Mathematical Problems in Engineering

Volume 2017, Article ID 5450297, 11 pages

https://doi.org/10.1155/2017/5450297

## Time-Varying Identification Model for Crack Monitoring Data from Concrete Dams Based on Support Vector Regression and the Bayesian Framework

^{1}State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China^{2}Key Laboratory of Earth-Rock Dam Failure Mechanism and Safety Control Techniques, Ministry of Water Resources, Nanjing 210029, China^{3}National Engineering Research Center of Water Resources Efficient Utilization and Engineering Safety, Hohai University, Nanjing 210098, China^{4}College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China

Correspondence should be addressed to Bo Chen; moc.621@uhhobnehc

Received 26 October 2016; Revised 20 December 2016; Accepted 19 January 2017; Published 19 February 2017

Academic Editor: Salvatore Caddemi

Copyright © 2017 Bo Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The modeling of cracks and identification of dam behavior changes are difficult issues in dam health monitoring research. In this paper, a time-varying identification model for crack monitoring data is built using support vector regression (SVR) and the Bayesian evidence framework (BEF). First, the SVR method is adopted for better modeling of the nonlinear relationship between the crack opening displacement (COD) and its influencing factors. Second, the BEF approach is applied to determine the optimal SVR modeling parameters, including the penalty coefficient, the loss coefficient, and the width coefficient of the radial kernel function, under the principle that the prediction errors between the monitored and the model forecasted values are as small as possible. Then, considering the predicted COD, the historical maximum COD, and the time-dependent component, forewarning criteria are proposed for identifying the time-varying behavior of cracks and the degree of abnormality of dam health. Finally, an example of modeling and forewarning analysis is presented using two monitoring subsequences from a real structural crack in the Chencun concrete arch-gravity dam. The findings indicate that the proposed time-varying model can provide predicted results that are more accurately nonlinearity fitted and is suitable for use in evaluating the behavior of cracks in dams.

#### 1. Introduction

Cracks are the most common structural threats to concrete dams. According to an investigative report issued by the International Committee on Large Dams, a total of 243 dams have collapsed as a result of cracking problems [1]. As an example, in the case of the Koelnbrein arch dam in Austria, diagonal cracks emerged at the heel of the dam, causing the grouting curtain to break. Under the enormous water pressure at the crack surfaces, these cracks expanded throughout the foundation and heel of the dam, which led to the full head of uplift pressure being applied over the entire surface of the base [2]. Therefore, the dam could no longer operate safely. In another example, soon after the pouring of the Phase II concrete for the Chencun dam in China, a crack of 300 m in length and more than 5 m in depth, which stretched across dam sections #5 to #28, emerged at the interface between the concrete layers poured in Phases I and II. Serious crack problems have also occurred during the operation of the Dworshak solid gravity dam in America, the Sayano-Shushenskaya arch-gravity dam in Russia, and the Xiaowan arch dam in China [3–5]. Cracks are destructive. Severe cracks can weaken the strength and rigidity of a dam, destroy its integrity and antipermeability, accelerate the corrosion and carbonation of the concrete, and endanger the safe operation of the dam.

The research on structural health monitoring of cracks is mainly classified into two categories [6, 7]. The first one focuses on the nondestructive detection and characterization of cracks, including location, size, depth, and other information. Numerous sensing techniques are proposed and applied, such as strain gauges [8], optical fiber sensor [9], piezoceramic sensor [10], carbon nanotube thread [11], and other vibration-based methods [12, 13]. The other one focuses on the assessment and prediction of crack propagation, as well as structural behavior, based on quantification monitoring data analysis. However, these data are still difficult to be obtained accounting for the limitations of sensing techniques, heterogeneity of concrete material, and bulkiness of dam concrete [7]. Luckily, there are still some real-time cracking series being obtained by later buried crack gauges in dam engineering, such as Chencun and Xiaowan dam [3].

To date, methods of propagation analysis and abnormality diagnosis for concrete crack behavior have mainly relied on fracture mechanics methods and applied mathematical methods. Currently, criteria such as the stress intensity factor criterion [14–16], the strain energy density factor criterion [17], the maximum energy release rate criterion [18], the mixed energy-stress criterion [19], and the dual-K fracture criterion [20] are frequently adopted to determine whether unstable propagation is occurring. But the variables on these criteria are difficult to monitor accurately and inconvenient to apply in practical engineering. The other approach [19, 21] is based on the consideration that the crack will lose its stability rapidly when the crack tip opening displacement (CTOD) reaches its critical value (CTOD_{C}). Several mathematical models based on the CTOD have been established based on the relationship between the CTOD and the crack aperture. However, CTOD theory is also difficult to use in the diagnosis of crack propagation behavior, mainly because the CTOD is extremely difficult to monitor. Moreover, the indicator, CTOD_{C}, must be measured in fracture tests using large-sized specimens, which is difficult to achieve in typical laboratories [22]. Therefore, these methods are subject to limitations in the diagnosis of structural behavior in dam engineering.

For research on dam health monitoring, especially with regard to major cracks, Wu et al. [23–25] considered the contributions of water pressure, temperature, and structural aging to establish a statistical model for crack monitoring data. In this model, the water pressure component is constructed based on the relationship between water level and COD derived from dam engineering theory; the structural aging component is derived based on the creep theory of concrete; and the temperature component is obtained by using the results of temperature field simulation or fitting a semiexperienced periodic function. Li and Gu [3–5, 26] suggested that the degree of abnormality of a crack in a dam can be diagnosed based on a time-dependent analysis. A semiparametric statistical model and a nonparametric statistical model [4, 26] were established by considering the relationship between the degree of abnormality of dam crack behavior and its statistical point of change to improve the fitting and explanation of crack monitoring data from concrete dams. Li et al. [3] analyzed the dynamic properties of crack monitoring series and proposed a dynamic structural diagnosis method for crack abnormality based on a fuzzy cross-correlation factor exponent. Li et al. [5] proposed a fluctuation-based method of regression coefficients for crack monitoring series, in which each abnormal event is detected from a cumulative sum of regression model residuals. These models play an important role in COD monitoring. However, the fitting and prediction capabilities of the models need to be improved accounting for the nonlinearity of crack monitoring series, and feasible crack monitoring criteria need to be proposed for more effective application in dam engineering. In pursuit of nonlinear modeling and prediction methods, Panizzo and Petaccia [27] developed predictive models based on random forests (RFs), boosted regression trees (BRTs), neural networks (NNs), support vector machines (SVMs), and multivariate adaptive regression splines (MARSs), and a comparison of the prediction accuracy of these models revealed that they showed poorer performance on average than did a conventional statistical model. Therefore, further research is needed to construct nonlinear modeling approaches for crack monitoring that are able to better fit and predict crack opening behavior and evaluate anomalous characteristics of crack propagation and structural health.

A technique based on the use of SVMs for regression, namely, support vector regression (SVR), is an advanced statistical method that has recently been used in the modeling of nonlinear systems [28–32]. In SVR, a nonlinear transformation is employed to map monitoring data into a higher-dimensional feature space, and the nonlinear relationships between the monitored values of dam behavior quantities and their influencing factors are linearized by searching for an optimized fitting function. With the introduction of the insensitive error loss function as proposed by Vapnik, SVR has emerged as one of the best methods of regression because of its improved robustness and generalization performance [33]. Several researchers have employed SVR in modeling methods for structural identification to verify its potential capabilities in this respect. Ranković et al. [34] established a nonlinear autoregressive SVR model that yields accurate results for the prediction of the tangential displacement of a concrete dam. Lee et al. [35] successfully predicted the strength of concrete based on its mix proportions using the SVR technique and NNs, as indicated by a comparison against experimental results, and concluded that the SVR method can be used to predict the compressive strength of concrete with higher estimation accuracy and within a shorter computation time compared with the NN method. Su et al. [36] developed a time-varying identification model for dam behavior before and after structural reinforcement based on the SVR method, and this model was found to yield more accurate fitted and forecasted results compared with classical statistical models. In these studies, one key factor in establishing a useful model is the selection of suitable SVR parameters, such as the penalty coefficient, the loss coefficient, and the width coefficient of the kernel function. There is an urgent need for the development of an appropriate algorithm for parameter optimization.

Based on the brief review presented above, it is clear that the modeling of crack monitoring data is still an important problem in research on concrete dams. In the present study, SVR is applied to establish a parametric statistical model for the safety monitoring of cracks in concrete dams, with the intent of improving the extraction of the nonlinear effects of the various influencing factors on COD. The BEF method is then introduced to optimize the model parameters. Finally, forewarning criteria based on the predicted components of the time-varying features of the crack monitoring model are proposed for the real-time diagnosis of abnormal dam behavior.

#### 2. Time-Varying Forewarning Criteria for the Crack Monitoring of Concrete Dams

Monitoring equipment such as crack measurement meters and optical fibers provides a large amount of COD data for concrete dams. The statistical model for crack monitoring data that is considered in this paper has been confirmed to be effective for analyzing the crack evolution process. However, there is still a need for suitable forewarning criteria to be proposed to characterize the relationship between the crack propagation process and abnormal structural behavior in dam engineering applications. In this section, the prediction of the components of the statistical model and their time-varying features are introduced to establish these criteria.

##### 2.1. Prediction Using the Statistical Crack Monitoring Model

The factors that affect the monitored crack quantities in a concrete dam can be attributed to external loads (such as water pressure and temperature) and time-varying effects. Therefore, the monitoring data for dam cracks can be decomposed into contributions from different influencing factors as follows:

In the formula above, is a measured value in the crack opening monitoring sequence; and are the crack opening components associated with the action of water pressure and temperature, respectively; and is the time-varying component, which characterizes the other effects on crack behavior that are induced by creep or structural changes in the concrete.

Once the factors influencing the COD have been decomposed, a regression model and a prediction model for the monitored crack displacement values can be established based on the measured sequence, and then the estimated values for each component, namely, , , and , can be calculated. The statistical prediction model for crack monitoring data is as follows:

In the formula above, is the predicted value of the COD at time ; , , and are the estimated water pressure, temperature, and time-varying components, respectively; is the residual standard deviation of the model; and is the variance.

##### 2.2. Forewarning Criteria for Dam Behavior

According to the prediction model presented above, the predicted value of the COD at a future time is calculated to be , and the measured value of the COD at that time is denoted by . If it is assumed that the residual standard deviation of the model satisfies a normal distribution, then, according to the mathematical theory of probability and statistics, the relationship between the distance from the predicted value to the measured value and the residual standard deviation of the model is as follows: the probability that falls into the interval is 95.5% and the probability that falls into is 99.7%. That is to say, both and are events that will occur with low probability. However, the time-dependent component of the COD reflects the irreversible changes to the concrete caused by creep and damage and is consequently an important index for characterizing the dam safety trend. Accordingly, by comparing between the measured values and the values predicted by the model and their eigenvalues and analyzing the trend of the development of the time-dependent component, time-varying forewarning criteria for the COD of a concrete dam can be established. Thus, a measured time sequence of data can be used to determine whether the structural state of the dam is changing. The forewarning criteria are as follows:

Ifthen the safety state of the concrete dam structure is normal.

Ifthen the safety state of the concrete dam structure is abnormal.

Ifthen the safety state of the concrete dam structure is significantly abnormal.

In the formulas above, and are the predicted and measured values, respectively, of the monitored COD; is the historical maximum value of the COD in the measured monitoring sequence; and is the time-dependent component extracted from the prediction model.

#### 3. Crack Prediction Model Based on B-SVR

The stepwise regression method is used in the traditional statistical modeling process. However, problems with modeling accuracy often arise because of the nonlinear characteristics of crack monitoring data. To address this problem, a nonlinear modeling approach, namely, the SVR method, is adopted here to establish a bridge between the monitoring data and the components of the statistical model. The Bayesian evidence framework is then applied to determine the optimal SVR modeling parameters for achieving the best possible accuracy of the crack monitoring model.

##### 3.1. Modeling of Crack Monitoring Data Based on SVR

Considering the nonlinear relationships between the effects on the monitored crack quantities and their influencing factors, let be the input sequence of the set of crack-influencing factors and let be the input sequence of monitored crack quantities. Then, the independent identically distributed sample set () of the crack monitoring sequence is subject to a certain distribution ,,, in the function space . Based on the method of support vector regression (SVR) [28], the basic idea used to solve the above nonlinear prediction problem in a multidimensional function space is as follows: the set of monitoring data is mapped from the input space to a high-dimensional feature space via a nonlinear transformation , and the hyperplane function () that yields the optimal classification of the linearized regression problem is found to fit the relationships between the influencing factors and the effects on the input crack quantities . Finally, the optimization problem for reproducing the monitored quantities based on the SVR model is established as follows:

In the formulas above, represents the risk function of the optimization problem; is the vector of the weights of all influencing factors; is a penalty parameter, which describes the tradeoff between the empirical risk and the model complexity; is a slack variable such that describes the deviation of the monitored crack quantity data with respect to the ideal conditions of the classification; is a constant bias term; and is the function used to map the input space to the high-dimensional feature space. The basic principle of the modeling of monitored crack quantities based on SVR to solve the corresponding nonlinear problem is illustrated in Figure 1.