The vibrationbased structural health monitoring has been traditionally implemented through the deterministic approach that relies on a single model to identify model parameters that represent damages. When such approach is applied for truss bridges, truss joints are usually modeled as either simple hinges or rigid connections. The former could lead to model uncertainties due to the discrepancy between physical configurations and their mathematical models, while the latter could induce model parameter uncertainties due to difficulty in obtaining accurate model parameters of complex joint details. This paper is to present a new perspective for addressing uncertainties associated with truss joint configurations in damage identification based on Bayesian probabilistic model updating and model class selection. A new sampling method of the transitional Markov chain Monte Carlo is incorporated with the structure’s finite element model for implementing the approach to damage identification of truss structures. This method can not only draw samples which approximate the updated probability distributions of uncertain model parameters but also provide model evidence that quantify probabilities of uncertain model classes. The proposed probabilistic framework and its applicability for addressing joint uncertainties are illustrated and examined with an application example. Future research directions in this field are discussed.
Steel truss bridges are commonly used in the highway system. Those truss bridges are typically composed of slender steel members connected at truss joints. The truss joints may take various types or configurations. During their longterm operations, steel truss bridges may become deteriorated (such as development of fatigue cracks and corrosions) due to increased volumes of traffic and adverse environment impacts. Such deterioration or damage could pose serious threats to the safe operation of bridges if its development cannot be identified in a timely manner. Researchers have explored various approaches for effectively detecting the development of deterioration or damages of truss structures at their early stages through implementing vibrationbased structural health monitoring (SHM), which typically relies on vibration measurements and structural models to identify model parameters that represent extents and locations of damages.
Gao et al. [
Some researchers, on the other hand, have attempted to explicitly consider the complexity of truss joint configurations in their models for damage identification of truss structures. Jones et al. [
Among two types of strategies for modeling joints of steel truss structures adopted in the previous research, one adopts oversimplified models of truss joint that could lead to model uncertainties due to the discrepancy between joint physical configuration and its model representation; the other could induce more uncertainties associated with model parameters due to difficulties in obtaining accurate model parameters of complex joint configurations. Both strategies could cause the deviation of the identified model parameters that represent damages from their true physical damages and lead to masking real damages or triggering false damage alarms. Even though progresses have been made in vibrationbased structural health monitoring of the truss or trusslike structures, most of existing methods rely on deterministic methods that cannot evaluate the impacts of uncertainties associated with the model of joint configurations on outcomes of damage identification. Because the in situ measurement data is always limited for a complex truss structure with many unknown or uncertain model parameters, deterministic approaches to damage identification often yield ill conditions (unidentifiable solution) or nonuniqueness (multiple solution) and cannot address inherent uncertainties associated with the structural model and model parameters.
With those uncertainties present, monitoring and assessing damages of steel truss bridges may better be conducted within a probabilistic framework. The recently resurgent Bayesian inference could provide an applicable computational framework for robustly addressing uncertainties associated with models and model parameters in damage identification. Within this framework, uncertainties associated with structural models are represented by the probability of model among competing candidate models, and unknown damage status of structural members is presented by the probability distributions of model parameters. The probabilities of competing candidate models and the probability distributions of uncertain model parameters can be first assumed based on prior knowledge and engineering judgment and then updated based on Bayes’ theorem by using available in situ measurement data. These updated model probabilities and probability distributions of model parameters not only identify the most suitable model among competing candidate models and estimate the most suitable model parameters that correspond to the damage location and extent but also quantify uncertainties in such identification and estimation, which are essential for assessing the reliabilitybased performance of the system for decision making. For example, the quantified uncertainties obtained from probabilistic inference can be used to make a statement such as that the probability of a specific truss member damaged with stiffness reduction ranging from 20% to 30% is 90%.
The general framework of Bayesian inference for structural model updating and damage identification was presented by Beck and his associates in the 90s [
This paper presents probabilistic computational framework for identifying structural damage of the truss members under uncertainties associated with modeling joint configurations by using vibration measurements. The framework integrates the advanced sampling algorithm of the transitional Markov chain Monte Carlo with structure’s finite element model. It can be used to effectively draw statistical samples that approximate the updated probabilistic distributions of uncertain model parameters and provide the model evidence that can be used to quantify probabilities of uncertain model classes. The paper in the subsequent sections is organized as follows. Firstly, the theoretic framework of Bayesian inference for damage identification is outlined. Secondly, computational procedures that integrate the structural model with advanced sampling algorithm are presented. Thirdly, an application of the proposed framework and its effectiveness are illustrated and examined through numerical simulation of damage identification of ninebay threedimensional steel truss model. Finally, future research directions in this field are discussed.
Vibration measurements have been used in SHM for identifying structural damages of engineering systems. The vibrationbased SHM approach is based on the premises that structural dynamic characteristics are functions of structural properties and boundary conditions, and can be reflected in the vibration responses of the structures under external dynamic excitations. As damages cause the change of structural properties, the measured vibration responses from the structure will also alternate. The process of the model updating or calibration is to determine the most suitable model parameters, which are related to damage status, structural properties, or boundary conditions, through minimizing the difference between the predicted structural response from the structural model and the measured structural response from sensor networks installed in the structure.
In model updating process, uncertain model parameters that need to be calibrated are unknown or cannot be predetermined accurately. In the context of damage identification, structural damages are usually parameterized in terms of the change of stiffness of structure members between damaged structures and undamaged structures, because the damage would cause the reduction of stiffness of an individual structural member or a group of structural members (i.e., a substructure). Thus, structural damages can be parameterized mathematically into the global stiffness matrix of a finite element model as follows:
In another similar formulation of damage identification, when measurement data are available for both before and after damage occurrence,
If there are uncertainties associated with model parameters for a specific model of the system
Bayesian inference essentially provides a probabilistic computational framework based on Bayes’ theorem to quantify uncertainties associated with model parameters and model classes by specifying the probability density function (PDF) (or probabilistic distribution) of uncertain model parameters and relative model probabilities of competing candidate model classes [
The key for applying Bayes’ theorem to update the probabilistic distributions of uncertain models and their parameters using the measurement data is to create a probability model
If the available inputoutput data
It is usually assumed that the prediction error vector
For the specific candidate structural model class
When there are uncertainties associated with the structural model of a specific system, classes of multiple competing candidate structural models may be considered for the system and denoted by
The model class with the largest posterior probability
The posterior PDF of model parameters as defined in (
One commonlyimplemented probabilistic simulation for drawing samples from the target PDF is the Markov Chain Monte Carlo (MCMC) algorithm proposed by Hastings [
In the subsequent descriptions of TMCMC procedures, the subscript
Briefly speaking, the method of TMCMC uses series of intermediate PDFs, in which the first PDF in the sequence is the prior PDF
Set the first intermediate PDF equal to the prior PDF, that is,
Choose the tempering parameter
Calculat the plausibility weight
Based on given
Stop, if
For the selected candidate structural model, the uncertain model parameters are selected as the parameters that need to be updated through Bayesian inference through drawing samples of parameters from the posterior PDF of parameters. While the candidate structural model is established by using finite element model, the TMCMC for sampling and Bayesian inference are coded and implemented in the MATLAB environment. To establish the computational framework, there is a need for the exchange of data between the finite element model and the MATLAB algorithm. To facilitate this data exchanging, the opensource program OpenSees is used to establish the finite element model for the structural system. OpenSees is the popular computational tool for studying dynamic responses of structures under the external excitations. It provides modular structures that allow users to develop and modify specific modules with relatively little dependence on other modules for the finite element analysis [
The applicability of the proposed framework is examined and illustrated for identifying damages of truss structures by using simulated measurements. The prototype truss structure is a ninebay steel truss structure as shown in Figure
Material and geometrical properties of the truss members.
Parameter  Columnbeam elements  Bolts  Balls 

Young’s modulus (N/m^{2}) 



Area (m^{2}) 

—  — 
Density (Kg/m^{3})  7850  7850  7850 
Mass (Kg)  2.835  0.05  3.868 
The Poisson ratio  0.3  0.3  0.3 
Moment of inertia 

—  — 
Moment of inertia 

—  — 
Simulation model of threedimensional prototype steel truss structure.
Prototype steel truss structure at the structure laboratory of Jackson State University
Simulation model in OpenSees
Details of prototype joint and finite element model of truss joints.
Details of prototype joint
Finite element model of truss joints
The finite element model for generating simulated measurements was established by using OpenSees program. All truss elements are assumed to be linearly elastic. This computational model consisted of 36 nodes and 100 truss elements. In addition, 200 zerolength elements developed in OpenSees program were adopted between the joint and the end of each truss element to simulate the complexity of truss joints. The rotational stiffness of the zerolength element is assigned with the value of
The simulated damage of truss structure was represented by the parameters of the stiffness reduction factor
To obtain simulated vibration measurements of the damaged truss structure, the excitations of the white noises are applied equally at the four top joints of the truss structure in the vertical direction. To examine the impact of quantities of measurements on the accuracy of damage identification, three different sensor deployment schemes, that is, foursensor measurement, eightsensor measurement, and tensensor measurement (see Table
Sensor deployment schemes.
Measurement scheme  Measurement type  Measurement nodes (locations)* 

Foursensor  Vertical acceleration  5, 6, 15, and 16 
Eightsensor  Vertical acceleration  5, 6, 7, 8, 15, 16, 17, and 18 
Tensensor  Vertical acceleration  5, 6, 7, 8, 15, 16, 17, 18, 22, and 31 
*Node locations can be referred to nodes in Figure
The proposed computational framework and its effectiveness are illustrated and examined through its application for damage identification of the truss structure using simulated accelerometer measurements as described above. For addressing uncertainties associated with the joint model, five different competing model classes were considered and compared for their accuracy in identifying truss member damages. For simplicity in the illustration, the selected five candidate model classes differ from each other only in modeling truss joints and are described in Table
Five competing model classes used for damage identification.
Model  Model description 

M1  Simple spatial truss structure model with fully pinned/hinge joints 
M2  Simple spatial truss structure model but with rigid joint connections 
M3  The segment connecting truss members and joints steel ball are represented by zerolength elements in OpenSees. The rotational stiffness of the zerolength elements is set equal to 
M4  The same model as 
M5  The same model as 
The simulated measurements from different sensor deployment schemes (see Table
For each model class with given set of measurements, the TMCMC algorithm was run with
The evolution of the TMCMC samples from the prior PDFs to the posterior PDFs for the model class M3 is illustrated in histograms of model parameter samples as shown in Figure
Histograms of PDFs of parameters from
The impact of measurement quantities on the accuracy of damage identification is revealed from results of damage identification based on the model class M3 using three different sensor deployments as specified in Table
Identified damages based on model class M3 with three measurement schemes.
Number of sensor measurements^{a}  Damage index 
Standard deviation of prediction error 







4sensor 
0.8880 
0.9202 
0.8612 
0.9075 
47.861 
8sensor 
0.8938 
0.8263 
0.8884 
0.9250 
56.838 
10sensor 
0.8370 
0.8527 
0.8602 
0.9507 
37.886 
True value  0.85  0.85  0.85  1.00  — 
Note: a: the measure point is referred to as sensor deployment schemes in Table
b: COV is the coefficient of variation of identified damage index and is listed in parenthesis below.
The impacts of model uncertainties on the damage identification are revealed through comparing identified damage obtained from different candidate model classes. Table
Identified damages based on five model classes with two measurement schemes.
Model class  Damage parameter  Using data from 4 measure points  Using data from 10 measure points  

Identified value (actual value)  COV 

Identified value (actual value)  COV 


M1 

0.9168 (0.85)  8.226%  −2139.6  0.8115 (0.85)  5.040%  −4560.6 

0.9540 (0.85)  8.327%  0.8378 (0.85)  7.626%  

0.9128 (0.85)  9.896%  0.7924 (0.85)  7.669%  

0.8972 (1.0)  9.042%  0.8373 (1.0)  10.032%  
M2 

0.7688 (0.85)  8.4173%  −1903.8  0.8266 (0.85)  5.707%  −1723.3 

0.9961 (0.85)  10.508%  0.8433 (0.85)  6.189%  

0.8469 (0.85)  5.8919%  0.9071 (0.85)  7.752%  

0.9504 (1.0)  6.3131%  0.9555 (1.0)  5.824%  
M3 

0.8880 (0.85)  11.189%  −1760.5  0.8370 (0.85)  7.181%  −1361.5 

0.9202 (0.85)  16.211%  0.8527 (0.85)  6.662%  

0.8612 (0.85)  11.318%  0.8602 (0.85)  8.306%  

0.9075 (1.0)  9.654%  0.9507 (1.0)  5.887%  
M4 

0.8959 (0.85)  6.7123%  −1752.5  0.8954 (0.85)  6.662%  −1543.6 

0.8313 (0.85)  10.538%  0.8729 (0.85)  9.782%  

0.9340 (0.85)  8.1471%  0.8848 (0.85)  5.845%  

0.8833 (1.0)  6.2183%  0.9320 (1.0)  8.620%  
M5 

0.8221 (0.85)  9.107%  −1760.1  0.8744 (0.85)  8.540%  −1375.7 

0.8452 (0.85)  8.291%  0.8779 (0.85)  7.072%  

0.9061 (0.85)  5.788%  0.8597 (0.85)  7.188%  

0.9537 (1.0)  6.637%  0.9333 (1.0)  8.371% 
Note: *log evidence here refers to the logarithm of the model evidence as defined in (
Table
With 10sensor measurements, the identified damages by using the model class M3 have the most accuracy, while the identified damages by using the model M1 have the poorest accuracy. This is because the model class M3 does not have model uncertainties and the model class M1 did not consider any extent of rigidity of joints by assuming that joints are simply hinged. While other model classes approximate the joint rigidity, they give the identified damage index close to the simulated damages. While the model classes M2, M4, and M5 have different model uncertainties, the damage indexes identified from those models are comparable. The accuracy of identified damages obtained from five candidate model classes by using 10sensor measurement scheme coincides with the actual extent of model uncertainties among five candidate model classes.
The model uncertainties among five candidate model classes and the accuracy of identified damage index
It also should be noted that even though the aforementioned model evidence provides effective index for selecting better models, it is only a relative index for comparison of competing models based on the same sets of measurement data. The model evidence may not provide the solid ground for comparing models under the different measurement conditions. For example, Table
With model evidence available from TMCMC, the relative probability of model can be evaluated based on (
When five competing model classes
Competing model classes and their prior and posterior probabilities.
Competing model classes  Prior model probability  Posterior model probability  

M1  M2  M3  M4  M5  M1  M2  M3  M4  M5  M1  M2  M3  M4  M5 
0.20  0.20  0.20  0.20  0.20  0.00  0.00  1.00  0.00  0.00  
M1  M2  M4  M5  M1  M2  M4  M5  M1  M2  M4  M5  
0.25  0.25  0.25  0.25  0.00  0.00  0.00  1.00  
M1  M2  M4  M1  M2  M4  M1  M2  M4  
0.33  0.33  0.33  0.00  0.00  1.00  
M1  M2  M1  M2  M1  M2  
0.50  0.50  0.00  1.00  
M2  M4  M2  M4  M2  M4  
0.5  0.5  0.00  1.00 
When four model classes M1, M2, M4, and M5 are considered without the model class M3, the prior model probability for each of the four model classes can be assumed to be 0.25, the posterior model probability of the model class M5 is near to 1.00, and the others are close to 0.00. The truss joints in the model M5 were modeled as the semirigid joint with the joint rotation stiffness that is larger than its true value that was used in the model class M3 to generate simulated measurements. Even though the model class M4 considers the semirigid joint with rotation stiffness that is half of the true value, its model probability is near to 0.00. This may be because the model class M5 has much better performance in modeling the truss structure than the model class M4.
However, when considering three model classes M1, M2, and M4 with absence of the models M3 and M5, the prior model probability for each of three model classes is 0.33, the posterior probability of model class M4 is equal to 1.00, and the others are close to 0.00 (see Table
If only two model classes M1 and M2 are considered and compared, the prior model probability for each of two model classes is 0.50, and the posterior probability of the model class M2 is 1.0 (see Table
In summary, the above application example reveals that the model evidence obtained from Bayesian model updating and the model probability obtained from Bayesian model class selection can be effectively used to identify the model which is most close to the “true” truss structure, when there are uncertainties associated with truss joint model and multiple competing candidate model classes are considered. When the true model class is not among the candidate model classes, the joint model that is more rigid than the “true” truss structure usually yields lager model probability than the joint model which is less rigid than the “true” truss structure. The accuracy of identified damage index may be poor if limited measurement data are available. However, it can be improved by increasing sensor channels. For all cases examined in the application example, the model with the largest model probability always much outperforms other candidate model classes. As a result, the model probability of the most plausible model class is near to 1.0, while the probability of other model classes is almost 0.0. In such cases, if Bayesian model averaging is conducted, the model averaging is actually predominated only by the most plausible model class.
This paper explores the applicability of proposed probabilistic framework into damage identification of truss structures under model uncertainties associated with truss joint details and measurement noises. However, the measurement data are numerical simulations obtained from structural finite element model with simulated damages of a truss structure. The complexity of joint configurations of the “true” truss structure is only represented by the semirigid rotational stiffness. The potentially damaged truss members are limited to four members. The further examination should be conducted using the measurement data obtained from the real physical truss structure with more induced damages occurring not only in truss members but also in truss joints. To make the probabilistic computational framework applicable for field implementation, the research needs and directions in this area should focus on two major aspects: one is to improve the computational efficiency of framework for reducing the computational time for implementation of the Bayesian probabilistic framework with more uncertain model parameters and the other is to address the impacts of environmental variations, such as change of temperature or humidity, on structural dynamic responses and distinguish them from those caused by structural damages.
This paper proposes and illustrates a new perspective for vibrationbased damage identification of truss structures under model uncertainties associated with truss joints based on Bayesian model updating and model class selection. The proposed probabilistic framework integrates the advanced sampling algorithm of transitional Markov chain Monte Carlo (TMCMC) with the finite element model created in OpenSees program to derive the probabilistic characteristics of associated uncertainties in terms of statistical samples of parameters of interest, model evidence, and model probabilities of competing model classes. The effectiveness of the framework is examined by using simulated measurements. The simulation results demonstrate that the model evidence obtained from TMCMC can effectively be used to quantify model probabilities of competing candidate model classes for assessing model uncertainties associated with truss joints, while statistical samples drawn from TMCMC can well approximate the updated probabilistic distributions of uncertain model parameters that represent damage in truss structures. It is also indicated that the model probability can provide effective index for describing the relative plausibility of the model class and selecting the model with fewer uncertainties among competing candidate model classes. The proposed framework can be used to identify the probabilistic characteristics of damage of truss members under the joint model uncertainties and measurement noise. The accuracy of identified damage index can be improved by using measurement data from more sensors.
Further examination of the effectiveness of the proposed framework should be conducted by using measurement data obtained from real physical truss structure with more damages occurring not only in truss members but also in truss joints. To make the probabilistic framework ready for practical implementation, the future research should focus on developing efficient algorithms for reducing the high computational demands of probabilistic simulation of large structures and distinguishing the impacts of environmental variations from those caused by damages on the measured structural dynamic response. The research direction in this regard should include (1) adopting efficient gradientbased sampling techniques that can replace random walking algorithm for proposing statistical samples; (2) implementing spectrabased Bayesian inference of damage identification based on outputonly dynamic response under ambient vibrations; (3) developing effective surrogate model or reduced model to represent the computationallyexpensive structure model; and (4) developing effective dynamic features that are extracted from the sensor measurements and are sensitive to structural damage and insensitive to environment variations for damage identification.
The authors gratefully acknowledge any support from the Institute for Multimodal Transportation at Jackson State University under Grant DTRT06G0049 and from National Science Foundation under Award NSF/DUE0837395. Any opinions, conclusions, or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of these funding agencies. The authors also thank Professor Jianye Ching from National Taiwan University for providing MATLAB codes of transitional Markov chain Monte Carlo simulation algorithm.