1. Introduction

    Structural health monitoring and safety assessment for long-span bridges became a dominant research topic and have received special attention of infrastructure authorities in recent years. Around the world, for many large-scale bridges, monitoring systems are installed to evaluate health condition and guarantee operational safety management [1]. The Finite Element Method (FEM) is the standard tool to model the structural behavior. However, due to a wide range of simplifying assumptions that rely on engineering judgment and a lot of uncertainties in structural properties (e.g., material properties, boundary conditions, etc.), the FE model is often only an approximate representation of the real structure. Therefore, the initial FE model should be corrected. If measured dynamic characteristics (such as natural frequencies, mode shapes, and modal damping ratio, etc.) are available, model updating can be applied to minimize the deviation between numerical prediction and experimental results. In recent decades, vibration-based bridge health monitoring (VBHM) has been more and more used to determine experimentally the physical behavior of large-scale bridges. Wu et al. [2] made use of the measurements with spatially-distributed optical fiber sensors during static loading tests to update a FE model of a highway bridge. The first order modal macro-strains and static long-gauge strains were included in the objective function. Feng et al. [3] updated a short-span steel railway bridge by applying a time-domain FE model updating approach based on in-situ dynamic displacements under ambient vibrations after train passage instead of using modal parameters (natural frequency, mode shape, etc.). They concluded that in the case of short-span railway bridges, it was difficult to extract dynamic behavior (natural frequencies) from measurements after train passing because the natural frequencies of such bridges were much higher than the frequencies of trainload’s excitation. Bayraktar et al. [4] updated a balanced cantilever bridge by a manual tuning procedure, using the peak picking method in the frequency domain. In their study, frequencies of the first ten modes were selected as objective function. El-Borgi et al. [5] applied the enhanced frequency domain decomposition technique and the Femtools software for model updating of a reinforced concrete bridge in Tunisia. They used both frequencies and mode shapes as objective function and concluded that Young’s modulus of concrete decreased dramatically during service life. Brownjohn et al. [6] updated a small highway bridge in a refurbishment and strengthening project in Singapore. They demonstrated that calculated natural frequencies of the first eight modes increased by approximately 50 % after model updating. Minshui et al. [7] applied a simple updating method to determine the physical behavior of a highway bridge. The objective function was to minimize the difference between the theoretical and experimental natural frequencies measured during ambient excitation. After model updating, the difference was reduced from 8.77% to 4.56% for the largest deviation. Parisa Asadollahi et al. [8] adapted a full-scale model of a long-span cable-stayed bridge through Bayesian FE model updating using long-term monitoring data of modal characteristics collected from a wireless sensor network during a one-year period. Zhang et al. [9], Ren et al. [10], BTemel Türker et al. [11], Teughels et al. [12], Schlune et al. [13], and many other researchers also presented examples of updating long-span bridges.

     Numerous researchers also applied optimization algorithms for model updating and obtained good results for different types of structures and fields, e.g. Tiachacht et al. [14], Samir et al. [15], Khatir et al. [16], Yuan et al. [17], Hou et al. [18]. Hamdia et al. [19-20]. Regarding application to bridges, Qin et al. [21] combined particle swarm optimization (PSO) algorithm with a surrogate model to update higher vibration modes for a continuous railway concrete bridge in Spain. They pointed out that combining PSO algorithm and a surrogate model reduced the computational time. They achieved a good accuracy between the numerical and experimental results. Jung et al. [22] applied hybrid genetic algorithm to update a small-scale bridge. The objective function included static deflections, mode shapes, and natural frequencies. Deng et al. [23] combined the response surface method and a Genetic Algorithm (GA) to update a simply supported beam. The response surface was selected as objective function and GA was employed to find the best solution. Qin et al. [1] updated a complex railway bridge by using GA combined with Kriging model. While Kriging model acted as a surrogate to reduce the deviation between structural parameters and responses, GA provided the opportunity for obtaining the global best solution. Liu et al. [24], Zordan et al. [25] and other authors also employed the mathematical power of optimization algorithms to update large-scale bridges.

This paper presents a study of updating a large-scale truss bridge. The first step is building an initial FE model, in which three possibilities of joint conditions (pin, rigid, and semi-rigid) are considered. The second step is to simulate of the effect of different joint assumptions. For steel truss structures, the joint stiffness’s are the most uncertain parameters that will influence the result of dynamic analysis. Therefore, this paper applies three scenarios of truss joints, namely pin, rigid, and semi-rigid, for the Nam O Bridge, and then analyzes and selects the scenario that reflects properly the dynamic behavior. This approach is also applied by some other authors [26-28]. However, most of aforementioned authors only evaluated different joint assumptions for static analysis, while others analyzed joint conditions for dynamic analysis of scale models in the laboratory. To the best of the authors’ knowledge, analyzing and evaluating the effect of different joint assumptions has never been applied to large-scale truss bridges. In this paper, we attempt to deal with this effect for the first time. The last step is employing optimization algorithm to update some uncertain parameters such as material properties and joint stiffness of the bridge. PSO and GA algorithms are employed to consider the efficiency of the optimization algorithm for model updating. Both PSO and GA are evolutionary algorithms based on the stochastic optimization technique, making use of a group of a random population to find the best solution through a fitness function. In GA, information of all particles is shared with each other after an iteration, whereas in PSO, only the best global position of particles, i.e. the best solution, is given out. In this paper, the efficiency of both optimization algorithms is compared based on computational cost and convergence level.
This paper is organized as follows: after an introduction, a detailed description of the bridge is presented. The next section describes the experimental campaign and the extracted modal parameters. Afterwards the Finite element model is presented followed by a section presenting the results of the model updating. Finally, conclusions are formulated.

2. Description of the bridge

     The Nam O Railway Bridge is a large-scale steel truss bridge, located in Da Nang city in the middle of Vietnam. The bridge plays a vital role in connecting train traffic from the North to the South. The Nam O Bridge is constructed in 2011, with funding from the Hanoi – Ho Chi Minh City Line Bridge Safety Improvement Project. The bridge includes 4 simply supported spans of equal length (75 m). The rail track is placed directly on the stringers of the bridge deck. The abutment on the Hai Van side is referred to as A-0, whereas the three piers are numbered as P-1, P-2, P-3, starting from the A-0 side. The last (forth) span goes from P-3 to the abutment A-1 on the Da Nang city side. Some views of bridge are given in Figure 1.

Figure 1. Some views of Nam O Bridge: (a) Upstream side; (b) Downstream side

   Truss members are made from steel with a variety of section types such as I, L, Box (Figure 2), and connected to each other by bolts.

Figure 2. Main structural elements

    The measured span (the first span from the Hai Van side) is put on rocker and pin bearings. Rocker bearings permit translation and rotation in one direction, while pin bearings only allow rotational movement. The characteristics of rocker and pin bearings (size, stiffness) are collected from catalogues of manufacturers to calculate the parameters of the equivalent springs.

3. Experimental measurements

3.1. The ambient vibration test

3.1.1. Test description

    The modal identification test was performed on the first span between abutment A-0 and pier P-1. Span length is l=75 m, maximum height at mid-span h=13 m. In total, there are 32 truss connections. The dynamic response was due to ambient wind forces or the free vibration of the bridge after train passage. In order to obtain sufficient data for vibration-based system identification as well as to be compatible with FE model analysis, ideally all nodes and all directions (longitudinal- x, transversal-y and vertical-z) from the bottom to the top of the arch-truss type should be included in the measurement grid. However, by neglecting deformation due to normal forces, several displacement components (DOFs) can be linked (slaved) to DOFs of other (master) nodes. Therefore, in the measurement layout 64 (= 32 x 2) DOFs are configuring in either of two directions (x and y or y and z depending on measurement positions: see Figure 3), of which 40 DOFs are real measurements and 24 DOFs virtual (slave) results. An overview of the sensor layout is shown in Figure 3.

Figure 3. The measurement grid: accelerometers at 40 DOFs;
red: reference points-106, 206, 302 and 402; blue: roving points.

3.1.2. Sensors placement

    On the bridge, ten accelerometers (PCB-393B12), with high sensitivity from 965 to 1,083 mV/m/s², were employed for response signal acquisition. However, the sensitivity of accelerometers needs to be carefully considered in this case. It is well known that modal properties, especially the natural frequencies, are influenced by the environmental conditions, mainly the temperature. Therefore, the calibration is valid for the (constant) temperature during the ambient vibration test. If the measurements would be repeated as part of a SHM program, i.e. to detect structural damage, definitely this dependency has to be taken into account. In practical, this kind of bridge will have a high amplitude vibration during the train passage. Therefore, using a high sensitivity sensor can lead to distortion or clipping the response. For this reason, besides the ambient response measurement, the vibration of bridge is only considered after a train passage. The vibration measurement grid was divided into 8 setups; each setup included maximum 10 accelerometers as in Table 1. From these 10, 4 served as references while the remaining accelerometers were roving over the bridge. The 4 reference sensors were present placed at lower and upper nodes of the two bays (see Figure 3). An optimal reference is a sensor where all lower modes of vibration are present. Therefore, modelling the structure beforehand is a reliable basis to allocate where the reference sensors should be located. Division of sensors in “reference” and “roving” is necessary when the number of available sensors is lower than the number of DOFs to be measured. In this case, a multi-setup measurement campaign is needed. Reference sensors are placed in nodes where all lower vibration modes have non-zero modal displacements. Here the position was selected based on the modal results of a preliminary FE model. It is also possible to generate “optimal positions” of the reference sensors when applying Optimal Sensor Placement algorithms [29]. In practical measurements on bridges, there will be as many reference sensors as possible dependent on the existing instrumentation. Preferably, there should be more than one reference sensor. In case of multiple setups, the other roving sensors cover all the remaining positions in the measurement grid.

Table 1. Overview of setups used for data acquisition and corresponding DOFs

In order to identify by model updating the real operational conditions of the bearings, 5 sensors were placed at bearings (four sensors at two nodes (100 and 300) in direction x and y, and the remaining one at node 308 in direction x). As the bearing at node 108 is a fixed one, no sensor was placed at this node.

3.1.3. Data acquisition process

    A 12-channel data acquisition system, using three NI 9234 modules from National Instruments was employed to record the voltage signals from the sensors and to convert these analog signals after conditioning to digital data. A portable computer is used to command the data acquisition system and to read and save the digital data.

Figure 4. Data acquisition process

The total acquisition time was at least ten to twenty minutes (approx. 900 - 1200 s) for one output-only setup at a sampling rate of 1651 Hz. It means each channel has 1485900 - 1981200 data points. The envisaged acquisition time was 20 minutes for the ambient vibration test. For some setups, this was shortened because of the passage of a train during the measurement. However, a measurement duration of 10 minutes was considered to be sufficient, considering the frequencies of the lowest modes. As the system identification is done for each setup separately, time lengths have not to be identical. The measurement campaign took place in two successive days. Figure 5 shows the installation of the equipment on site.

Figure 5. Field measurement instrumentation: (a) DAQ system (Compact DAQ Chassis NI 9178 and 3 vibration modules NI 9234) and portable computer; (b) Transversal accelerometer (PCB-393B12) at truss connection.

3.2.  System identification by MACEC

3.2.1. Data pre-processing

   In this stage, MACEC software Edwin Reynders et al. [30] is employed to process the measured data. Some parameters in MACEC have to set up for treating the acquired acceleration data from 8 setups in a systematic manner. The procedure of data pre-processing is as follows:

• The first step is to construct a grid of measured nodes, then connecting these nodes by lines to create a visualization of the structure.
• Input required parameters e.g., labels, data types, sensitivities, amplification factors and measurement units, corresponding to each channel.
• A pure ambient response measurement in a period of approximately ten minutes is extracted.
• There is always an offset on the measured signal data, therefore, the DC component needs to be removed from the time series of all channels.
• A “FILT-FILT” function is applied to pass signals with a frequency higher than a cutoff frequency of 0.5 Hz and attenuates signals with frequencies lower than 0.5 Hz. The purpose of this step is to remove low frequency blurring or noise by highlighting frequency trends, i.e. higher than 0.5 Hz [31].
• The frequency range of interest in bridge structure often lies between 0-20 Hz. Therefore, to reduce the data and so, to facilitate the System Identification, first a digital filtering is applied to the measurement signals, followed by a resampling them to achieve a Nyquist frequency of 20 Hz by the “DECIMATE” function with a decimation factor of 40.

3.2.2. Covariance based system identification (SSI-COV)

    After the pre-processing stage, a measurement model of structure will be identified. The stochastic subspace identification (SSI) method is often employed to perform system identification for the output-only or operational modal analysis (OMA) of structures. There exist two implementations of the SSI: the data-driven (SSI-data) option and the covariance (SSI-cov) option. [31] pointed out that the implementation of the SSI-cov is more straightforward, as well as, computationally less expensive than of the SSI-data. In comparison with SSI-data, the SSI-cov implementation also obtains a similar accuracy. Therefore, the dynamic system identification of the tested bridge was performed by SSI-cov.

    The System Identification is started with specifying the number of blocks that the raw time data is divided in. The number of blocks is used for computing sample covariance of the output correlation matrices. In general, half the number of block rows i, can be chosen based on the relationship between the lowest frequency of interest and the Nyquist-frequency. In practice, the value of i has a significant influence on the quality of the identified system model. Its value should be as large as possible, however the excessiveness of calculation time and memory usage should be considered [32]. For this case, the value of i is chosen as 250.

    Another parameter needs to be considered is the maximum system order. In the theoretical aspect, observing the number of non-zero singular values of the block Toeplitz can identify the system order n. In practice, it is not easy to inspect this number of non-zeroes because of the noise from modelling inaccuracies, measurement noise…etc., the higher singular values do not equal zero exactly. Therefore, a maximal “gap” between two successive singular values becomes an important evidence to find the system order. Peeters et al [33] stated that the gap is not clear to find out, especially in large structures. For system identification of the Nam O bridge, the considered system order is ranging from 2 to 140 in increasing steps of 2 i.e., [2:2:140].

3.3.  Modal analysis

   To obtain a clear stabilization diagram when model orders range from 2 to 140, some criteria need to be specified. The criteria are: 1% for frequency stabilization, 5% for damping ratio stabilization, 1% for mode shape stabilization. These values are selected based on experience with many other similar structures. The stable poles appear systematically in certain frequency sub-intervals, from 1 to 15 Hz. The stabilization diagram with SSI-cov of setup 1 is shown in Figure 6 for illustration purpose.

  Figure 6. The stabilization diagram of setup 1 in the interval from 0 to 20 Hz.

     Theoretically, a bridge has a multitude of vibration modes. However, all the modes of vibration do not contribute equally to the response of a structure. Natural frequencies and mode shapes are determined based on stable poles. The spectrum is just a visual aid that is not used to select the poles. The poles (also closely spaced) are identified by the SSI algorithm in MACEC [30]. Normally only the first few modes, which have higher participation factors are considered to get the dynamic response of structures. Mainly those main lower modes are enough to solve the model updating problem. In this case, only the first 5 modes were used within the frequency interval from 1.45 Hz to 6.05 Hz, as shown in Figures 7 to 11. For a detailed explanation about the construction and the interpretation of the stabilization diagram in Figure 6, the reader is referred to [30] and [33].

Figure 7. The identified mode 1, first lateral, f = 1.45 Hz, xi=0.82%

Figure 8. The identified mode 2, first torsion, f = 3.11 Hz, xi=0.19%

Figure 9. The identified mode 3, second lateral, f = 3.28 Hz, xi=0.27%

Figure 10. The identified mode 4, first vertical bending, f = 4.62 Hz, xi=2.54%

Figure 11. The identified mode 5, second torsion, f = 6.05 Hz, xi=0.28%

4. Finite element model

    In order to predict structural dynamic behavior, and compare it with that obtained from measurement, a finite element model of Nam O bridge was built by using the MATLAB toolbox StaBil [34] (see Figure 12).

Figure 12. FE model of the Nam O Bridge

The following details describe the FE model:

• The bridge is modeled by 137 nodes, 227 elements, and 9 section types of truss members are used (see Table 2).
• Main structural members are shown in Figure 2: upper chords, lower chords, vertical chords, diagonal chords, stringers, upper wind bracings, lower wind bracings, and struts, are modeled using three dimensional (3D) beam elements. This element has six degrees of freedom at each node including translations in the x, y, and z directions and rotations around the x, y, and z It is based on Timoshenko beam theory, which includes shear-deformation effects. The element provides options for unrestrained warping and restrained warping of cross-sections. The transverse girders (of the deck system) are also modeled using beam elements.
• The global X-axis is in the longitudinal direction of the bridge; the Z-axis is in the transverse direction (to the river flow direction) and the Y-axis is in the vertical direction. The left main truss is on the downstream side. Likewise, the right main truss is on the upstream side. Both trusses represent the side view of the bridge (Figure 1).
• Bearings were modeled by using spring elements.
• Mass is added to the lower chords: the rail track and non-structural components such as power lines, hand rails, maintenance paths and protections were estimated as 247.2 kilogram per meter.
• Section properties and material properties of structural members are listed in Table 2 and Table 3.

Table 2. Cross-sectional properties of truss members

Note: I_y is the moment of inertia of strong axis (the same direction with global Y), I_z is the moment of inertia of weak axis (the same direction with global Z).

Table 3. Material properties of truss members

• The connection between truss joints: due to the uncertainty of the actual joint stiffness, three possible models of link types are considered: a) pin, b) rigid, and c) semi-rigid. This process can be considered as a trial step to analyze and determine some unknown parameters (uncertainties in material properties, section properties, or joint conditions, etc.).

Case 1: Pin connection

    In order to simplify the calculation, researchers as Duerr [35], Deng et al. [36], Saka [37] assumed truss members linked to each other by pin connections (Figure 13a). In this case, the influence of rotational stiffness is neglected, and no moments are transferred between truss members. In static analysis of truss structures, this link is often applied for node joints, which has little effects on the result of force of truss members because moment transfer is insignificant with axial forces (compression and tension). Assuming a pin connection results in a statically determinate structure that can easily be calculated by hand.

Figure 13. Connection types of truss joints: (a) Pin connection; (b) Rigid connection; (c) Semi-rigid connection (rotational springs)                  

Case 2: Rigid connection.

   Duerr [35], and other authors have also applied rigid links (Figure 13b) for truss joints. Basically, a rigid joint can transfer axial forces (compression, tension) as well as moments between members.

Case 3: Semi-rigid connection.

   This link type has recently been applied by many researches. Luong el al [26] determined the stiffness of node joints, when updating a truss structure in laboratory, and found that a semi-rigid connection represent the most accurately the dynamic characteristics of the considered truss structure. Dubina et al. [27], Csébfalvi [28], and other researchers applied semi-rigid links (rotational springs) for truss structures. However, most of aforementioned authors only applied this type of joint in static analysis, whereas a few others employed it to analyze dynamic analysis of scale models in the laboratory. Rotational springs (Figure 13c) are often used to present semi-rigid link at truss joints. This study also applied the aforementioned three scenarios of joint conditions to predict structural dynamic behavior of the Nam O Bridge. Linear elastic rotational springs with three degrees of freedom at each node: rotations around the x, y, and z directions are used for vertical chords and diagonal chords as Figure 13c. The original stiffness of rotational springs is estimated according to reference [26]. The results are given in Table 4 and Figure 14.

Table 4. The natural frequencies from the FE model for three connection cases, and from measurement.

Figure 14. MAC values of mode shapes before model updating: (a) Rigid connection, (b) Semi-rigid connection

      Table 4 demonstrates that the FE model of truss bridge with pinned connection do not predict the behavior of the Bridge properly. Frequencies of the first five modes are lower than those of measurement. The result of the FE model with rigid connection is also not satisfactory in comparison to the experimental one. Specifically, there are deviations between frequencies calculated from FEM and measurement (from 26% to 36%). All frequencies from the FE model are higher than those from the experiment. Rigid links make the structure stiffer than in reality, while the truss members in Nam O Bridge are linked with each other by bolts of welded. Additionally, MAC values (Figure 14a) lower than 0.9 indicate the difference between mode shapes of FE model and measurement. The model with semi-rigid connection provides improved simulated modal results in comparison to the experimental ones. There is a small deviation between frequencies calculated from FEM and measurement (around 1%, apart from mode 5 having 13%). The MAC values (Figure 14b) higher than 0.9 indicate consistent correspondence between the numerical model and measurement [38-39]. However, it is necessary to update some uncertain parameters such as: Young’s modulus, the stiffness of springs at bearings and truss joints to get the best correspondence between theoretical and experimental results.

5. Model updating

5.1. PSO algorithm

  In 1995, Eberhart et al. [40] developed particle swarm optimization (PSO) algorithm referenced from animal (particle) behaviors such as: Birds flocking and Fish schooling. Each particle “fly” or “swarm” randomly through the search space, records and communicates with other particles about their best solution (the best local position) that they have discovered. Therefore, it can assist in looking for the best global position (the best solution) easier and faster. PSO has already been applied in many optimization issues. Khatir et al. [41] used PSO to predict the location and severity of damage of a steel cantilever beam and a two-dimensional frame based on inverse problem. Wei et al. [42] employed an improved PSO to detect damage of different structures (a frame, beam, and a truss) with a high accuracy. Shabbir et al. [43], Shao et al. [44], Qiing et al. [45], Pau et al. [46],  Mangiatordi et al. [47] and other researchers also successfully used PSO to deal with other optimization problems. The PSO algorithm is based on two equations. The first equation updates the position of a particle:

The second equation updates the velocity of a particle:

    Where xⁱ(t), and xⁱ(t+1) represent the position vectors of particle i at time t and t+1, respectively, v is the velocity vector of particle, w indicates the inertia weight parameter, C₁, C₂ represent for the cognition learning factor and the social learning factor, respectively, r₁ and r₂ are random numbers in the range of (0,1), pⁱ (t) is the best position of each particle (the local best), and G_best is the best position of all particles (the global best). Each particle is featured by a velocity vector and its physical position in the space. In the moving process, particles can remember the best local position pⁱ(t), and communicate with others to look for the global best position (G_best). The algorithm develops iterations to determine the fitness of each particle based on the objective function. When the objective function is minimum, the global best position (G_best) is achieved. Because only the best global position of particles (G_best) is given out, and a few parameters to adjust, compared with other optimizations algorithms, PSO tends to converge to the global best faster, reduces the computational time, and results in a good correspondence between the FE model and the real structure.

5.2. Genetic Algorithm

  Genetic Algorithm (GA) is an evolutionary optimization method, used efficiently for different kind of optimization problems in the last decades [48-49]. In our study, different individuals, also called chromosomes or populations, represent the rigidity of springs to update using experimental results of natural frequencies and mode shapes. The population evolves iteratively toward a better solution in a process inspired from a natural evolution, where they are allowed to cross among themselves in order to obtain favorable solutions. For a review of the approach that is searching for the global best GA, the reader is referred to [50]. The fitness is the objective function value, calculated in Equation 3. The best feasible solutions have higher probability to be chosen as parent of new individuals, where the properties of the parents are combined by exchanging chromosomes parts, to produce two new designs. Afterwards, the mutation is performed by randomly replacing the digits of a randomly selected chromosome. These basic operators are repeated to create the next generations till the maximum number of iterations.

5.3. Model updating of Nam O Bridge

   A FE model updating was applied in the Nam O Bridge. Eight uncertain parameters, including Young’s modulus of truss members (E), the stiffness of 6 springs under bearings (k₁,k₂,k₃,k₄,k₅,k₆) as shown in Figure 15, and the stiffness of rotational springs at truss joints (k₇) were chosen to update. The original stiffness of rotational springs is estimated based on reference [26], whereas the stiffness of springs at the bearings is calculated based on bearing types. The stiffness of springs is listed in Table 6.  

Figure 15. Uncertain structure parameters are selected to update in model: (a) The Springs at truss joints; (b) The springs at bearings.

  The objective function was built based on both mode shapes and natural frequencies as follows:

  On the right hand side of Equation 3, the first part is the discrepancy between first five mode shapes in terms of the MAC values and the second part presents the deviation between numerical and experimental first five natural frequencies, ( ) are analytical and experimental frequencies, mode shapes, respectively, “i” is the modal order, and T denotes a transposed matrix. PSO and GA algorithm is utilized to look for the minimum (convergence) of the objective function (fitness) performed as illustrated by the diagram in Figure 16.

Figure 16. Methodological approach for model updating by using GA and PSO

   In PSO, a population size of 50 individuals is used. The inertia weight parameter (w) is 0.3, and the values of the cognition learning factor and the social learning factor are C₁ = 2 and C₂ = 2. In order to compare with PSO, for GA, the population size of 50 individuals is also applied, crossover and mutation coefficients are 0.8 and 0.1, respectively. The stop criteria of loops in both PSO and GA are established as follows: the deviation of objective function (fitness) value between two consecutive iterations is lower than 10^-7, or the maximum number of iterations is 100.

Figure 17. Fitness tolerance (a) GA; (b) PSO

  Figure 17 shows that the convergence rate of PSO is faster than GA. With the same population, PSO can find the result of the global best (the best solution) after only 3 iterations, Whereas GA needs approximately 20 iterations to obtain the global best. Besides, the convergence level of PSO outperforms than that of GA. The tolerance of objective function of PSO is lower 0.1, while the result of GA is about 1.2. That means that the deviation between numerical model and measurement after model updating using PSO was lower than GA. This result can be explained based on the approach to find the best solution in the two algorithms. While in PSO, only the best global position of particles (the best solution) is given out, in GA, information of all particles is shared with each other after an iteration. Therefore, the populations in PSO may not only look for the global best faster but also avoid a local best.

5.3. Comparision between the updated model and experimental results

    A summary of the analysis and the experimental results is given in Table 5 and Figure 18. A good correspondence is obtained between the measured and the calculated frequencies and mode shapes of GA and PSO.

• The results of mode 1, mode 2 and mode 3 calculated by FEM and measurement do perfectly match.
• There are only small discrepancies in mode 4 and mode 5 (from 0.8% to 7.6%)
• The MAC values from 0.99 to 0.90 (Figure 18) demonstrate a close correspondence between the mode shapes of FE model and measurement.
• The results of both calculated frequencies and mode shapes applying PSO are closer to those of the measurement than GA.
• After model updating, the FE model is applied to validate higher modes that were not included in the objective function. Table 5 and Figure 18 show that the model updating also reduces the deviation between the measured and calculated frequencies, mode shapes of the higher modes (mode 6 to mode 10).

Table 5. The modal frequencies from the FE model after model updating compared to the measurement

Figure 18. MAC values of mode shapes after model updating: (a) GA; (b) PSO

Table 6 provides the range of variation of the uncertainty parameters based on experience or estimated according to reference [26]. Model updating process also adjusts uncertain parameters of the bridge (Table 7). The changes in Table 7 show that the parameters before and after updating are not too much different. This is easy to explain since the Nam O Bridge is just in operation since about 7 years. Therefore, its properties still remain close to original. The stiffness of bearing and rotational springs k₁, k₂, k₃, k₄, k₅, k₆, and k₇ have a decreasing trend with lower levels for GA compared to that of PSO. This proves that the stiffness of bearings and rotational springs were overestimated. Therefore, to get consistent correspondence between theoretical and experimental results, the stiffness of bearings and rotational springs should decrease.

Table 6 The range of variation of the uncertainty parameters

Note: unit of k₁,k₂,k₃,k₄, is E10 N/m, unit of k₅,k₆ is E7 N/m, unit of k₇ is E5 N.m/rad, unit of E is E10 MPa   

 ^[1]

Table 7 Value of uncertain parameters before and after updating.

Note: unit of k₁,k₂,k₃,k₄, is E10 N/m, unit of k₅,k₆ is E7 N/m, unit of k₇ is E5 N.m/rad, unit of E is E10 MPa