Finite element model validation of bridge based on structural health monitoring—Part I: Response surface-based finite element model updating

Accepted Manuscript Finite element model validation of bridge based on structural health monitoring—Part I: response surface-based finite element model updating Zhouhong Zong, Xiaosong Lin, Jie Niu PII:

S2095-7564(15)00054-9

DOI:

10.1016/j.jtte.2015.06.001

Reference:

JTTE 26

To appear in:

Journal of Traffic and Transportation Engineering (English Edition)

Please cite this article as: Zong, Z., Lin, X., Niu, J., Finite element model validation of bridge based on structural health monitoring—Part I: response surface-based finite element model updating, Journal of Traffic and Transportation Engineering (English Edition) (2015), doi: 10.1016/j.jtte.2015.06.001. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Original research paper

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Finite element model validation of bridge

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based on structural health

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monitoring—Part I: response

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surface-based finite element model

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updating

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Zhouhong Zonga,*, Xiaosong Linb, Jie Niua a

School of Civil Engineering, Southeast University, Nanjing 210096, China

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School of Civil Engineering, Fujian University of Technology, Fuzhou 350002, China

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Abstract

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In the engineering practice, merging statistical analysis into structural evaluation and assessment is a tendency in the future. As a combination of mathematical and statistical techniques, response surface (RS) methodology has been successfully applied to design optimization, response prediction and model validation. With the aid of RS methodology, these two serial papers present a finite element (FE) model updating and validation method for bridge structures based on structural health monitoring. The key issues to implement such a model updating are discussed in this paper, such as design of experiment, parameter screening, construction of high-order polynomial response surface model, optimization methods and precision inspection of RS model. The proposed procedure is illustrated by a prestressed concrete continuous rigid-frame bridge monitored under operational conditions. The results from the updated FE model have been compared with those obtained from online health monitoring system. The real application to a full-size bridge has demonstrated that the FE model updating process is efficient and convenient. The updated FE model can relatively reflect the actual condition of Xiabaishi Bridge in the design space of parameters and can be further applied to FE model validation and damage identification.

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Keywords:

response surface methodology; finite element model updating; structural health monitoring; parameter screening; finite element model validation; continuous rigid frame bridge model

36 *Corresponding author. Tel.: +86 25 83792550; fax: +86 25 83792550. E-mail addresses: [email protected] (Z. Zong), [email protected] (X. Lin), [email protected] (J. Niu). 1

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1 Introduction

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According to the definition by Thacker et al. (2004), model verification and validation (V&V) is an enabling methodology for the development of computational models that can be used to make engineering predictions with quantified confidence. Verification is the process of determining that a model implementation accurately represents the developer’s conceptual description of the model and its solution. Validation is the process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model. Both verification and validation are processes that accumulate evidence of a model’s correctness or accuracy for a specific scenario. Model V&V procedures are currently needed by government and industry to reduce time, cost, and risk associated with full-scale testing of products, materials, and weapon systems. For example, AIAA and ASME have issued guidelines for the V&V of computational fluid dynamics and solid mechanics respectively (AIAA, 1998; ASME, 2006). Some researchers from Los Alamos National Laboratory (LANL) (Thacker et al., 2004), Sandia National Laboratories (SNL) (Oberkampf et al., 2003) and Lawrence Livermore National Laboratory (LLNL) (Logan and Nitta, 2002) have built the general philosophy, definitions, concepts, and processes for conducting a successful V&V program. The expected outcome of the model V&V process is the quantified level of agreement between experimental data and model prediction, as well as the predictive accuracy of the model. This is consistent with the requirements of structural health monitoring (SHM) in civil engineering. The damage identification and damage prognosis are the theme issues on structural health monitoring (Farrar and Lieven, 2007; Farrar and Worden, 2007). In the past 20 years, there is a large volume of literatures relating to the dynamic- and static-based methods for damage identification and prognosis using the model updating strategy (Farrar et al., 2007; Friswell and Mottershead, 1995; Weber and Paultre, 2010; Xia and Brownjohn, 2004). In general, there are two kinds of model updating methods. Sensitivity-based FE model updating methods rely on the parametric model of a structure and the minimization of certain objective function based on the errors between measured data and prediction from the model. This method is time-consuming and has limited freedom domain. An alternative meta-model, which is fast-running and less parameter involved, can replace the sensitivity-based model updating, and response surface is one of the commonly used meta-models (Zong and Ren, 2012). Response surface methodology (RSM) was firstly proposed by Box and Wilson (1951). The basic idea of RSM is to build a model relating inputs and outputs using a small number of data sets even for large-scale structures by providing a way of

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rigorously choosing a few points in the design space to efficiently represent all possible points. In this sense, RSM has been applied in the fields of experimental design and analysis (Guo et al., 2006; Rijpkema et al., 2001), optimization (Jones, 2001) and structural safety and reliability analysis (Bucher and Bourgund, 1990; Cheng et al, 2007), etc. Box and Draper (2002), Myers and Montgomery (2002) summarized the fundamentals of this methodology, as well as the relevant research, applications and developments. Related to the model updating problem, RSM might reconstruct the relationship of the design and objective variables in the design space of parameters. The optimization iteration can be conducted in the reconstructed RSM, so that errors between FE model and test model are minimized. In recent years, some researches have been conducted into the application of RSM for model updating. Among them, Marwala (2004), Guo and Zhang (2004) first introduced the use of response surface method to structural model updating. Marwala (2004) used the multilayer perception (MLP) to approximate the implicit function between the response and parameters. Guo and Zhang (2004) constructed two high-order polynomial RS models to update an H-shaped structure using the central composite design (CCD) and the D-optimal design. In their study, stiffness and frequencies were chosen as the input and response features, respectively. They found that the RSM-based method was much more cost-efficient than the traditional sensitivity-based FE model updating considering likewise accurate predictions. Ren and Chen (2010) used the factorial design (FD) to quantitatively identify the relative significance of each updating parameter with respect to the responses, and the CCD was also employed to construct the RS models for updating a tested full-scale box-girder bridge in structural dynamics. With the response surface to replace the original FE model, the model updating process becomes efficient and converges quickly compared with the traditional sensitivity-based model updating method. Deng and Cai (2010) presented a new practical and user-friendly FE model updating method using the response surface method and genetic algorithm. Natural frequencies from modal analysis and strains/deflections from static tests are used as responses in the objective function. CCD was adopted to conduct the experimental design. Numerical examples of a simply supported beam and an existing bridge were used to demonstrate the proposed method. Results show that this method works well and achieves reasonable physical explanations for the updated parameters. Ren et al. (2011) proposed a response surface based FE model updating method using measured static responses of structures. The proposed method is verified against a numerical beam and an experimental full-scale continuous box-girder bridge. It is demonstrated that the proposed response surface-based FE model updating in structural statics has the advantages of easy implementation, high cost-efficiency, and

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adequate updating accuracy. Furthermore, some work has been performed to extend the concepts of RSM in damage identification problems. Cundy (2003) has explored the feasibility of applying RSM to structural damage identification (SDI) by using FD and CCD to construct response surface models for identifying the damage existing in a simulated mass–spring–damper system and a tested cantilever beam. In addition, Fang and Perera (2011) presented a damage identification method achieved by response surface based model updating. In this study, firstly D-optimal designs were used to establish response surface models for screening out non-significant updating parameters, and then first-order response surface models were constructed to substitute for FE models in predicting the dynamic responses of an intact or damaged physical system. Three case studies of a numerical beam, a tested reinforced concrete frame and a tested full-scale bridge have been used to verify the proposed method. It has been observed that the proposed method gives enough accuracy in damage prediction of not only the numerical but also the real-world structures with single and multiple damage scenarios, and the first-order response surface models based on the D-optimal criterion are adequate for such damage identification purposes. Based on structural health monitoring, these two serial papers present FE model updating and validation methods with aid of the response surface methodology. The key issues such as design of experiments, parameters screening, construction of a high-order polynomial response surface model, optimization methods, and precision inspection of RS model, are discussed in this paper. The proposed procedure is verified against a prestressed concrete continuous rigid-frame bridge monitored under operational conditions. It is demonstrated that the proposed response surface–based finite-element-model updating in structural dynamics has the advantages of easy implementation, high cost-efficiency, and adequate updating accuracy. The updated finite element model can be used to the further model validation and damage identification, and to be helpful for the bridge health monitoring and safety evaluation.

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The procedure of the third-order RS-based method to update the FE model is shown in Fig. 1 (Zong and Ren, 2012).

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Damage identification method based on RSM updating and EMSE index

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Fig. 1 Flowchart of response surface-based model updating

2.1

Design of experiment

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Design of experiment (DOE), which selects appropriate trial points from system variable parameter space, is a crucial emphasis of RSM. DOE is the strategy of designing optimized substitutive model for multi-subject purposes, which would determine the number of sample points and their distribution of the substitutive model. Suitable DOE method is vital for the purpose of attaining perfect RS model with the least sample points. The ideal DOE should satisfy the following requirements (Montgomery, 2006): (1) system parameters must be firstly defined, and the minimum number of design parameters under assurance of the modeling accuracy is required; (2) appropriate levels of system parameters will be determined instead of too many levels; (3)

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parameter values should be reasonable and easy to be calculated; (4) distribution of sample points within the possible parameter spaces should be fair and reasonable; (5) it does not need too many experiments or testing for the real physical system. The usual DOE methods used in RS method refer to full-factorial experimental design, central composite experiment design, D-optimality, orthogonal design, and uniform design, etc. Normally, orthogonal design and uniform design are usually applied to low-order RSM. Full-factorial experimental design needs too much computation though it could lead to relatively more accurate results. For the purpose of FE model updating in structural dynamics, central composite experiment design and D-optimality can be used for large-scale high-order RSM and achieve almost the same accuracy in the creation of polynomial response surface (Guo and Zhang, 2004). The regressed model in the D-optimality can be defined as follows (Box and Wilson, 1951):

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ya = β1 f1 ( xa ) + β 2 f 2 ( xa ) + L + β m f m ( xa ) + ε a , a = (1,L, N )

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(1)

The least-square estimation bT = (b1 , b2 ,L, bm ) of parameter β T = ( β1 , β 2 ,L , β m ) can be

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obtained by least square method. The serried ellipsoid of (b1 , b2 ,L, bm ) is defined to be

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the region surrounded by m-dimensional homogeneous distribution with the same mean-value and correlogram. The volume of the serried ellipsoid is the index for

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distribution and concentration level of (b1 , b2 ,L, bm ) . D-optimality design is an

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experimental plan, which can minimize the volume of the serried ellipsoid, i.e.

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(b1 , b2 ,L, bm ) is the highest concentrative. In this paper, D-optimality design was applied

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to design experiment and obtain sample points in the process of FE model updating and validation based on third-order RSM.

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The problem of screening and tuning of parameters to optimize a system’s performance has been studied extensively in areas of industrial engineering, using results obtained from the design and analysis of experiments. In the traditional FE model updating, parameter updating were usually selected in the empirical way and should be able to clarify the ambiguity of the model. It is also necessary for the model response to be sensitive to these selected parameters. In the frame of response surface-based FE model updating, the selected updating parameters can be evaluated from the sampled data by performing a hypothesis test according to

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Parameter screening

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statistical variance analysis. The F-test method can be employed to carry out the hypothesis test to check the significance of parameter: Fm =

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SSE ( x1 , x2 ,L, xm −1 )/m − F0.05,m ,n − m −1 SSE ( x1 , x2 ,L, xm −1 , xm )/(n − m − 1)

(2)

where Fm and F0.05,m ,n − m −1 denote the F-test value of an input parameter and the

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chosen F-test criterion, SSE ( x1 , x2 , L , xm −1 ) and SSE ( x1 , x2 ,L , xm −1 , xm ) are the sums of

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squares from the model and the residual under m-1 variables and m variables, respectively, n is the total number of all the independent variables of the regression

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model. If Fm ≥ F0.05,m ,n − m −1 , it indicates the input parameter xm significantly contributes to

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the total variance of the model and must be included in model updating. Compared with the traditional sensitivity analysis, the analysis of variance (ANOVA) method is derived from the global perspective on the entire design space to select sensitive features with the significant influence of design parameters, so it overcomes deficiencies which exist in the sensitivity analysis method.

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The choice of RS function type should satisfy the following two requirements: (1) the expression of mathematical function is as simple as possible, which can describe/reflect the true relationship between the input parameters and the output response of the system; (2) the number of undetermined coefficient in the expression is as less as possible to minimize the testing or calculation, the increase of the number of items would considerably increase the number of design space. For the majority of engineering applications, a polynomial function with interaction is adequate for representing real structural system. Order of the function is determined by the number of the points used to “train”. The third-order polynomial function is adopted to establish the response surface model in this paper:

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Selection and fitting of high-order RS functions

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i =1

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y = β0 + ∑βi xi + ∑βii xi 2 + ∑βiii xi 3 + ∑∑βij xi x j +

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where y is the response characteristic as output variable, xi (i = 1, 2,L, k ) is the

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selected design parameters by variance analysis, and xi ∈ [ xil , xiu ] , xil , xiu are upper

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limit and lower limit of xi , respectively, xi x j or xi x j xk denote the interaction effects

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among the i , j and k , ο is the error item, β 0 , β i , β ii , β iii , β ij , β iij , β ijk are regressive

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coefficients of corresponding parameters, respectively, determined by least-squares estimation. If the number of input parameters is 3, the number of regressive coefficients in the third-order polynomial expression will be 20, and if number of input parameters is 7, the number of the regressive coefficients in third-order polynomial expression will be 120.

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Before the regressive response surface is used to conduct FE model updating, it should be verified whether the regressed response surface has enough accuracy. R2 criterion and root mean squared error (RMSE) criterion are usually used for multi-RSM and complicated model (Box and Draper, 2002; Fang and Perera, 2011).

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Verification of RS model

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( j ) − y ( j )) 2

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the the the the

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RS

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where yRS(j) is the response value from RSM, y(j) is true value from the measurements, y is the average value of y, N is the number of confirmation sample points in design space. R2 is a statical correlation ratio, which is a measure for the reduction of variability in the output feature through the use of the input variables chosen for the model. The larger the value R2 is, the more accurate the regressed response surface is. RMSE is the square root of the mean square error, where the distance vertically from the point is taken to the corresponding point on the curve fit. The smaller the value of RMSE is, the closer the matching of the curve fits.

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Multiobjective optimization is concerned with the minimization of a vector of objective

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functions F ( x) subject to a number of constraints or bounds. A multiobjective

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optimization problem is formulated as follows.

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Optimization methods

min F ( x) = min{ f1 ( x), f 2 ( x),L , f p ( x)}

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subject to gi ( x) = 0, i = 1, 2,L, m , h j ( x ) ≤ 0, j = 1, 2,L, n

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(6)

T where x = ( x1 , x2 ,L , xk ) ( k ≥ 2 ) is the vector of design parameters, fi(x) ( i = 1, 2,L, p ) is

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problems. In this method, a vector of design goal F ( x) = { f1 ( x), f 2 ( x),L , f p ( x)} , which

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* * * * is associated with a set of objective function F ( x) = { f 1 ( x), f 2 ( x),L, f m ( x)} , is

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the initial objective functions, the vector function gi(x) and hj(x) return the values of the equality and inequality constraints. As a result of the multiobjective optimization problem, all individual for the objective functions is often not commensurable. The most optimal solution of multiobjective optimization problem, where each sub-objective function achieves the most optimal solution as far as possible, was called Pareto optimality (Censor, 1977). In general, Pareto optimality of the multiobjective optimization problem is an optimal solution set. The goal of the multiobjective optimization is to construct non-dominated set, and make the non-dominated set approach to approximate optimal solution set, ultimately to achieve the most optimal solution. The goal-attainment method (Chankong and Haimes, 1983; Gembicki and Haimes, 1975) is a powerful tool to find the best-compromising solution in multiobjective

expressed. The relative degree of under- or over-attainment of the desired goals is

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controlled by a vector of weights: ϖ = {ϖ 1 ,ϖ 2 ,L ,ϖ p } . In the optimization process, the

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following nonlinear programming problem is solved: min λ

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subject to F ( x) + ϖ i λ ≥ F * ( x)

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where x is a set of desired parameters which can be varied, λ is a scalar variable which introduces an element of slackness into the system, Ω is a feasible solution region that satisfying all the parametric constraints, and

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Λε = ϖ ∈ R n , s. t. ϖ i ≥ ε , ∑ϖ i = 1, ε ≥ 0  .

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Fig. 2 shows the geometrical illustration of two-dimensional goal-attainment method. The multiobjective optimization is concerned with the generation and selection of non-inferior solution points (Chu, 1970) to characterize the objectives, where an improvement in one objective requires degradation in the other objectives. By varying

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ϖ over Λε , the set of non-inferior solutions is generated. In the two-dimensional

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representation of Fig. 2, the set of non-inferior solution lies on the curve AB. The weight vector ϖ enables the designer to express a measure of the relative trade-offs

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between the objectives. Given the vectors ϖ and F ( x) , the direction of the vector

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F ( x ) + ϖ i λ is determined. A feasible point on this vector in function space, which is

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closest to the origin, is then searched. The first point λ0 at which F ( x) + ϖ i λ

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intersecting the feasible region H in the function space would be the optimal non-inferior solution. During the optimization, λ is varied, which changes the size of the feasible region. The constraint boundaries converge to the unique solution point

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(f

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Fig. 2 Geometrical illustration of two-dimensional goal-attainment method

If ϖ i = 0 , it means that fi is the maximum limit of objective f i* ( x) . Starting with an

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initial feasible solution, the objective constraints are checked. If any of these

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* constraints is violated (i.e. fi ( x) > fi for any ϖ i ≠ 0 ), a new solution is sought. The

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* function λ = max i ( fi ( x) − fi )/ϖ i for any ϖ i ≠ 0 forces the new solution to satisfy the

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constraint F ( x ) + ϖ i λ ≥ F * ( x) by choosing the largest value of λ , and an optimization

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technique is applied to minimize the value λ , The simulated annealing technique (Kirkparick et al., 1983) is employed as the optimization technique in the present work. It is important to check whether the solutions obtained from the goal attainment method are Pareto optimal points or not. For this purpose, the optimization is carried out with different goals and weights to the objective functions. The goals and weights are given numerical values by user. Each time the algorithm gives the vector of design variables and values of objective functions. In this paper, the optimization process is

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accomplished by using optimization toolbox of MATLAB 6.5.1.

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3 Structural health monitoring of Xiabaishi Bridge

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Xiabaishi Bridge (Fig. 3) is a long-span PC rigid-frame bridge in the highway from Fuzhou City to Wenzhou City, consisting of four spans (145 m+2×260 m+145 m). The bridge is located in the line of vertical curve, the radius of which is 52,000 m. The width of the upper decks is 24.50 m, including two parallel box-girders, connected with four transverse beams in the position of main piers. The cross section of box girder is the single-box-girder, the height of which varies from 14 m on the top of the pier to 4.2 m in the middle of main spans and present-casting segments. The KPZ series basin type rubber bearings were adopted in the bridge, and the double thin-wall pier were used to support the box girder. The load class of the bridge is Steam super-20, hang-120. The bridge was open to traffic in July 2003.

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Description of Xiabaishi Bridge

Fig. 3 Xiabaishi Bridge

3.2 Structural health monitoring of Xiabaishi Bridge

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The structural health monitoring system (SHMS) was implemented successfully in Xiabaishi Bridge in 2007 (Zong et al., 2009). From then on, the SHMS of Xiabaishi Bridge demonstrates normal operation and stable performance, and provides accurate and reliable monitoring data. The contents of real-time monitoring include the vertical, transversal and longitudinal vibration responses of the bridge (Fig. 4). The output-only data modal parameter identification was carried out by using the peak picking of average normalized power spectral densities (ANPSDs) in frequency-domain and stochastic subspace identification in time-domain (Ren and Zong, 2004) to extract the modal characteristics of Xiabaishi Bridge. It is obvious that the dynamic characteristics derived from the ambient vibration

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testing are related to not only the mass and boundary conditions of the bridge but also the ambient factors, such as the ambient temperature. Particularly, temperature has significant effects on the long-span bridge structures. In this study, on the basis of modal analysis of the SHMS data of the Xiabaishi Bridge, a temperature updating model was utilized to exclude the model frequency variation form the baseline temperature field (20 °C).

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Fig. 4 Acceleration time-history curves of Xiabaishi Bridge (7-1: vertical; 7-2: transverse; 7-3: longitudinal)

The first six-order of vertical, first two-order of transverse, and first one-order of longitudinal modal frequencies were obtained by modal analysis of 276 field measured samples, which were chosen from one year’s monitoring data of Xiabaishi Bridge. The temperature updating model is formulated as follows.

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ync = y% nc − ( yˆ c (Tn ) − yˆ c (T0 ))

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c c where yn is the cth-order modal frequency after temperature updating, y% n is the field

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measured value of cth-order modal frequency on the n time interval, yˆ c (⋅) is the

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distributed model about temperature of cth-order modal frequency, Tn indicates the temperature vector value in the measured regression model on the n time interval, T0 indicates the reference baseline temperature (20 °C ). The statistical values of field measured frequencies after temperature model updating are shown in Table 1.

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Table 1 Statistical values of field measured frequencies after temperature updating Mean value µ

Standard deviation σ

Coefficient of variation σ/µ (%)

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0.418

0.013

3.107

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0.527

0.013

2.467

1

0.659

0.023

3.490

2 3 4

0.813 1.261 1.404

0.019 0.033 0.028

2.337 2.617 1.994

Modal order

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Vertical

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Longitudinal

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1.675 1.892

0.022 0.011

1.313 0.581

1

1.684

0.025

1.485

4 Response surface-based model updating of Xiabaishi Bridge

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Based on the configuration of the bridge, the initial three-dimension FE model of Xiabaishi Bridge was established based on ANSYS platform. Several types of finite elements were used to model the different structural components. The concrete box girders (including flange, web, and diaphragms) in box girder were modeled by Solid45 element, and reinforce steel was modeled by Link8 element. The pavement concrete was modeled by the concentrated mass elements (Mass21). Bridge bearings were modeled by a set of Combine14 spring element connecting the superstructure, the piers and the platform. It was also recognized the need of spring elements (Combine14) to account for the longitudinal restrained action of expansion joints from the adjacent span at both ends. Because the prestress force of internal prestressing tendons has no obvious effects on vibration frequencies (Dall’Asta and Dezi, 1996), the prestressing force effects were not considered in the FE model in this paper. In the initial FE model, the initial material characteristics of Xiabaishi Bridge are selected as follows: the main box girder of bridge uses C60 concrete strength grade with the initial

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elastic modulus as Ec = 3.65 × 104 MPa ; the main pier/column of the bridge uses C50

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concrete strength grade with the initial elastic modulus as Ec = 3.50 × 10 4 MPa ; the

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concrete density of main bridge is 2450 kg/m3, and the Poisson ratio is 0.167; the 3-D FE model of Xiabaishi Bridge has 29,250 elements and 45,975 nodes, respectively (Fig. 5).

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Initial FE model of Xiabaishi Bridge

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Fig. 5 FE model of Xiabaishi Bridge

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4.2

Feature extraction and parameter screening

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The success of FE model updating largely depends on the feature extraction and parameters selection. Model frequencies are generally extracted to represent the response characteristics in the dynamic-based model updating. According to the health monitoring results, the first six-order of vertical, first two-order of transverse and first one-order of longitudinal modal frequencies were selected as the dynamic characteristics of Xiabaishi Bridge in this study. In practice, parameters errors and uncertainties are the main reasons which cause the discrepancies between simulation and experiments. The updating parameter selection should be made with aim of considering the errors and uncertainties in the model. Physical and geometric properties of the finite element, as well as the stiffness of connection, are normally selected as the updating parameters. In this study, total seven parameters, representing elastic modulus of concrete material of bridge box-girder, the stiffness of connection elements at the supporting bearing, expansion joints and adjacent constraints, were chosen as the updating parameters. The first is the multiple of relative initial value of the elastic modulus of concrete material which is represented by N. The vertical, longitudinal and transverse spring element stiffness of support bearings and expansion joints are represented by RV1, RL1, RH1, respectively. The vertical, longitudinal and transverse spring element stiffnesses between two adjacent box-girder are represented by RV2, RL2, RH2, respectively. According to engineering experience and trial calculation, the initial values of the selected parameters and their low and upper bounds are listed in Table 2. Although it is very difficult to accurately estimate the variation bound of the parameters during the updating, a quite large variation in the parameters of the connection elements is expected due to more possible uncertainties involved in the bridge bearing and adjacent constraints.

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365

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Table 2 Parameter range under updating

Parameter

Minimum value

Maximum value

RV1 RV2 RL1 RL2 RH1 RH2 N

40.00 40.00 10.00 0.05 8.00 1.00 0.85

240.00 320.00 150.00 15.00 80.00 20.00 1.25

Initial value 100.00 80.00 50.00 3.00 15.00 5.00 1.00

392

Note: unit of RV1, RV2 and RL1 is 107 N/m, unit of RL2, RH1 and RH2 is 106 N/m, and unit of N is 3.65×104 MPa.

393

4.3 Design of experiments

394 395

Once the model parameters and their bounds are selected, the sample parameters points can be determined by design of experiments. D-optimality design is used to

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402

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obtain the sample points of the parameter values. There are 120 undetermined coefficients in the third-order response surface polynomial function, which includes 7 input parameters. Total 130 times of experiment design were conducted and listed in Table 3. With these sampled parameters values, corresponding bridge natural frequencies (response values) were calculated from FE modal analysis, and listed in Table 4. Table 3 Parameter combinations using D-optimality design Times

RV1

RV2

RL1

RL2

1

24.00

32.00

1.00

5.07

2

24.00

4.00

1.00

3

24.00

32.00

15.00

4

24.00

4.00

1.00

15.00

5

4.00

32.00

6

4.00

4.00

7

4.00

4.00

8

24.00

22.67

9

24.00

32.00

M

RH2

32.00

13.67

15.00

22.67

13.67

0.80

0.10

4.00

1.00

1.30

SC

RH1

N

0.80

1.00

1.05

20.00

0.97

4.00

1.00

1.30

32.00

20.00

1.30

32.00

7.33

1.30

32.00

20.00

0.97

M

M

M

M

1.00

5.07

32.00

7.33

1.30

4.50

3.83

11.00

5.75

1.18

M AN U

32.00

22.67

15.00

0.10

15.00

15.00

15.00

15.00

15.00

15.00

15.00

15.00

M

M

M

122

24.00

32.00

123

9.00

11.00

124

24.00

4.00

8.00

15.00

4.00

20.00

0.80

125

24.00

4.00

1.00

0.10

32.00

20.00

1.30

126

24.00

32.00

15.00

10.03

4.00

13.67

0.80

127

19.00

11.00

4.50

3.83

25.00

5.75

1.18

128

4.00

13.33

15.00

15.00

13.33

1.00

0.80

129

24.00

32.00

1.00

0.10

22.67

13.67

0.80

8.00

0.10

32.00

1.00

0.80

403

EP

130

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396 397 398 399 400 401

4.00

4.00

Table 4 Frequency response sample values using D-optimality design

AC C

Times

H1

H2

V1

V2

V3

V4

V5

V6

L1

1

0.341

0.466

0.562

0.691

1.092

1.128

1.273

1.567

1.589

2

0.343

0.464

0.532

0.674

1.058

1.100

1.312

1.703

1.566

3

0.398

0.510

0.681

0.876

1.334

1.471

1.645

1.971

1.988

4

0.352

0.437

0.567

0.726

1.077

1.159

1.198

1.534

1.512

5

0.381

0.524

0.600

0.764

1.251

1.297

1.462

1.998

1.783

6

0.388

0.475

0.630

0.816

1.222

1.373

1.596

1.943

1.982

7

0.433

0.593

0.630

0.833

1.373

1.384

1.632

2.114

1.982

8

0.417

0.562

0.668

0.866

1.314

1.460

1.704

2.009

1.986

9

0.381

0.524

0.602

0.768

1.297

1.303

1.499

1.999

1.785

M

M

M

M

M

M

M

M

M

M

122

0.415

0.558

0.685

0.854

1.253

1.355

1.413

1.840

1.801

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0.393

0.523

0.624

0.804

1.185

1.310

1.569

1.884

1.692

124

0.353

0.478

0.535

0.686

1.134

1.177

1.348

1.846

1.871

125

0.428

0.584

0.627

0.810

1.282

1.323

1.368

1.972

1.957

126

0.347

0.474

0.575

0.717

1.142

1.209

1.395

1.756

1.846

127

0.394

0.524

0.624

0.806

1.194

1.322

1.578

1.892

1.699

128

0.321

0.408

0.555

0.701

1.049

1.148

1.340

1.566

1.721

129

0.339

0.466

0.552

0.681

1.091

1.100

1.127

1.567

1.588

130

0.312

0.392

0.503

0.662

0.982

1.099

1.299

1.551

1.538

RI PT

123

As mentioned in the section of parameter screening, the F-test method was employed to carry out the hypothesis test to check the significance of parameter, in which variables were accepted or rejected. Table 5 shows the results of F-test of model frequency regression equations. It can be concluded that the selected 7 parameters are all sensitive to the response characteristics.

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Table 5 F-test of each regression equation model frequency

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404 405 406 407 408

F value (α=0.05)

Modal frequency 1 2 1 2 3 4 5 6 1

Transverse

Vertical

4665.86 3724.12 450.40 573.96 169.19 89.61 102.48 169.18 80.59

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Longitudinal

F critical value 1.71 1.71 1.72 1.72 1.72 1.69 1.74 1.74 1.69

Significance Noteworthy Noteworthy Noteworthy Noteworthy Noteworthy Noteworthy Noteworthy Noteworthy Noteworthy

4.4 Fitting of response surface function

411 412 413 414 415 416 417 418 419 420 421 422 423

Using least-square fitting with these sample values, three degree polynomial response surface can be constructed with regard to the parameters N, RV1, RL1, RH1, RV2, RL2 and RH2. For the construction of a third-order polynomial response surface in FE model updating in structural dynamics, some of the interaction effects in the surface expression are less significant on the structural natural frequencies. Therefore, some cross-three items in the third-order polynomial can be omitted to increase the efficiency of structural FE model updating. The final regression coefficients of the created third-order polynomial surface without the cross-three terms are shown from Eqs. (9) to (17). Figs. 6-14 illustrate the created response surface shapes of the vibration frequencies. The horizontal axes are the selected parameters (N, RV1, RL1, RH1, RV2, RL2 and RH2), while the vertical axis gives a value of each order of natural frequency (response) at any point or location when one parameter is changed and the other parameters are fixed to zero.

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H1 = 0.12836 + 3.68744 × 10-5 RV 1 + 1.34831 × 10-3 RV 2 + 2.96994 × 10-4 RL1 + 9.68159 × 10-4 RL 2 + 1.38378 × 10-4 RH 1 + 4.29834 × 10-3 RH 2 + 0.25287 N + 1.85393 × 10-5 RV 1RV 2 + 6.71734 × 10-6 RV 1RL1 - 5.91576 × 10-6 RV 1RL 2 + 3.51574 × 10-6 RV 1RH 1 + 2.21751 × 10-6 RV 1RH 2 -

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1.49944 × 10-4 RV 1N + 3.83066 × 10-6 RV 2 RL1 -1.73559 × 10-5 RV 2 RL 2 - 4.88902 × 10-6 RV 2 RH 1 -

5.09800 × 10-5 RV 2 RH 2 -1.09696 × 10-4 RV 2 N + 7.07091 × 10-6 RL1RH 1 + 5.28595 × 10-5 RL1RH 2 +

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2.12060 × 10-6 RL 2 RH 1 - 5.64181 × 10-4 RL 2 N + 3.57521 × 10-6 RH 1RH 2 - 3.12208 × 10-4 RH 1N +

(9

5.87581 × 10-4 RH 2 N -1.13489 × 10-6 RV21 - 6.11041 × 10-5 RV2 2 - 2.60453 × 10-5 RL21 -1.33915 ×

10-5 RL22 - 2.46532 × 10-4 RH2 2 - 0.045157 N 2 - 3.26295 × 10-7 RV 1RV 2 RL1 - 3.83127 × 10-7 RV 1RL 2 RH 1 +

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1.46619 × 10-5 RV 1RL 2 N + 1.63818 × 10-5 RV 2 RL 2 N + 5.57860 × 10-6 RV 2 RH 1 N - 5.42570 × 10-7 RL1RH 1RH 2 6.01549 × 10-7 RV21RV 2 + 4.81849 × 10-7 RV2 2 RH 2 + 1.01505 × 10-6 RV 2 RH2 2 -1.51473 × 10-6 RL1RH2 2 +

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1.08585 × 10-4 RH 1N 2 + 1.00055 × 10-6 RV3 2 + 5.32210 × 10-6 RH3 2

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)

H 2 = 0.14739 + 4.65791 ×10-4 RV 1 + 3.08172 ×10-3 RV 2 + 1.47273 ×10-3 RL1 + 4.39581 ×10-4 RL 2 + 5.17602 ×10-4 RH 1 + 9.43536 ×10-3 RH 2 + 0.33316 N - 3.62392 ×10-5 RV 1RL2 + 9.48577 ×10-6 RV 1RH 1 + 3.35135 ×10-5 RV 1RH 2 - 3.26771 ×10-4 RV 1 N -1.92362 × 10-4 RV 2 RH 2 + 2.59884 ×10-4 RV 2 N + 2.89071 ×10-5 RL1RH 2 + 9.41724 ×10-6 RL 2 RH 1 - 4.39415 ×10-4 RL 2 N 1.23476 ×10-3 RH 1 N + 5.66419 ×10-3 RH 2 N -1.64387 ×10-5 RV21 - 9.79355 ×10-5 RV2 2 -

(10)

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1.67647 ×10-4 RL21 - 6.15479 ×10-4 RH2 2 - 0.078286 N 2 - 8.94690 × 10-7 RV 1RL 2 RH 1 + 3.24939 ×10-5 RV 1RL2 N + 8.46271 ×10-7 RV21RL 2 -1.41037 ×10-6 RV 1RH2 2 + 1.95663 ×10-6 RV2 2 RH 2 + 3.61577 ×10-6 RV 2 RH2 2 + 6.30462 ×10-4 RH 1 N 2 -1.44150 ×10-4 RH2 2 N + 1.11073 ×10-6 RV3 2 +

EP

5.30087 ×10-6 RL31 + 1.43195 ×10-5 RH3 2

V1 = 0.13129 + 5.73808 ×10-3 RV 1 + 7.38600 ×10-3 RV 2 + 1.19852 ×10-3 RL1 + 4.79507 ×10-3 RL2 -

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2.53723 ×10-4 RH1 -7.54579 ×10-4 RH 2 + 0.52529N + 5.23490 ×10-6 RV1RV 2 + 3.24251×10-5 RV1RL1 9.20517 ×10-5 RV 1RL2 + 4.98462 ×10-5 RV 1RH1 -0.010424RV 1N -1.00039 ×10-4 RV 2 RL1 1.85561×10-4 RV 2 RL2 -3.63220 ×10-4 RV 2 N + 2.45305 ×10-5 RL1RH1 + 3.67255 ×10-5 RL1RH 2 -

427

1.90291×10-5 RL2 RH1 -3.35615 ×10-3 RL2 N + 9.67559 ×10-6 RH1RH 2 + 6.26131×10-4 RH 2 N 1.23964 ×10 R - 2.87910 ×10 R - 6.61254 ×10 R + 3.21630 ×10 R - 0.12816N -5

2 V1

-4

2 V2

-5

2 L1

-4

2 L2

1.67875 ×10-6 RV1RV 2 RL1 + 9.78457 ×10-5 RV 1RL2 N + 7.32314 ×10-5 RV 2 RL2 N 2.27216 ×10-6 RL1RH1RH 2 -1.75245 ×10-6 RV21RH1 + 4.72319 ×10-3 RV 1N 2 + 3.08056 ×10-6 RV22 RL1 + 2.76506 ×10-6 RV22 RL2 + 4.49751×10-6 RV32 -1.64446 ×10-5 RL32

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V 2 = 0.29902 + 2.02101×10-3 RV 1 + 6.01879 ×10-3 RV 2 -5.33754 ×10-3 RL1 + 2.24536 ×10-3 RL2 -8.10087 ×10-6 RH1 -1.76598 ×10-3 RH 2 + 0.46314N 7.19153×10-5 RV 1RL1 + 1.89217 ×10-5 RV1RH 2 -3.53695 ×10-5 RV 2RL1 -

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6.52969 ×10-5 RV 2RL2 -1.22920 ×10-5 RV 2Rh2 - 2.54611×10-4 RV 2 N -

4.41122 ×10-5 RL1RL2 + 1.07313 ×10-5 RL1RH1 + 1.11763 ×10-4 RL1RH 2 + 428

0.012520RL1N + 6.40920 ×10-5 RL2RH1 + 7.89101×10-5 RL2RH 2 -1.44008 ×

(12)

10-3 RL2 N - 4.37047 ×10-7 RH1RH 2 + 1.60919 ×10-4 RH1N + 7.52860 ×10-3 RH 2 N 7.12371×10-5 RV21 - 2.88161×10-4 RV22 + 2.99364 ×10-4 RL21 -3.50979 ×10-4 RH2 2 -

SC

0.10318N 2 + 1.54592 ×10-6 RV 2RL1RH 2 -3.12041×10-6 RV 2RL2RH 2 + 9.84828 × 10-5 RV 2RL2 N -1.58220 ×10-6 RL1RH1RH 2 -1.58737 ×10-4 RL1RH 2 N - 6.96568 ×

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10-5 RL2RH1N + 4.52949RV1RL22 -5.55204 ×10-4 RL22 N -1.97762 ×10-4 RH2 2 N + 5.34482 ×10-6 RV32 + 1.52862 ×10-5 RH3 2

V 3 = 0.14130 + 0.013065RV 1 + 4.87851×10-3 RV 2 - 4.76474 ×10-3 RL1 - 3.68559 ×10-3 × RL2 + 0.013302RH1 -0.010482RH 2 + 1.23064N -6.45008 ×10-5 RV 1RV 2 -1.38034 ×10-4 × RV 1RH1 + 9.09935 ×10-5 RV1RH 2 - 2.67340 ×10-3 RV 1N - 2.46843×10-4 RV 2 RL2 - 4.57226 × 10-5 RV 2 RH1 - 4.84696 ×10-5 RV 2 RH 2 + 0.016679RL1N + 2.07447 ×10-4 RL2 RH1 -9.76996 × 10-5 RL2 RH 2 + 7.31869 ×10-3 RL2 N + 9.61645 ×10-5 RH1RH 2 - 0.022237RH1N + 2.87593×

TE D

429

10-3 RH 2 N -5.98964 ×10-4 RV21 -5.39517 ×10-5 RV22 + 6.89815 ×10-4 RL21 -5.20268 ×10-5 ×

(13)

RH2 1 + 1.60012 ×10-3 RH2 2 -0.41457N 2 + 3.33802 ×10-6 RV1RV 2 RH1 - 4.59049 ×10-6 RV1RH1 × RH 2 + 1.60554 ×10-4 RV 1RH1N - 4.06312 ×10-6 RV 2 RL2 RH 2 + 8.00631×10-6 RL2 RH1RH 2 -

EP

3.03692 ×10-4 RL2 RH1N -1.00311×10-4 RH1RH 2 N + 8.44822 ×10-6 RV22 RL2 -1.09368 ×10-3 × RL21N + 0.011127RH1N 2 +1.00486 ×10-5 RV31 - 4.53038 ×10-5 RH3 2

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V 4 = 0.69458 - 0.013769 RV 1 + 2.91761 × 10-3 RV 2 + 0.021807 RL1 + 0.016919 RL 2 + 1.02268 × 10-3 RH 1 + 4.26179 × 10-3 RH 2 + 0.33418 N - 7.05290 × 10-5 RV 1RV 2 -1.10221 × 10-4 RV 1 RL 2 1.36091 × 10-4 RV 1RH 2 + 0.020348 RV 1 N -1.10090 × 10-4 RV 2 RL1 -1.68729 × 10-4 RV 2 RH 2 -

430

1.27008 × 10-3 RL1RL 2 -1.9606 × 10-4 RL1RH 1 + 1.87004 × 10-4 RL1RH 2 - 5.30115 × 10-4 RL1 N 5.33443 × 10-5 RL 2 RH 1 - 8.43685 × 10-4 RH 1 N + 4.92256 × 10-4 RV21 - 7.10294 × 10-4 RL21 9.61342 × 10-4 RL22 -1.79379 × 10-4 RH2 2 + 5.81090 × 10-6 RV 1RV 2 RH 2 + 9.45837 × 10-6 RV 2 RL1RH 2 + 2.13375 × 10-4 RL1RH 1 N - 6.37962 × 10-4 RV21 N + 1.07926 × 10-5 RV 1 RL22 + 2.00641 × 10-5 RL21 RL 2 + 5.85444 × 10-5 RL1 RL22 - 2.37292 × 10-5 RL1RH2 2 + 1.50689 × 10-5 RH3 2

(14)

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V 5 = 0.51483 - 0.027666 RV 1 - 5.07835 × 10 -3 RV 2 + 0.052584 R L 1 + 0.038083 R L 2 + 0.014612 R H 1 - 0.019467 R H 2 + 1.00627 N + 7.43678 × 10 -4 RV 1 R H 1 + 6.41748 × 10 -4 RV 1 R H 2 + 0.039969 RV 1 N - 4.08529 × 10 -4 RV 2 R L 2 + 4.38266 × 10 -4 R L1 R H 1 + 1.36293 × 10 -3 R L1 R H 2 + 0.048688 R L 1 N +

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1.08149 × 10 -4 RV 2 R H 1 + 1.93965 × 10 -4 RV 2 R H 2 - 4.66399 × 10 -3 R L 1 R L 2 5.41780 × 10 -4 R L 2 R H 1 - 6.90889 × 10 -4 R L 2 R H 2 + 5.88101 × 10 -3 R L 2 N +

2.35352 × 10 -4 R H 1 R H 2 - 0.039849 R H 1 N - 1.61930 × 10 -3 R H 2 N - 7.09685 × 10 -5 RV2 1 + 1.0 0851 × 10 -4 RV2 2 - 6.58672 × 10 -3 R L21 - 1.92149 × 10 -3 R L2 2 +

7.83902 × 10 -4 R H2 2 - 0.35156 N 2 - 1.67963 × 10 -5 RV 2 R L 2 R H 2 - 5.29671 ×

SC

10 -6 RV 2 R H 1 R H 2 + 1.89557 × 10 -5 R L 1 R L 2 R H 1 - 1.46410 × 10 -5 R L1 R H 1 R H 2 +

5.19193 × 10 -4 R L 1 R H 1 N - 6.83898 × 10 -4 R L 2 R H 1 N + 4.55404 × 10 -4 R L 2 R H 2 N 2.60168 × 10 -5 RV2 1 R H 1 - 1.39966 × 10 -3 RV2 1 N - 2.65119 × 10 -5 RV 1 R H2 2 +

432

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4.29065 × 10 -5 RV 2 R L2 2 + 1.52692 × 10 -4 R L21 R L 2 - 2.78364 × 10 -3 R L21 N +

6.79337 × 10 -5 R L 1 R L2 2 - 5.57440 × 10 -5 R L 1 R H2 2 + 3.64253 × 10 -5 R L2 2 R H 2 +

(15)

0 .018090 R H 1 N 2 + 4.00634 × 10 -5 RV3 1 + 2.67539 × 10 -4 R L31

V 6 = 0.17393 + 0.046914 RV 1 -1.71877 × 10-3 RV 2 + 0.079641RL1 + 0.022388RL 2 + 6.84027 × 10-3 RH 1 + 2.29904 × 10-3 RH 2 + 1.67968 N + 2.29278 × 10-5 RV 1RV 2 9.00909 × 10-5 RV 1RL1 - 7.60421 × 10-5 RV 1RH 1 - 0.086439 RV 1 N + 3.63379 × 10-4 RV 2 RL1 -

TE D

8.74840 × 10-5 RV 2 RL 2 - 4.81979 × 10-5 RV 2 RH 1 - 7.89853 × 10-5 RV 2 RH 2 + 4.67556 × 10-3 RV 2 N 1.98389 × 10-3 RL1RL 2 + 3.37975 × 10-5 RL1RH 1 - 2.14922 × 10-5 RL1RH 2 + 0.014304 RL1 N -

433

1.30863 × 10-4 RL 2 RH 1 - 2.35293 × 10-4 RL 2 RH 2 - 0.015566 RL 2 N - 3.47239 × 10-4 RH 1RH 2 +

(16)

8.24885 × 10 RH 2 N - 9.03441 × 10 R + 4.17636 × 10 R -1.73466 × 10 R + -3

-3

2 L1

-4

2 L2

-4

2 H1

EP

9.30319 × 10-4 RH2 2 - 0.49926 N 2 - 7.42703 × 10-6 RV 1RV 2 RL1 + 9.05858 × 10-6 RV 1RL1RH 1 + 1.56108 × 10-5 RV 2 RL1RH 2 - 4.37601 × 10-4 RV 2 RL1 N + 4.95934 × 10-6 RV 2 RL 2 RH 1 + 1.92160 × 10-5 RL1RL 2 RH 2 + 1.25917 × 10-3 RL1RL 2 N - 8.53312 × 10-6 RL1RH 1RH 2 -

AC C

1.52100 × 10-3 RL1RH 2 N + 0.040453RV 1 N 2 + 5.98245 × 10-5 RL21RL 2 - 5.90937 × 10-5 RL1RL22 + 7.83998 × 10-5 RH 1RL22 + 1.24273 × 10-5 RH2 1RH 2 + 2.81115 × 10-4 RL31 - 4.49747 × 10-5 RH3 2

L1 = 0.87213- 6.08630 × 10-3 RV 1 -1.10891 ×10-3 RV 2 + 0.015154 RL1 - 0.010143RL 2 + 7.12400 × 10-4 RH 1 + 0.023293RH 2 + 0.56298N + 8.36542 × 10-5 RV 1RV 2 + 1.73274 × 10-5 RV 1RL1 + 1.01724 × 10-3 RV 1RL 2 2.39897 × 10-4 RV 1RH 2 + 1.66952 × 10-4 RV 2 RL1 + 5.36725 × 10-3 RL1RL 2 - 2.90155 × 10-3 RL1RH 2 +

434

0.010776RL1 N + 1.35968 ×10-4 Rl 2 RH 1 - 3.93724 × 10-4 RL2 RH 2 -1.41990 ×10-5 RH 1RH 2 + 2.46759 × 10-4 RV21 - 2.88372 ×10-4 RL21 + 3.82519 × 10-4 RL22 -1.15058 × 10-5 RV 1RV 2 RL1 + 2.20692 × 10-5 RV 1RL1RH 2 + 1.68380 × 10-5 RV 1RL 2 RH 2 + 5.65130 × 10-5 RL1RL 2 RH 2 1.65453 ×10-5 RL 2 RH 1RH 2 - 4.34197 ×10-5 RV21RL2 - 2.56745 × 10-4 RL21RL 2 + 9.15882 × 10-5 RL21RH 2 - 8.18328 ×10-5 RL1RL22

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(a)

453 454

(c)

455 456

(e)

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435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452

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(d)

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459 460 461 462

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463 464

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465 466

(e)

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(g)

Fig. 6 Response surface of 1st transverse frequency. (a) H1-RV1. (b) H1-RV2. (c) H1-RL1. (d) H1-RL2. (e) H1-RH1 (f) H1-RH2. (g) H1-N.

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(b)

(d)

(f)

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469 470 471 472

(a)

473 474

(c)

475 476

(e)

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(g)

Fig. 7 Response surface of 2nd transverse frequency. (a) H2-RV1. (b) H2-RV2. (c) H2-RL1. (d) H2-RL2. (e) H2-RH1 (f) H2-RH2. (g) H2-N.

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(d)

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479 480 481 482 483

(a)

484 485

(c)

486 487

(e)

Fig. 8 Response surface of 1st vertical frequency.

(b)

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(a) V1-RV1. (b) V1-RV2. (c) V1-RL1. (d) V1-RL2. (e) V1-RH1 (f) V1-RH2. (g) V1-N.

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490 491 492 493 494

(a)

495 496

(c)

497 498

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Fig. 9 Response surface of 2nd vertical frequency.

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Fig. 10 Response surface of 3rd vertical frequency. (a) V3-RV1. (b) V3-RV2. (c) V3-RL1. (d) V3-RL2. (e) V3-RH1 (f) V3-RH2. (g) V3-N

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Fig. 11 Response surface of 4th vertical frequency. (a) V4-RV1. (b) V4-RV2. (c) V4-RL1. (d) V4-RL2. (e) V4-RH1 (f) V4-RH2. (g) V4-N

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Fig. 12 Response surface of 5th vertical frequency. (a) V5-RV1. (b) V5-RV2. (c) V5-RL1. (d) V5-RL2. (e) V5-RH1 (f) V5-RH2. (g) V5-N.

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Fig. 13 Response surface of 6th vertical frequency.

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Fig. 14 Response surface of 1st longitudinal frequency. (a) L1-RV1. (b) L1-RV2. (c) L1-RL1. (d) L1-RL2. (e) L1-RH1 (f) L1-RH2. (g) L1-N.

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Response surface model verification

549 550 551 552 553 554 555 556 557

It is necessary to verify the reliability of response surface function fitting. Compared the calculated results of RS model with those from monitoring, R2 criterion and RMSE criterion given in Eqs. (3) and (4) were used to verify the response surface model. The final results are shown in Table 6. The values of R2 are very close to 1, at the meantime, the values of RMSE are very close to 0. This indicates that the precision of the regressed RS model is very high, and the RS model can well reflect the relationships between characteristics and parameters. Therefore, it is feasible to employ the regressed RS model instead of the initial FE model in the following model updating process.

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Table 6 Accuracy inspection of each RS

H1

H2

V1

V2

V3

V4

V5

V6

L1

R2

0.9994

0.9993

0.9921

0.9943

0.9807

0.9565

0.9742

0.9836

0.9487

RMSE

0.000376

0.000413

0.000765

0.000573

0.004326

0.008587

0.006326

0.003386

0.01358

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FE model updating

560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579

Subsequently, taking the field test frequencies as the corresponding objective values of each response surface function, the goal attainment method carried out on the created third-order polynomial response surface within the value range of selected updating parameters. The optimization process was completed based on MATLAB toolbox. Then, the updated frequencies and vibration modes were obtained by FE analysis using the updated values of parameters to replace the initial values. Comparisons of updated frequencies and measured frequencies are shown in Table 7. The maximum error between FE and testing frequencies is reduced below 3.65%. This indicates that the overall correlations between FE calculated and experimental natural frequencies have been improved after performing the response surface-based FE model updating. It can be concluded that the updated FE model can relatively reflect the actual situation of Xiabaishi Bridge in the design space of the selected parameters. After updating, the changes in the selected three updating parameters of the bridge are shown in Table 8. It can be seen that the updated value for concrete Young’s modulus has an increase of nearly 25%, but a significant correction on the initial estimations of the connection stiffness of adjacent box-girder constraints and the transverse spring element stiffness of support bearings and expansion joints has turned out. Therefore, the simulation of support conditions and adjacent box-girder constraints in the FE model updating of a real bridge is very important, especially for the bridge in service.

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Table 7 Natural frequency differences of Xiabaishi Bridge before and after updating Monitored frequency (Hz)

H1 H2 V1 V2 V3 V4 V5 V6 L1

0.418 0.527 0.659 0.813 1.261 1.404 1.675 1.892 1.684

Initial frequency (Hz) 0.363 0.483 0.573 0.743 1.100 1.220 1.456 1.740 1.813

Updated frequency (Hz) 0.409 0.526 0.658 0.838 1.249 1.366 1.615 1.946 1.714

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Initial relative error (%) -13.158 -8.342 -13.097 -8.647 -12.799 -13.077 -13.069 -8.029 7.643

Updated relative error (%) -2.153 -0.188 -0.146 3.114 -0.967 -2.728 -3.612 2.854 1.793

Table 8 Selected parameters of Xiabaishi Bridge before and after updating

Parameter

Initial value (106 N/m)

Updated value (106 N/m)

RV1

80.000

154.550

93.19

RV2

80.000

251.900

214.88

RL1 RL2 RH1 RH2 N

50.000 3.000 8.000 5.000 1.000

44.490 1.202 31.997 3.582 1.244

-11.02 -59.93 299.96 -28.36 24.40

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As the basement of finite element model validating, this paper presents an application of the response surface method for finite element model updating of bridge structures. A third-order polynomial response surface is constructed and then employed to improve the computation efficiency in the model updating. The proposed procedure is illustrated on a full-size long-span prestressed continuous rigid-frame bridge based on the on-line structural health monitoring system. With the response surface to replace the original FE model, the model updating process becomes efficient and converges fast compared to the traditional sensitivity-based model updating method. Once the response surface is constructed, there is no finite element calculation involved in each iterative optimization. As a result, the FE model updating can be easily implemented in practice with available commercial finite element analysis software. Through this study, it has been found that the D-optimal design is a powerful tool for parameter screening since it gives quantitative evaluation of the significance of updating parameters. For more complex civil engineering structures, a high-order response surface function would be necessary to consider more input parameters. It can be concluded that the updated FE model has high precision and practicality. The finite element model updating based on third-order response surface method is feasible and useful for the further model validation process. It is still a challenge to clarify the response surface method for the finite element model updating of more complex civil engineering structures, where the relationship between the unknown parameters and the response quantities of interest is more complex. It should be noted that the real bridge has many uncertainties and potential damages, which would lead to more difficulties, such as parameter selection, type of response surface functions, iteration algorithm, response prediction, and damage identification.

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Acknowledgments

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This research was supported by the National Natural Science Foundation of China (No. 51178101,

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References

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AIAA, 1998. Guide for the verification and validation of computational fluid dynamics simulations. AIAA-G-077-1998, American Institute of Aeronautics and Astronautics, Reston. ASME, 2006. Guide for verification and validation in computational solid mechanics. ASME V&V 10-2006, American Society of Mechanical Engineers, New York.

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51378112), National Scientific and Technological Supporting Plan (No. 2011BAK02B03), Scientific Research and Development Foundation of Fujian University of Technology (No. GY-Z10085).

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Box, G. E. P., Draper, N. R., 2002. Empirical Model Building and Response Surfaces, Wiley Series in Probability and Statistics. John Wiley & Sons, New York. Box, G. E. P., Wilson, K. B., 1951. On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society. Series B (Methodological), 13(1), 1–45. Bucher, C. G., Bourgund, U., 1990. A fast and efficient response surface approach for structural reliability problems. Structural Safety, 7(1), 57–66. Censor, Y., 1977. Pareto optimality in multiobjective problems. Applied Mathematics & Optimization, 4(1), 41–59. Chankong, V., Haimes, Y. Y., 1983. Multiobjective Decision Making: Theory and Methodology. North-Holland, New York. Cheng, J., Zhang, J., Cai, C. S., et al., 2007. A new approach for solving inverse reliability problems with implicit response functions. Engineering Structures, 29(1), 71–79. Chu, K. C., 1970. On the noninferior set for the systems with vector-valued objective function. IEEE Transactions on Automatic Control, 15(5), 591-593. Cundy, A. L., 2003. Use of Response Surface Meta-models in Damage Identification of Dynamic Structures. Master Thesis, Virginia Polytechnic Institute and State University, Blacksburg. Dall'Asta, A., Dezi, L., 1996. Discussion of “prestress force effect on vibration frequency of concrete bridges” by M. Saiidi, B. Douglas, and S. Feng. Journal of Structural Engineering, 122(4), 458-458. Deng, L., Cai, C. S., 2010. Bridge model updating using response surface method and genetic algorithm. Journal of Bridge Engineering, 15(5), 553-564. Fang, S. E., Perera, R., 2011. Damage identification by response surface based model updating using D-optimal design. Mechanical Systems and Signal Processing, 25(2), 717-733. Farrar, C. R., Lieven, N. A., 2007. Damage prognosis: the future of structural health monitoring. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 365(1851), 623-632. Farrar, C. R., Worden, K., 2007. An introduction to structural health monitoring. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 365(1851), 303-315. Farrar, C. R., Worden, K., Todd, M. D., et al., 2007. Nonlinear system identification for damage detection. No. LA-14353-MS, Los Alamos National Laboratory, Los Alamos. Friswell, M., Mottershead, J. E., 1995. Finite Element Model Updating in Structural Dynamics. Springer Science & Business Media, Berlin. Gembicki, F., Haimes, Y. Y., 1975. Approach to performance and sensitivity multiobjective optimization: The goal attainment method. Automatic Control, IEEE Transactions on, 20(6), 769-771. Guo, Q. T., Zhang, L. M., 2004. Finite element model updating based on response surface methodology. In Proceedings of the 22nd IMAC, Dearborn, 2004. Guo, Q. T., Zhang, L. M., Fei, Q. G., 2006. Response surface method and its experimental design for deterministic computer simulation. Acta Aeronautica et Astronautica Sinica, 27(1), 55-61. Jones, D. R., 2001. A taxonomy of global optimization methods based on response surfaces. Journal of Global Optimization, 21(4), 345-383. Kirkparick, S., Gelatt, C.D. Vecchi, M. P., 1983. Optimization by simulated annealing. Science, 220(4598), 671–680. Logan, R. W., Nitta, C. K., 2002. Verification& validation (V&V) guidelines and quantitative reliability at confidence (QRC): basis for an investment strategy. No. UCRL-ID-150874, Lawrence Livermore National Laboratory, Livermore. Marwala, T., 2004. Finite element model updating using response surface method. Collection of Technical Papers—AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 7, 5165–5173. th Montgomery, D. C., 2006. Design and Analysis of Experiments, 6 ed. John Wiley & Sons, New York. Myers, R. H., Montgomery, D. C., 2002. Response Surface Methodology: Process and Product Optimization Using nd Designed Experiments, 2 ed. John Wiley & Sons, New York. Oberkampf, W. L., Trucano, T. G., Hirsch, C., 2003. Verification, validation, and predictive capabilities in computational engineering and physics, No. 2003-3769, Sandia National Laboratories, Albuquerque. Ren, W. X., Chen, H. B., 2010. Finite element model updating in structural dynamics by using the response surface method. Engineering Structures, 32(8), 2455-2465. Ren, W. X., Fang, S. E., Deng, M. Y., 2011. Response surface–based finite-element-model updating using structural static responses. Journal of Engineering Mechanics, 137(4), 248-257. Ren, W. X., Zong, Z. H., 2004. Output-only modal parameter identification of civil engineering structures. Structural Engineering and Mechanics, 17(3/4), 429-444. Rijpkema, J. J. M., Etman, L. F. P., Schoofs, A. J. G., 2001. Use of design sensitivity information in response surface and kriging metamodels. Optimization and Engineering, 2(4), 469-484. Thacker, B. H., Doebling, S. W., Hemez, F. M., et al., 2004. Concepts of model verification and validation. LA-14167-MS, Los Alamos National Laboratory, Los Alamos.

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Weber, B., Paultre, P., 2009. Damage identification in a truss tower by regularized model updating. Journal of Structural Engineering, 136(3), 307-316. Xia, P. Q., Brownjohn, J. M., 2004. Bridge structural condition assessment using systematically validated finite-element model. Journal of Bridge Engineering, 9(5), 418-423. Zong, Z. H. Ren, W. X., 2012. Finite Element Model Updating and Model Validation of Bridge Structures. China Communications Press, Beijing. Zong, Z. H., Sun, J. L., Xu, L. Q., et al., 2009. Design and Study of Structural Health Monitoring System of Xiabaishi Bridge. Journal of the China Railway Society, 31(5), 65-71.

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