Concurrent multi-scale modeling of civil infrastructures for analyses on structural deteriorating—Part II: Model updating and verification

Finite Elements in Analysis and Design 45 (2009) 795 -- 805

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Concurrent multi-scale modeling of civil infrastructures for analyses on structural deteriorating—Part II: Model updating and verification T.H.T. Chan a,b , Z.X. Li c, ∗ , Y. Yu a , Z.H. Sun c a

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong School of Urban Development, Faculty of Built Environment and Engineering, Queensland University of Technology, Queensland, Australia c College of Civil Engineering, Southeast University, Nanjing, PR China b

A R T I C L E

I N F O

Article history: Received 2 July 2008 Accepted 8 June 2009 Available online 14 July 2009 Keywords: Model updating Model verification Multi-scale modeling Sensitivity analysis

A B S T R A C T

This paper is a continuation of the paper titled “Concurrent multi-scale modeling of civil infrastructure for analyses on structural deteriorating—Part I: Modeling methodology and strategy” with the emphasis on model updating and verification for the developed concurrent multi-scale model. The sensitivity-based parameter updating method was applied and some important issues such as selection of reference data and model parameters, and model updating procedures on the multi-scale model were investigated based on the sensitivity analysis of the selected model parameters. The experimental data of modal properties as well as static response in terms of component nominal stresses and hot-spot stresses at the concerned locations were used for dynamic response- and static response-oriented model updating, respectively. The updated multi-scale model was further verified to act as the baseline model which is assumed to be finite-element model closest to the real situation of the structure available for the subsequent arbitrary numerical simulation. The comparison of dynamic and static responses between the calculated results by the final model and measured data indicated the updating and verification methods applied in this paper are reliable and accurate for the multi-scale model of frame-like structure. The general procedures of multi-scale model updating and verification were finally proposed for nonlinear physical-based modeling of large civil infrastructure, and it was applied to the model verification of a long-span bridge as an actual engineering practice of the proposed procedures. © 2009 Elsevier B.V. All rights reserved.

1. Introduction The finite-element (FE) model of a structure is usually constructed by simplifying the original structural from engineering drawings and designs that may not exactly represent all the physical aspects of the original structure. Under this situation, the finite-element model would need to be calibrated by modification of inaccuracies or uncertainties quantitatively expressed by parameters in order to eliminate the discrepancies as much as possible. This process is usually termed as “model updating.” Once the finite-element model is calibrated according to the reference data, commonly measured dynamic characteristics in term of natural frequencies and mode shapes, the model can be used for aerodynamic and/or seismic response predictions. Furthermore, the updated finite-element model still needs to be verified by comparing the calculated results of the updated model with experimental data so as to address the extent of differences between the final model and the true value of the structure. Therefore,

∗ Corresponding author. Tel.: +86 25 83790268; fax: +86 25 83792247. E-mail address: [email protected] (Z.X. Li). 0168-874X/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2009.06.008

the FE model after updating and verification is assumed to be the one closest to the real situation of the structure as a baseline model and numerical analyses could be then carried out for the purpose of structural health assessment. From this point of view, the model updating and verification procedure are absolutely necessary for all the FE models to minimise the discrepancies between FE predictions and reference data to approach the real situation of the structure in numerical simulation as much as possible. Model updating is a step in the model validation process that modifies the values of parameters in an FE model in order to bring the FE model prediction into better agreement with experimental data. Model verification is another step in the process of model validation. In this step, the model is verified so that it is modeled according to the initial requirement on the model. In the viewpoint of model validation, model verification should provide a verified model which can be updated to match the experimental data by only modifying the parameters of the model. The FE model updating method [1,2] emerged in the 1990s as a subject of great importance for mechanical and aerospace structures. Several methods of structural model updating have been proposed and the topic is still under active study in various areas. A comprehensive survey on the model updating techniques [3] and a literature review on the FE model updating

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techniques and their applications to damage detection and structural health monitoring (SHM) [4,5] could be found. The effectiveness of some of these techniques has been verified on simple structures such as simply supported beams, cantilever beams and space truss structures. However, this updating technology can be hard to apply as an engineering tool for civil engineering structures, because of the difficulties in prototype testing and experimental data analysis resulting from the nature, size, location and usage of these structures. Only recently, the civil engineering community has begun to adopt this advanced technology. Cantieni [6] investigated model updating of a concrete arch bridge while Pavic et al. [7] and Reynolds et al. [8] applied the technique to footbridges and concrete floors. These investigations marked the beginning of the successful application of the model updating technology to civil engineering structures. The application of the model updating technology to even more complex structures is still a challenge to researchers. In particular, for a complex structure with a large number of degrees of indeterminacy, such as a large-span bridge, model updating becomes difficult since it inevitably involves too many uncertainties in material and geometrical properties and boundary conditions. Moreover, the measured data may also be contaminated by the noise. Generally, the finite-element modeling gives a detailed description of the physical and modal characteristics of the civil infrastructure while the field test serves as a valuable source of information. The results by finite-element model and the field test show reasonable correlation usually in terms of the natural frequencies and the mode shapes. However, significant discrepancies can still be found between the predicted dynamic properties of a structure and those by measurement directly from the structural vibration. These discrepancies could be attributed to several reasons [9]: (1) inaccuracy in the FE model discretisation; (2) uncertainties in geometry and boundary conditions; (3) variations in material properties; (4) environmental variability (such as temperature and wind) and variability in operational conditions (such as traffic flow) during measurement and (5) errors associated with measured signals and post-processing techniques. Among these possible sources of discrepancy, the first three are related to the assumptions and the inputs of the FE model. The fourth one is a reflection of the fact that the structural properties (and hence the measured modal properties) are not temporally invariant. If we can ignore this temporal invariance of modal properties and assume that the measured modal properties are very close to the actual behaviour of the structure under a certain measurement condition, then a challenging issue becomes how we can update the FE model so that the predicted modal properties can match those obtained from the direct measurements. Model updating procedures can be classified as being one-step procedures (global methods) or iterative (local methods). Global methods directly reconstruct the updated global mass and stiffness matrices from the reference data (measured frequencies and mode shapes). Local methods are based on corrections applied to local physical parameters of the FE model. They have a good physical interpretation of the obtained modifications and preserve the symmetry, positive-definiteness and sparseness of the stiffness matrix. The effective and most popular local methods for model updating are generally based on the sensitivity analysis, so it is usually called as “Sensitivity-based updating method”. This method had been widely used in the civil infrastructure such as large-span bridges so that the detailed work could be found [10–14]. This paper is the second part of the serial papers which aimed to develop the method for a nonlinear and physical-based modeling of civil infrastructure at the different scale levels for the purpose of concurrent analyses on structural deteriorating process. The strategy and method for developing the Concurrent Multi-Scale Model of Structural Behaviour (CMSM-of-SB) had been studied in the first part, in which global structural behaviour and nonlinear features of

local details in the complex structure could be concurrently analysed at both spatial scale levels for the purpose of structural state evaluation as well as structural deteriorating, and the CMSM-of-SB of a typical longitudinal truss section selected from steel bridge decks was developed as a case study. And the CMSM-of-SB of the bridge deck system of the Runyang Yangtse Cable-stayed Bridge (RYCB) was also accordingly constructed as a practical application of the developed method for concurrent multi-scale modeling. These developed models should be updated and verified in order to get closer to the real situation of the structure as much as possible. Obviously, some specific issues generated in the process of the model updating and verification for CMSM-of-SB have to be studied since the developed model is quite different from traditional models of civil infrastructures such as trusses and bridges. Therefore, the second part of the serial papers is aimed at studying how to update the developed CMSM-of-SB model of a civil structure and how to perform the verification of the model for the case-studied truss structure and the Runyang Yangtse Bridge. The sensitivity-based parameter updating method will be first adopted because of its clear physical meaning coupled with application in real civil structures. The model parameters needed to update will be selected according to the candidate parameter sensitivity, and model updating will be carried out by using the laboratory experimental data. The baseline model closest to the real structure could be eventually obtained after the verification of the updated multi-scale model. The general procedures for multi-scale model updating and verification were finally proposed for nonlinear physical-based modeling of large civil infrastructure considering the structural deteriorating process, and then it was applied to the verification of the CMSM-of-SB of the RYCB as an actual engineering practice of the proposed procedures. 2. Formulation of sensitivity-based model updating with constraints For a discretised system, the ith eigenvalue i and the corresponding eigenvector (mode shape) i can be obtained from the n degrees of freedom finite-element model by solving the eigen-equation Ki i = i Mi

(1)

where K and M are structural stiffness and mass matrices, respectively. Typically, the stiffness and mass matrices are functions of structural parameters, including geometrical and material properties as well as boundary conditions. If a set of structural parameters (pj,a , j = 1, . . . , np ) can be estimated for the finite-element model and represented by a vector Pa Pa = {pj,a |i = 1, 2, . . . , np }T

(2)

where np is the total number of structural parameters, then a set of eigenvalues (a ) can be obtained from the model as

a = {i,a |i = 1, 2, . . . , na }T

(3)

where na is total number of the computed modes. The subscript a is used to indicate that the corresponding properties are related to the “prediction” by the finite-element analysis. Also, the modal characteristics of the structure can be obtained from experiments as

e = {i,e |i = 1, 2, . . . , ne }T

(4)

where e is the vector of measured eigenvalues, and ne the total number of measured modes. The subscript e is used to indicate that the corresponding properties are obtained from the measurements. In general, the experimental (e ) and the predicted modal properties (a ) do not necessarily correlate well due to inaccuracy in both the FE modeling and the measurements. As mentioned before,

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any inaccuracy in connection with a finite-element model may come from three possible sources—model structural errors, model order errors and model parameter errors. If it is assumed that the measured modal properties are very close to the actual behaviour of the structure and that the model parameter errors are the main contributor to the inaccuracy of the finite-element model, then the model errors could be reduced or corrected through the model updating procedure. Let P represent the vector of structural parameters after updating P = {pi |i = 1, 2, . . . , np }T

(5)

It may be assumed that the total number of measured modes is the same as the total number of predicted (or calculated) modes (ne = na ). The functional relationship between the measured and the initial predicted eigenproperties can be approximated by a firstorder Taylor series expansion with respect to the structural parameters as follows [2]:

797

the parameters updated in the previous iteration. The eigenvalue sensitivities can be approximated by Sij =

i pj i pj

(11)

where Sij is the dimensionless sensitivity of the ith eigenvalue i with respect to the jth parameter pj . pj is the perturbation of parameter pj and i the change of eigenvalue i due to pj . The convergence criteria for the iteration are set as     (k) (12) fa − fexp  ⱕ tolerance    (k+1) − (k)   a a   ⱕ tolerance    (k)

(13)

a

(k)

where fexp is the vector of the measured natural frequencies and fa

(k) a

where  is the eigenvalue residual vector and P the perturbation vector of structural parameters defined, respectively, as

and are the vectors of the analytical natural frequencies and eigenvalues of the kth iteration, respectively. The fast criterion (12) relates to the global convergence of the analytical natural frequencies to the measured natural frequencies while the second criterion (13) represents the convergence during iterations. The iteration would be terminated if either one of these two criteria is satisfied.

 = e − a

(7)

3. Analyses of the sensitivity on model parameters

P = P − Pa = {pi |i = 1, 2, . . . , np }T

(8)

 = SP

(6)

where S is the sensitivity matrix, containing the first derivative of the eigenvalues with respect to the structural parameters evaluated at the initial estimate Pa or, in the following iterative scheme, at the current parameter estimate. The higher order terms in the Taylor series expansion are neglected under the assumption that the changes in the structural parameters between successive iterations are small. The perturbation of structural parameters can be obtained by solving Eq. (6). Many approaches have been proposed to solve this inverse problem and basically can be classified according to whether there are more structural parameters than measured modal properties or vice versa. When there are more measured modal properties than structural parameters (ne  np ), an optimal solution may be derived in a sense that it minimises a least-squared error function. However, it is quite possible that the use of parameter perturbation to minimise the objective function can result in relatively large variations. These extreme values not only violate the assumption of the first-order Taylor series approximation, but also produce an updated result that may be physically meaningless. In order to guarantee the physical significance of updated parameters and avoid physically impossible updated parameter values, inequality constraints for the structural parameters are introduced as follows: Bl ⱕ P ⱕ Bu

(9)

where Bl and Bu are the vectors of lower and upper bounds for the structural parameters P, respectively. Subsequently, the structural parameter perturbations are bounded by bl ⱕ p ⱕ bu

(10)

where bl and bu are lower and upper bounds of the perturbation, respectively. Thereby, the parameter updating can be achieved by minimising the objective function subjected to the constraints in the form of Eq. (10). The iterative procedure starts with the selection of a proper set of parameters for adjustment together with the upper and lower bounds of each parameter. The parameter used in the initial finiteelement model is taken as the starting point for the iteration. In each iteration step, an eigenvalue sensitivity analysis is performed using

3.1. Selection of parameters and modes The differences between the predicted and the measured results of natural frequencies are mainly due to inaccurate estimation of structural parameters such as material properties, geometric parameters and boundary conditions. Theoretically speaking, all possible structural parameters can be selected for adjustment in the updating procedure. However, it would take computational costs if too many parameters are to be included. Therefore, the set of parameters to be updated should be carefully selected. For the most of civil infrastructure, some geometric parameters such as the lengths of the components and cross-sectional dimensions can be measured accurately, which should not be the candidate parameters for updating. By contrast, the exact boundary condition is usually complicated and uncertain, and there exist uncertainties and disunities because of either the material itself or the processing technology such as the material properties around the concerned area nearby the welding toe with respect to welding thickness and elasticity modulus. These uncertain factors affect significantly global structural behaviour and local damage evolution for the civil infrastructure under service loading, so that they should eventually be selected as structural parameters for model updating. In this work, the constraint stiffness parameter for boundary condition at different direction, Ki , and the parameter for the welding thickness and elasticity modulus, t and E were selected as model parameters to be studied. Ideally, it would be better if we can match as many modes as possible between the measurement and FE prediction. However, the request is not always true when applied in practical engineering. In the area of health monitoring and damage detection of complex structures, it is found that the global mode is usually at the low order while the higher modes are more sensitive to local damage. Therefore, it is not practical and also unnecessary to include modes (especially those higher modes) when the emphasis is focused on the dynamic properties at the global level. In this study, it had been decided to select the first 10 modes to be matched between the updated finite-element analysis and the measured results. The selected modes include four lateral-dominant, four vertical-dominant and two torsional-dominant modes. The frequencies calculated from the initial FE model (denoted as fFEM ) and corresponding values obtained from measurement (denoted as fexp ) are summarised in

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Table 1 Analytical (initial FEM) and experimental natural frequencies. Mode no.

Mode shape description

Experimental frequency fexp (Hz)

Analysed frequency by i initial FEM fFEM (Hz)

1 (V1) 2 (L1) 3 (L2) 4 (L3) 5 (T1) 6 (L4) 7 (T2) 8 (V2) 9 (V3) 10 (V4)

V asym. L sym. L asym. L asym. T sym L sym. T, L sym. V asym. V asym. V sym

15.0 35.0 70.2 89.6 159.2 211.3 263.5 273.6 322.1 338.6

20.2 45.8 89.8 106.0 170.2 209.6 256.5 296.0 316.1 318.1

fi −f FEM exp fexp

(%)

34.89 30.90 27.92 18.27 6.90 −0.80 −2.65 8.21 −1.86 −6.05

V, L, T—vertical, lateral and torsion bending modes, respectively. sym—symmetric, asym—asymmetric.

A0 B0

1

C0

2

D0

3

4 5

6

7

uy

E0

8 9

10

14

uz

11 15

A

y

12

900

C

ux

13

16

B

z

x

17

900

D 900 4200

E 900

Fig. 1. The tested steel truss and the decomposed boundary parameter.

Table 1. It can be seen from Table 1 that the differences between the computed and the measured frequencies range between −0.8% (mode L4) and 34.89% (mode V1). 3.2. The sensitivity of eigenvalues on the selected parameters The reduced scale specimen of the steel truss section, as Fig. 1 shows, had been experimentally studied and the CMSM-of-SB has been developed in the previous paper [15]. It is still to be used as the case study of model updating and verification. The boundary condition of the truss in the laboratory was adopted to be restrained at both ends of the top chords while the bottom chords are free. The accurate values of constraint parameters in the developed model should be identified and updated. In order to investigate how boundary conditions affect the mode shapes, the boundary was decomposed into six DOFs. First of all, the ideal conditions under the fixed support and the simple support were simulated, respectively, to obtain the corresponding value of frequencies of the 1–10th modes as the bound figures of true parameter value as tabulated in Table 2. Then dynamic properties of the truss under different values of DOFs in three displacement (ux , uy , uz ) and

rotational (x , y , z ), respectively, were calculated, in which the parameter of each DOF ranges from 1 to 1×1012 N/m in order to study the influence of the parameter in each direction on the final frequency value. The detailed procedure for the calculation could be found in Sun [16]. The sensitivity of eigenvalues for the selected 10 modes on the selected model parameters was calculated by using Eq. (11). The dimensionless sensitivity indicates the effect of the given model parameter on a particular frequency as in Fig. 2, in which Ki (i = ux , uy , uz , x , y , z ) is the parameter to measure the constraint stiffness on the boundary at the ith direction, respectively, and tw , Ew represent, respectively, the welding thickness and elasticity modulus at the welding area while tb , Eb are those at the boundary region between welding and parent material as shown in Fig. 8 in Part I of the serial papers [15]. Analysed results of the sensitivity of eigenvalue show that Kuz affects mainly the order of mode 3, 4, 6 while Kz is sensitive to order 5, 7 and Ky to the first four orders while there are barely effects of geometric and material properties at the welding toe on the frequency results. Therefore, only the four parameters, namely Kuz Kx Ky Kz , need to be considered for updating if the objective function is focused on the structural dynamic property.

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799

Table 2 Calculated frequencies by initial FEM under two ideal support conditions. Boundary conditions 1 simulation

Mode no. 2

3

4

5

6

7

8

9

10

BC1 BC2

20.2 0.1

45.8 22.3

48.2 34.5

89.8 79.1

104.3 68.1

106.0 80.5

124.5 104.3

133.1 132.8

133.3 132.4

147.0 141.9

BC1: ideal condition of fixed support (ux = uy = uz = x = y = z = 0). BC2: ideal condition of simply support (ux = uy = uz = 0).

Normalized Sensitivity

2

Kux Kuy Kuz Kx

1.5

Ky

Ew

Kz tw tb

Eb

1 0.5 0 - 0.5 1

2

3

5 7 4 6 Mode Order Number

8

9

10

Fig. 2. Sensitivity of eigenvalue to the selected model parameters.

Kux

Ky

tb

Kuy

Kz

Ew

Kuz

tw

Eb

It was again taken as the study objective, and the eigenvalue sensitivity on this hot-spot location with respect to selected parameters in term of distance from the welding toe was investigated and the results are shown in Fig. 4. The results of the calculated value of the sensitivity of hot-spot stresses on the model parameters indicate that the local geometric configuration and material property have shown a great effect on the hot-spot stress situation, so that the emphasis should be laid on the those parameters for the purpose of model updating with the objective function of hot-spot stress. Generally speaking, the proper selection of structural parameters serves the goals of objective function, on the contrary, different objective functions should be appropriately chosen for the purpose of different updated parameters. In this study, on the basis the aforementioned sensitivity analysis carried out for the selected structural parameters, the measured frequencies combined with measured nominal stress situation should be chosen as the objective function for boundary condition parameters updating while measured local hot-spot stress situation for geometric and material parameters updating. 4. Model parameters updating with the measured responses

Kx

Normalized Sensitivity

0.06 0.04 0.02 0 -0.02 -0.04 -0.06 0

5 10 15 Measurement Location Number

20

Fig. 3. Sensitivity of nominal stress to the selected model parameters.

3.3. The sensitivity of nominal stresses on the selected parameters Apart from the widely used natural frequency and mode shape, the sensitivity of the static response in term of nominal stress and local hot-spot stress at the concerned area of welding were also investigated. The dimensionless value of sensitivity of nominal stress on the selected parameters is shown in Fig. 3. It can be seen that, results of the sensitivity of nominal stresses due to the parameter Kuy and Kz have relatively large influences on the nominal stress of the truss. 3.4. The sensitivity of hot-spot stresses on the selected parameters The welding toe and adjacent area at point C (as shown in Fig. 1) on the bottom chord had been concerned about in Part I of these serial papers [15], which is plotted in Fig. 4a with a red line.

The selected structural parameters are updated using the sensitivity-based updating method described in Section 2. The initial value of the constraint stiffness, Ki (i = ux , uy , uz , x , y , z ), was estimated according to the values performed in the sensitivity analysis, and those of geometric and material property tw , Ew , tb , Eb are based on the experimental measurement which could be found in details [17]. It has been shown that the faulty setting of the bounds of the parameter would significantly affect the updating result, so in this study the bounds of variation was estimated based on the initial parameter estimates, and judgment from their physical meaning. For values of the constraint stiffness, Ki in boundary condition, since there exists too many uncertainties, the variation of 1000% was assigned for every parameter. As for the model parameters of thickness and elasticity modulus, tw , Ew , tb , Eb , the variation of 20% is permitted. The initial estimates of the parameters used in the initial model are taken as the starting point for the iterative process. In each iterative step, the tolerance value is 10% and an eigenvalue sensitivity analysis is performed using the parametric values updated in the previous iteration. In the current study of model updating, the majority of reference data is the experimental results such as measured frequencies and mode shapes which are generally the information from the dynamic property of civil structure. In this sense, the model updating method could be classified as “Dynamic property-oriented model updating”. In other words, the eigenvalues of dynamic properties in terms of natural frequencies and mode shapes are the objectives during the entire updating process. However, the updating technology is still difficult to apply as a standard and matured engineering tool for civil engineering structures, especially for the more complex civil infrastructure, because of the difficulties in conducting the tests and subsequent experimental data analysis and the greater uncertainties in material physical properties. In particular, for the field test of a bridge, the tester is working under the circumstance of

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Normalized Sensitivity

800

3 2.5 2 1.5 1 0.5 0 -0.5

Kux

Ky

tb

Kuy

Kz

Ew

Kuz

tw

Eb

Kx

-0.1

-0.05 0 0.05 Distances from the toe of weld (m)

Fig. 4. (a) The concerned location of welding area. (b) Sensitivity of hot-spot stress at concerned location to the selected model parameters.

Table 3 Initial and updated values of boundary constraint parameters. Constraint stiffness (N/m)

Kux

Kuy

Kuz

Kx

Ky

Kz

5×107 4.1×107

2×107 2×107

5×106 5×106

3×106 3×106

8×107 8×107

7

Initial estimates Updated values (A0 ) Updated values (E0 )

1×10 1×107 2×107

Table 4 Comparison of measured frequencies with calculated values before and after model updating in the first 3 modes out of plane. Mode no.

Measured frequency (Hz)

Calculated value by initial model (Hz)

Differences before updating (%)

Calculated value by updated model (Hz)

Differences after updating (%)

1 2 3

15.0 35.0 70.2

20.2 45.8 89.8

34.89 30.90 27.92

15.6 36.4 75.0

4.00 3.94 6.87

4.1. Updating the constraint stiffness of boundary with the measured modal data and nominal stress Based on the preliminary analyses on parameter sensitivity, modal property and nominal stress situation are relatively

6 Stress Value (MPa)

varying environmental conditions not limited to temperature and traffic. Without the benefit of the relatively controlled and repeatable conditions of laboratory testing, combined with errors associated with measurement data and post-processing techniques, the identified response parameters are likely to change with poor assurance. Moreover, when relying on ambient excitation, it is necessary to recognise whether a response is not a feature of the structure but of the excitation based on the skill of the analyst. Therefore, it is fact that the structural properties (and hence the measured modal properties) are not temporally invariant and the phenomena can only be minimised instead of eliminated. From this point of view, to some extent, the measured reference data are not accurate enough for the model updating to get closer to the real situation as much as possible. Besides the measured modal data, the information of static response even dynamic response obtained from the more controlled test on the reduced scale specimen, constructed based on the prototype according to the law of similitude in laboratory, could also be used to provide the reference data for the model updating, which is better than mere dependence on the measured modal data. This kind of model updating method based on the static response information could thus be categorised as “static response-oriented model updating”. Those two model updating strategies could mutually supplement and be applied in the subsequent model updating procedure.

Tested Value

4

Before updating After Updating

2 0 -2 -4 -6 0

5 10 15 Measurement Location Number

20

Fig. 5. Comparison of the calculated results of nominal stress by the initial and updated parameters, respectively, with those measured at measurement points.

sensitive to the boundary condition, so the measured modal data and structural nominal stress were taken as the objective function when updating the parameter of the constraint stiffness. The first 3 modes data out of the plane of the truss and structural nominal stress under the loading case of 10 kN acting on point D shown in Fig. 1 were chosen as the reference data in the model updating with the sensitivity-based updating method. The updated values of boundary constraint parameters listed in Table 3 indicated that the values were between the ideal fix supported and simply supported conditions, and the identified values at transitional DOF at X, Y which is in good agreement with the measurement data [17] revealed not identical supporting conditions at both ends in laboratory simulation. The comparison of measured frequencies with calculated ones

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801

Table 5 Updated values of parameter for geometric and material property. Locations

The area of welding toe

The boundary area between welding and parent material

Parameters

Thickness, tw (mm)

Elasticity modulus, Ew (GPa)

Thickness, tb (mm)

Elasticity modulus, Eb (GPa)

Initial estimation Updated values Difference (%)

6 7 16.7

206 223 8.3

6 6.1 1.7

206 211 2.4

Tested Value Before Updating After Updating

-2 -4

Table 6 Comparison of frequencies calculated by using initial and updated model with those from measurement. Mode no.

-6 -8 -10 -0.2

-0.1 0 Distances from the toe of weld (m)

0.1

Fig. 6. Comparison of the calculated results of the local hot-spot stress by the initial and updated parameters, respectively, with those measured around welding.

by the initial and updated parameter had been tabulated in Table 4. It is found that the differences were significantly reduced in term of average error from 31.2% to 4.94%. A tiny reduction of component nominal stresses at the measurement points were observed in Fig. 5. 4.2. Updating parameters of geometric and material property near the welding toe with the measured hot-spot stresses In a similar way, the measured information about the local hotspot stress situation at the concerned locations, in particular, a welding at the bottom chord as shown in Fig. 4 was focused on during the process of updating parameters of the geometric and material property. As shown in Table 5, the updated values were bigger than the initial estimation and the changing magnitude at the welding toe was larger than that in the transition area. It can be seen in Fig. 6 that the calculated values of updated parameters got much closer to the measurement data than those by the initial model and there is almost hardly any changes when far away from the welding.

1 (V1) 2 (L1) 3 (L2) 4 (L3) 5 (T1) 6 (L4) 7 (T2) 8 (V2) 9 (V3) 10 (V4)

Frequencies calculated by using i Initial model fFEM (Hz)/ (error in percentage compared with fexp )

u Updated model fFEM (Hz)/ (error in percentage compared with fexp )

20.2 45.8 89.8 106.0 170.2 209.6 256.5 296.0 316.1 318.1

15.6 36.4 75.0 87.6 160.7 226.1 282.4 295.6 316.1 318.1

(34.89) (30.90) (27.92) (18.27) (6.90) (−0.80) (−2.65) (8.21) (−1.86) (−6.05)

(4.00) (3.94) (6.87) (−2.23) (0.91) (7.01) (7.18) (8.05) (−1.86) (−6.05)

Measured frequencies fexp (Hz)

15.0 35.0 70.2 89.6 159.2 211.3 263.5 273.6 322.1 338.6

Tested Value Calculated by the initial model Calculated by the updated model

6 Stress Value (MPa)

Stress Value (MPa)

0

4 2 0 -2 -4 0

5 10 15 Measurement Location Number

20

Fig. 7. Nominal stress calculated by the initial and updated model compared with the measured values under the point loading on C.

5. Verification of the model for the case-studied steel truss After the model updating on the selected parameters with respect to the boundary conditions combined with geometric and material property, the updated multi-scale model would be closer to the prototype of the structure and the resulting values obtained from those updated parameters taken as the objective function would also further approach the true value. In some sense, the updated model should be superior to the initial one. However, as for the results predicted by those parameters that had not been considered as a factor in objective function, the superiority of the updated model is questionable. Under this circumstance, the updated model should be further verified on the basis of the measured data of structural response other than those applied in the previous updating process. 5.1. Verifying the model by modal properties The measurement data on modal properties are still the most commonly used and provides effective reference information in both the updating and verification process. Verification on the updated multi-scale model had been carried out and the result listed in Table 6 indicated that except for the first 3 modes, the frequency differences between computed values by updated model in higher

modes are relatively bigger compared with that before updating. However, overall the frequency differences are significantly reduced so that all the error values are less than 10% of the average error range from 13.23% to 5.02%. 5.2. Verifying the model by static response Apart from the validation based on modal property information, the measured data of static response could be used for further verification. Since the static measurement information had also been applied in the previous updating process, data chosen for verification should be different from those used in model updating with respect to different loading case or measurement locations. Here, the loading case of applying loads on point C as shown in Fig. 1 for nominal stress analysis and hot-spot stresses studied at some selected points was chosen for verification. Nominal stress calculated by the initial and updated model compared with the measured values under the point loading on C is shown in Fig. 7, while Fig. 8 gives hot-spot stresses calculated by the initial and updated model compared with the measured values at the selected points. It can be seen that, the static response in components of the truss and local welded area

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T.H.T. Chan et al. / Finite Elements in Analysis and Design 45 (2009) 795 -- 805

8 Stress Value (MPa)

a

Tested Value Calculated by the initial model

b

C

Calculated by the updated model

c

6

d e

4 T

2

1

D

T

0 a

b

c

d

e

2

Corresponding Measurement Locations Fig. 8. Hot-spot stresses calculated by the initial and updated model compared with the measured values at the selected points.

Table 7 Frequency comparison between the measured values and prediction by structural global model and CMSM-of-SB. Mode no. fe 1 2 3 4 5 6 7

(V1) (AV1) (V2) (AV2) (L1) (T1) (AL2)

Mode shape description f0

f1

Frequencies (Hz)

0

1

Error (%)

1

V sym. V asym. V sym. V asym. L sym. T sym. L asym.

0.325 0.425 0.700 0.775 0.800 1.125 1.900

0.301 0.415 0.739 0.823 0.773 1.207 2.001

0.304 0.416 0.741 0.825 0.776 1.202 2.004

−7.38 −2.35 5.57 6.19 −3.38 7.29 5.32

−6.46 −2.12 5.86 6.45 −3.00 6.84 5.47

1.00 0.24 0.27 0.24 0.39 −0.41 0.15

V, L, T—vertical, lateral and torsion bending modes, respectively. sym—symmetric, asym—asymmetric. fe —experimental frequency from the field test, 0 =

f0 −fe fe

× 100%.

f0 —calculated frequency by global model of bridge structure, 1 = f1 —calculated frequency by CMSM-of-SB of bridge structure, 1 =

f1 −fe fe f1 −f0 f0

× 100%. × 100%.

under the selected situation of loading showed good fitting with the measured data under the different boundary condition and loading case. In this sense, the updated multi-scale model is much closer to the true value and it could be used in the further analyses of structural behaviour. 6. CMSM-of-SB model verification of the Runyang cable-stayed bridge (RYCB) After model updating and verification, the multi-scale model of the studied structure can be used as a baseline model close to the prototype of the structure for the subsequent numerical simulation. The methodology of model updating and verification implemented for the cased studied truss provides the reference or even guide for modeling of other civil infrastructure. Based on the above study on the method for model updating and verification performed with the steel truss section, the updating and verification procedures for the civil infrastructures could be summarised and outlined accordingly. 6.1. The procedure of model updating and verification with SHM data On the process of modeling civil infrastructure, the model updating and verification have to be carried out on the basis of the measured data from the field test and the structural health monitoring system instead of the data from experimental studies in the laboratory, which is the major difference between large civil infrastructure and the truss as a case study in this work. Considering this point of

view, the multi-scale model updating procedure was generally summarised as follows: (1) Selection of responses data as reference which are customarily the measured data, such as measured data of frequencies and mode shapes from the ambient vibration in the field test mainly for dynamic property-oriented model updating. As for the large civil infrastructure, the static or dynamic response data directly obtained from the structure testing upon the completion of construction can also used for static response-oriented model updating. Usually, each structure, such as a long-span bridge, undergoes static and dynamic load tests before the bridge is open to traffic. The data from completed bridge tests reflect the initial health status when the bridge is finished. These data provide useful reference for the model updating and verification. (2) Adoption of a proper model updating method. In this study, the sensitivity-based parameter updating method was eventually selected because of its clear physical meaning of the selected parameters and successful application to the civil structure. (3) Selection of parameters to update, and the corresponding responses should be sufficiently sensitive. The number of parameters should be kept small and it would be most effective in producing a genuine improvement in structural modeling. Sensitivity analysis was thus conducted for choosing the structural parameter with consistently large values. Other important factors such as selected modes, initial estimation and bound limitations of parameters, and convergence criteria for

T.H.T. Chan et al. / Finite Elements in Analysis and Design 45 (2009) 795 -- 805

2nd Vertical Mode (V2)

1st Vertical Mode (V1) Normalized Mode Shape

803

1.5

1.5

1 1

0.5 0

0.5

-0.5 0

-1

-0.5 -378 -278 -178

-78

22

122

222

322

-1.5 -378 -278 -178

-78

22

122

222

322

222

322

222

322

Bridge Coordinates (m)

Normalized Mode Shape

3rd Vertical Mode (V3)

4th Vertical Mode (V4)

1.5

1.5 1

1

0.5 0

0.5

-0.5

0

-1

-0.5 -378 -278 -178

-78

22

122

-1.5 222 322 -378 -278 -178 Bridge Coordinates (m)

Normalized Mode Shape

1.2

0.9

0.8

0.6

0.4

0.3

0

0

-0.4

-0.3

-0.8 -78

22

122

22

122

2nd Lateral Mode (L2)

1st Lateral Mode (L1) 1.2

-0.6 -378 -278 -178

-78

222

322

-1.2 -378 -278 -178

-78

22

122

Bridge Coordinates (m) 1st Torsional Mode (T1) Normalized Mode Shape

1.5 1

Test Values Calculated Values

0.5 0 -0.5 -1 -1.5 -378 -278 -178 -78 22 122 222 Bridge Coordinates (m)

322

Fig. 9. Comparison of the mode shape calculated by the developed model with the measured data.

iteration combined with objective function should be also carefully considered during the updating process. Model updating is a process by iteration steps to modify the selected parameters based on the reference data from before so that a closer-to-real model can be achieved. The verification procedure serves the purpose of measuring how many discrepancies were left between the FE model predictions and reference data after the model updating. The generalised form of model verification procedures could be described as follows: (1) First of all, the dynamic characteristics predicted by the updated model, with respect to natural frequencies and mode

shapes, would be closer to the measured data from the ambient vibration test or structural health monitoring data directly obtained from the installed sensors than those before updating. (2) Secondly, the updated model could be further verified at the global scale level, in particular, the aspect of component nominal stress by comparison of static response between the predicted results from the updated model and the data from health monitoring or field tests available at the selected points. (3) Verification would be finally carried out focused on the local scale level if the monitoring data or experimental data are available at the concerned hot-spot area for comparison.

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T.H.T. Chan et al. / Finite Elements in Analysis and Design 45 (2009) 795 -- 805

Upstream T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11

Downstream T12

U1

U2

U3

U4

U5

U6

U7

U8

U9

U10

U11

U12

B4

B5

B1

B2

B3

B6

B7

B8

Fig. 10. Locations of strain gauge sensors installed in the mid-span section of the bridge deck in the loading test before the RYCB is open to traffic.

Table 8 Comparison of nominal stresses calculated by the developed model with those measured at the selected measurement points. Sensor location

a

e

 (%)

Sensor location

a

e

 (%)

Sensor location

a

e

 (%)

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12

−17.5 −17.0 −18.8 −16.7 −17.9 −20.8 −21.0 −17.8 −16.9 −18.3 −16.9 −17.4

−16.6 −16.6 −17.4 −17.7 −18.2 −19.1 −19.2 −18.6 −18.0 −19.0 −18.0 −17.2

5.4 2.3 8.1 −5.5 −1.7 8.7 9.4 −4.0 −5.8 −3.4 −6.1 1.2

U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12

−9.3 −10.1 −14.1 −10.6 −11.4 −9.5 −9.8 −12.5 −11.1 −15.0 −10.9 −9.8

−9.9 −9.6 −14.4 −10.4 −9.4 −9.9 −9.7 −11.3 −11.3 −14.4 −11.1 −11.3

6.1 −4.9 1.8 −1.6 −21.1 3.9 −1.2 −10.2 1.7 -4.2 1.4 13.3

B1 B2 B3 B4 B5 B6 B7 B8 – – – –

27.8 24.7 22.8 20.9 22.0 21.8 24.9 27.8 – – – –

29.8 28.3 26.2 25.8 23.0 23.4 24.6 26.0 – – – –

6.7 12.7 13.1 18.9 4.3 6.7 -1.1 -6.8 – – – –

The CMSM-of-SB of the Runyang cable-stayed bridge deck had been developed by using multi-stage substructuring based on the proposed methodology and strategy for concurrent multi-scale modeling [15]. Here, the verification of the developed model will be studied since it is definitely necessary to prove its accuracy and efficiency before the model is to be used for the analysis of structural deterioration. The model verification for the Runyang cable-stayed bridge is carried out on the basis of the measured data from the static and dynamic load tests before the bridge is open to traffic. The relevant detailed information on the static and dynamic load tests of the RYCB could be found in Sun's thesis [16]. 6.2. Verifying the model with the measured dynamic parameters The ambient vibration test of the RYCB was carried out to obtain the measured values of dynamic characteristics in term of natural frequencies, mode shapes and damping ratios. Then, with the condition of corresponding mode shape matching, the natural frequencies comparison of measured values and the calculated ones by structural global model and CMSM-of-SB was performed as in Table 7. The comparison of the calculated natural frequency with the test data revealed that there were only small error between the predictions of two models and measurement, where the maximum error was 7.38% for the global model and 6.84% for the CMSM-of-SB. Furthermore, as for the errors between the two finite-element models, it was even smaller with less than 1% error. All of those indicated that both of the predicted values by two finite-element models agreed well with measured ones while the calculated values by two models were very close to each other to some extent. In addition, the results proved the accuracy of the modeling was close to the real structure and made it reasonable because on the basis of the global structural model, CMSM-of-SB was modified at the concerned locations of limited number that will not change the global dynamic characteristics with respect to natural frequencies. As for the mode shape correlation extent, the result calculated by the global structural model was plot and compared with those from the test as shown in Fig. 9, which could be considered as a good agreement within the allowable engineering errors.

6.3. Verifying the model with the measured stresses in the deck Besides the model verification at the aspect of dynamic characteristics, further model verification on the structural response under the specified loading can be done by using the stress distribution obtained from the measurement in static load tests before the RYCB is open to traffic. Under the specified loading case [16], the strain data measured by the strain gauge installed in the mid-span section of bridge deck, are illustrated in Fig. 10, was extracted and converted into the nominal stresses for the comparison with those calculated by the developed CMSM-of-SB of the RYCB at the selected measurement points are tabulated in Table 8 and plotted in Fig. 11 shows. The comparison of the calculated nominal stresses with the measured data showed a good fit with each other and overall differences were only 6.4%, which was accurate enough for the engineering problem. 7. Conclusions The studies on the model updating and verification of the concurrent multi-scale model have been carried out in this paper. The entire procedures were implemented on the concurrent multi-scale model of a typical longitudinal steel truss section as a case study which had been developed based on the modeling methodology and strategy proposed in the previous part of this companion paper. On the basis of the results obtained from this study, the following issues are concluded from this paper. The sensitivity-based parameter updating procedure can be applied to the model updating of Concurrent Multi-Scale Model of Structural Behaviour of civil infrastructure, which has been successfully implemented by the steel truss as a case study presented in this work. The model updating procedure mainly contains three important issues, namely selection of measured responses as reference data, selection of sensitive model parameters to be updated, and model parameter updating with the measured data of structural response as the target values. • Besides the natural frequencies and mode shapes obtained from ambient vibration tests, the measured data of static or dynamic

T.H.T. Chan et al. / Finite Elements in Analysis and Design 45 (2009) 795 -- 805

Stress Values (MPa)

25

preliminary implementation procedures had been conducted on the reduced scale specimen of longitudinal truss section studied in the laboratory. As an important actual engineering practice, CMSMof-SB of the Runyang cable-stayed bridge had been successfully constructed by employing the proposed strategy and methods, and the developed model was further verified to be reliable and accurate enough for the actual engineering problem by comparing the measurement data from the field test and predictions by constructed model. Therefore, the proposed theoretical framework and relevant implementation methods with regard to the concurrent multi-scale modeling will be applied to further studies on the nonlinear analysis on structural deterioration of large civil infrastructure, which will be soon presented in our next paper on damage analyses as the application of concurrent multi-scale modeling of civil infrastructures.

Tested Values Calculated Values

20 15 10 5 0 T1

T3

T5

T7

T9

T11

Stress Values (MPa)

18 Tested Values

15

Calculated Values

12 9

Acknowledgements

6 3

The work described in this paper was substantially supported by the grant from the National Natural Science Foundation of China (10672038 and 90715014) and the Research Grants Council of the Hong Kong Special Administrative Region, China (PolyU 5134/03E), which are gratefully acknowledged.

0 U1

U3

35 Stress Values (MPa)

805

U5

U7

U9

U11

Tested Values Calculated Values

30 25

References

20 15 10 5 0 B1

B2

B4 B5 B6 B3 Sensor Installed Locations

B7

B8

Fig. 11. Nominal stress comparison between the calculated values with those measured at the selected locations with strain gauges.

responses of the structure could also be used for model updating of CMSM-of-SB for civil infrastructure. The combined methods of model updating-based dynamic characteristics and static response provide a new method better than the model updating based on mere dynamic property. • The verification and updating processes were mutually beneficial and it embodied the accuracy extent of the FE modeling through addressing the differences between result predictions by the updated model with reference data. The model can be considered as the baseline model to be the closest to the prototype of the structure after updating and verification. • After justified to be effective and accurate for the engineering application, the updating and verification procedures proposed in this paper has been extended from the laboratory scaled structure to civil infrastructure, and the model verification of CMSM-ofSB for the Runyang cable-stayed bridge has been successfully performed as an actual engineering practice of the proposed procedures. The concurrent multi-scale modeling methodology and strategy proposed in the first part of the companion paper constitute the theoretical basis of nonlinear physical-based modeling for structural health monitoring considering the structural deterioration process. The modeling updating and verification procedures described in this paper were then applied to obtain the baseline model for subsequent analysis by numerical simulation. The whole theoretical framework was therefore considered to be completed and

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