek (2007).  4  Model-based methods imply the utilization of an analytical model, usually a finite element model (FE), to parameterize the structural system under consideration. Liu (1995) presented a methodology for identification of truss properties, formulated as an optimization problem. The FE model was utilized to derive the error norm of the eigenvalue equation to be minimized. Given the sufficient amount of measured data, a closed form solution was developed to obtain the elements properties, specifically their mass and stiffness, which can be used to assess damage in trusses. Vibration-based methods utilize the dynamic properties of the structure, such as natural frequencies and mode shapes. Pothisiri and Hjelmstad (2003) presented a similar algorithm where the parameters of the parameterized structure can be found as well for the case of sparse measured data, by means of a grouping scheme in the FE model. In this case the definition of the error function is suitable for least squares minimization and a gradient based search algorithm was applied. Damage was assessed by comparing the statistical distribution of a Monte Carlo sample of estimated parameters for the damaged and undamaged or baseline planar truss, which incorporates to some extent the probabilistic nature of the problem. The development of probabilistic approaches to explicitly account for the uncertainties inherent to SHM problems is somewhat more novel. Doebling et al. (1996) introduced the idea of most likely damage case: Penny et al. (1993) presented a method for locating the most likely damage scenario by simulating the frequency shifts that would occur for all damage cases under consideration. The measured frequencies were then fitted to the simulated frequencies for each simulated damage case in a least-squares sense. The damage scenario was indicated by the minimal error in this fit. Friswell et al. (1994) presented the results of an attempt to identify damage on the basis of a known catalogue of likely damage 5  scenarios by using two different measures of goodness of fit. Nevertheless, these papers are part of the methods applied to solve “the forward problem”, specifically, the initial state of damage detection. Xia et al. (2002) presented a damage detection method by directly comparing the measured dynamic responses before and after damage of two laboratory specimens, specifically a cantilever beam and portal frame. By assuming that the measured data and the FE model contain some normally distributed random noises, the statistics of the stiffness parameters in the damaged configurations were calculated with Monte Carlo simulation. The probability of damage existence was then defined and utilized to estimate the chances of damage in the structural members. Probabilistic approaches based on the “inverse problem” of model identification, i.e. determining the characteristics of the structure, including its damage, from the observed modal measurements, were presented by Papadopoulos and Garcia (1998) and Vanik et al. (2000). The authors discussed the uncertainty in whether or not changes in identified model parameters reflect damage in a structure. The uncertainty arises from the characterization of the structure trough an analytical model, presence of measurement noise and the selection of damage sensitive parameters. The latter presented a Bayesian probabilistic technique (Beck and Katafygiotis 1999; Katafygiotis and Beck 1998) for continual online SHM so that gradual deterioration as well as damage from sudden events can be detected. The structural model was represented by a linear FE model where the model parameters represent a nominal stiffness contribution of each substructure considered in such model. Then, the authors derived an explicit joint probability density function for the model parameters conditional to the damage state. The marginal distributions are approximated by a Gaussian distribution and used to calculate the probability that a certain stiffness parameter, in a possible damage state, 6  has been reduced by some amount. Then, changes in the estimated probabilities are studied to detect structural damage. The method is illustrated using simulated data from a shear building model and with up to 20 percent stiffness loss in the fifth storey. Results are most favorable when the damage measured is calculated using the current data set. Addition of prior data tends to create a bias towards the undamaged state. Applications were also presented by Yuen et al. (2004) and Ching and Beck (2004) as part of a two stage SHM procedure for a frame structure adopted as benchmark study by IASC-ASCE. In addition, more illustrative examples of this methodology using responses to seismic excitation are found in Beck and Yuen (2004) and Zonta et al. (2007). Other similar Bayesian approach has also been proposed by Sohn and Law (1997) for damage detection of frame structures. The most likely locations and amount of damage are determined by formulating the relative posterior probability of an assumed damage event and applying a branch and bound search scheme. The damage parameters are defined to model the stiffness contribution of each substructure considered in the FE model. Moreover, the parameters are assumed independent and to have a uniform distribution between zero and one. Using the mentioned scheme the authors avoid the explicit expression of an updated or posterior probability and simplify the calculation of it, which by means of the Bayes’ theorem involves the examination of all hypotheses or damage events. Through mathematical simplifications the authors calculate the sought probability minimizing an error function. In this methodology damage is defined when the damage parameter is less than a certain threshold in stiffness reduction. The authors applied the described methodology to a reinforced concrete column using as data sets both frequencies and modal vectors (Sohn and Law 2000). They used two simple analytical models to somewhat include the non-uniqueness 7  issue in model updating. Damage was defined as a 10 per cent reduction on stiffness, that is, 0.9 was used for the damage threshold. In both cases, the Bayesian approach was able to converge to the actual damage location. Another application was presented by Sohn and Law (2000) to demonstrate the use of load dependent Ritz vectors as an alternative to modal parameters and the flexibility of the approach. Based on numerical examples of an eight-bay three-dimensional truss structure and a five-storey frame structure, the authors claim that load-dependent Ritz vectors are in general more sensitive to damage than the corresponding modal vectors and that structural elements of interest can be made more observable using the Ritz vectors generated from particular load patterns. Finally, a comprehensive review on recent developments on Bayesian class selection in civil engineering structures was presented by Yuen (2010). The research done in this Thesis accounts for the uncertainties inherent to damage detection problems incorporating the randomness associated with the basic structural variables. Therefore, the variability of the responses is conditional not only on the damage state but also on such randomness. The damage states are a-priori defined as a reduction in the stiffness contribution of each of the substructures considered in the FE model. The goal of the Bayesian methodology is to produce as results, the relative likelihoods of the damage cases considered.  1.5  THESIS ORGANIZATION  The developments in this Thesis are organized into four chapters. Chapter 2 provides an introduction to the probabilistic concepts that are applied in subsequent chapters. Particular  8  emphasis is placed on the Bayesian inference of damage state probability based on observations. Chapter 3 provides the theoretical formulation of the new methodology. The generic nature of the approach is first described, followed by an explanation of the particular application to damage detection. In Chapter 4, the methodology is applied to a case study of a frame structure. Chapter 5 studies the robustness of the methodology, considering different issues that influence its effectiveness in predicting a most likely damage scenario. This is done by means of parametric studies that validate the method and the analysis of results. Finally, conclusions and comments on suggested further work are presented in Chapter 6.  9  2 PROBABILISTIC FORMULATION The objective in this chapter is to describe a probabilistic formulation for the problem of damage state detection. In particular, the Bayesian approach is adopted to update prior probabilities assigned to each damage state using evidence from measured structural response parameters. This chapter presents first a review of Bayes' rule and then describes its application to the problem in this Thesis.  2.1  BAYES' RULE  Thomas Bayes was a 19th century English priest-statistician whose original contributions led to the development to what is now called the Bayesian approach to statistical inference and probability theory (Box and Tiao 1973). The basics concepts of Bayesian statistics are now briefly reviewed and the significance of Bayesian inference is presented. Consider Figure 2.1, which shows a set of ND mutually exclusive events Di, and an event, E, which could occur jointly with any of the events Di. Then, knowing the conditional probabilities, P(E|Di), of E occurring given each Di, and the prior probabilities, P(Di), Bayes' rule allows the calculation of the probabilities P(Di|E). This implies that the probability of each event Di is updated given that the event E is known to have occurred. These updated probabilities are called the posterior probabilities and are given by Bayes' rule:  PDi | E    PE | Di  PDi  i  1, 2,, N D PE   (1)  10  This rule can be derived by combining the multiplication rule and conditional probability rule and stems from the equality of the joint probabilities, PE  Di   PDi  E  (Bolstad 2007). The conditional probability, P(E|Di), is defined as the likelihood that E will occur given Di. The denominator in Eq. (1) can be further expressed as,         ND  PE   PE | D1  PD1     P E | DN D P DN D   PE | D j  PD j   (2)  j 1  where ND is the number of mutually exclusive and collectively exhaustive events Di, as shown in Figure 2.1. Finally, introducing Eq. (2) into Eq. (1): PDi | E    PE | Di  ND   PE | D  PD  j 1  j  PDi   (3)  j  which gives the final form of Bayes' rule as it is used in this study.  Figure 2.1: Bayes' rule for events  In summary, it can be observed that the probability of Di appears in both sides of Eq. (3), meaning that Bayes’ rule provides an updating calculation in the light of new information, specifically the occurrence of E. In doing so, the Bayes' theorem represents a way to revise the belief distribution from the prior probabilities, given the data E. 11  2.2  APPLICATION OF BAYES' RULE TO DAMAGE DETECTION  In this Thesis, the Bayesian probabilistic approach is applied to the calculation of the probability assigned to damage states. These probabilities are updated when information is obtained in the form of measured dynamic characteristics. Thus, the different damage states take the place of the events Di shown in Figure 2.1, while the event E is given by the occurrence of a set of measurements. One of the possible states is the undamaged or virgin state. The prior probabilities assigned to each state could be initially given arbitrarily, with the constraint that they must all add up to one. Response information collected allows the updating of these probabilities, using Bayes’ rule, and these updated or posterior probabilities become the new priors to be used when a new information set becomes available. The damage state with the highest posterior probability would be deemed to be the most likely one and, in this sense, the Bayesian updating process should converge to the actual damage state after a series of input information. For convenience of notation let us define r as a measured dynamic response instead of the general event E in Eq. (1). On the other hand, let the events Di represent the different damage states. Each Di is considered to include a discrete combination of damage magnitudes, occurring at different locations in the structure. Therefore, Bayes’ rule from Eq. (3) then reads,  PDi | r    Pr | Di  ND   Pr | D  PD  j 1  j  PDi   (4)  j  12  This damage state characterization requires the a-priori definition of the possible damage states, ND. The likelihoods P(r|Di) can now be expressed in terms of the conditional probability density functions (PDF) f(r|Di), as,  Pr | Di   f r | Di  dr  (5)  from which Eq. (4) becomes, PDi | r    f r | Di  ND   f r | D  PD  j  j 1  PDi   (6)  j  The application of Eq. (6) is represented in Figure 2.2 where r* indicates a particular measurement of the response, r. The diagram shows how the prior probabilities P(Di), represented in the upper figure, are updated by means of the likelihoods f(r|Di) for the specific r* value of the measured response. The posterior probabilities are shown in the lower figure and the most likely damage state, at the current evaluation, is the damage state with higher probability P(Di|r). Then, these probabilities become the priors for the next evaluation. In our case, the measured response r will be primarily the vibration frequencies n, so that Eq. (6) becomes, P  Di |  n    f  n | Di  ND   f  j 1  n  | D j  P D j   P  Di   (7)  The likelihoods in Eq. (7) represent the relative chance of observing a specific value of a frequency given the occurrence of a certain damage state. Given such state, the uncertainty in 13  the frequencies is due to the remaining random variables in the structure, for example, the modulus of elasticity of the components.  Figure 2.2: Bayesian updating formulation scheme  Secondly, the response r could be a particular component of a mode shape. For example, when considering the n-th component of the first mode shape,  n1 , Eq. (7) becomes,  PDi | n1    f n1 | Di  ND   f  j 1  n1  | D j  PD j   PDi   (8)  14  In a third definition for the response, r, it could be taken as a vector of measured dynamic responses. Thus, r={r1, r2,..., rn}, could be either a vector of natural frequencies ={1,2,...,n} or of the mode shapes n={1, 2,..., n}, each with n components. In this third case of using multiple response measurements, the likelihoods correspond to conditional joint PDF. Thus, in the case of measured frequencies,  Pω | Di   f 1 , 2 ,  , n | Di  d1 d2  dn  (9)  Or in the case of mode shapes,  Pφ n | Di   f 1n , 2n ,  , nn | Di  d1n d2n  dnn  (10)  In Eqs. (9) and (10) the likelihoods f are n-variate conditional PDF. As before, these must be calculated for each damage state considered in the analysis.  2.3  LIKELIHOOD FUNCTIONS AND THEIR CALCULATION  The structural response vector r is obtained from measurements, and the corresponding likelihood function f(r|Di) provides the probability that the outcome r be measured when the structure has a particular damage state Di. It is assumed in this Thesis that r represents measurements of dynamic responses. However, since structural responses to static conditions can also be influenced by the damage state, they could also be used for damage assessment. Dynamic responses could be measured during ambient (Wenzel and Pitchler 2005) or forced vibration testing. This Thesis, however, does not specifically differentiate the source of the dynamic measurements.  15  What is important to consider is that the actual response r* of the structure differs from r because of random measurement errors. It is assumed that the calculated response r equals, on average, the true value r*, but that there is an error distribution about r* as follows:  r  r * 1.0     (11)     RN  (12)  where  in which RN is a Standard Normal variable and  its standard deviation. In the case of multiple measurements, e.g. when r includes all measured frequencies, it is possible that each component of r be affected by a different error. Complete calibration of the statistics for the error  would require a thorough evaluation of the differences between actual responses and corresponding measurements. This aspect of the problem is not covered in this Thesis, since no experimental data or measurement details are included. However, the methodology proposed here to introduce uncertainty from measurement errors is general and could be directly applied should quantification of the error statistics be separately obtained. Here, for simplicity, it will be assumed that all components of r are subject to the same error. Thus, only one random variable RN and standard deviation need to be considered as per Eq. (12) to obtain the calculated response r. This assumption implies that if a system is measured many times independently, most measurements will give data around the actual response r*, and there are about 95 percent measurements that will fall into the interval of the mean value plus or minus 2 standard deviations .  16  The likelihoods, f(r|Di), must be obtained a priori by calculation using a model for the structure, under the assumptions that it has undergone a state of damage defined by Di. The structural model needs to be calibrated so that it will produce the accurate responses (within the measurement errors) for the case in which no damage is present. One of the possible methods to obtain the likelihood functions is by using data sampling as described in Sections 2.3.1 and 2.3.2 both for the case of univariate r or multivariate r.  2.3.1  UNIVARIATE LIKELIHOOD FUNCTION BY SAMPLING  The univariate likelihood functions, f(r|Di), are a direct result of the variability of individual dynamic parameters, either frequencies or mode shapes components, with respect to the basic random variables of the problem and the measurement error . For different realizations of these variables and for a given damage state, different outputs are calculated using the structural model. The obtained discrete results are ranked assigning a probability level, pi, to each sample, pi   i  0 .3 N  0 .4  (13)  where i indicates the corresponding ranking of a sample and N is the total number of samples. Many other choices exist to define the plotting position given by Eq. (13), for example i/N. Since probability plotting is only used for a visual check on the type of distribution chosen to represent the variability of r, either option is as good as any. Eq. (13) is used in this Thesis as it produces good comparisons with distributions estimated by maximum likelihood (Bury 1999).  17  Figure 2.3 represents a generic example of this procedure making use of data observed later in the case study. It shows that the points never reach one or zero to recognize the existence of results being greater than the sample maximum or smaller than the sample minimum, respectively.  Figure 2.3: Generic sample cumulative distribution function  The sample data in Figure 2.3 is then fitted with an appropriate probability distribution function to obtain the cumulative distribution F(r|Di), or CDF. The corresponding PDF, conditional on the damage Di, is then obtained from differentiation, dF r | Di   f r | Di  dr  (14)  Different types of distribution functions could be used in the fitting. In general, a simple visual inspection is sufficient to choose a distribution that will provide a good estimate of the sample values. Figure 2.4 presents four different distribution types plotted with the sample data and shows that the Normal and Lognormal distributions are the best estimates for this case. Moreover, a Lognormal distribution would be appropriate to satisfy the condition that the frequencies are all positive. On the other hand, a Normal distribution could be used for  18  simplicity when the mean value is sufficiently away from zero and of a magnitude several times the standard deviation of the data. This was the approach followed in this Thesis, which provided consistency with the jointly Normal distribution described in Section 2.3.2.  Figure 2.4: Sample and different types of CDF  Once the distribution type is adopted, the distribution parameters could be obtained by minimization of the sum of the square of the errors, i.e. least squares, or by the method of maximum likelihood. The choice of distribution type, number of samples and the calculation of distribution parameters will be further discussed in the case study presented in Chapter 4.  2.3.2  MULTIVARIATE LIKELIHOOD FUNCTION BY SAMPLING  The multivariate likelihood functions, f(r|Di), represent the variability of r with respect to the basic random variables of the problem and the measurement error variable . In this case, the measured response r is a vector of more than one dynamic parameter. For example, all measured frequencies or all components of a vibration mode could be used simultaneously. Therefore, the objective is now to find the n-variate joint PDF from the sampling results.  19  In general, to determine the most suitable joint PDF for any set of parameters is not an easy task. The assumption of the n-variate Normal distribution is commonly found in the literature and is used here (Adhikari 2007). This is more justified as the number of parameters increase, as a consequence of the central limit theorem. The Normal joint PDF is then given by,  f r, r, Σ   1 Σ 2   d   1 T exp  r  r  Σ 1 r  r     2  (15)  in which r is the calculated response vector of dimension d, with mean r and covariance matrix . The covariance matrix arises from the fact that the components of r, for example, all the frequencies, correspond all to the same structure and are therefore correlated. This Normal joint PDF approach will be used in this Thesis. The covariance matrix is estimated by sampling. In order to describe in detail this calculation, let us consider only the case in which r contains the n measured natural frequencies. The vector of mean values corresponds to ω  1,2 ,,n  where its components are calculated by,  1 N n  nk N k 1  (16)  in which N is the number of samples. Each sample corresponds to a combination of the structural random variables and a random value for the error variable RN. The covariance matrix is then obtained from, ΣD R D  (17) 20  with D being the diagonal matrix of standard deviations:  s1 0 0    D 0  0   0 0 s n   (18)  where  sn  2    1 N  n  n N  1 k 1 k    2  (19)  R is the correlation matrix defined as:  12  1  R  symm  13  23   (20)  1   where  i j     Cov i ,  j    si s j  (21)  and      Cov i ,  j   1 N   ik  jk  N i  j  N  1  k 1   (22)  in which  is the correlation coefficient between the i-th and j-th frequencies, and Cov[ , ] represents the corresponding covariance between these two variables. A similar approach is used when using all the calculated components of a vibration mode.  21  2.3.3  UNIVARIATE LIKELIHOOD FUNCTION BY FORM  Another method to obtain the univariate likelihoods is through First Order Reliability Analysis (FORM) (Hasofer and Lind 1974). In this case, the CDF is constructed by using the limit state function defined as, g  r  r0  (23)  where r0 indicates a threshold value within the range of interest and r is the calculated dynamic response dependent upon the basic variables, the measurement error and the given damage state. In order to perform such calculations, a reliability analysis program needs to be linked to the probabilistic structural model. In this Thesis, the computer program RELAN (Foschi et al. 2000) was used and the eigenvalue solution subroutine for the structure under evaluation was written in the same programming language (FORTRAN). In the end, every FORM analysis, executed for different threshold values and a fixed damage state gives directly the probability previously defined by Eq. (13), conditional on the chosen damage state: F r0   P r  r0   (24)  The results are then similarly fitted with an appropriate CDF and differentiated to obtain the required conditional PDF. The slope of the CDF could have been obtained numerically if the number of FORM calculations were sufficiently large. This approach however was not used. In this Thesis, the results obtained for the univariate case using either sampling or FORM are shown in Chapter 4 for the case study. The results were very close to each other, and the two  22  methods could be considered equivalent. The integrated methodology, as described in Chapter 3 and implemented in MATLAB (The Mathworks 2008) computer program, only included the sampling procedure to obtain the likelihood functions.  23  3 METHODOLOGY: STRUCTURAL MODEL AND DAMAGE STATES This chapter discusses how the different damage states Di are defined in terms of the structural analysis model. The structure is assumed to include M sub-structures susceptible to damage. These sub-structures could be members, connections or structural panels. Individually, they contribute a component stiffness matrix Kj (j = 1, M) to the global matrix K used for elastic, linear analysis. For each damage state Di it is assumed that the individual contribution Kj are each degraded by a damage indicator θij, representing stiffness degradation of the corresponding sub-structure from its virgin state. The indicators take values between 0 and 1, corresponding respectively to either complete damage or to the undamaged case.  3.1  ESTABLISHING THE STRUCTURAL MODEL  The underlying model is one for the structure under consideration, for example, a FE model. This uses several parameters as input and produces a structural response. In this Thesis the responses of interest are dynamic, in particular, natural frequencies and vibration mode shapes. The response vector r is initially obtained as an implicit function of random variables, r  h D i , X   (25)  where h(.) represents the structural model itself, X={X1, X2, ..., XN} is the vector of basic random variables involved, and Di indicates the i-th damage state. Here, for completeness, a brief review of the dynamic analysis model is provided. 24  This model implements a solution to the general equation of motion for the structure as derived from D'Alembert's principle. For a single degree of freedom (SDOF) elastic system with displacement, u(t), velocity, u (t ) , and acceleration, u(t ) subjected to a load p(t), the differential equation that describes the motion of the system is, M ut   C u t   K u t   p t   (26)  in which M is the applied mass, C the damping coefficient, and K the stiffness of the system. For a multiple degree of freedom system (MDOF) the equation is given in matrix form as, t   C u t   K u t   F t  Mu  (27)  where now, for n degrees of freedom, M is the mass matrix of the structure (n x n), C is the damping matrix (n x n), K is the stiffness matrix (n x n), u is the vector of degrees of freedom (displacements and/or rotations) (n x 1), and F is the vector of loads (n x 1). The simplest method to find the solution to Eq. (27) for a MDOF system is to first find the vibration modes and associated frequencies for an undamped, free vibration case. These modal parameters can then be used to de-couple the problem into n SDOF systems. The individual solutions are then combined to form the complete MDOF response. This procedure is known as modal analysis. The modal parameters, i.e. the natural frequencies ={1,2,...,n} and the mode shapes n, are obtained by solving the eigenvalue problem.  K    2 n    M φn  0   K   n2 M  0  (28)  That is, the natural frequencies n are those that make possible to find vibration modes n other than zero, and are obtain by finding the roots of the n x n determinant in Eq. (28). 25  Since the properties in the structural model are random, the calculated modal parameters will be random outputs which, in turn, permit the generation of the likelihood functions to be used in the Bayesian updating. The calculated response vector r is finally obtained as, r  h D i , X ,    (29)  where  is the variable related to measurement errors as per Eq. (12). It is important that the structural model be capable of generating the correct modal parameters when the input data is known. This implies that the structural or FE model must be properly calibrated. In principle, the calibration is done so that the model, with calibrated statistical properties for the random variables, produces a probability distribution of responses for the undamaged structure, which agrees with the measured distribution, within the measurement error. A full description of details of such calibration procedure is not within the scope of this Thesis.  3.2  DEFINING POTENTIAL DAMAGE STATES  As previously discussed, the damage states need to be defined prior to the calculation of the likelihood functions. In this case, each of the damage states Di is defined by a vector i, with components ij modifying the stiffness contribution of the j-th substructure included in the damage state. Hence, the structure’s stiffness matrix is for the state Di represented as, M  K i   ij K j j 1  i  1, N D  (30)  26  where Ki is the stiffness matrix for the i-th damage state, Kj is the stiffness matrix corresponding with the j-th substructure in that state, ij is the damage parameter indicator associated with the j-th substructure, 0  θ ij  1 , ND is the number of damage states considered and M is the total number of substructures being monitored for damage. Finally, a possible damage state for the structure under evaluation is defined as,  Di  θi   i1 , i 2 ,, iM  ,  0   ij  1 j  1,, M  (31)  As a result, the components of i implicitly indicate the spatial distribution of the damage. In essence, this representation implies that the overall stiffness matrix of the structure changes due to the presence of the damage indicators, each modifying the contribution of the corresponding individual substructures. It is assumed that, at each state of damage, the dynamic properties, which correspond to the damaged structure, are the same as those for a modified structure according to the damage indicators in Eq. (30). The individual matrices Kj remain the unmodified elastic stiffness matrices for each substructure. As described in Chapter 1, a similar approach has been used by other authors (Sohn and Law 1997), differing from this work in that here, different damage indicators  are defined a-priori, specifying the set of different damage hypotheses to be incorporated in the Bayesian analysis. The next chapter applied this methodology and the Bayesian analysis described in Chapter 2 to a specific case study.  27  4 CASE STUDY In this chapter, the Bayesian analysis from Chapter 2 and the damage quantification described in Chapter 3 are applied to the damage monitoring of a simple three-storey frame. Although the methodology is applicable to problems with many degrees of freedom, only three are considered in this study, namely, the horizontal displacements at each storey level. In addition, the set of columns in each of the three inter-stories constitute the substructures of the model and are considered susceptible to damage. As an example, the dynamic responses are found solving the eigenvalue problem described in Section 3.1 for the undamaged or virgin state and using the mean value of the basic variables. Also for the undamaged state, the different procedures to obtain the likelihood functions presented in Chapter 2 are applied, and their results discussed. Finally, this chapter introduces the integrated methodology as was implemented in MATLAB software and from which results are subsequently presented in this Thesis.  4.1  STRUCTURAL MODEL  The structure being analyzed and its model representation are shown in Figure 4.1. Three substructures, the set of columns in each of the three inter-stories, are considered susceptible to damage, i.e. M = 3. Three horizontal rigid beams are supported by elastic columns for which the axial deformations are neglected. The masses are assumed concentrated at each level and the basic random variables are the stiffness of each storey, including the two columns per storey.  28     u1  1    m1  1.8  10 5 kg    u2  1   u3  1    3 k11  107  10 3   k12  107  10   k13  0    u1   kN   k1  107  10 m  1   3  m 2  2.7  10 5    1   u2   k 2  214  10 3    m3  3.6  10 5    3 k 21  107  10 3   k 22  321 10    u3    k 23  214  10 3    1   k 3  321  10   3  k 32  214  10 3    k 31  0    k 33  535  10 3    Figure 4.1: Three storey frame  As a result, the FE model is a three-degree-of-freedom system, one for each horizontal floor displacement, with stiffness matrix K, and mass matrix, M.  k11 K   k 21  k 31  k12 k 22 k 32  k13 k 23 k 33       (32)   m1 M   0  0  0 m2 0  0 0 m3       (33)  Figure 4.1 shows the definition of the basic random variables, and the mean individual storey stiffnesses, 12(EI/L3), in the undamaged state. Also shown are the coefficients kij entering into the corresponding stiffness matrix K. This structural example has been extracted from the literature (Clough and Penzien 2003). Thus,  29  0   107  107 kN  K    107 321  214   10 3 m  0  214 535   (34)  The undamaged case, D1, is by definition, the first damage state of interest to the methodology. In essence, it represents the full stiffness, adding the contributions from the three substructures of the FE model. The mass matrix M is, 0   1.8 0  M   0 2.7 0   10 5 kg  0 0 3.6   (35)  This mass matrix is assumed to remain the same throughout all possible damage states whereas the stiffness matrix is redefined for each damage state, based on the damage indicators associated with each storey. The elasticity modulus for the columns, their area moment of inertia and height at each storey may be considered random. In this case study, however, the stiffness 12(EI/L3) for each storey is taken as one random variable per storey (k1, k2 and k3), with mean values  k1  107 103 kN m , k 2  214 103 kN m and k3  321103 kN m respectively, as shown in Figure 4.1. Coefficients of variation associated with these variables are all assumed equal to 0.10. When simulating data, the distribution of these variables was assumed to be Lognormal. They were treated as independent. If the columns were of the same material they will all share the same modulus E, introducing a level of correlation between the stiffness of the components. This refinement was not included in the analysis since it will not affect the methodology applied to the case study.  30  The Lognormal distribution parameters  and , for each stiffness variable, are obtained from the mean and coefficient of variation as, k i ~ LN  ,     (36)    ln 1   2   (37)  where  and   ln    0 .5   2  (38)  in which  represents the mean value of the stiffness variable, in turn k 1 , k 2 , k 3 , and  is the coefficient of variation assumed to be equal to 0.10. Introducing numerical values, the Lognormal distribution parameters for the random stiffness at each storey are graphically shown in Figure 4.2 and given by, k1 ~ LN 11.576 ,0.099  , k 2 ~ LN 12.269 ,0.099  , k 3 ~ LN 12.674 ,0.099   (39)  Figure 4.2: Lognormal CDF for the basic stiffness random variables 31  The structural dynamic parameters are found by solving the eigenvalue problem from Eq. (28). Thus, in this case, 1 0  1 B KN   1 3  1.5B  2  K   M  107  10  m 2 5  2 B   0 2  3  (40)  with  B  2  (41)  594.44  For this particular case, setting the determinant of Eq. (40) equal to zero yields, B 3  5 .5 B 2  7 . 5 B  2  0  (42)  For the undamaged case, using the mean values for the basic random variables, the eigenvalues or actual frequencies are obtained from the roots of Eq. (42). Then,   12  2  2  32     0.351       594.4  1.61    3.54     1  ω   2  3    14.5     31.1  rad    s   46.1   (43)  The modes shapes are found choosing the first component of the mode equal to one and replacing the calculated frequencies. That is in Eq. (28) ,  1  Bn  1   0  1 3  1 .5 B n 2  0 2 5  2 Bn        1   2n  3 n   0   0      0   (44)  Finally, the undamaged mode shapes result: 32   1.000   1.000   1.000      φ1   0.644  , φ 2    0.601  , φ 3    2.571   0.300    0.676   2.470   (45)  These mode shapes are shown in Figure 4.3. It is important to emphasize that these results are only valid for the undamaged state and for the mean values of the storey stiffnesses. For different realization of the basic variables and damage states, all the mode shapes are normalized with the first component equal to one. Consequently, the first component becomes a deterministic variable and it is not part of the Bayesian updating.    1.000     0.601  0.644    1.000    1.000   2.571   0.676  0.300    Mode 1   1  14.5  rad   s  2.470    Mode 2    2  31.1  rad   s  Mode 3    3  46.1  rad   s  Figure 4.3: Mode shapes for the three storey frame  Stiffness contributions from each storey will vary according to their assumed uncertainty, and the solution of the eigenvalue problem will reflect the changes in global stiffness as a result of the evolution of damage. This variability and the randomness due to measurement errors are represented later by the likelihood functions.  33  4.2  DAMAGE STATES  Following the characterization of a damage state Di, as describe in Section 3.2, it is necessary to introduce the corresponding vector of damage indicators i, which correspond to each of the three substructures. These indicators are chosen between 0 and 1, namely with four specific levels: 1.0, 0.7, 0.4 and 0.1, giving a total of 64 damage states (43). More damage states could be incorporated by reducing the intervals between damage indicators, but the number used here serves to illustrate the applicability of the methodology. In this case study, each storey of the structure shown in Figure 4.1 constitute a substructure susceptible to damage, or M = 3. Then Eq. (30) becomes for this example, 3  K i   ij K j j 1  i  1, ,64  (46)  3  where Ki is the global stiffness matrix for the damage state Di, and   j 1  ij  K j represents the  damaged assembly of the stiffness matrices from each substructure. In detail,   1  1 0 0 0 0  0 0 0     K i   i1 k1  1 1 0   i 2 k 2 0 1  1   i 3 k 3 0 0 0  0 0 0 0  1 1  0 0 1  (47)  Here, the parameters kj identify the basic stiffness random variables, lognormally distributed, which are sampled to build the global matrix corresponding to the damage state Di. This matrix, in turn, permits the calculation of the dynamic parameters, e.g. frequencies, for that  34  damage state. Then, the calculated frequencies to be used in the generation of the likelihood functions are found by incorporating a Standard Normal error  as discussed in Chapter 2.  D1    11  1 2  D   22   21 Di  θ i   2           D64   64 1  64 2  13   2 3   1.0 1.0 1.0  1.0 1.0 0.7             64 3  0.1 0.1 0.1  (48)  The 64 different damage states are arranged as per Eq. (48) in terms of the damage indicators and shown in Table 4.1. This arrangement of damage states should be considered when analyzing the Bayesian updating results, in order to associate the damage state number with the corresponding damage indicators. This association facilitates the assessment of the level of damage in the structure. Table 4.1: Definition for the 64 damage states in terms of the damage indicators  Di  i1  i2  i3  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.7 0.7  1 1 1 1 0.7 0.7 0.7 0.7 0.4 0.4 0.4 0.4 0.1 0.1 0.1 0.1 1 1  1 0.7 0.4 0.1 1 0.7 0.4 0.1 1 0.7 0.4 0.1 1 0.7 0.4 0.1 1 0.7 35  Di  i1  i2  i3  19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59  0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1  1 1 0.7 0.7 0.7 0.7 0.4 0.4 0.4 0.4 0.1 0.1 0.1 0.1 1 1 1 1 0.7 0.7 0.7 0.7 0.4 0.4 0.4 0.4 0.1 0.1 0.1 0.1 1 1 1 1 0.7 0.7 0.7 0.7 0.4 0.4 0.4  0.4 0.1 1 0.7 0.4 0.1 1 0.7 0.4 0.1 1 0.7 0.4 0.1 1 0.7 0.4 0.1 1 0.7 0.4 0.1 1 0.7 0.4 0.1 1 0.7 0.4 0.1 1 0.7 0.4 0.1 1 0.7 0.4 0.1 1 0.7 0.4 36  Di  i1  i2  i3  60 61 62 63 64  0.1 0.1 0.1 0.1 0.1  0.4 0.1 0.1 0.1 0.1  0.1 1 0.7 0.4 0.1  Samples of the calculated frequencies are shown in Figures 4.4, 4.5 and 4.6, for each of the 64 damage states. Observing a particular damage state, the variability in these figures arises from the dependence described by Eq. (29). These figures show that natural frequencies may not vary substantially from one damage state to another. This is more evident in Figures 4.5 and 4.6 for the second and third frequency. This characteristic implies that similar dynamic parameters can be expected for different damage states, which makes it rather difficult to select a damage state, with high confidence, by a simple inspection of the measured data.  1 rad s  16.0 14.0 12.0 10.0 8.0 6.0 4.0 1  2  3  4  5  6  7  8  9  10   11   63 12  64  Di  Figure 4.4: First frequency variation with the damage state  37  2 rad s  36.0 30.0 24.0 18.0 12.0 6.0 1  2  3  4  5  6  7  8  9  10   11  63 12  64  Di  63 12  64  Di  Figure 4.5: Second frequency variation with the damage state   3 rad s  52.0 46.0 40.0 34.0 28.0 22.0 16.0 10.0 1  2  3  4  5  6  7  8  9  10   11   Figure 4.6: Third frequency variation with the damage state  The samples shown in these figures are used in the generation of the likelihood functions. This is discussed in the following section for the case of univariate and multivariate PDFs. 38  4.3  LIKELIHOOD FUNCTIONS  The univariate likelihood functions were determined by sampling or FORM, as described in Chapter 2, for the calculated natural frequencies and for the components of the first vibration mode. The results presented in this section aim to exemplify and demonstrate the applicability of both methods. Ultimately, the sampling procedure was employed as part of the integrated Bayesian updating implemented in MATLAB software.  4.3.1  UNIVARIATE LIKELIHOOD FUNCTIONS FOR THE CALCULATED RESPONSES  Different realizations of the random stiffnesses k1, k2 and k3 permit, for each damage state, the corresponding calculation of the global stiffness matrix according to Eq. (47). This, in turn, permits solving the new eigenvalue problem to obtain the set of actual dynamic responses corresponding to the damage state. For this responses, a realization of the measurement error variable  permits to obtain a sample of the calculated frequencies and components for the first vibration mode. Following the procedure detailed in Chapter 2, the sampling results corresponding to the calculated frequencies were fitted with different probability distributions. The sample CDFs for the calculated frequencies, obtained as per Eq. (13), are shown in Figure 4.7 for the undamaged case and using 50 samples. Several standard distributions were applied and the goodness of the fit evaluated. As an example, Figure 4.8 to Figure 4.10 show Normal and Lognormal distributions fitted for the first, second and third natural frequency of the undamaged structure, respectively.  39  The fits obtained with both the Normal and Lognormal distribution are satisfactory, indicating that either distribution is suitable to represent the variability in the natural frequencies.  pi 1.0 0.8 0.6  1  3  2  0.4 0.2 0.0 12.0  16.0  20.0  24.0  28.0  32.0  36.0  40.0  44.0  48.0  rad 52.0 s   Figure 4.7: Sample CDFs for the calculated frequencies  F 1 | D1   1.0 0.8 0.6  Sample 0.4  Normal Lognormal  0.2 0.0 12.0  12.5  13.0  13.5  14.0  14.5  15.0  15.5  16.0  16.5  rad s  117.0  Figure 4.8: Fitted distributions for 1  40  F  2 | D1  1.0 0.8 0.6  Sample 0.4  Normal Lognormal  0.2 0.0 26.0  27.0  28.0  29.0  30.0  31.0  32.0  33.0  34.0  35.0  rad s   2 36.0  Figure 4.9: Fitted distributions for 2  F  3 | D1  1.0 0.8 0.6  Sample 0.4  Normal Lognormal  0.2 0.0 38.0  40.0  42.0  44.0  46.0  48.0  50.0  52.0  rad s   354.0  Figure 4.10: Fitted distributions for 3  The parameters of the fitted distributions, shown in Table 4.2 to Table 4.4, were found using RELAN, minimizing by least squares the difference between the simulated values and the theoretical ones. Another analogous method, namely maximum likelihood, is used in the integrated Bayesian updating implemented in MATLAB software. The emphasis in these 41  methods is to optimize the fit over the entire set of available data rather than just using the sample statistics. All the fitted parameters depend on the total number of samples used. Both distributions are included in the Bayesian updating. However, for consistency with the assumption of a Normal joint PDF, the results presented later in this Thesis correspond with the fitting of a Normal distribution. For the purpose of running the Bayesian updating for predictions, a higher number of samples should be used to minimize the fitting error and increase the accuracy of the derived PDFs. Table 4.2: Parameters of fitted distributions for 1 with RELAN  1|D1  Mean  Standard deviation  FE (error)  Normal  14.3071  0.7222  0.0036  Lognormal  14.3091  0.7307  0.0026  distribution  Table 4.3: Parameters of fitted distributions for 2 with RELAN  2|D1  Mean  Standard deviation  FE (error)  Normal  30.5515  1.5778  0.0026  Lognormal  30.5555  1.5919  0.0020  distribution  Table 4.4: Parameters of fitted distributions for 3 with RELAN  3|D1  Mean  Standard deviation  FE (error)  Normal  45.5142  2.3603  0.0040  Lognormal  45.5212  2.3883  0.0031  distribution  An alternative approach, as discussed in Chapter 2, is to implement FORM directly rather than using sampling. In this Thesis the results from FORM were obtained using the software 42  RELAN. The FORTRAN code for the case of the natural frequencies is included in Appendix A . Here a comparison is shown between the results using sampling or FORM to demonstrate the applicability of both methods. Figure 4.11 shows the results obtained from FORM for all three calculated frequencies. The analysis was implemented for 50 values of the threshold r0 in Eq. (23), within the range of each of the natural frequencies. F  n | D1  1.0 0.8 0.6  1  0.4  3  2  0.2 0.0 12.0  16.0  20.0  24.0  28.0  32.0  36.0  40.0  44.0  48.0  rad 52.0 s   Figure 4.11: CDFs obtained using FORM for the calculated natural frequencies  In all cases the results obtained with FORM are very similar to the fitted distributions using sampling data. The comparison is presented in Figure 4.12 for the third natural frequency and the sampling results correspond with the presented ones in the previous section. The third natural frequency is shown because it is the one that showed the largest discrepancy.  43  F  3 | D1  1.0 0.8  FORM 0.6  Fitted NORMAL to  sample distribution Fitted LOGNORMAL to  sample distribution  0.4 0.2 0.0 38.0  40.0  42.0  44.0  46.0  48.0  50.0  52.0  rad s   354.0  Figure 4.12: Comparison for the third natural frequency  A similar analysis was done for the components of the vibration modes. Figure 4.13 shows the sample CDF for each calculated component of the first mode of vibration obtained, again, for a total of 50 samples.  pi 1.0 0.8 0.6   31  0.4   21  11  0.2 0.0 0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  Figure 4.13: Sample CDFs for 1 components  44  The mode shape components are normalized for a unitary roof floor displacement. Thus, the first component of every mode shape vector is chosen to be equal to 1.0 and therefore it becomes a deterministic variable. Consequently, only two of the components of each mode shapes are random variables and were also fitted with standard distributions. As an example, Figure 4.14 and Figure 4.15 show the fitting results using either a Normal or a Lognormal distribution, for the calculated components of the first mode shape. F  21 | D1  1.0 0.8 0.6  Sample 0.4  Normal Lognormal  0.2 0.0 0.50  0.55  0.60  0.65  0.70  0.75   21 0.80  Figure 4.14: Fitted distributions for 21  The parameters of the fitted distributions, shown in Table 4.5 and Table 4.6, were found using the RELAN software. Either the Normal or the Lognormal distributions are suitable representations for their components. However, since for higher modes some of the components may be of different sign, the representation should use the corresponding absolute values. The Normal distribution was used here to implement the integrated Bayesian updating, without loss of generality and to maintain consistency with the assumption of joint Normal distribution for the multi-variate case as discussed in Section 2.3.2.  45  F  31 | D 1  1.0 0.8 0.6  Sample 0.4  Normal Lognormal  0.2 0.0 0.20  0.25  0.30   31 0.40  0.35  Figure 4.15: Fitted distributions for 31  Table 4.5: Parameters of fitted distributions for 21 with RELAN  21|D1  Mean  Standard deviation  FE (error)  Normal  0.6457  0.0476  0.0077  Lognormal  0.6459  0.0485  0.0053  distribution  Table 4.6: Parameters of fitted distributions for 31 with RELAN  31|D1  Mean  Standard deviation  FE (error)  Normal  0.3025  0.0326  0.0207  Lognormal  0.3026  0.0331  0.0201  distribution  As discussed for the case of natural frequencies, FORM can similarly be applied. The distributions obtained with 50 thresholds are shown in Figure 4.16, which can be fitted with either Normal or Lognormal distributions as shown in Figure 4.17. It can be observed that the agreement is again very good, even for 50 samples. 46  F  n1 | D1  1.0 0.8 0.6   31  0.4   21  0.2 0.0 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  Figure 4.16: CDFs obtained using FORM for the calculated first mode shape components  F  21 | D1  1.0 0.8 0.6  FORM  0.4  Fitted Normal to  sample distribution  0.2  Fitted Lognormal to  sample distribution  0.0 0.50  0.55  0.60  0.65  0.70  0.75   21 0.80  Figure 4.17: Comparison for 21  The results presented in this section aim to illustrate the applicability of both sampling and FORM methods to obtain the likelihood functions. In the end, the sampling procedure was implemented in the integrated Bayesian updating algorithm, for the calculated frequencies, all three mode shapes, and each of the 64 damage state defined. 47  4.3.2  MULTIVARIATE LIKELIHOOD FUNCTIONS FOR THE CALCULATED RESPONSES  When analyzing the structure for a given realization of the random stiffnesses and a specific realization of the measurement error variable , the output includes all natural frequencies and corresponding vibration mode shapes. As described in Chapter 2, the use of more than one frequency or more than one component of the vibration modes requires the calculation of the corresponding covariance matrix in the assumed joint Normal PDF. For the vector of frequencies , from Eq. (15) in Chapter 2,  f ω, ω, Σ   1  Σ 2   d   1 T exp  ω  ω  Σ 1 ω  ω    2   (49)  where ω is the vector of mean values, and  is the covariance matrix. Those components define the parameters of the sought distribution. The results that follow correspond to a total of 50 samples in the undamaged case, and Table 4.7 presents typical samples of the frequency vector components. Table 4.7: Realizations of calculated   |D1 Sample # 1 2 3    50  1    3  14.915 13.939 15.437  30.347 30.523 34.067  49.170 44.312 48.099    14.124    30.899    44.096  The undamaged state is assumed only for the purpose of developing the data shown, and as a typical calculation, which must be repeated for all damage states. A larger number of samples  48  is used when developing the results implemented in the Bayesian updating to minimize the fitting error and increase the accuracy of the derived likelihood function. The vector of the mean values for the calculated frequencies in the undamaged state resulted in ω  14.309, 30.555, 45.523 and the correlation matrix, following the procedure outlined in Chapter 2, was calculated as,  1.000 0.869 0.965 R  0.869 1.000 0.768 0.965 0.768 1.000  (50)  The correlation structure results from a significant linear dependency between the natural frequencies. Figure 4.18 to Figure 4.20 show scattered plots and the corresponding correlation coefficients between every pair of calculated frequencies. The diagonal matrix of standard deviations was found to be,  0 0  0.725  1.568 0  D 0  0 0 2.372  (51)  Finally, the covariance matrix required by the joint PDF of Eq. (49) results,  0.525 0.989 1.660 Σ  0.989 2.460 2.858 1.660 2.858 5.625  (52)  49   2 rad s  36.0 34.0 32.0 30.0 28.0 26.0 12.0  13.0  14.0  15.0  16.0  1 rad 17.0 s   Figure 4.18: Correlation between first and second natural frequencies   3 rad s  53.0 50.0 47.0 44.0 41.0 38.0 12.0  13.0  14.0  15.0  16.0  1 rad 17.0 s   Figure 4.19: Correlation between first and third natural frequencies   3 rad s  53.0 50.0 47.0 44.0 41.0 38.0 26.0  28.0  30.0  32.0  34.0   2 rad 36.0 s   Figure 4.20: Correlation between second and third natural frequencies 50  A similar analysis was performed for the components of the vibration mode shapes in the undamaged state. Once more, the results correspond to a total of 50 samples, with some typical values included in Table 4.8 to Table 4.10. Table 4.8: Realizations of calculated 1  1|D1 Sample # 1 2 3  21  31  0.652 0.758 0.641  0.273 0.362 0.332        50  0.568  0.274  Table 4.9: Realizations of calculated 2  2|D1 Sample #  22  32  1 2 3  -0.652 -0.422 -0.573  -0.652 -0.832 -0.663        50  -0.412  -0.707  Table 4.10: Realizations of calculated 3  3|D1 Sample # 1 2 3  23  33  -2.240 -1.752 -2.870  2.350 1.308 2.620        50  -2.041  1.586  The vectors of mean values, for a total of 50 samples and for the undamaged state were found to be: φ1  0.646, 0.303 , φ 2   0.614,  0.684 , and φ3   2.648, 2.573 . 51  The corresponding correlation matrices were obtained as,  1 .000 0.612   1 .000  0 .417   1.000  0.957  R  2131   , R  2232   , R 2333      0.612 1 .000    0 .417 1 .000    0.957 1.000   (53)  which indicate the linear dependency between every pair of random mode shape components as represented in Figure 4.21 to Figure 4.23. The diagonal matrices of standard deviations and the resulting covariance matrices are found to be, 0  0  0  0.048 0.080 0.508 D2131   , D2232   , D2333     0.033 0.049 0.632  0  0  0  0 .0023 0 .0010  ,  0 .0064 Σ  2131   Σ  2232    0 . 0010 0 . 0011     0 .0017   0 .0017  0 .258 , Σ  2333   0 .0024    0 .307   0 .307  0 .399   (54)  (55)  For this bi-variate case, the resulting joint PDF can be plotted and are shown in Figure 4.24 to Figure 2.1 for the three mode shapes in the undamaged case.  52   31 0.40 0.35 0.30 0.25 0.20 0.50  0.55  0.60  0.65  0.70  0.75   21 0.80  Figure 4.21: Correlation between first mode shape components   32 ‐0.50 ‐0.60 ‐0.70 ‐0.80 ‐0.90 ‐0.90  ‐0.80  ‐0.70  ‐0.60  ‐0.50   22 ‐0.40  Figure 4.22: Correlation between second mode shape components   33 5.00 4.00 3.00 2.00 1.00 ‐4.00  ‐3.50  ‐3.00  ‐2.50  ‐2.00   23 ‐1.50  Figure 4.23: Correlation between third mode shape components 53  100  50  0 0.4   31  0.3  0.7 0.2  0.6   21  0.8  0.5  Figure 4.24: Bi-variate Normal distribution for the first mode shape  40  20  0 ‐0.5 ‐0.5  ‐0.7  ‐0.7   32  ‐0.9  ‐0.9   22  Figure 4.25: Bi-variate Normal distribution for the second mode shape  1.5 1 0.5 0 6.0 4.0   33  2.0 0  ‐5.0  ‐4.0  ‐3.0  ‐2.0  ‐1.0  0   23  Figure 4.26: Bi-variate Normal distribution for the third mode shape 54  4.4  BAYESIAN UPDATING  The Bayesian updating formulation was implemented as described in Chapter 2, Eq. (7) and Eq. (8). This implementation makes use of the likelihood functions and the prior probabilities assigned to each of the damage states. The procedure was programmed using the software MATLAB and it is referred to in this Thesis as the integrated Bayesian updating. The program code and a brief description for the case of measured natural frequencies are included in Appendix B The initial prior probabilities represent the initial degree of belief in each of the hypotheses, i.e. damage states. If all states are assumed initially equiprobable, then the initial values for the prior probabilities are, P Di    1 .0 ND  (56)  where ND is the total number of damage states considered or ND = 64. Thus, initially, P(Di) = 0.01562 for all states, as shown in Figure 4.27.  Figure 4.27: Initial prior probabilities  55  Measured data are needed in order to be able to run the Bayesian updating algorithm and estimate the posterior probability of each damage state. No actual measurements are used in this Thesis. Rather, to test the algorithm, a damage state scenario is assumed and measured responses are randomly simulated consistent with that chosen damage state as discussed in Section 2.3. If the algorithm for updating works well, repeated draws of simulated measurements should allow the repeated probability updating of all damage states, with the result that the larger probability would be eventually assigned to the chosen damage scenario. The damage scenario chosen first in this case study was the damage state number seven, D7, with the corresponding vector of damage indicators, 7 = {1.0, 0.7, 0.4}. This scenario implies reductions of 60% and 30% in the stiffness contribution of the first and second storey, respectively while keeping the roof level undamaged. To generate random data sets of measured responses corresponding to this damage scenario, first, random values were chosen for the stiffness variables and then, for each frequency or mode shape component, a random value was chosen for the corresponding error term. Essentially, the measurements are for one unique structure defined by the realization of the stiffness variables and the variation from measurement to measurement is due to the corresponding measurement error as per Eq. (11). The error term parameter, , adopted in the simulation of measured data sets was 5% both for the natural frequencies and the mode shapes. The results for a random generation of 5 measured data sets corresponding with k1 = 119.35 kN/m, k2 = 243.36 kN/m, and k3= 294.41 kN/m are shown in Table 4.11 for the frequency vector and in Table 4.12 to Table 4.14 for  the mode shapes. The normalized first components of the mode shapes are all equal to one and therefore are not part of the integrated Bayesian updating. 56  Table 4.11: Measured data sets for  and D7 as damage scenario  data set |D7  1 2 3  1  2  3  4  5  9.926 25.615 38.902  11.321 29.214 44.368  10.154 26.203 39.795  11.096 28.634 43.487  10.291 26.555 40.330  Table 4.12: Measured data sets for 1 and D7 as damage scenario  data set |D7  21 31  1  2  3  4  5  0.802 0.550  0.825 0.566  0.885 0.607  0.891 0.611  0.803 0.551  Table 4.13: Measured data sets for 2 and D7 as damage scenario  data set |D7  22 32  1  2  3  4  5  -0.106 -0.755  -0.102 -0.726  -0.113 -0.805  -0.108 -0.772  -0.103 -0.736  Table 4.14: Measured data sets for 3 and D7 as damage scenario  data set |D7  23 33  1  2  3  4  5  -1.562 0.828  -1.565 0.830  -1.604 0.851  -1.563 0.829  -1.677 0.889  The updated or posterior probabilities of occurrence for each of the damage states, after a number of measure evaluations, are now discussed. As previously mentioned, a sufficiently large number of samples should be used to minimize the error when fitting a standard distribution. Since the structural model presented for this case study in Section 4.1 is computational inexpensive, the number of samples used to obtain the likelihood functions was chosen for all cases equal to 1000. 57  The first results presented in the following correspond to the integrated Bayesian updating using only the natural frequencies as individual inputs. The results for each of the three frequencies are shown in Figure 4.28 after NE = 1, 5 or 10 evaluations. In general, observing the results from the last evaluation, the updating is able to distinguish a few damage states as possible damage scenarios. However, the use of the individual frequencies is not able to discriminate the damage scenario D7, with a substantial probability and therefore, more evaluations are required to select the damage scenario as the most likely one. .  58  1.0 0.8  P D i |  1  NE   0.6  1  5  10  0.4 0.2 0.0 1 1.0 0.8  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  P D i |  2  NE   0.6  1  5  10  0.4 0.2 0.0 1 1.0  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  P D i |  3   0.8  NE   0.6  1  5  10  0.4 0.2 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  Figure 4.28: Bayesian updating results, posterior probabilities of each damage state using individual natural frequencies 59  Similar to the case of the individual natural frequencies, the integrated methodology was run using a single mode shape component as input. Figure 4.29 to Figure 4.31 show the posterior or updated probabilities using, the second or the third components of the first, second and third mode shapes respectively. The probabilities are shown for after 1, 5, 10, or 20 evaluations. When using the second component of the first mode shape, the results are similar to the case of employing individual frequencies as measurements. The updating is not able to identify, with a substantial probability, a specific damage state. In contrast, when using the third mode component of the first mode shape, the damage state distinguished by the updating was D3. However, this was not the damage scenario proposed. Essentially, this damage state does not account for damage in the second floor. Similar conclusions can be drawn for the case of using the individual components of the second mode shape, as shown in Figure 4.30. A more detailed analysis of the causes that may influence these results is discussed in Chapter 5. This is due to the similarity between the likelihood functions obtained for several damage states. Specifically, the probability of observing the measured component in several damage states is very similar. The most favorable results were found when using the individual components of the third mode shape as measurements, and are shown in Figure 4.31. The Bayesian updating assigned a substantial probability to the damage scenario proposed in both cases after 20 evaluations.  60  P Di |  21  1.0 0.8 0.6  NE   0.4  1  5  10  20  0.2 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  P Di |  31  1.0 0.8 0.6  NE   0.4  1  5  10  20  0.2 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  Figure 4.29: Bayesian updating, posterior probabilities using individual first mode components  61  P Di |  22  1.0 0.8 0.6  NE   0.4  1  5  10  20  0.2 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  P Di |  32  1.0 0.8 0.6  NE   0.4  1  5  10  20  0.2 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  Figure 4.30: Bayesian updating, posterior probabilities using individual second mode components  62  P Di |  23  1.0 0.8 0.6  NE   0.4  1  5  10  20  0.2 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  P Di |  33  1.0 0.8 0.6  NE   0.4  1  5  10  20  0.2 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  Figure 4.31: Bayesian updating, posterior probabilities using individual third mode components  63  Using the more complete information available from the measurements at the time of an evaluation, the integrated Bayesian updating was run using the joint likelihood function for the vector of natural frequencies, . The damage scenario to be detected remains D7 and again 1000 samples were used when calculating the parameters in the 3-variate jointly Normal likelihood function. The measured data are shown in Table 4.11. The results were obtained after one or two evaluations and are shown in Figure 4.32. This figure also shows the initial probabilities, which correspond to the uniform distribution and are indicated as evaluation zero. It can be observed that using more information, in this case all three frequencies, permits the integrated methodology to efficiently recognize the damage scenario D7 after few evaluations.  Utilizing the complete information in the case of the mode shapes, means to run the integrated Bayesian updating using the joint likelihood function for each mode shape vector, n. The results for the same damage scenario D7 and using now the bi-variate jointly Normal PDF for each of the three mode shapes are presented in Figure 4.33 to Figure 4.35. Similarly to the results previously mentioned for the vector of frequencies, the results are very satisfactory, namely the methodology is able to effectively distinguish the most probable damage state as the proposed damage scenario. Furthermore, only two evaluations are needed to obtain probabilities of around 0.9 when using the second or third mode as measurements.  64  1.0  P D i | ω   0.8  NE   0  1  2  0.6 0.4 0.2 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  Figure 4.32: Bayesian updating, posterior probabilities of each damage state using the vector of natural frequencies  1.0  P D i | φ 1   0.8  NE   0  1  2  0.6 0.4 0.2 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  Figure 4.33: Bayesian updating, posterior probabilities of each damage state using the first mode shape vector 65  1.0  P D i | φ 2   0.8  NE   0  1  2  0.6 0.4 0.2 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  Figure 4.34: Bayesian updating, posterior probabilities of each damage state using the second mode shape vector  1.0  P D i | φ 3   0.8  NE   0  1  2  0.6 0.4 0.2 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  Figure 4.35: Bayesian updating, posterior probabilities of each damage state using the third mode shape vector 66  5 PARAMETRIC STUDIES The methodology presented in this Thesis is now further discussed in consideration to the following questions related to the method robustness:   What is the influence of the input measurement type? Bayes' updating was formulated in Chapter 2 allowing the use of the entire data set of information. Thus, the methodology accepts as input either a single dynamic parameter as measured data or a set of dynamic characteristics. The objective is now to determine which type of input leads to a better prediction for the same number of observations.    Are the results from Chapter 4 dependent on the choice of damage scenario?    What is the number of evaluations required to achieve a certain confidence in the discrimination of the most likely damage state? After the initial evaluation, the methodology revises the results previously obtained by means of the updating procedure. The number of evaluations, needed to have prediction results that can be used with confidence in the assessment of damage is an issue central to the effectiveness of the method.    What is the influence of different initial prior probabilities? In general applications, Bayes' updating rule is rather independent of the chosen initial prior probabilities and results for different initial conditions converge to very similar solutions. However, since the SHM application is related to the collection of data (extracted from measurements subject to error), optimizing the updating procedure starting conditions acquires significance.  67  These questions will now be considered. Chapter 4 has shown that results are obtained very efficiently when the measured input data include a combination of dynamic characteristics. On the other hand, when using individual frequencies as input the results are not as conclusive for some of the damage scenarios presented to the integrated methodology. More evaluations and therefore measured data sets are needed to make a distinction amongst the damage states. The same conclusions are drawn analyzing the results when using the individual mode shapes components. Thus, using the vector of frequencies is better than using just one measured frequency, and using one entire mode shape is better than using only one of its components. Are these results corresponding to an initial assumption of a D7 damage scenario valid for other damage states? In order to investigate this question, the updating results for a set of 6 different assumed damage scenarios are analyzed in the following. These damage scenarios were: D1  θ 1  1.0, 1.0, 1.0: the undamaged case. D 22  θ 22  0.7, 0.7, 0.7: 30 percent of reduction in all three storey stiffnesses. D 23  θ 23  0.7, 0.7, 0.4 : 30 percent of stiffness reduction in the second and third floors  and 60 percent loss in the stiffness contribution from the first storey. D 28  θ 28  0.7, 0.4, 0.1: a damage scenario with progressively damaged floors. D 38  θ 38  0.4, 0.7, 0.7 : a scenario with more damaged in the third floor.  D 43  θ 43  0.4, 0.4, 0.4: 60 percent of reduction in all three storey stiffnesses.  68  The corresponding posterior probabilities to the above damage scenarios are represented in Figure 5.1 and Figure 5.2. using different color coding. All the results correspond to the Bayesian updating after one evaluation, in order to be able to compare the relative efficiency of the method when the damage scenario is changed. It can be observed in Figure 5.1 that when using the complete vector of frequencies, namely accounting the three measured natural frequencies simultaneously, the methodology not only distinguished the damage scenario but also provided conclusive results in just one evaluation. In all cases, the updated probabilities for the most likely damage state, coinciding with the input damage scenario, were above 50 percent. The results of employing the mode shape vectors, is presented in Figure 5.2 . This figure uses the third mode shape as an example. Since the first component of each mode shape is not relevant because of the normalization, the information accounted in this type of evaluation includes only two inputs and is less than the three frequencies in the complete vector of frequencies. Therefore, the results are not as accurate as in the previous case and the damage scenarios with similar levels of damage in all the floors are the most difficult to detect using this information. In general, more than 10 evaluations are needed to obtain distinguishing results for the damage scenarios assumed.  69  1.0  P D i | ω   Damage scenario   1  22  23  28  38  43  0.8 0.6 0.4 0.2  NE  1  0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  Figure 5.1: Bayesian updating, posterior probabilities for different damage scenarios after the first evaluation, using  as input 1.0  P D i | φ 3   Damage scenario   1  22  23  28  38  43  0.8 0.6  NE  1 0.4 0.2 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  Figure 5.2: Bayesian updating, posterior probabilities for different damage scenarios at the first evaluation, using 3 as input 70  The results shown here illustrate the performance of the integrated methodology in terms of the number of inputs that is required in order to achieve a conclusive evaluation. In general, the lesser the number of inputs, and therefore the smaller the measured data sets, more evaluations are needed when compared to using as input either the vector of frequencies or any of the complete mode shapes. The effect of the latter, more comprehensive data, is shown in Figure 4.32 to Figure 4.35 where with just two evaluations the methodology identifies the correct damage scenario. On the other hand, Figure 4.28 to Figure 4.31 show that more evaluations are needed if less information is used, namely considering only one frequency or a mode shape component. In the cases shown in the figures, 10 or even 20 evaluations are needed to decisively detect the damage scenario. The updated probabilities of occurrence of each damage state, after a certain number of evaluations, is influenced by the magnitude of the error in the measurements. Here is now consider this effect, using the damage scenarios D7 = 7= {1.0, 0.7, 0.4} and D22 = 22= {0.7, 0.7, 0.7} for a structure defined by the realization of stiffness variables k1 = 119.35 kN/m, k2 = 243.36 kN/m, and k3= 294.41 kN/m. Figure 5.3 shows the results obtained for the vector of natural frequencies with different error characteristics defined by  = 0.02, 0.05 or 0.10, after the second evaluation measurement. As expected, the predictions improved as the standard deviation  decreased, resulting in a higher updated probability at the same evaluation step.  71  1.0  P D i | ω   0.8     0.02  0.05  0.10  0.6 0.4  Damage scenario  7 0.2  NE 1 0.0 1 1.0  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  P D i | ω   0.8     0.02  0.05  0.10  0.6 0.4  Damage scenario  22 0.2  NE 1 0.0 1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63  Di  Figure 5.3: Bayesian updating, posterior probabilities for different errors at the first evaluation, using  as input  72  6 CONCLUSIONS AND FURTHER WORK This Thesis has studied the use of a Bayesian approach to estimate the probability of occurrence of different damage states in a structure. The approach utilizes the measurement of dynamic characteristics, natural frequencies or mode shapes, and implements the hypothesis that these characteristics will degrade as the damage appears and increases over time. The approach takes initial, or prior, probabilities for each damage state and updates them after a set of dynamic characteristics have been measured or observed. The Bayesian updating then proceeds forward, using the updated probabilities as the new priors, as new measurements become available. This formulation requires the development of likelihood functions, giving the probability of observing a certain dynamic input conditional on a particular damage state. The Thesis has considered these functions in detail, including strategies for their calculation whether the input measurements consist of a single dynamic input or a set of such inputs. Damage has been defined as degradation in stiffness. This could occur in individual substructures or components, and the approach is based on the introduction of damage parameters, or indicators, which essentially modify the stiffness matrix contribution from each component to the global stiffness matrix. Different damage states can be defined as different vectors of the damage indicators, which could then represent intensity and spatial distribution of damage. A case study, using a simple 3-storey frame structure has been used to evaluate the performance of the Bayesian approach and to study different issues regarding its efficiency. 73  These issues relate to the following questions: 1) is the proposed approach able to identify a damage state that has been purposely introduced a-priori?; 2) does the performance of the proposed approach depend on which type of damage state was purposely introduced?; 3) is it better to use as many dynamic measurements as possible or is it enough to use just a single input?; 4) is it better to rely on frequency inputs or on measured mode shapes?; 5) how many measurements are required to achieve a certain confidence in identifying a given damage state?; 6) since measurements are susceptible to errors, how do the magnitude of these errors affect the probabilities assigned to each damage state? Answers to these questions have been discussed in detail, pointing to a generally very efficient approach. The probability updating methodology has been implemented in MATLAB software, which has been included in Appendix B for the case of natural frequencies. The use of only one structural dynamic response parameter is generally not sufficient to lead the updating to conclusive results as to the probabilities of the different damage states. Such is the case when using only one of the individual frequencies or an individual mode shape component. On the other hand, using as input a set of several dynamic parameters for the updating, leads to better and conclusive results with a fewer number of required measurements. The usefulness of the proposed updating methodology resides not so much in assigning a probability level to a given damage state but, through a ranking of those probabilities, guiding an engineer as to when and where to trigger a visual or direct inspection, with a corresponding decrease in structural maintenance costs.  74  The work presented here used discrete damage state indicators at each of the damage locations considered. In the future, it would be interesting to formulate the updating methodology by also considering probability distributions for these indicators, probabilities which would have to be bounded by 0.0 and 1.0. Future work could also include the incorporation of the causes for the stiffness degradation: is it caused by corrosion, by crack extension? Although the simple structure used for the case study satisfies the purpose of evaluating the performance of the proposed methodology, further work could also include the implementation of the updating approach using actual data from structural monitoring. This type of data is beginning to be collected from ambient vibration testing of bridges in British Columbia.  75  References Adhikari, S. (2007). "Joint statistics of natural frequencies of stochastic dynamic systems."  Comput.Mech., 40(4), 739-752. Beck, J. L., and Katafygiotis, L. S. (1999). "Updating models and their uncertainties. I: Bayesian statistical framework." J.Eng.Mech., 124(4), 455. Beck, J. L., and Yuen, K. (2004). "Model selection using response measurements: Bayesian probabilistic approach." J.Eng.Mech., 130(2), 192-203. Bolstad, W. M. (2007). Introduction to Bayesian statistics. Wiley-Interscience, . Box, G. E. P., and Tiao, G. C. (1973). Bayesian inference in statistical analysis. Addison Wesley, . Bury, K. V. (1999). Statistical distributions in engineering. Cambridge University Press, Cambridge, UK. Ching, J., and Beck, J. L. (2004). "Bayesian analysis of the Phase II IASC-ASCE Structural Health Monitoring experimetnal benchmark data." J.Eng.Mech., 130(10), 1233-1244. Clough, R. W., and Penzien, J. (2003). Dynamics of structures. Doebling, S. W., Farrar, C. R., Prime, M. B., and Shevitz, D. W. (1996). "Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review." Rep. No. LA-13070-MS, Los Alamos National Laboratory, NM (United States), . Farrar, C. R., and Worden, K. (2007). "An introduction to structural health monitoring."  Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 365(1851), 303. Figueiredo, E., Park, G., Figueiras, J., Farrar, C., and Worden, K. (2009). Structural Health  Monitoring Algorithm Comparisons using Standard Data Sets, . 76  Foschi, R. O., Li, H., Zhang, J., and Yao, F. (2000). "RELAN: General software for reliability analysis." Department of Civil Engineering. University of British Columbia., . Friswell, M. I., Penny, J. E. T., and Wilson, D. A. L. (1994). "Using vibration data and statistical measures to locate damage in structures." Modal Analysis: The International  Journal of Analytical and Experimental Modal Analysis, 9(4), 239-254. Hasofer, A. M., and Lind, N. C. (1974). "Exact and Invariant Second-Moment Code Format."  J.Eng.Mech., 100(1), 111-121. Katafygiotis, L. S., and Beck, J. L. (1998). "Updating models and their uncertainties. II: Model identifiability." J.Eng.Mech., 124(4), 463-467. Liu, P. (1995). "Identification and damage detection of trusses using modal data." Journal of  Structural Engineering New York, N.Y., 121(4), 599-607. Papadopoulos, L., and Garcia, E. (1998). "Structural damage identification: A probabilistic approach." AIAA J., 36(11), 2137-2145. Penny, J. E. T., Wilson, D. A. L., and Friswell, M. I. (1993). "Damage location in structures using vibration data." Proceedings of the 11th International Modal Analysis Conference,  Febrary 1, 1993 - Febrary 4, Publ by Society of Photo-Optical Instrumentation Engineers, Kissimmee, FL, USA, 861-867. Pothisiri, T., and Hjelmstad, K. D. (2003). "Structural damage detection and assessment from modal response." J.Eng.Mech., 129(2), 135-145. Sohn, H., Farrar, C. R., Hemez, F. M., and Czarnecki, J. J. (2004). "A review of structural health monitoring literature: 1996-2001." Los Alamos National Laboratory, . Sohn, H., and Law, K. H. (1997). "A Bayesian probabilistic approach for structure damage detection." Earthquake Engineering and Structural Dynamics, 26(12), 1259-1281.  77  Sohn, H., and Law, K. H. (2000). "Bayesian probabilistic damage detection of a reinforcedconcrete bridge column." Earthquake Engineering and Structural Dynamics, 29(8), 1131115 2. Sohn, H., and Law, K. H. (2000). "Application of load-dependent Ritz vectors to Bayesian probabilistic damage detection." Prob.Eng.Mech., 15(2), 139-153. The Mathworks. (2008). "MATLAB: the language of technical computing." . Turek, M. E. (2007). "A method for implementation of damage detection algorithms for civil structural health monitoring systems." PhD thesis, University of British Columbia, Vancouver. Vanik, M. W., Beck, J. L., and Au, S. K. (2000). "Bayesian probabilistic approach to structural health monitoring." J.Eng.Mech., 126(7), 738-745. Wenzel, H., and Pitchler, D. (2005). Ambient vibration monitoring. John Wiley & Sons, Ltd. Xia, Y., Hao, H., Brownjohn, J. M. W., and Xia, P. (2002). "Damage identification of structures with uncertain frequency and mode shape data." Earthquake Engineering and  Structural Dynamics, 31(5), 1053-1066. Yuen, K. (2010). "Recent developments of Bayesian model class selection and applications in civil engineering." Struct.Saf., 32(5), 338-346. Yuen, K., Au, S. K., and Beck, J. L. (2004). "Two-stage structural health monitoring approach for phase I benchmark studies." J.Eng.Mech., 130(1), 16-33. Zonta, D., Pozzi, M., and Zanon, P. (2007). "Bayesian approach to condition monitoring of PRC bridges." Damage Assessment of Structures VII, Trans Tech Publications Ltd, Laubisrutistr.24, Stafa-Zuerich, CH-8712, Switzerland, 227-232.  78  Appendices Appendix A RELAN (Foschi et al. 2000) is a general reliability analysis software developed at the University of British Columbia, which includes: FORM, SORM, Monte Carlo Simulation, and Reduce Variance Simulation amongst its features. Both of the subroutines implemented to obtain the CDFs presented in Section 2.3.3, for the case of measured frequencies, are included in this appendix. The subroutine DETERM is employed to define any deterministic variables related to the problem. In this case, involves the definition of the standard deviation of the error variable, , and the mass matrix of the frame analyzed in Chapter 4.  79  The subroutine GFUN is used to define the limit state function and in this case employs another standard subroutine, named JACOBI1 to solve the eigenvalue equation.  80  Appendix B The integrated Bayesian updating involves first the definition of the basic random variables. Recalling from Section 4.1, the basic random variables are three, that is, the stiffness contribution of each substructure of the FE model, which are assumed to be lognormally distributed. Then, the definition of damage states is done by generating the different combination of damage indicators as mentioned in Section 4.2. The sampling procedure was implemented and it was run for every damage state, assembling the new structure’s stiffness matrix and solving the eigenvalue problem, with the dynamic parameters found and saved. In particular, the code file written for the use of the frequencies is included in this appendix. Two distribution types were included to fit the univariate sample distributions, namely a Normal or Lognormal distribution. However, the assumption of a jointly Normal distribution was included for the multivariate case and the results presented in this thesis are consistent with the Normal distribution. Finally, a damage state was selected as the damage scenario to be detected by the Bayesian updating and the updated probabilities calculated. The code presented in the following is sufficient to reproduce the case study example, for the case of the measured natural frequencies.  81  %% % ** Integrated Bayesian Updating ** % Program to calculate P(D|wi) using Bayes' Rule % ** The likelihoods f(wi|D)are calculated fitting a known distribution % to Monte Carlo sampling results % ** When using w vector, a joint Normal distribution is assumed % ** The structure is a 3-story frame modeled as a 3DOF system % ** wi: natural frequency (i=1,2,3) % ** D: damage scenario defined by the damage indicators (theta)* % % Written by Laura Quiroz % August 2011 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% DATA INPUT disp(' ************ DATA INPUT ************ ') nss = 3 ; %number of substructures nds = 64 ; %number of damage states % Definition of Damage States DamageS = zeros(nds,nss) ; cont = 0 ; for i = 1:-0.3:0.1 for ii = 1:-0.3:0.1 for iii = 1:-0.3:0.1 cont = cont +1 ; DamageS(cont,1) = i ; DamageS(cont,2) = ii ; DamageS(cont,3) = iii ; end end end % Samples for Monte Carlo Sampling nsamples = 1000 ; % Damage state # to generate plot file with sampling results state = 1 ; % DAMAGE SCENARIO to be detected by Bayesian Updating DScenario = 7 ; % # according to above definition % Data Simulation ndata = 5 ; %number of simulated data sets epsilon_std = 0.05 ; % error term % % Mass Matrix MM = 1.80*[1.0 0.0 0.0 0.0 1.5 0.0 0.0 0.0 2.0] ; % Stiffness Matrix of one element or substructure % (will be multiplied later by the variable "k") KM1 = zeros(nss) ; KM0 = [1.0 -1.0 -1.0 1.0] ; % % RANDOM VARIABLES (Stiffnes parameter of each level) meank(1) = 1070.0 ; %premultiplication constant covk(1) = 0.10 ; %coefficient of variation meank(2) = 2140.0 ; covk(2) = 0.10 ; 82  meank(3) = 3210.0 ; covk(3) = 0.10 ; distrKey1 = 'Lognormal' ; % stiffness parameters' distribution distrKey2 = 'Normal' ; % fitting omega's distribution jointKey = 'yes' ; % Assuming multivariate normal distribution % if strcmp(distrKey1,'Lognormal') zeta = zeros(nss,1) ; lambda = zeros(nss,1) ; end % k = zeros(nss,1) ; % %% MONTE CARLO SAMPLING FOR EACH DAMAGE STATE disp(' ******* MONTE CARLO SAMPLING ******* ') disp(' * FITTING DISTRIBUTION * ') XOMEGA = zeros(nsamples,nss,nds) ; XOMEGAM = zeros(nsamples,nss,nds) ; % for ds = 1:nds theta = zeros(1,nss) ; disp('DAMAGE STATE # '), disp(ds) for i = 1:nss theta(i)= DamageS(ds,i) ; end % for samples = 1:nsamples KM = zeros(nss) ; switch distrKey1 case 'Lognormal' for i = 1:nss zeta(i) = sqrt(log(1+covk(i)^2)) ; lambda(i) = log(meank(i))-zeta(i)^2/2.0 ; k(i) = lognrnd(lambda(i),zeta(i)) ; end % Assembling Structure's Stiffness Matrices KM0 KM(1,1) = theta(1) * k(1)* KM0(1,1) ; KM(1,2) = theta(1) * k(1)* KM0(1,2) ; KM(2,1) = theta(1) * k(1)*KM0(2,1) ; KM(2,2) = theta(1) * k(1)* KM0(2,2) + theta(2)* k(2) * KM0(1,1); KM(2,3) = theta(2) * k(2) * KM0(1,2) ; KM(3,2) = theta(2) * k(2) * KM0(2,1) ; KM(3,3) = theta(2) * k(2) * KM0(2,2) + theta(3)* k(3) * KM0(1,1); % case 'Normal' for i = 1:nss k(i) = randn ; k(i) = meank(i)+k(i)*(meank(i)*covk(i)) ; end % Assembling Structure's Stiffness Matrices KM0 KM(1,1) = theta(1) * k(1)* KM0(1,1) ; KM(1,2) = theta(1) * k(1)* KM0(1,2) ; KM(2,1) = theta(1) * k(1)*KM0(2,1) ; KM(2,2) = theta(1) * k(1)* KM0(2,2) + theta(2)*k(2)* KM0(1,1) ; KM(2,3) = theta(2) * k(2) * KM0(1,2) ; KM(3,2) = theta(2) * k(2) * KM0(2,1) ; KM(3,3) = theta(2) * k(2) * KM0(2,2) + theta(3)* k(3) * KM0(1,1); 83  % end omegasq = eig(KM,MM) ; omega = sqrt(omegasq) ; % Saving calculated frequencies to XOMEGAM matrix Rn = randn ; while (1.0 + epsilon_std * Rn) < 0 Rn = randn ; end XOMEGAM(samples,:,ds) = omega(:)*(1.0 + epsilon_std * Rn) ; % end % % Fitting a known distribution for the calculated frequencies % Calculating distribution's parameters for j = 1:nss switch distrKey2 case 'Lognormal' alpha = 0.01 ; [parmhat,parmci] = lognfit(XOMEGAM(:,j,ds),alpha) ; PzetaWm(ds,j) = parmhat(2) ; PlambdaWm(ds,j) = parmhat(1) ; % [M,V] = lognstat(parmhat(1),parmhat(2)) case 'Normal' alpha = 0.01 ; [muhat,sigmahat,muci,sigmaci] = normfit(XOMEGAM(:,j,ds),alpha) ; PmuWm(ds,j) = muhat ; PsigmaWm(ds,j) = sigmahat ; end % end % if strcmp(jointKey,'yes') % Calculating statistics for joint Normal PDF % Mean vector, std vector, coefficient of variation vector MeanWm(ds,:) = mean(XOMEGAM(:,:,ds)) ; StdWm(ds,:) = std(XOMEGAM(:,:,ds)) ; CovWm(ds,:) = StdWm(ds,:)./MeanWm(ds,:) ; RWW0m(:,:,ds) = corrcoef(XOMEGAM(:,:,ds)) ; % switch distrKey2 case 'Normal' % Correlation matrix, Covariance matrix RWWm(:,:,ds) = RWW0m(:,:,ds) ; SigmaWWm(:,:,ds)=diag(StdWm(ds,:))*RWWm(:,:,ds)*diag(StdWm(ds,:)) ; case 'Lognormal' disp('when jointKey = yes then distrKey2 = Normal') disp('or distrKey2 = Lognormal then jointKey = no ') disp('modify this in DATA INPUT section') disp(' ******* END OF PROGRAM ******* ') return end % end % end disp(' ************* END MCS ************** ') 84  % %% PRINT MCS SELECTED RESULTS TO FILE idf = int2str(state) ; name = strcat('w1m_ds',idf,'.fit') ; fileid = fopen(name,'w') ; fprintf(fileid,'%10.5f\n',XOMEGAM(:,1,state)) ; name = strcat('w2m_ds',idf,'.fit') ; fileid = fopen(name,'w') ; fprintf(fileid,'%10.5f\n',XOMEGAM(:,2,state)) ; name = strcat('w3m_ds',idf,'.fit') ; fileid = fopen(name,'w') ; fprintf(fileid,'%10.5f\n',XOMEGAM(:,3,state)) ; fclose('all') ; % %% BAYESIAN UPDATING FOR DAMAGE SCENARIO % Prior probabilities for each Damage State (uniform distribution) Prob = zeros(nds,nss,ndata+1) ; %ndata+1 stores the posterior for ndata if strcmp(jointKey,'yes') JProb = zeros(nds,ndata+1) ; end % for i = 1:nds if strcmp(jointKey,'yes') JProb(i,1) = 1.0/nds ; %initial joint probability data set #1 end for j = 1:nss Prob(i,j,1) = 1.0/nds ; %initial probabilities data set #1 end end % % Calculations for Damage Scenario disp(' ********* DATA SIMULATION ************ ') % Simulating Measured Data datam = zeros(nss,ndata) ; omegam = zeros(nss,1) ; % theta(:)= DamageS(DScenario,:) ; % Defining structural properties switch distrKey1 case 'Lognormal' for i = 1:nss zeta(i) = sqrt(log(1+covk(i)^2)) ; lambda(i) = log(meank(i))-zeta(i)^2/2.0 ; k(i) = lognrnd(lambda(i),zeta(i)) ; end case 'Normal' for i = 1:nss k(i) = randn ; k(i) = meank(i)+k(i)*(meank(i)*covk(i)) ; end end % Assembling Structure's Stiffness Matrix (given Damage State) KMD(1,1) = theta(1) * k(1)* KM0(1,1) ; KMD(1,2) = theta(1) * k(1)* KM0(1,2) ; KMD(2,1) = theta(1) * k(1)* KM0(2,1) ; KMD(2,2) = theta(1) * k(1)* KM0(2,2) + theta(2) * k(2)* KM0(1,1) ; KMD(2,3) = theta(2) * k(2)* KM0(1,2) ; 85  KMD(3,2) = theta(2) * k(2)* KM0(2,1) ; KMD(3,3) = theta(2) * k(2)* KM0(2,2) + theta(3) * k(3)* KM0(1,1) ; % k(:); % Calculating eigenfrequencies omegasq = eig(KMD,MM) ; omega = sqrt(omegasq) ; % Simulating ndata sets of Measurements for ii = 1:ndata Rn = randn ; while (1.0 + epsilon_std * Rn) < 0 Rn = randn ; end omegam(:) = omega(:)*(1.0 + epsilon_std * Rn) ; datam(:,ii) = omegam(:) ; end % %% BAYES' RULE disp(' ******** BAYESIAN UPDATING ********** ') fileid = fopen('Bayes Results w1.OUT','w') ; fprintf(fileid,'\n ********** BAYESIAN UPDATING ********** \n') ; % for ndat = 1:ndata if ndat==1||ndat==10||ndat==20||ndat==30||ndat==40||ndat==ndata disp('Data Set'), disp(ndat) end for ds = 1:nds if strcmp(jointKey,'yes') % JOINT PROBABILITY switch distrKey2 case 'Normal' %Calculating PDF jlikely = mvnpdf(datam(:,ndat)',MeanWm(ds,:), SigmaWWm(:,:,ds)) ; %Calculating normalizing constant (denominator) NormC = 0.0 ; for dj = 1:nds jpdens = mvnpdf(datam(:,ndat)',MeanWm(dj,:),SigmaWWm(:,:,dj)) ; NormC = NormC + jpdens*JProb(dj,ndat) ; end %Updating probabilities JProb(ds,ndat+1) = jlikely/NormC * JProb(ds,ndat) ; %posterior % case 'Lognormal' disp(' ******* END OF PROGRAM ******* ') return end % end % % for ii = 1:nss % MARGINAL PROBABILITIES switch distrKey2 case 'Lognormal' %Calculating PDF pzeta = PzetaWm(ds,ii) ; plambda = PlambdaWm(ds,ii) ; xomega = datam(ii,ndat) ; 86  likely = lognpdf(xomega,plambda,pzeta) ; %Calculating normalizing constant (denominator) NormC = 0.0 ; for dj = 1:nds xz = PzetaWm(dj,ii) ; xl = PlambdaWm(dj,ii) ; pdens = lognpdf(xomega,xl,xz) ; NormC = NormC + pdens*Prob(dj,ii,ndat) ; end % if ii==1 fprintf(fileid,'\n DS= %i \t likely= %8.5f \t NormC= %8.5f \n',ds,likely,NormC) ; end % case 'Normal' %Calculating PDF mu = PmuWm(ds,ii) ; sigma = PsigmaWm(ds,ii) ; xomega = datam(ii,ndat) ; likely = normpdf(xomega,mu,sigma) ; %Calculating normalizing constant (denominator) NormC = 0.0 ; for dj = 1:nds xm = PmuWm(dj,ii) ; xs = PsigmaWm(dj,ii) ; pdens = normpdf(xomega,xm,xs) ; NormC = NormC + pdens*Prob(dj,ii,ndat) ; end end %Updating probabilities Prob(ds,ii,ndat+1) = likely/NormC * Prob(ds,ii,ndat) ; %posterior % if ii==1 fprintf(fileid,'\n Prior= %10.5f \n',Prob(ds,ii,ndat)) ; fprintf(fileid,'\n Post= %10.5f \n',Prob(ds,ii,ndat+1)) ; end % end % end if strcmp(jointKey, 'yes') plot(JProb(:,ndat), 'DisplayName', 'JProb', 'YDataSource', 'JProb'); figure(gcf) end % plot(Prob(:,:,ndat),'DisplayName','Prob','YDataSource','Prob');figure(gcf) end % fclose('all'); disp(' ******** END BAYESIAN UPDATE ******** ') % %% PLOTTING FINAL RESULTS % plot(Prob(:,:,ndata+1), 'DisplayName', 'Prob', 'YDataSource', 'Prob'); figure(gcf) % hold all if strcmp(jointKey, 'yes') 87  plot(JProb(:,ndata+1), 'DisplayName', 'JProb', 'YDataSource', 'JProb'); figure(gcf) end % disp(' ******** END OF PROGRAM ******** ')  88  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0063203/manifest

Comment

Related Items