Modal analysis of the Jalon Viaduct using FE updating

Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4 Modal analysis of the Jalon Viaduct using FE updating Chaoyi Xia 1,2, Guido De Roeck 2 1 Department of Civil Engineering, Beijing Jiaotong University, 100044 Beijing, China 2 Department of Civil Engineering, Faculty of Engineering, KU Leuven, B3001 Leuven, Belgium email: xiacy88@163.com, Guido.DeRoeck@bwk.kuleuven.be ABSTRACT: Due to differences in stiffness, mass, damping, internal connections, external boundary conditions between a real structure and a FE model, differences will exist in the vibration characteristics including natural frequencies and mode shapes. To solve this discrepancy, FE updating has been widely used in civil engineering. The uncertain parameters in the FE model are adapted to minimize the differences. The Jalon Viaduct along the high-speed railway line between Madrid and Barcelona is selected as a case study. The viaduct is a continuous box girder supported on high piers. From ambient vibration, mode shapes have been derived in vertical, lateral and longitudinal direction. For the updating the natural frequencies and the threedimensional mode shapes of the first five lateral and first five vertical modes of the viaduct are used. For this calibration, 14 updating parameters including 12 spring stiffnesses and 2 Young s moduli are selected. The fact that also longitudinal components of the mode shapes are considered helped the updating process. Due to the existence of some local minima of the objective function induced by interfering modes, a Multistart Solver is used to obtain the global solution. It is shown that the relative weight of some residuals has to be adjusted in order to have better results for the longitudinal components of the vertical modes. The updating results in a much better agreement between the measured and the predicted natural frequencies and mode shapes. KEY WORDS: FE updating; dynamic properties; bridge structure; updating parameter; multistart solver. 1 GENERAL GUIDELINES FE model updating method has been widely used in civil engineering, especially for structural identification based on measured modal parameters. Because of differences in stiffness, mass damping, connections and boundary conditions between the real structure and the FE model, dissimilarities will exist in the vibration characteristics including natural frequencies and mode shapes [1]. The uncertain parameters in the FE model are adapted to minimize these differences. This optimization procedure is also called FE-updating. Levin and Lieven indicated that there are two methods to adapt the FE model, one is via updating the mass and stiffness matrices, the other is parametric updating, an indirect but physically meaningful way [2]. In the past decades, the form of the objective function and the solution strategy in parametric optimization have been studied by many researchers. Jaishi presented a finite element (FE) model updating method for a real bridge structure under operational condition using modal flexibility, in which the FE mass matrix together with the Guyan technique is used for the mass normalization of the mode shapes extracted from the ambient modal test to calculate the modal flexibility [3]. During the optimization procedure, some parameters are considered as updating variables, to make the calculated dynamic properties closer to the experimental ones. Spring stiffnesses at the boundaries and material properties, are commonly chosen as updating parameters. Before starting the optimization, initial values of the updating parameters should be selected. This selection might influence the result of the updating if local minima exist. Methods such as Globalsearch, Multistart, simulated annealing, and coupled local minimizers could be used in the optimization procedure [4]. In this paper, a FE model updating analysis for the prestressed concrete Jalon Viaduct in Spain is executed using a solid model, based on the data from 303 measurement points. 14 updating parameters are chosen, including the values of spring stiffnesses at the supports and Young s moduli of the materials. The updating procedure is performed by using the command lsqnonlin from Matlab, and a multistart solver is used to obtain the global optimum. 2 UPDATING METHOD The FE model of the system can be treated as an objective function F(p) that generates the correct measured model frequencies (f) and mode shapes (ϕ) of the structure upon providing the actual physical parameters as ideal inputs to the model (p). Thus, the FE model updating is formulated as an optimization problem, where the difference between the initial experimental structural model and the predictive FE model is minimized. The minimization of the objective function is stated by the following nonlinear least square problem. p = arg min F( p) = arg min 1 [ wr ( )] 2 i i p (1) p p 2 i The residual vector R(p) can be split into two parts: the frequency residual vector, R f (p) R, and the mode shape residual vector R ϕ (p) R. The residual vectors for frequency and mode shape are shown as follows. Residual of mode shape: 2311

( p % %) conj( φ i( p)) φi( ) φi φ i R( φ, p) = / ( % φi ) conj( φi( p) % φi % φi) ( φi( p) % φi % φi) i i (1, 2, n) (2) where conj(a ) represents the non-conjugated transposed matrix of A. Residuals of frequency: R( f, p) = ( f ( p) f % )/ f % j (1, 2, m) (3) 2 2 2 j j j where, respectively, f and f % represent the predicted and experimental modal displacements; f and f % represent the predicted and experimental frequencies; w f and w f represent weight factors of mode shape and frequency; n is the number of mode shapes, and m the number of frequencies. During the optimization procedure, the predicted and experimental results will match well when the objective function is close to a minimum. To evaluate the consistency quality between the measured and the FE-computed mode shape, the modal assurance criterion value is used, which is expressed as (b) Figure 1. Configuration of the Jalon Bridge To determine the dynamic characteristics of the bridge, a vibration measurement campaign was carried out. In total, vibrations at 303 nodes were measured in 38 setups by using twelve triaxial wireless sensors [5]. Shown in Figure 2 are the locations of the 303 measurement nodes, which are arranged in three lines inside the box girder. It can be seen the measurement nodes are arranged at left, right and off-center positions on the bottom slab of the viaduct. The measured vibration data are analyzed by using the Modal Analysis of Civil Engineering Constructions (EC) toolbox [5], by which both ambient vibration data and free vibration data after the train passage are analyzed. k ()() φ % j φ j j = 1 = k k ()() j j ()() % % j= j= j j φ φ φ φ 1 1 2 (4) where, f and f % represent, respectively, the FE computed and experimental modal displacements at the measurement point j. The value is between 0 and 1. The updating work is based on the trust region Newton method, which is a gradient-based iterative algorithm. The procedure is performed by using the command lsqnonlin from Matlab. 3 MODAL ANALYSIS OF THE JALON BRIDGE 3.1 Description of the Jalon Bridge The Jalon viaduct is a six-span continuous railway bridge located in Spain. The bridge consists of (35+45+45+45+45 +35) m span PC box girders, supported by concrete piers with round section, as shown in Figure 1(a) and (b). Figure 2. Arrangement of the measurement nodes 3.2 FE model of the Jalon Bridge A finite element model with solid elements is built in ANSYS. Beam4 elements for the piers and Solid45 elements for the box girder and the ballast are used, as shown in Figure 3. In total, there are 22886 nodes, 35 beam elements and 15812 solid elements in the FE model. This model is used for the updating calculation. Figure 3. Solid element model of the Jalon Bridge In the FE Model, the connections between girder and piers are set to be rigid in lateral, vertical and rotational directions, as shown in Figure 4. (a) 2312

Figure 4. Rigid in plane connection between the girder and the pier In this model, Combine 14 elements are used to simulate the lateral stiffnesses at the fixed and the sliding end of the viaduct. Other Combine14 elements are built at the foundation of the piers to simulate the vertical stiffness and the rotational stiffness along the longitudinal direction. The other degreesof-freedom at the foundation are fixed. 3.3 Modal results of the Jalon Bridge By FE calculation, the dynamic properties such as mode shapes and frequencies of the bridge are obtained, and the results are compared with the experimental ones [5], as shown in Table 1. Table 1. Modal results by FE analysis and experiment Frequency Mode Nr (Hz) shape FEM Exp Description result results value 1 0.601 0.648 0.995 1st lateral bending 2 1.182 1.238 0.993 2nd lateral bending 3 2.201 2.230 0.989 3rd lateral bending 4 3.279 3.279 0.990 Vertical bending 5 3.595 3.536 0.973 4th lateral bending 6 3.778 3.758 0.990 Vertical bending 7 4.094 - Vertical bending 8 4.421 4.471 0.961 Vertical bending 9 5.211 4.917 0.969 5th lateral bending 10 5.015 5.056 0.967 Vertical bending 11 5.717 5.309 0.865 Torsion 12 5.966 6.036 0.980 Vertical bending In the table, is the modal assurance criterion value defined in Eq. (4). If the experimental and the FE computed mode shapes match perfectly, the value will be 1.0. Shown in Figure 5 are the lateral and the longitudinal components of the experimental and the FE modal displacements of the first lateral mode. Mode shapes are normalized to unit maximum modal displacement. In the figure, the lines with blue asterisks represent the experimental modal displacement, and the red lines represent the modal displacements obtained by ANSYS. The red and blue lines match quite well, which means that the lateral mode shapes obtained by the FE model are in good accordance with the experimental ones. Figure 5. Comparison of the experimental and the FE modal displacements of the first lateral mode Shown in Figure 6 are the vertical and the longitudinal components of the experimental and the FE modal displacements of the 4 th vertical mode. Figure 6. Comparison of the experimental and the FEM modal displacements of the fourth vertical mode A good correspondence can be found from the figures between the vertical components of the modal displacements, while in the longitudinal direction there are obvious differences in modal displacements. The further from the fixed end, the bigger the deviation. 4 FE UPDATING OF THE JALON BRIDGE As seen from Table 1, there exist clear differences between the calculated and the measured frequencies and mode shapes. Therefore, the FE model of the Jalon Bridge should be updated by correcting the uncertain parameters of the bridge structure. From Table 1, one can see that the values of the 1st, 2nd, 3rd, 4th, 5th, 6th, 8th, 9th, 10th and 12th modes are 2313

higher than 0.96, therefore the updating process will be carried out toward improving the frequencies and mode shapes of these modes. 4.1 Choice of updating parameters To update the FE model, fourteen uncertain parameters of the bridge structure are selected in the updating calculation, as listed in Table 2. The layout of the spring elements in the model is shown in Figure 7. In the figure, the x-axis, y-axis and z-axis represent, respectively, the longitudinal, lateral and vertical directions. 4.2 Multistart solver The FE model is updated using the optimization procedure defined in Eqs. (1)-(3). The predicted and the experimental results will match well when the objective function is close to a minimum. The optimization procedure will stop after reaching a convergence criterion. To illustrate the possibility of the occurrence of local minima and hence the need of a multistart solver, a simulation example is presented. Only one updating parameter P 1 will be considered. The initial value P init is set to 10% of the true ( experimental ) result, so the scale factor θ 1 =10 corresponds to the true ( experimental ) results. Figure 8 shows the square value of the objective function as a function of θ 1. The peak A in the function curve of Figure 8 will be analyzed. Figure 7. Layout of the spring elements in the bridge model Table 2. Description of the updating parameters Nr Parameter Details P 1 stiffness longitudinal springs at the bottom of the cross-section at the fixed end of the girder P 2 stiffness lateral springs at the bottom of the cross-section at the fixed end of the girder P 3 stiffness longitudinal springs in the center of the cross-section at the fixed end P 4 stiffness lateral springs in the center of the cross-section at fixed end P 5 stiffness longitudinal springs between the girders and the piers P 6 stiffness lateral springs at the foundation P 7 stiffness longitudinal springs at the foundation P 8 stiffness rotational springs at the foundation along the vertical direction P 9 stiffness rotational springs at the foundation along the lateral direction P 10 stiffness rotational springs at the foundation along the longitudinal direction P 11 stiffness longitudinal springs at the bottom of the cross-section at the sliding end of the girder P 12 stiffness lateral springs at the bottom of the cross-section at the sliding end of the girder P 13 elastic modulus box girder P 14 elastic modulus ballast Figure 8. Local minima of the objective function The values (at θ 1 =1.9) between the experimental and the predicted mode shapes of the 1 st, 2 nd, 3 rd, 4 th and 5 th modes are shown in Table 3. Table 3. values between the experimental and predicted modes FEM Experimental Nr Nr 1 2 3 4 5 1 1.000 0.000 0.000 0.000 0.000 2 0.000 1.000 0.000 0.000 0.000 3 0.000 0.000 1.000 0.000 0.000 4 0.000 0.000 0.000 0.992 0.000 5 0.000 0.000 0.000 0.972 0.000 6 0.000 0.000 0.000 0.000 1.000 There are two predicted modes (the 4 th and 5 th modes) that can be matched with the 4 th experimental mode. The detailed comparisons are shown in Figure 9. (a) The 4 th mode (b) The 5 th modes Figure 9. Experimental (blue) and predicted (red) modal displacements of the 4 th and 5 th modes 2314

In Figure 9(a) and (b) are the vertical modal displacements, with the blue dotted lines representing the 4 th experimental mode, the red line in Figure 9(a) the 4 th predicted mode, and the red line in Figure 9(b) the 5 th predicted mode. By comparing the modal displacements of the 4 th experimental mode and the 4 th and 5 th predicted modes, the local minimum of the objective function can be explained by an interference effect of a longitudinal and a vertical mode. In a next simulation, local minima due to a change of P 1 and P 3 (both are stiffnesses of longitudinal springs at the fixed end) will be explored. Firstly, let q = P (5) P init P should converge to the true value of this updating parameter. P init is the initial value of the updating parameter in the initial FE-model. P 1init and P 3init are chosen, and their values are set as 0.1P 1 and 0.1P 3. The calculated results correspond to different values of P 1 (θ 1 ) and P 3 (θ 3 ). The exact solution for θ 1 and θ 3 will be 10. A 3D plot in Figure 10 shows the relationship between the objective function and the updating parameters. Figure 10. Objective function composed of mode shapes and frequencies of the first five lateral and vertical modes From Figure 10, a number of local minima can be seen in the undulation of the blue area. To solve this problem, a method named Multistart solver is adopted. This method considers several starting points in the hope that at least one of them can reach the global minimum. These starting points can be set manually, and they can also be chosen at random by using the function of x0rndgen in the CLM toolbox [4]. The convergence paths of these different starting points are shown in Figure 11. Figure 11. Convergence process for different starting points In this figure, the red Asterisks represent the starting points, the black Asterisks represent different points during iteration, and the red circles represent the final solution of the optimization process. The convergence paths for different starting points are connected by black lines. It shows that some of the starting points converge to several local minima of the objective function, but most points converge to the global minimum. Although the Multistart solver costs more computational time, the feasibility of this method has been proved. Therefore, this method will be adopted in the updating analysis using the real experimental data. The calculation results are shown in Table 4. It can be seen that there is a clear improvement for the natural frequencies, especially of those modes which are affected by the foundation stiffness. Table 4. Comparison of experimental results, initial FEM results and updated results Experimental Initial FEM Updated results results results Nr Freq Freq Freq (Hz) (Hz) (Hz) 1 0.648 0.601 0.9877 0.647 0.9905 2 1.238 1.182 0.9827 1.232 0.9802 3 2.230 2.201 0.9782 2.242 0.9774 4 3.279 3.279 0.9773 3.255 0.9906 5 3.536 3.595 0.9708 3.573 0.9714 6 3.758 3.778 0.9802 3.778 0.9461 8 4.471 4.421 0.9283 4.446 0.9103 9 4.917 5.211 0.9487 4.916 0.9306 10 5.056 5.015 0.9327 5.085 0.9463 12 6.036 5.966 0.8874 6.083 0.9846 For the values of the mode shapes that are improved compared to the results without updating, the reduction of the differences between predicted and experimental results are mainly due to the longitudinal components of the vertical modes. Normally, the updating work is focused on natural frequencies and lateral and vertical components of the mode shapes, as the longitudinal modal displacements are normally not measured. Therefore, the results in the Table 4 can be treated as good results. 2315

4.3 Sensitivity test of the updating parameters The sensitivity of the frequencies and mode shapes to the parameters provides useful information for the updating process. The frequency sensitivity with respect to the selected value of any of the first 12 parameters p i, is computed. S ij p Δω i j = Δpi ω j i, j (1,2,3, ) (6) where, S ij is the sensitivity, i represents the number of the considered updating parameter, j represents the mode number, p is the value of updating parameter, and ω is the value of the measured frequency of mode j. The sensitivity will be almost zero, when the value of the updating parameter is close to the minimum of the objective function. In this study, the values of p i are 90% of the final values of the updating parameters, the values of p i are 10% of p i. The calculated sensitivity values for the most influencing parameters 5 and 10 in Table 2 are shown in Figure 12. From this figure, it can be found that the frequencies of the 1 st, 2 nd, 3 rd, 5 th and 9 th modes are sensitive to the updating parameters P 5 (longitudinal springs between girders and piers) and P 10 (rotational springs at foundation along the longitudinal direction). Moreover, these two parameters mainly influence the frequencies of the lateral 1 st, 2 nd and 3 rd modes. Sensitivity Sensitivity 0.095 0.075 0.055 0.035 0.015-0.005 0.095 0.075 0.055 0.035 0.015 Parameter 5 1 2 3 4 5 6 7 8 9 10 11 12 Mode number Parameter 10 (rotational springs at the foundation along longitudinal direction) influences the lateral modes. The influence of small increments of other updating parameters to the values is not obvious. 4.4 Adjustment of weight factors The objective function is composed by the mode shapes and frequencies of the 1 st, 2 nd, 3 rd, 4 th, 5 th, 6 th, 8 th, 9 th, 10 th and 12 th modes. In the previous calculation, the weight factors for all frequencies (w f ) and all mode shapes (w ϕ ) were set as 1.0. In the next analysis, the weight factors for the longitudinal modal displacements of the 4 th, 6 th and 8 th vertical modes are set as 3.0, and those for the other modal displacements and all frequencies are set as 1.0. The comparison of the experimental to the initial and updated FEM results is shown in the Table 5. It is found that although the predicted frequencies are less good than the ones without adjustment of weighting factors, most of the values are clearly improved. Table 5. Comparison of experimental results, initial FEM results and updated results with different weights Experimental Initial FEM Updated results results results Nr Freq Freq Freq (Hz) (Hz) (Hz) 1 0.648 0.601 0.9877 0.644 0.9928 2 1.238 1.182 0.9827 1.226 0.9845 3 2.230 2.201 0.9782 2.239 0.9796 4 3.279 3.279 0.9773 3.256 0.9946 5 3.536 3.595 0.9708 3.576 0.9723 6 3.758 3.778 0.9802 3.758 0.9870 8 4.471 4.421 0.9283 4.502 0.9890 9 4.917 5.211 0.9487 4.906 0.9251 10 5.056 5.015 0.9327 5.113 0.9714 12 6.036 5.966 0.8874 6.154 0.9848 Detailed comparisons of the 4 th and 12 th experimental, initial and updated FEM mode shapes are shown in Figures 13 and 14. -0.005 1 2 3 4 5 6 7 8 9 10 11 12 Mode number Figure 12. Sensitivity values of the frequencies with respect to the first 12 parameters Then, a similar study can be done to investigate how the variations in any of the 14 identified parameters impact the FE-computed mode shapes. In the analysis, the values of the lower and upper bounds are 90% and 110% of the final values of p i. The results show that the most influencing updating parameter is parameter P 1 (stiffness of the longitudinal spring at the fixed end of the girder). An increment to it will result in the changes of all values, especially for the 2 nd, 3 rd, 4 th, 5 th, 6 th, 8 th and 10 th vertical modes. The updating parameter P 10 Figure 13. Comparison of the experimental (blue), initial (black) and updated (red) FEM modal displacements: vertical mode 4 2316

ACKNOWLEDGMENTS This study is sponsored by the Natural Science Foundation (51308035), the State Fundamental Research Funds 973 Program of China (2013CB036203), the 111 Project (B13002) and the Fundamental Research Funds for the Central Universities (2013JBM011) of China, and the Flanders (Belgium)-China Bilateral Project (BIL 07/07). Figure 14. Comparison of the experimental (blue), initial (black) and updated (red) FEM modal displacements: vertical mode 12 In these figures, the blue lines represent the experimental modal displacements, the black lines represent the modal displacements obtained by the initial FE model, and the red lines represent the predicted modal displacements after updating. 5 CONCLUSION A FE model updating analysis for the dynamic properties of a prestressed concrete viaduct is performed by using the command lsqnonlin from Matlab. The following conclusions can be drawn: (1) A good correspondence has been obtained between the measured and the calculated dynamic characteristics, especially for the natural frequencies and the vertical and the longitudinal components of the vertical mode shapes, by adapting 12 spring stiffnesses and 2 Young s moduli as updating parameters in the FE model. (2) Among these parameters, the stiffness of the longitudinal springs between the girder and piers and the stiffness of the rotational springs at the foundation along the longitudinal direction are the most influencing parameters on the frequencies of the first five lateral and vertical modes. The updating with regard to the frequencies of the first 12 modes is mainly performed through adjustments of the foundation stiffnesses. (3) The stiffness of the longitudinal springs at the fixed ends of the girder is the most influencing parameter of the values of both lateral and vertical modes. Another influencing parameter is the stiffness of the rotational spring at the foundation along the longitudinal direction. (4) Because of the expected local minima of the objective function due to interfering modes which appear with the change of stiffness of the springs in the longitudinal direction, it is difficult to select an initial searching point that will converge directly to the global minimum. (5) The relative weight factors play an important role in the updating process. To have better results for the longitudinal components of the vertical modes, some weights have to be adjusted. REFERENCES [1] Bakir P.G., Reynders E. and De Roeck G.. Sensitivity based finite element model updating using constrained optimization with a trust region algorithm. Journal of Sound and Vibration, 2007, 305, 211 225. [2] Levin R. and Lieven N. Dynamic finite element model updating using simulated annealing and genetic algorithms, Mechanical Systems and Signal Processing, 1998, 12(1), 91 120. [3] Jaishi B., Kim H.J., Kim M. K., Ren W.X. and Lee S.H. Finite element model updating of concrete-filled steel tubular arch bridge under operational condition using modal flexibility. Mechanical Systems and Signal Processing, 2007, 21(6), 2406-2426. [4] Badsar A., Schevenels M., Reynders E., Lombaert G. and Degrande G. User guide of CLM toolbox for Matlab, K.U. Leuven, 2007. [5] He L.Q., Qin S.Q., Bui Tien T., Reynders E., Museros P. and De Roeck G. Operational Modal Tests of the Jalon Viaduct, Technical Report, K.U. Leuven, 2011. [6] Reynders E., Schevenels M. and De Roeck G. EC 3.1: a Matlab toolbox for experimental and operational modal analysis. Report BWM-2010-05, K.U. Leuven, 2010. [7] He L.Q., Simoen E., Reynders E. and De Roeck G. Vibration-based damage assessment of the Boirs Viaduct by using finite element model updating. The 5th European Conference on Structural Control, 2012. [8] Christodoulou K., Ntotsios E., Papadimitriou C., Panetsos P. Structural model updating and prediction variability using Pareto optimal models, Comput. Methods Appl. Mech. Engrg. 2008, 198, 138-149. 2317