Ambient vibration testing and updating of the finite element model of a simply supported beam bridge

Abstract

An ambient vibration test on a concrete bridge constructed in 1971 and calibration of its finite element model are presented. The bridge is characterized by a system of post-tensioned and simply supported beams. The dynamic characteristics of the bridge, i.e. natural frequencies, mode shapes and damping ratios were computed from the ambient vibration tests by using the Eigensystem Realization Algorithm (ERA). Then, these characteristics were used to update the finite element model of the bridge by formulating an optimization problem and then using Genetic Algorithms (GA) to solve it. From the results of the ambient vibration test of this type of bridge, it is concluded that two-dimensional mode shapes exist: in the longitudinal and transverse; and these experimentally obtained dynamic characteristics were also achieved in the analytical model through updating. The application of GAs as optimization techniques showed great versatility to optimize any number and type of variables in the model.

Keywords

modal analysis / parameter identification / ambient vibration test / Eigensystem Realization Algorithm (ERA) method / finite element method

Introduction

In the last decades, one of the main research topics in structural engineering was the development and application of new and powerful numerical methods for the correct analysis and design of large and complex structures. In this context, the growing developments of the finite element method and the advances in the field of computers have been the reason for such success.

However, it is necessary to make sure that the finite element model, FEM, of the structure corresponds to the built structure. For civil engineering structures, some uncertainties in the FEM are referred, for example, to the value of the elasticity modules of the used materials, the stiffness values used to idealize the conditions of supports (soil-structure interaction), the stiffness of joints between elements, among others. Thus, an update process should be performed, where the most important structural properties (static and dynamic) are verified with the aid of experimental analysis.

As regards to modal identification, the early researches consisted of a forced vibration test (Input-Output) [1]. Nevertheless, the difficulty of carrying out this type of dynamic test in large civil structures in a controlled way made the scientific community seek other identification methods that could best be applied. The technological progress in transducers and signal converters, analog to digital, opened a new and promising research line for the modal identification of large structures. The basic idea is the use of exclusive measures of structural response to excitation produced by the environment (Only-Output) and the application of stochastic modal identification methods. These are the so-called ambient vibration tests, which determine the response of the structure under the excitation of wind, traffic or micro-earthquakes without any artificial excitation, from these tests; we can obtain the more relevant dynamic properties of the structure (natural frequencies and modes of vibration) [2,3]. These parameters will be used to perform the update of the FEM of the structure.

A class of updating technique of FEMs is based on sensitivity analysis and by using either optimization software or manually [4,5]. The procedure consists in establishing the relevant parameters in the response of the structure and, then, performing an optimization process to minimize the error between the experimental response and the response obtained by an analytical model. In Ref. [6]. was used a method presented by Ref. [7]. to calibrate the FEM of a bridge, the method minimizes the difference between theoretical and experimental natural frequencies by updating a few variables of the FEM.

On the other hand, the difficulty level of an optimization problem increases with the number of optimization variables and the type of objective function. The success of conventional optimization techniques might depend on the correct choice of initial values for the variable to be optimized. For the model updating problem, it is necessary to have a methodology that permits the handling large quantities of parameters and also, that it can overcome the difficulties associated with highly nonlinear objective function. Unconventional techniques could show better results as in Ref. [8,9].

This paper presents the updating process of the FEM of a simply supported concrete bridge by using information from ambient vibration tests. The modal analysis experiment will provide the dynamic characteristics (frequencies, mode shapes and damping rates) of the structure. Genetic algorithms were used as optimization technique.

Description of the bridge

Garcia Cadena Bridge is a post-tensioned reinforced concrete structure, which was built in 1971 (Figs. 1–3). In 1993, the bridge was expanded by the addition of four beams and it was reinforced in order to satisfy architectural, functional, structural, and safety requirements. Currently, the bridge has the length of 261.65 m and the width of 25 m (six-way traffic). At the time of its construction, the bridge had only the width of 17.20 m. A total of eight spans (seven piers) are observed, where the central and extreme spans have the length of 33 and 31.825 m, respectively.

Analytical model of the bridge

Two types of elements were used to generate the finite element model of the bridge: shell-type elements for modeling the slab of the superstructure and frame-type elements for the other members. The beams in the superstructure were connected to the slab through the nodes conforming each element. This connection consists in matching all of the translational and rotational degrees of freedom (constraint equal or body type) of both elements. The latter ensures that the slab translates and deforms equally to the beams, guarantying the continuity of the movement between the two elements. The support of the main beams with the beam head of the piers were modeled using nlink elements, which simulate the conditions of semi-restricted movements in the degrees of freedom either by the allocation of a stiffness value or completely restricted. In the supports, horizontal displacements in the longitudinal direction and rotations around the transverse axis that were allowed. For each adjacent span beam, a different nlink was generated, which simulated the independence between the span beams.

Figure 4 shows the first four vibration modes from the finite element model of a single span simply supported without the inclusion of piers (span B7). It was observed that the fundamental frequencies were less than 5 Hz and that the bridge has the vibration components in the transverse and longitudinal directions.

The definition of a numerical model of the structure plays an important role before the ambient vibration test is carried out as it permits obtaining approximated natural frequencies and configurations of the main vibration modes. The values of the frequencies are essential for the defining of sensor types, the data acquisition time and the sampling frequency. In turn, mode shapes help making the decision on the choice of the sensor locations along the span.

Ambient vibration test

The ambient vibration tests were carried out by using two force-balanced triaxial episensors (FB ES-T, Kinemetrics) and a K2 multichannel data acquisition system (AltusDigital Recorder, Kinemetrics Company). The K2 has 9 channels for expansion and other 3 channels that are occupied by an internal triaxial episensor. Also, a laptop was used to manage the data acquisition software, which allows the capturing and displaying of the information gathered.

Acceleration measurements were made in different points of the structure spans and along the east, west and central axis in the three orthogonal directions. The measurement configuration consisted in defining eight sections along the span with seven internal measurement points in each section. Then, the test was performed by locating the internal episensor of the K2, called reference sensor, in a specific measurement point and mobilizing the other two episensors on the other measurement points. The location of the reference sensor was selected based on the analytical mode shapes, which should not be on a nodal point for some of the first mode shapes. Also, the piers cannot be used as measurement points as there is a possibility of measuring frequencies other than those of the beams. Using the selected measurement point, it was only possible to characterize the vibration modes of the deck. A total of three sensor arrays, called setups, were configured, as shown in Fig. 5.

On the other hand, Cantieni [10] proposed that the data acquisition window should be between 1000 and 2000 times the analytical fundamental perio, thereby a value of 1500 was chosen for that parameter in this research. Thus, acceleration time series were collected for each setup during 5 to 6 min as the analytical period of the single span was 0.2 s. The sampling frequency was 200 Hz, a value that might be excessive for the expected vibration frequencies; but one that avoids the aliasing problem in the signal. In general, the total measurement time for each span was estimated in 20 min including the equipment transporting time.

Modal identification method

The modal identification was performed by using the Eigensystem Realization Algorithm (ERA) in the time domain [11]. This method uses the estimated correlation functions between signals (Markov parameters) as input parameters in the construction of the Hankel matrix. The singular value decomposition (SVD) is applied to the Hankel matrix, and then the matrices of state space model in discrete time are determined. The modal parameters of the system are obtained from the state-space matrices.

The ERA method can be summarized in the following steps:

1) Build the Hankel matrix for the time k = 1, [H(\mathrm{0})] and time k = 2, [H(\mathrm{1})] through the Markov system parameters [Y_k]. To do that, the dimensions of the Hankel matrix determined by the parameters am\times \beta r have to be defined, where \alpha is the number of Markov parameters in a column of [H(\mathrm{0})] and \beta is the number of Markov parameters in a row.

2) Decompose [H(\mathrm{0})] in singular values to obtain the matrices [{\sum }_{\mathrm{2}n}], [R_{\mathrm{2}n}]and[S_{\mathrm{2}n}].

(1)[H(\mathrm{0})]=[S_{\mathrm{2}n}][{\sum }_{\mathrm{2}n}]{[R_{\mathrm{2}n}]}^T.

3) Determine the order of the system (2n) by examining the singular values of [H(\mathrm{0})].

4) Compute the matrices of discrete state space by:

(2)[A]={[{\sum }_{\mathrm{2}n}]}^{-\mathrm{1}/\mathrm{2}}{[R_{\mathrm{2}n}]}^T[H(\mathrm{1})][S_{\mathrm{2}n}]{[{\sum }_{\mathrm{2}n}]}^{-\mathrm{1}/\mathrm{2}},

(3)[B]={[{\sum }_{\mathrm{2}n}]}^{\mathrm{1}/\mathrm{2}}{[S_{\mathrm{2}n}]}^T[E_r],

(4)[C]={[E_m]}^T[R_{\mathrm{2}n}]{[{\sum }_{\mathrm{2}n}]}^{\mathrm{1}/\mathrm{2}}.

Matrices [A], [B] and [C] represent a minimal realization of the system.

5) Transform the state matrices in modal coordinates and extract the modal parameters.

(5)\begin{array}{l}A=\mathrm{\Psi }{\mathrm{\Lambda }}_d{\mathrm{\Psi }}^{-1},\hfill \\ V=C\mathrm{\Psi }\mathrm{.}\hfill \end{array}

The matrices A and C are sufficient to compute the modal parameters: discrete poles {\mathrm{\Lambda }}_d and the observed mode shapes V.

Bridge dynamics characteristics

The frequency spectra of the signals that were captured in each setup independently were used to implement the ERA. Either Markov parameters or functions of correlation between the signals were obtained by using the inverse Fast Fourier transform (IFFT). As the structure is under-damped, a maximum time of 5 s for creating all correlation functions was used. Figure 6 shows one of the correlation functions found computed.

Then, the correlation functions permitted computing the Hankel matrix for the times k = 1, [H(\mathrm{0})] and k = 2, [H(\mathrm{1})], which is one of the most important issues in the ERA implementation. The choice of the matrix dimensions has a direct relationship with the number of frequencies to identify. For the analyzed structure, it was determined after some trials that Hankel matrix had 270 rows and 180 columns after some trials. The person carrying out the analysis should be careful in the choice of the dimensions of the Hankel matrix dimensions: on one hand, if a very large dimension is chosen, the identification degenerates in the attainment of a higher number of vibration modes that do not exist in the structure and on the other hand a very small dimension could not be enough to guarantee the desired number of vibration modes to be identified. Finally, the dimension also depends on maximum time selected of correlation function.

For the selection of the model order of state-space matrices, bar charts are constructed relating the singular values of the Hankel matrix for each setup to the order of the model (Fig. 7). In most cases, the order of the models ranged between 30 and 50.

Stability diagrams were constructed after of the configuration of the above parameters. These diagrams represent a variation of the order of the model and allow checking if the poles found in the matrix of state space satisfy some stability criteria, as indicated in Table 1.

Figure 8–10 show the stability diagrams for setup 1 on each side of the span B7. These diagrams were superimposed with ANPSD (Normalization Average Potential Spectrum Density), in order to show the correspondence between the spectrum peaks and the frequencies identified with the ERA method (see Figs. 9–11). It is important to notice that only vertical vibrations were considered.

The final selection of frequencies and vibration modes that were identified by the ERA method depends on the definition of a specific model order and on the stability level of the frequencies. So, Table 2 shows the frequencies and damping rates identified by the ERA method.

The assessment of vibration modes using this methodology requires the following steps: 1) construction stability diagrams for all setups, 2) selection of a model order, 3) evaluation the eigenvectors of the matrix state space identified for each setup and, 4) normalization of each eigenvectors respecting the reference point. Table 3 shows the six first mode shapes identified by the ERA method.

As commented before, ambient vibration measurements were made only in the longitudinal direction of the bridge. The results for the configuration of sensors showed that in the east, west and central sides there is identical mode shapes for the same frequencies and also identical mode shapes for different frequencies in the same longitudinal axis. The results obtained in span B7 were also obtained in the other spans on the bridge.

To achieve fully defined vibration modes for this type of bridge, which has a big wide board; it is essential to carry out cross-sectional measurements. In this study, it was not possible to do these cross measurements as the equipments used were connected by wires. This implied in the necessity to stop traffic in one way to locate the sensors, which was not permitted by the local government authorities because of the huge quantity of vehicles crossing the bridge. In spite of that, from the finite element models were found the same mode shapes that those experimentally measured only in the longitudinal direction.

Update methodology of finite element model

The procedure to calibrate the finite element model involves an iterative method, which tries to match the experimental and analytical modal parameters by introducing changes in the structural parameters chosen. The following steps have to be carried out before performing the calibration process:

1) Select the type and quantity of experimental modal information to be compared with theoretical results taken from the finite element model (number of natural frequencies, mode shapes, frequency response functions, etc.).

2) Formulate an objective function that quantifies the differences between the solutions found by the optimization algorithm and the experimental data.

3) If it is necessary, the correspondence between the experimental and analytical degrees of freedom has to be checked. This requirement can be easily met by planning the experimental measurements from a preliminary finite element model.

4) Define the finite element model parameters to be adjusted, such as the elasticity modules of groups of elements, joints and/or support stiffness.

The objective function used was that proposed by [12].

(6)f={\displaystyle \sum _{j=\mathrm{1}}^r{W}_{mj}{\left({\displaystyle \frac{\delta w_{mj}-\delta w_{aj}}{w_{mj}}}\right)}^{\mathrm{2}}}+{\displaystyle \sum _{j=\mathrm{1}}^r{W}_{{\varphi }_j}(\mathrm{1}-MAC({\varphi }_{mj},{\varphi }_{aj}))},

where the subscript m refers to experimental data and the subscript a to those obtained from an analytical model. \delta w corresponds to variations in natural frequencies, this expression reduces the effect of modeling errors and w_j and ø_j are the jth natural frequency and mode shape, respectively. The terms W are weighting factors for each of the dynamic characteristics and were considered to be one. r is the number of mode shapes used for comparison. The term MAC is the Modal Assurance Criterion, which is a method for estimating the degree of correlation between a natural mode of vibration of the model and the corresponding experimental. MAC provides a correlation factor between 0 (low correlation) and 1 (maximum correlation) and is expressed as follows:

(7)MA{C}_{m,j}={\displaystyle \frac{{({\varphi }_m^T.{\varphi }_a)}^{\mathrm{2}}}{({\varphi }_m^T.{\varphi }_m)({\varphi }_a^T.{\varphi }_a)}},

where {\varphi }_m and {\varphi }_a are the values of the experimentally measured mode shapes and their corresponding analytical values, respectively. This term is used to avoid using scaled mode shapes.

Optimization algorithm: genetic algorithms

Genetic algorithms (GA) are stochastic processes that can find optimal or near optimal solutions to optimization problems, by analogy with the process of natural selection where the fittest survives. This technique was developed by Holland at the University of Michigan with two main objectives: to explain rigorously the adaptive process of natural systems and to create computer programs based on mechanisms of natural systems [13].

The algorithm originally proposed by Holland is basically described as follows: First, define an initial population of random individuals that must be coded either binary or real and correspond to possible solutions to the problem. Then, the adaptation of individuals of the first generations the environment is assessed from the objective function defined for the problem. This adjustment determines the probability of achieving a given individual survive and reproduce with another individual. The stage of reproduction will produce new individuals, which have high possibilities of having better properties that those individuals of previous generation. In addition, these new individuals may be subjected to a process of mutation that slightly changes its characteristics. The above procedure is performed iteratively to find the best value in the objective function or until a predetermined number of generations. In summary, each generation of individuals requires three operations: selection, crossover and mutation. For a better understanding of each of these processes is recommended that the reader is directed to classic texts such as [13,14].

A final process usually applied to a GA is elitism, which guarantees that the best individual of the current population is not lost due to the processes of crossover and mutation. A copy of the said individual is made and the next generation replaces the worst individual obtained after the application of mutation operator. This operator only applies when none of the individuals is better than the previous generation.

During the calibration process, the dynamic parameters of the different solutions found by the GA are necessary to be computed, which implies in a high computational cost. Therefore, it is very important to use high performance structural analysis software that allows a quick computing of the dynamic parameters from a large quantity of different structures. In this research the free software called OPENSEES© was utilized, which uses notepad or WordPad as tools to establish the characteristics of the structure and type of analysis to develop. The above files should contain mainly the distribution of nodes, material properties, cross-sections, defining the characteristics of the elements, constraints, analysis type and a base file that concatenates the information to be executed with Opensees.exe.

Basically, the optimization process consists of: first, randomly establishing the parameters to optimize the structure. Determine the dynamic characteristics for the different solutions using OPENSEES © [15]. These dynamic characteristics are compared with the characteristics experimentally identified through the objective function. Then, the values of the objective function are returned to the GA to produce new solutions. This process is repeated iteratively until either minimal error is satisfied in the objective function or a maximum number of generations in the GA is reached. The implementation of this process was developed in MATLAB © [16] as well as the data communication between the program OPENSEES© and the GA. Figure 11 shows a flow chart of the implementation of GA as an iterative algorithm, using the OPENSEES© tool.

Updated results of finite element model

After we carried out the experimental modal analysis in the Garcia Cadena Bridge, it was observed that each span in the bridge has an independent dynamic behavior. This was notably expressed in the repetition of mode shapes and frequencies between spans. Therefore, it was decided to perform the calibration of only one span, in this case the B7 span, and the piers were not included. Moreover, the structure was considered as simply supported since the mode shapes found experimentally have important rotations at the supports (see Table 3). Consequently, the simulation of stiffness in the support system was not included in the updating process.

After building the reference finite element model in the OPENSEES©, the algorithm shown in Fig. 11 was applied, using the parameters and genetic operators given in Table 4 and the objective function expressed in Eq. (6).

In the updating process of natural frequencies and mode shapes in a FEM, stiffness matrix for each structural element can be parameterized by obtaining the modulus of elasticity. The modulus of elasticity of structural elements has been a widely used parameter for both updating analytical models [3] also in detecting structural damage [17]. Therefore, in the calibration the elasticity modules of the following elements were optimized: the main beam before the expansion of the bridge and the beams that are part of the extension, the slab and braces beam: in total four variables were optimized. It is important to note that the calibrated elastic modules do not belong only to concrete; the modules found represent the stiffness of the contributions of the concrete, steel and post-tensioning. Consequently, the modules of elasticity found cannot be directly compared to the concrete elasticity modules. The mass considered for the beams and slab was of 2400 kg/m³ and an additional mass of 15 cm of asphalt of 1300 kg/m³, distributed on the slab.

Establishing all of the above considerations, we proceeded to perform the calibration of the structure span. Three different runs of the GA were performed and each took between 45 min to 1 h in a HP computer with an AMD processor, Athlon X2 Dual Core 3GHz and 800MHz DDR2 memory.Leer fon©ticamente

Table 5 shows the elasticity modules calibrated by the GA and Fig. 13 shows an example of the error evolution of the objective function as the generation increases.

From the elasticity modules found in the different runs of the GA, the best solution was selected and used to evaluate the frequencies and mode shapes of the finite element model.

The results are shown in Table 6, which compares the frequencies of the calibrated model and experimental frequencies identified by the ERA method. The mode shapes obtained from the finite element model calibration are presented in Fig. 12.

It is important to note that the mode shapes found were incomplete, since no measurements were made in the transversal direction to the structure due to the inconveniences described previously. However, within the objective function was controlled in a way that the modes shapes were the same in the longitudinal direction, in different axes (east, west and central). It is clear that a calibration with least uncertainties had been carried out with complete mode shapes.

The results of the calibration of finite element model with GA are considered satisfactory, although not achieving the frequency of 14.58 Hz and also no coincidence in the mode shape for the frequency of 12.50 Hz to that found experimentally.

Conclusions

From the results of ambient vibration tests, it was concluded that the dynamic behavior of the bridge superstructure presents mode shapes as a structure of type slab with components longitudinal and transverse vibration. This was evidenced by the repetition of mode shapes in the longitudinal direction for different natural frequencies, which were reproduced in the calibrated finite element model showing the behavior described.

The use of GAs as optimization technique in the calibration of finite element model showed great robustness to the adjustment of any amount and type of variables in the model calibration. However, what was achieved denote various aspects that influence decision-making in obtaining good results: 1) structural idealization that was considered in the finite element model, for example to model the behavior of some elements as shell or frame type, the type of conditions of joint and/or support, 2) the quantity and quality of the modal information that could be identified experimentally and 3) the use of this, in the calibration.

Calibration of a finite element model is also a procedure that allows the identification of structural damage based on future dynamic measurements. This requires an adequate study on a better configuration of the architecture of the GA in terms of the selection, reproduction, crossover and mutation processes. In addition, exploring the performance of other heuristic techniques such as Particle Swarn Optimization and Neural Networks, we consider that the implementation of these techniques should explore the experimental and not just theoretical identification of damage, due to the difficulties found in the identification of modal parameters.

References

[1] Cunha Á, Caetano E, Magalhães F, Moutinho C. From input-output to output-only modal identification of civil engineering structures. SAMCO Final Report (FEUP), Portugal, 2006

[2] Ren W, Wael Z, Issam E. Harik. Ambient vibration-based seismic evaluation of a continuous girder bridge. Engineering Structures, 2004, 26(5): 631–640

[3] Gentile C, Gallino N. Ambient vibration testing and structural evaluation of an historic suspension footbridge. Advances in Engineering Software, 2008, 39(4): 356–366

[4] Catbas F N, Ciloglu S K, Hasancebi O, Grimmelsman K, Aktan A E. Limitations in structural identification of large constructed structures. Journal of Structural Engineering, 2007, 133(8): 1051–1066

[5] Brownjohn J M W, Xia P Q. Dynamic assessment of curved cable stayed bridge by model updating. Journal of Structural Engineering, 2000, 126(2): 252–260

[6] Gentile C, Saisi A. Dynamic-based F.E. Model updating to evaluate damage in masonry towers. In: Proceedings of the 4th International Seminar on Structural analysis of Historical Constructions. Padova, Italy, 2004: 439–449

[7] Douglas B M, Reid W H. Dynamic tests and system identification of bridges. Journal of the Structural Division, 1982, 108(10): 2295–2312

[8] Levin R I, Lieven N A J. Dynamic finite element model updating using simulated annealing and genetic algorithms. Mechanical Systems and Signal Processing, 1998, 12(1): 91–120

[9] Levin R I, Lieven N A J. Dynamic finite element model updating using neural networks. Journal of Sound and Vibration, 1998, 210(5): 593–607

[10] Cantieni R. Experimental methods used in system identification of civil engineering structures. 2° Workshop: Problemi di vibrazioni nelle strutture civili e nelle costruzioni meccaniche. Perugia, 2004, 10–112

[11] Juang J N. Applied System Identification. New Jersey: Prentice Hall, 1994

[12] Friswell P J. Garvey S. A combined genetic eigensensitivity algorithm for the location of damage in structures. Computers & Structures, 1998, 69(5): 547–556

[13] Goldberg D. Genetic Algorithms in Search Optimization, and Machine Learning. Addison-Wesley: Publishinc Company, United States, 1989

[14] Michalewicz Z. Genetic Algorithms + Data Structures = Evolution Programs, 2nd ed. Berlin: Springer-Verlag, 1994

[15] Mazzoni S, McKenna F, Scott M H, Fenves G L, Jeremic B. OPENSEES© Command Language Manual. Berkeley, 2004

[16] MATLAB©. Users Manual Version 6.5. The Math Works Inc. Natick, MA, 2002

[17] Koh B H, Dyke S J. Structural health monitoring for flexible bridge structures using correlation and sensitivity of modal data. Computers & Structures, 2007, 85(3-4): 117–130

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