Using this PDF, the marginal PDF of the stiffness parameter can be calculated for each substructure given the data. In such an approach, the model parameters that minimize the margin of error between the measured and model modal values would be used as the "best" model of the structure.
List of Figures
List of Tables
Chapter 1 Introduction
- A Need For Structural Health Monitoring
- Structural Health Monitoring
- Local Structural Health Monitoring
- Global Structural Health Monitoring
- Local and Global Method Working Together
- What Does This Work Present?
Finally, the authors make no mention of the effects of noise in the data on their results. The objectives of the method presented in this study are defined in the problem statement in Table 1.1.
Chapter 2
Theoretical Development
Introduction
It can reproduce the results of some of the deterministic methods, albeit with a slightly different interpretation. Furthermore, it provides the machinery by which a measure of the likelihood of damage can be established.
Bayesian Probability
In case the initial PDF reflects that c provides more information than a, it is called informative. A possible informative prior could arise by letting b and c include successive data sets and using the updated PDF to a obtained by using c as the initial PDF when data set b is considered.
Outline of Bayesian SHM
However, uncertainties in observable parameters and structural model parameters must be included. In this case, the updated PDF will be formulated only for the parameters of the structural model.
Formulating the Updated PDF
- Measured Data
- Structural Model Class
- Probability Model Class
- Updated PDF for the Model Parameters
All the measured modal parameters of the nth time domain data set are referred to by i'n. These relationships will be used extensively in the development of the updated PDF for the model parameters.
Bayesian SHM Framework
- The Undamaged and Potentially Damaged PDFs
- Observations on the Marginal Distributions
- Defining the Probability of Variation
- Considerations in the Use of pvar
- Sounding an Alarm
- Applying pvar for SHM
Changes in fJ between an undamaged reference state and the actual state will be used to determine the health of the structure. Finally, some comments are made on the ability of the defined SHM strategy to determine the degree of damage. In this phase, the measured modal data is used by the SHM algorithm to monitor the damage condition of the structure.
The damage measure, defined at a glance, uses changes between the potentially damaged PDFs and the undamaged PDF to monitor the state of the structure. These observations will be used to qualitatively discuss the behavior of the damage measure defined in the next section. Based on the observations in the previous sections, the behavior of the sequence fl/,d can be seen as a random walk about pud.
2.5. 7 Degree of Damage
Calculating the Cumulative Distributions
In this study, two procedures are proposed to approximate the value of the integral. To use this method, the maxima of the updated PDF with respect to the model parameters must be found. As noted earlier, the shape of the updated PDF (2.72) lends itself well to approximation by a multivariable joint Gaussian distribution function.
First find the maximum of the PDF with respect to all the structural parameters. This serves as a starting point for the calculation of the marginal PDF for each of the structural parameters, and will ensure that the maxima are not missed. For the results presented in this study, the Hessian matrix of the MOF always has large enough eigenvalues that the rough approximation is sufficient for all calculations.
Summary of the SHM Method
For a permanently instrumented structure, this will facilitate the implementation of the SHM method as an automatic monitoring system. The stiffness matrices of the substructure model the contributions of part of the structure to the overall stiffness matrix. Expressing the dependence of the stiffness matrix on the structural parameters in this form is useful for the mathematical analysis.
Bayes' theorem is used to find the probability of the structural and expansion parameters given these measured data and the PDF over the observable data. Since the PDF tends to be very high as a function of O:r for a number of different models, the approximation is very good. For each parameter, an asymptotic solution of the expansion for the integral over the other parameters is obtained at a series of discrete values of the nonintegrated parameter.
Chapter 3
Simulated Data Testing
Introduction
- Common Definitions
- Brief Descriptions of the Examples
- Picturing Damage
Regarding the development in Chapter 2, the tabulated values of Ew~ and E'I/Jr. However, when multiple data sets are available, the choice of initial distribution becomes less important, as the data will control the shape of the updated distribution. For the 2-DOF case, the dependence of the procedure performance on the alarm function, pialarm(k), the level and location of the damage, and the number of identified modes is described.
Many of the results in the 2-DOF and 10-DOF examples to be presented are explained using a graphical representation of the probability of variation. The probability of variation, pvaT, in a given monitoring cycle, tmon = 34, is plotted as a function of the monitoring cycle window, k. Thus, in the probability of variation plot, the maximum value on the axis of the monitoring cycle window cannot exceed the number of the monitoring cycle, but will often be less.
Two DOF Shear Structure
- Results of the Testing Stabilizing the PDF
- Observations on the 2-DOF Example
7 iJ1 as a function of the number of modal data sets used to form the undamaged PDF: Test case 4. It is also noted that no apparent consistent trends in the probability of variation curves are associated with the occurrence of the false alarm. Also shown in Tables 3.6 to 3.8 are the degree of damage measures for the parameter of the substructures.
True alarms are distinguished from false alarms by considering the behavior of the variation probability plots. Another way would be to reduce the degree of fluctuation of the damage probability at small values of the selection parameter. Investigating the reduction of fluctuations in the variation probability curves is therefore an avenue of future work.
Ten DOF Shear Structure
- Initialization Phase
- Monitoring Phase
- Concluding Remarks on the 10-DOF Example
In the first case, M2, only two modes were measured, but the structure was instrumented at all floors. In the second case, M5, five modes were measured, but the structure was instrumented only at floors. This was believed to occur because the fluctuations in pvar for this case were not as large as in the 2-DOF example.
However, test case 4 also triggered a damage alarm in the first story, which appeared to be a real alarm. The sample cases presented in this section have shown that the behavior observed in the 2-DOF case carries over to the 10-DOF case. However, as the problem size increased, some problems emerged that were not a significant problem in the 2-DOF case.
Chapter 4
Conclusions and Future Work
Future Work
- Additional Testing
- Improving the Method
- Automation
For example, in the 2-DOF displacement model, if only the two frequencies are measured, then there are two local maxima for the PDF. Research into the application of the method when multiple maxima are present will thus be useful. Having a better method for selecting the alarm function, which reduces the number of data measurements required in the initialization phase of the SHM technique, would be a significant improvement.
Also, a finer subtree network can be maintained since not all subtree parameters need to be updated when data is acquired. In light of the fact that the ultimate goal of this work is to create a system for use on real structures, many operational questions need to be answered. For example, online determination of modal parameters from measured data is a fundamental requirement if the SHM system is to be fully automated.
Limitations of The SHM Method
Criteria for establishing the maximum size of the monitoring cycle window used to determine the order of probabilities of variation must also be established. One of the fundamental assumptions for performing global SHM, as mentioned in Section 1.3, is that changes in the structure will affect some measured data to a sufficient extent to be able to characterize the changes. Unless measurements are taken in the specific area of the member, the damage may go unnoticed.
The highly redundant structure example brings up another limitation of SHM methods based on global models: The extent to which damage can be localized depends on the nature of the measured data. Assume that a significant portion of the variation in modal parameters when there is no damage is due to the identification technique rather than the underlying measured data. This effect is believed to be caused by a combination of loosening of concrete, soil and non-structural elements in the structure.
Conclusions
Over time, some frequencies may increase slightly, but generally remain lower than the original values. To answer this question, probabilistic descriptions of the model's stiffness parameters were required, given different data. The results of the tests showed that the damage measure derived in Chapter 2 exhibited a number of distinctive characteristic behaviors based on whether the structure was damaged or not.
Although the probability of variation is used in this work to detect stiffness reductions, it can also be applied more generally to determine increases in stiffness parameters. Another contribution of this work is the consideration of issues related to the automated application of the SHM method. Therefore, designing the method that lends itself well to automated application and addressing some of the issues associated with online monitoring are important contributions.
Appendix A
Probability Model Details
Since the second derivative of f(x, y, TJ) with respect to TJ is clearly positive, the TJ given in equation A.l minimizes the norm with respect to TJ. Proof: Expand the square term on the right-hand side in equation (2.28) and manipulate the result. fr is multiplied and a new function p* fr p is defined to simplify the notation.
Appendix B
Calculating the Minimum and Hessian
- Derivatives of J(B, a)
- Some Final Notation
- Minimizing the MOF
- Derive the Hessian
The term Cr ( ¢r ) is never actually used in any calculations, so its form is not expressed here. The term C is not a function of () and is never used explicitly, so its form is not listed. The minimum of J(O) must be found to determine the maximum of the updated PDF.
Furthermore, any algorithm that finds the minimum of the MOF can be used to find the minimum, so the choice of a routine is not critical for the SHM. Extensive use of the method (Beck and Yanik 1996) for cases where the minimum was known has shown this. As with the minimization iteration scheme, this is not guaranteed to converge to a fixed point.
Bibliography
Proceedings of the Conference on Optical Fiber Sensor-Based Smart Materials and Structures: April Lancaster, PA. I Proceedings of Nodestructive Evaluation of Civil Structures and Materials, University of Colorado, Boulder, Colorado.
