Statistical analysis of modal parameters of a suspension bridge based on Bayesian spectral density approach and SHM data
Introduction
Vibration-based structural health monitoring (SHM) techniques have become valuable methods for evaluating structural integrity, and reliability throughout the life-cycle of structures. The SHM techniques are often based on modal parameters such as natural frequencies, damping ratios and mode shapes [1]. However, in field conditions, when structures are subject to changing environmental conditions, the modal parameters of the structure exhibit non-stationary behavior [2], [3], [4]. Meanwhile, since the ambient vibration and operational conditions such as wind and traffic may affect the measured signals, the identified modal parameters will exhibit uncertainties. Therefore, the modal parameters identification methods need to provide estimation of the parameter uncertainty.
During the past few decades, modal identification of civil engineering structures based on ambient vibration measurements has been widely investigated, and a variety of output-only modal identification methods have become available in both time and frequency domains [5]. Among the frequency domain methods, the peak picking (PP) and frequency-domain decomposition (FDD) techniques are most widely used [6]. Although many methods are available in literature for identification of modal parameters, there are relatively few studies on the statistical bounds of the identified parameters [7]. While time domain methods have some advantages in identifying the closed modes and damping ratios, they have some drawbacks. The advantage of stochastic subspace identification (SSI) methods is that they are numerically robust and non-iterative. However, most of SSI algorithms do not provide any direct estimation of the variance of the identified modal parameters [7].
Bayesian inference provides a promising and feasible identification solution for the purpose of structural health monitoring. The Bayesian approaches offer a powerful tool for system identification that explicitly addresses uncertainty. A significant advantage of the Bayesian inference is that not only the optimal estimates can be determined but also their associated uncertainties can be quantified in the form of probability distributions [8]. Recently, there is a rising interest in computing the uncertainties of modal parameters using the Bayesian approaches including the Bayesian spectral density approach (BSDA) [9], Bayesian time domain approach, and Bayesian FFT approach [10], [11], [12]. These methods provide rigorous means for obtaining modal properties as well as their uncertainties for given measured data and modeling assumptions. Although there exist many Bayesian approaches, computational difficulty has severely hindered their wider applications. In [11], a two-stage fast Bayesian Spectral Trace Analysis (BSTA) method is developed, which can address the computational challenges of the conventional Bayesian approach. In the BSTA method, the interaction between the spectrum variables (i.e., frequency, damping ratio as well as the spectral density of modal excitation and prediction error) and the spatial variables (i.e., mode shape components) can be decoupled completely. Although the theory and computational methods provide a rigorous and viable platform, practical and important issues regarding data limitation, interpretation of results, trade-off between identification precision and modeling error risk need to be addressed for application of the Bayesian spectral density approach in real structures [13].
In the case of bridges, environmental factors such as temperature, wind profile and traffic are known to increase the variance of a modal frequency. The influence of varying environmental conditions on structural modal properties has been extensively investigated through field measurements and dynamic tests [14], [15], [16], [17]. Results from monitoring of the Humber Bridge showed that the temperature and the wind were the two most significant influences on the bridge behavior [18]. Several studies have shown that operational modal parameters obtained using vibration-based structural monitoring systems are not fixed and can vary within 5% of their range from their mean value [2], [15]. However, the relationships between the modal frequencies and the environmental and loading effects (e.g. temperature, wind, traffic) are complex. In [19], it was observed that the dynamic behavior of the suspension bridges is dependent on the magnitude of the environmental effects. For instance, the fundamental frequency of a suspension bridge reduces as the wind speed increases. On the other hand, the modal damping increases when the wind velocity increases and exceeds a certain level. According to long term observation and estimation results of the operational modal parameters, the uncertainty distributions of these parameters can be fitted by normal distributions [20]. Although a lot of field measurements and observations were conducted regarding the relation between the modal frequency and the environmental influences, very few studies addressed the statistical distribution over a long period time. Moreover, the variation of damping parameters was rarely studied.
This paper presents a statistical analysis for the modal parameters of the Runyang Suspension Bridge (RSB) based on its SHM data. RSB is a long-span suspension bridge constructed in 2005 over Yangtze River, China. Since 2005, a long-term structural monitoring system installed on the bridge provides large data sets. As there are a large amount of data sets recorded by the SHM system of the bridge, a robust and automatic identification method is necessary. To process large data sets for the long-term structural monitoring system, the Bayesian spectral density algorithm is adapted to address the uncertainty of modes extraction from output-only response. Furthermore, the extraction of modal parameters is fully automated by introduced the Bayesian spectral trace analysis method, which can address the computational difficulties of conventional BSDA. Moreover, statistical studies of the operational modal parameters over a long-term period were conducted to study the parameter distributions and variation. In addition, a period data sets during a strong wind excitation is presented to research the wind-induced variation of modal parameters.
Section snippets
Dynamics of linear systems
Consider a linear dynamic system with degrees of freedom and equation of motion:where , and are the mass, damping and stiffness matrix, respectively; is a force distributing matrix. The external excitation can be modeled as zero-mean Gaussian white noise with spectral intensity matrix .
Using modal analysis, one obtains the uncoupled modal equations of motion:where ; , , and are the
Description of Runyang suspension bridge
The Runyang Suspension Bridge, as shown in Fig. 1, is one of the most critical traffic links spanning the Yangtze River, China. At the time of its completion in 2005, it was the longest suspension bridge in China. RSB consists of a 1490 m main span and two 470 m side spans. The cross section of the deck of RSB is an aerodynamically shaped closed steel box girder, which carries two carriageways and three traffic lanes on each carriageway. The width and the height of the steel box girder are 36.3 m
Statistical analysis on long-term extracted results
A long-term modal parameters analysis was performed based on BSTA method using the SHM system data sets over a 3-year period in this bridge’s initial life (2005–2007). This section first presents identification of modal parameters based on the vibration data with a duration of one year. Then the effects of the environmental conditions on the model identifications were studied.
Wind-induced variation of modal parameters
The wind-induced vibration plays an important role for long-span bridges. In wind sensitive slender structures, such as long-span bridges, the model parameters might exhibit significant variations under wind. During the period from August 6 at 16 pm to August 7 at 5 am, 2005, the Typhoon Matsa passed through the Runyang Suspension Bridge. The Typhoon Matsa was the second of eight Pacific tropical cyclones to make landfall on China during the 2005 Pacific typhoon season. When the strong wind
Conclusions
In this paper, a Bayesian spectral density method was successfully applied to identify modal parameters of a long-span suspension bridge based on its large sensor data sets. There are two main sources of uncertainty in the identification problem treated, which are due to the estimation and the variable environmental and operational conditions. The first uncertainty can be quantified by the BSTA. The second uncertainty was treated by the statistical method. Through Bayesian inference, not only
Acknowledgements
The authors would like to express thanks to Professor Aiqun Li and Professor Xiaolin Han of Southeast University and to Professor Wangan Zheng of the AZCRAS company. This study was funded by the National Natural Science Foundation of China (Grant Number 51208252) and the National Key Basic Research Program of China (Grant No. 2013CB036300).
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