AN APPROACH OF SOLVING MOVING LOAD PROBLEMS WITH APPLICATION TO AN EXPERIMENTAL CASE Jing Yang, Huajiang Ouyang University of Liverpool – School of Engineering, Liverpool L69 3GH, UK email:
[email protected]
Dan Stancioiu Liverpool John Moores University, Mechanical Engineering and Materials Research Centre, Liverpool John Moores University, Liverpool L3 3AF, UK Many engineering applications should be treated as moving load problems and have been studied for long time. Various methods have been explored by enormous researches to solve moving load problems, but the easy way to implement those methods still needs to be studied. This paper presents an approach of solving moving load problems by using ABAQUS and MATLAB. Taking bridge-train interaction for example, the basic idea is to build a Finite Element (FE) model of the bridge in ABAQUS and obtain its numerical modes which then, are exported into a MATLAB code which implements the Modal Superposition (MS) method. The approach is validated by experimental results and found to be effective and efficient.
1.
Introduction
Many problems in engineering should be treated as moving load problems, such as the dynamic interaction between bridges and trains, working cranes, bullets shooting out from guns and so on [1, 2]. Among those, the train-bridge coupled dynamics has drawn numerous researchers’ attention [37]. The methods to solve the train-bridge dynamic interaction problem are fairly mature, but easy ways to implement the methods still need to be explored for engineers to use. To calculate the dynamic responses of bridges excited by moving trains, Modal Superposition (MS) method is generally considered to be very efficient, as only the first several modes of the bridges are needed to reach good accuracy [8]. However, analytical modes of bridges normally are hard to obtain [9]. In this case, their numerical modes can be obtained alternatively by applying the FE method to the bridges. An easy way adopted in this paper to obtain the numerical modes of the bridges is to use an FE software package like ABAQUS to do modal analysis of the bridges, saving tedious coding work by researchers. The equation of motion of the moving train can be established by treating it as rigid bodies with spring and dashpot connections [10, 11]. In this paper, an iterative method is adapted and implemented in MATLAB to solve the equation of motion of the bridge and that of the train through compatibility of the displacements of the train and the bridge and their contact force equilibrium at their interface [12, 13]. Numerical simulations of train-bridge dynamics have been studied by lots of researchers, but few people study this problem experimentally. Zhai et al. [14] validated the simulation software (TTBSIM) by comparing simulation results with experimental results of the Yangcun Bridge in the Beijing-Tianjing high-speed railway and the Yellow River Bridge in Beijing-Shanghai high-speed railway. Xia et al. [5, 15] conducted field measurements on the Antoing Bridge in the Paris-Brussels high-speed railway line and the PC box girders in Qin-Shen Special Passenger Railway in China. Stancioiu et al. [16] studied the vibration of a multi-span structure excited by moving masses nu1
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merically and experimentally. The mode shape and vibration of a composite seven-span railway bridge due to the passage of high-speed trains were studied by Liu [17]. This paper presents an easily implemented approach of predicting the vibration of an infrastructure subjected to moving loads by using an FE software package (ABAQUS) and MATLAB. Experimental measurements are carried out on a four-span plate with rails subjected to a moving car and compared with its simulation results.
2.
Experimental setup
The first span view of experimental setup is shown as Figure 1. The plate is rested on top of five simple supports which split the plate to 4 equally long spans. On top of the plate are two rails which guide the moving track of the car. The plate is supported by four elastic supports provided by four shakers at mid-span locations as well. Four laser transducers are used to measure the displacements of the plate at corresponding locations. The whole plate is 3.6 m long, 101.67 mm wide and 3.16 mm thick. The rail is 6.85 mm wide and 8.52 mm high. Ramps are located before and after the plate. The mass of the moving car is 4.335 kg. The stiffness of the elastic support is measured to be 8548 N/m around.
Figure 1: First span of the rig.
3.
Parameter identification
A FE model of the test structure (plate with rails) is built in ABAQUS to predict its mode shapes. Before that parameter identification of the structure needs to be done for the purpose of getting accurate numerical predictions. There are many methods to identify the parameters of a system, like geometric parameters, material parameters, and support stiffness, etc. While some parameters can be measured directly, some parameters can only be measured indirectly. The geometric parameters of the plate can normally be measured directly, but for complicated geometry, the technology of image processing may be made use of. The Young’s modulus of an Euler-Bernoulli beam can be identified by 𝛽𝑘2 𝐸𝐼 √ 𝑓𝑘 = 2π𝑙 2 𝜌𝐴
2
(1)
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where 𝑓𝑘 is the kth frequency of the beam; 𝛽𝑘 is the kth wave number of the beam and it depends on of the boundary conditions of the beam; 𝑙 is the length of one span of the beam. 𝑓𝑘 can be obtained by modal testing and 𝛽𝑘 can be calculated by using Euler beam theory according to its boundary conditions and 𝑙, 𝜌, A and I can be identified directly or indirectly, so E can be calculated inversely from Eq. (1). The Young’s modulus of the rails and the plate were identified by using Eq. (1) too. Modal tests are done for the rail and the plate to measure their frequencies. As the width and height of the rail are very small, it is not easy to carry out modal tests for the rail by using accelerometers. In this case, a scanning laser vibrometer from Polytech is used by taking advantage of its non-contact measurement. The length of the clamped rail for Modal Testing is 0.407 m shown in Figure 2. The first four frequencies of the rail are obtained by Modal testing and the ratios of 𝐸𝐼/𝜌𝐴 are calculated inversely by Eq.(1) and given in Table 1. It can be seen from the table that the first three ratios are quite close, which indicates Euler-Bernoulli beam theory can be applied to the rail in this case.
Figure 2: Clamped rail ready for modal testing. Table 1: Experimental frequencies and the ratio of 𝐸𝐼/𝜌𝐴 of the clamped rail.
Mode k 1 2 3 4
𝛽k 1.875104 4.694091 7.854757 10.995541
Fre. by exp. (Hz) 29.84 186.41 519.06 1000.94
𝐸𝐼/𝜌𝐴 77.2610 76.7708 75.9221 73.5214
To know the area and second moment of area of the rail section, a clear picture of the rail section is taken and the profile of the rail in the picture is detected by using ‘outline trace’ of CorelDRAW and saved in a CAD drawing. Figure 3 shows the picture of a rail section and the CAD drawing of its profile. The two figures look similar but not exactly the same, which could be caused by the error of edge detecting. The area and second moment of area of the rail section can be obtained by ‘MassProperty’ command in CAD to be around 𝐴r = 28.0 mm2 and 𝐼r = 207.0472 𝑚𝑚4 . The density and Young’s modulus of the rail are identified to be around 8356.534 kg/m3 and 86.582 GPa.
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Figure 3: The picture of a rail section and its CAD drawing.
Figure 4 shows the specimen of the plate in the apparatus in Figure 1. The density of the plate is identified to be 7700 kg/m3 . The area and second moment of area of the plate are 𝐴p = 3.213 × 10−4 m2 and 𝐼p = 2.673 × 10−10 m4 respectively. Modal testing was done in [16] on a four-span plate which is made of the same material as the plate specimen. Each span of plate is 0.91 m long. The Young’s modulus of the plate can be calculated from the ratio of 𝐸𝐼/𝜌𝐴 in Table 2: 𝐸p = 183.4 GPa.
Figure 4: The specimen of the plate. Table 2: The ratio of 𝐸𝐼/𝜌𝐴 of the plate calculated from [16].
Mode k 1 2 3 4
4.
Fre. by [16] 8.37 9.80 13.25 17.21
𝛽𝑘 3.141593 3.393231 3.926602 4.463324
𝐸𝐼/𝜌𝐴 19.4705 19.6121 19.9935 20.2047
Average 𝐸𝐼/𝜌𝐴 19.8202
FE model of the plate with rails
A 3D FE model of the plate with rails and elastic supports is built in ABAQUS. 480 shell elements (S4R) for the plate and 160 beam elements (B31) for the two rails are used for the model. 4 spring elements are adopted to model the four elastic supports below four spans. The supports of the plate are modelled to be simply supported. Tie function is used in ABAQUS to fix the rail elements on top of the plate elements. Different offset ratios for shell elements are tried to make the systems’ frequencies obtained from the modal analysis of the model close to its experimental ones. It is found that the ratio of 1.87 makes the simulated frequencies closest to experimental ones. The difference between the two sets of frequencies for the first eight ones does not exceed 5%, which verifies the FE model of the system. It can be found that using five or more modes of the system does not make a big difference to the dynamic response of the plate excited by a moving car. Therefore, five numerical modes of the plate with rails are obtained by modal analysis in ABAQUS and ex4
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ported into MATLAB. The frequencies used to calculate the dynamic response of the plate is the experimental ones. Figure 5 gives the first mode shapes of the plate with rails obtained from ABAQUS. These mode shapes look very similarly with those of a four-span continuous beam.
Figure 5: First six mode shapes of the structure from ABAQUS.
The equation of motion of the structural FE model can be expressed as 𝐌𝐗̈(𝑡) + 𝐊𝐗(𝑡) = 𝐅(𝑡)
(2)
where 𝐌 and 𝐊 are mass and stiffness matrixes of the FE model of the structure respectively, 𝐗(𝑡) is the nodal displacement of the structure, 𝐅(𝒕) is the nodal force vector acting on the structure and changes with the movement of the car. Applying MS method to Eq. (2) leads to 𝐪̈ (𝑡) + diag[𝜔𝑖2 ] ⋅ 𝐪(𝑡) = 𝚽 T 𝐅(𝑡)
(3)
where 𝐪(𝑡) is the modal coordinate vector in time domain, 𝜔𝑖 is the 𝑖th natural frequency of the structure, 𝚽 is the mass-normalised mode matrix of the structure. Experimental frequencies are used in Eq. (3), while numerical modes are adopted in computation.
5.
Car model
The car is treated as a 2 DOFs rigid body with degree of freedoms of heave and pitch. The car model parameters are: 𝑚v = 4.335 kg , moment of inertial 𝐼v = 0.012045 kg m2 , its wheel span 𝑠 = 0.126 m and the car body length 𝐿c = 0.208 m. The initial car position is set to be at the place where the front wheels is at the start point of the plate and the rear wheels is located on the ramp as the stage 1 in Figure 6. The ramp is assumed to be rigid. Therefore, the displacement of the wheel at the ramp is zero.
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Figure 6: Different stages of the car moving process.
The equation of motion of the car from stage 1 to stage 3 can be expressed as: ̈ = −𝑚v 𝑔 − 𝑘1 [𝑧(𝑡) − 𝜃(𝑡) 𝑠 − 𝑤(𝑥1 , 𝑦1 , 𝑡)] − 𝑘2 [𝑧(𝑡) − 𝜃(𝑡) 𝑠 − 𝑤(𝑥2 , 𝑦2 , 𝑡)] − 𝑚v 𝑧(𝑡) 2
2
𝑠
𝑠
2
2
𝑘3 [𝑧(𝑡) + 𝜃(𝑡) − 𝑤(𝑥3 , 𝑦3 , 𝑡)] − 𝑘4 [𝑧(𝑡) + 𝜃(𝑡) − 𝑤(𝑥4 , 𝑦4 , 𝑡)]
(4)
̈ = 𝑘1 [𝑧(𝑡) − 𝜃(𝑡) 𝑠 − 𝑤(𝑥1 , 𝑦1 , 𝑡)] 𝑠 + 𝑘2 [𝑧(𝑡) − 𝜃(𝑡) 𝑠 − 𝑤(𝑥2 , 𝑦2 , 𝑡)] 𝑠 − 𝑘3 [𝑧(𝑡) + 𝐼v 𝜃(𝑡) 2 2 2 2 𝑠
𝑠
𝑠
𝑠
𝜃(𝑡) 2 − 𝑤(𝑥3 , 𝑦3 , 𝑡)] 2 − 𝑘4 [𝑧(𝑡) + 𝜃(𝑡) 2 − 𝑤(𝑥4 , 𝑦4 , 𝑡)] 2
(5)
where 𝑧(𝑡) is the heave displacement of the car with upward positive direction, 𝜃(𝑡) is the pitch angle of the car with positive direction of clockwise rotation, 𝑤(𝑥𝑖 , 𝑦𝑖 , 𝑡) is the deflection of the rail at the contact point with 𝑖th wheel, 𝑘𝑖 is the contact spring stiffness at 𝑖th wheel (the contact between car wheels and rails is treated as contact springs). It should be noticed that the deflection of the ramp is zero due to its rigid-body assumption, which means for period between stage 1 and stage 2 𝑤(𝑥3 , 𝑦3 , 𝑡) = 0
(6)
𝑤 (𝑥4 , 𝑦4 , 𝑡) = 0
(7)
The car is given a push at the start and allowed to travel along the rails freely. The time instants when it passes 4 mid-span points are recorded by 4 electromagnetic sensors. It is found that the travelling speed of the car can be taken to be constant.
6.
Experimental validation of numerical results
The structural model and car model are solved separately by the iterative method [13], satisfying the contact condition between car wheels and rails. The time step is set constant as 0.001 s. The velocity is measured to be around 0.7 m/s. Figure 7 shows the displacements at four measurement points where laser displacement transducers are located obtained by simulation and measurement. It can be seen that the amplitude of simulation results basically match that of experimental ones, but slightly bigger in positive direction. This is probably due to the modelling error of the boundary conditions of the structure. It can also be found that the peaks of simulation lines at 1st and 2nd points lag behind those of measurement lines. On the contrary, the peaks of simulation lines at 3rd and 4th points appear ahead of those of measurement lines. This is because of the assumption of constant car speed. Actually, the car moves fast at beginning, then slow down due to the friction between wheels and rails. 6
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1st point 0.5
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2nd point
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Figure 7: Comparison between simulation results and measurement results at 4 points.
7.
Conclusions
The vibration of bridges excited by moving trains is an important issue and needs to be predicted accurately by efficient methods. However, no general-purpose commercial software package can deal with this problem efficiently due to the difficulty in tackling the dynamic interaction between trains and bridges. One way to deal with this problem easily by taking advantage of existing powerful Finite Element (FE) software packages, like ABAQUAS is to build a structural FE model in ABAQUS and obtain its numerical modes, if geometric and property parameters and boundary conditions of the structure can be obtained accurately, among which Young’s modulus can be identified from modal test data. The numerical modes of the bridge obtained from ABAQUS can be exported into MATLAB where the equation of motion of the bridge expressed by Modal Superposition (MS) method and the equation of motion of the train can be solved efficiently by an iterative method. The simulated vibration results of a four-span continuous beam/plate with two rails on top excited by a moving car determined by this approach are found to be close to experimental results, which verify the good performance of this approach.
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6 Baeza, L. and Ouyang, H. Dynamics of a truss structure and its moving-oscillator exciter with separation and impact-reattachment, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, (2008). 7 Zhai, W., Wang, K. and Cai, C. Fundamentals of vehicle-track coupled dynamics, Vehicle System Dynamics, 47 (11), 1349-1376, (2009). 8 Tanabe, M., Wakui, H., Matsumoto, N., Okuda, H., Sogabe, M. and Komiya, S. Computational model of a Shinkansen train running on the railway structure and the industrial applications, Journal of Materials Processing Technology, 140, 705-710, (2003). 9 De Salvo, V., Muscolino, G. and Palmeri, A. A substructure approach tailored to the dynamic analysis of multi-span continuous beams under moving loads, Journal of Sound and Vibration, 329 (15), 3101-3120, (2010). 10 Marchesiello, S., Fasana, A., Garibaldi, L. and Piombo, B. A. D. Dynamics of multi-span continuous straight bridges subject to multi-degrees of freedom moving vehicle excitation, Journal of Sound and Vibration, 224 (3), 541-561, (1999). 11 Dinh, V. N., Kim, K. D. and Warnitchai, P. Dynamic analysis of three-dimensional bridge-high-speed train interactions using a wheel-rail contact model, Engineering Structures, 31 (12), 3090-3106, (2009). 12 Feriani, A., Mulas, M. G. and Candido, L. Iterative procedures for the uncoupled analysis of vehiclebridge dynamic interaction, Proceedings of ISMA 2010 including USD2010, (2010). 13 Yang, J., Ouyang, H. and Stancioiu, D. An approach of solving moving load problems by Abaqus and Matlab using numerical modes The 7th International Conference on Vibration Engineering, (2015). 14 Zhai, W., Wang, S., Zhang, N., Gao, M., Xia, H., Cai, C. and Zhao, C. High-speed train–track–bridge dynamic interactions – Part II: experimental validation and engineering application, International Journal of Rail Transportation, 1 (1-2), 25-41, (2013). 15 Xia, H., De Roeck, G., Zhang, N. and Maeck, J. Experimental analysis of a high-speed railway bridge under Thalys trains, Journal of Sound and Vibration, 268 (1), 103-113, (2003). 16 Stǎncioiu, D., Ouyang, H., Mottershead, J. E. and James, S. Experimental investigations of a multi-span flexible structure subjected to moving masses, Journal of Sound and Vibration, 330 (9), 2004-2016, (2011). 17 Liu, K., Reynders, E., De Roeck, G. and Lombaert, G. Experimental and numerical analysis of a composite bridge for high-speed trains, Journal of Sound and Vibration, 320 (1-2), 201-220, (2009).
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