With the characteristics of heavy and concentrated loads, the influence of moving loads on the dynamic response of the bridges is significant. Therefore, in this paper, the dynamic response of a large-scale truss bridge is studied to consider the effect of the various parameters of moving loads. The considered main parameters consist of moving mass, moving velocity, and type of moving loads. The nonlinear dynamics of the bridge based on time history analysis are obtained using the Wilson-  method. four time history – based dynamic analysis method including modal superposition in frequency domain, modal superposition in time domain; direct time integration, and direct solution in the frequency domain are employed to analysis the obtained results. To compare the effectiveness of the aforementioned method. A large-scale railway truss bridge is employed for dynamic response analysis. The obtained results give more insight into the nature of the problem and help to determine the significant parameters of moving load affecting the bridge response

NGUYEN, Hai Manh

NGUYEN, Long Ngoc

NGUYEN, Quyet Huy

NGUYEN, Tung Xuan

TIEN, Thanh Bui

TRAN, Hoa Ngoc

English

With the characteristics of heavy and concentrated loads, the influence of moving loads on the dynamic response of the bridges is significant. Therefore, in this paper, the dynamic response of a large-scale truss bridge is studied to consider the effect of the various parameters of moving loads. The considered main parameters consist of moving mass, moving velocity, and type of moving loads. The nonlinear dynamics of the bridge based on time history analysis are obtained using the Wilson-  method. Four time history – based dynamic analysis methods including modal superposition in frequency domain, modal superposition in time domain; direct time integration, and direct solution in the frequency domain are employed to analysis the obtained results. To compare the effectiveness of the aforementioned method. A large-scale railway truss bridge is employed for dynamic response analysis. The obtained results give more insight into the nature of the problem and help to determine the significant parameters of moving load affecting the bridge response

Université Mouloud Mammeri de Tizi Ouzou (UMMTO): Research Review of Sciences and Technologies

 JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 9 (2022) 447–454 447     * Corresponding author.  E-mail address: ngxuantung@utc.edu.vn  e-ISSN: 2170-127X,  Research Paper Dynamic response analysis of truss bridges under the effect of moving vehicles Tung Nguyen-Xuan a,*, Quyet Nguyen-Huu b, Hai Nguyen-Manh b,Long Nguyen-Ngoc b, Hoa Tran-Ngoc b, Thanh Bui-Tien b a Structural Engineering Department; Faculty of Civil Engineering, University of Transport and Communications, Hanoi, Vietnam; b The Department of Bridge Engineering & Underground Infrastructure, Faculty of Civil Engineering, University of Transport and Communications, Hanoi, Vietnam;  A R T I C L E  I N F O Article history: Received : 15 November 2022 Revised : 19 December 2022 Accepted : 21 December 2022  Keywords: Dynamic response analysis Truss bridge Moving-load  Velocity Time history analysis Wilson-𝜃 method            A B S T R A C T With the characteristics of heavy and concentrated loads, the influence of moving loads on the dynamic response of the bridges is significant. Therefore, in this paper, the dynamic response of a large-scale truss bridge is studied to consider the effect of the various parameters of moving loads. The considered main parameters consist of moving mass, moving velocity, and type of moving loads. The nonlinear dynamics of the bridge based on time history analysis are obtained using the Wilson-𝜃 method. four time history – based dynamic analysis method including modal superposition in frequency domain, modal superposition in time domain; direct time integration, and direct solution in the frequency domain are employed to analysis the obtained results. To compare the effectiveness of the aforementioned method. A large-scale railway truss bridge is employed for dynamic response analysis. The obtained results give more insight into the nature of the problem and help to determine the significant parameters of moving load affecting the bridge response.  F. ASMA & H. HAMMOUM (Eds.) special issue, 4th International Conference on Sustainability in Civil Engineering ICSCE 2022, Hanoi, Vietnam, J. Mater. Eng. Struct. 9(4) (2022) 1 Introduction The effect of moving loads of trucks, trains, or other types of transport vehicles often causes complex vibrations for bridges. Depending on the velocity of movement, the changes in the position of the loads over time, forced vibrations, resonance vibration, and fatigue failure can occur and exert adverse effects on bridges. These effects can cause unsafety for people and vehicles on the bridge as well as affect the life of the structure. The problem of analyzing the impact of moving 448   JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 9 (2022) 447–454  loads on the bridges is challenging because it depends on many random parameters such as the types of transport vehicles, the characteristics of the structure, the moving speed of the vehicles, and so on [1-3]. Previously, the linear elastic analysis method or the vibration spectrum analysis method was employed to analyze the structural dynamic response. However, while the first method applies too many quantitative requirements, the last one does not consider power dissipation. This may reduce the accuracy of obtained results. To deal with this shortcoming, over the last decades, many researchers have proposed methods based on real-time data (time history method) to analyze the dynamic response of the structure. Some of the most popular methods are the Newmark method [4] , Wilson-θ method [5],  Hiber-Hughes-Taylor (HHT) method [6], and so on. For example, Bamer et al [7] analyzed a two-dimensional frame structure subjected to four different excitation functions using Wilson-θ method. Mohammadzadeh et al [8] proposed solutions to the traditional Wilson-θ method. The authors found that the proposed approach not only provides a high degree of accuracy but also achieved controllable amplitude decay. Ozkul [9] presented an approach to analyze the dynamic behaviors of shells by utilizing the Wilson-θ method. Mohseni et al [10] applied the HHT method for dynamic response analysis of a skewed bridge caused by moving loads. Lui et al [11] used the Runge-Kutta method when analyzing the dynamics response of a semi-rigid frame structure. Pasetto et al [12] proposed the Waveform Relaxation Newmark (WRNβ) method to overcome the disadvantages of Newmark methods to analyse the dynamic behavior of 2-dimensional flat plates. However, most aforementioned authors only applied the time history method to solve dynamic analysis problems of structures in the laboratory or small structures. Therefore, in this paper, we propose using the most effective method based on the time history method, termed, the Wilson-𝜃 method for the dynamic response analysis of a real large-scale steel truss bridge. In addition, in this research, four time history – based dynamic analysis method including modal superposition in frequency domain, modal superposition in time domain; direct time integration, and direct solution in the frequency domain are employed to analysize the obtained results.  2 Methodology The basic assumption of the Wilson-𝜃method is that the acceleration of the structure changes linearly from time 𝑡 to time 𝑡′ = 𝑡 + 𝜃. ∆𝑡; 𝜃 (𝜃 ≥ 1) is determined based on optimizing the stability and accuracy of the calculation results. From the time interval 𝑡to 𝑡 + 𝜃. 𝛥𝑡, we have:   .   t σ t t' tσa a a at (1)  2. .2.    t σ t t t' tσv v a σ a at (2)  321. . . .2 6.     t σ t t t t' tσd d σ v σ a a at (3) with 0 ≤ 𝜎 ≤ 𝜃. 𝛥𝑡; at time 𝑡 + 𝛥𝑡, we have:   .2   t t t t t ttv v a a  (4)  2. 2 .6    t t t t t t ttd d t v a a  (5) Substituting Eqs. (1), (2) and (3)into the equations of dynamic equilibrium with 𝜎 = 𝜃. 𝛥𝑡, we have:               . . . .  t' t' t' t' M a  C v  K d P  (6) Where 𝑀, 𝐶, 𝐾, and 𝑃 are the mass, stiffness, and damping matrices, forces, respectively. In this paper, we assume that the system is linear, in which 𝑀, 𝐶, and 𝐾 are constant.  Solving Eq.(6)with a single unknown 𝑎𝑡′, then substituting Eq.(4)and(5), we get the values of displacement, velocity, and acceleration at time 𝑡′.  449   JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 9 (2022) 447–454  3 Dynamic analysis of a steel truss bridge 3.1 Description: Chuong Duong bridge Chuong Duong Bridge (Fig.1) is a steel truss bridge built in 1985. The bridge connects Hoan Kiem district with Long Bien district, Hanoi capital (Vietnam), consisting of 11 truss spans. The length of each span is almost equal (90m). In this paper, we focus on considering the dynamic response of 11th span (see Fig.2)  under the effect of moving loads.   Fig.1 – Chuong Duong bridge (Vietnam) Fig.2 – Thegeneral view of 11th span 3.2 Finite Element Model (FEM) The FEM of 11th span of the bridge (Fig. 3) is build by using Matlab.   Fig. 3 – The FEM of 11th span of the bridge  Some information about the model: The number of nodes: the FEM includes69 nodes, in which each node contains 6 degrees of freedom (DOF) corresponding to translational and rotational displacements in the 𝑋, 𝑌, and 𝑍 axes.  The number of elements: 192 elements are used, including stringers, floor beam, lower chord, upper chord, vertical, diagonal, portal bracing, lower lateral bracing, upper lateral bracing, and so forth. Boundary conditions: The span are put on rocker and pin bearings. The first bearing permits translation and rotation in one direction, whereas the second one only allows rotational movement. The input parameters of the material such as elastic modulus (𝐸), density (𝑤), Poisson's ratio (𝜐) as well as the section (area, moment of inertia) are referenced from the as-built records. The material properties of the truss members are depicted in Table 1. 450   JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 9 (2022) 447–454  Table 1 - Material properties of truss members Components Value Unit Young’s modulus  2 × 1011 Pa Volumetric mass density 7850 Kg/m3 Poisson’s ratio  0.3 /  The geometric properties of the truss members are shown in Table 2 as follows: Table 2 - Geometric properties of truss members. Components Area (𝑨) Moment of inertia (𝑰𝒛) Moment of inertia (𝑰𝒚) (𝒎𝟐) (𝒌𝒈. 𝒎𝟐) (𝒌𝒈. 𝒎𝟐) Other components (lower chord, upper chord, vertical, diagonal, and so on) 0.067 0.01368 0.02194 Upper lateral bracing 0.008 0.00038 0.00071 Lower lateral bracing 0.0081 0.00036 0.00065 3.3 Dynamic analysis To analyze the dynamic responses of the structure, the Wilson-𝜃 method is employed. Three moving loads are surveyed to evaluate the influence of moving mass, moving velocity, and type of moving loads (see Table3 and Table 4).  Fig.4 – The position of the nodes determines dynamic displacements following the 𝒛 −axis Table3 - Specification of moving loads. Specification Axle load: 𝑃1 Axle load: P2 Axle load: P3 Axis distance 𝑃1 –  𝑃2: 𝑎1 Axis distance 𝑃2 –  𝑃3: 𝑎2 Time step 𝛥𝑡: 0.001𝑠 451   JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 9 (2022) 447–454  Table 4 - Types of typical load. No. 𝑷𝟏 (ton) 𝒂𝟏 (m) 𝑷𝟐 (ton) 𝒂𝟐 (m) 𝑷𝟑 (ton) ∑𝑷 (ton) I 4.76 3.70 7.62 1.350 7.62 20.00 II 5.40 3.80 10.30 1.385 10.30 26.00 III 7.00 3.80 11.50 1.85 11.50 30.00 The dynamic analysis of the bridge under the load of three moving loads with the average speed of 40, 50, and 60 km/h are conducted. Four methods including, modal superposition in frequecy domain, modal superposition in time domain, direct solution in the frequency domain, and direct time integration are employed. The results are shown in Fig.5-Fig. 11 and Table 5- Table 8. Modal superposition in frequency domain    Fig.5 – Excitation time history Fig.6 – Modal transfer function   Fig.7 – Modal response frequency domain Fig.8 – Dynamic displacement Table 5 - Dynamic displacement according to modal superposition in frequency domain  Velocity 40 km/h 50 km/h 60 km/h Loads I II III I II III I II III Node (m) 2 0.248 0.324 0.372 0.259 0.340 0.386 0.270 0.354 0.394 6 0.381 0.499 0.565 0.426 0.565 0.635 0.433 0.574 0.630 7 0.341 0.446 0.507 0.385 0.511 0.574 0.396 0.526 0.576 108 0.230 0.302 0.345 0.263 0.351 0.394 0.276 0.367 0.400 11 0.058 0.077 0.088 0.070 0.094 0.105 0.074 0.099 0.106  452   JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 9 (2022) 447–454  Modal superposition in time domain  From obtained results of Fig.5 - Fig.12 and Table 5 - Table 8, we can see that the obtained results from the four methods are almost the same. Specifficially, the displacement at the mid-span (node 6) is the largest in all cases. With the same speed, the maximum displacement along the vertical axis (𝑧-axis) is proportional to the load of the vehicles. In the case of the 3rd moving load, the displacement of the structure when the vehicle moves at a speed of 50km/h is greater than the speed of 60km/h. This demonstrates that the speed of the moving load is not completely proportional to the displacement of the structure.    Fig.9 – Modal response time domain Fig.10 – Dynamic displacement Data 1 and data 2 in the Fig.9- Fig.10 present the times when moving vehicle enters and move out of the bridge. Direct time integration Direct solution in the frequency domain   Fig. 11 – Dynamic displacement Fig.12 – Dynamic displacement Table 6 – Dynamic displacement according to modal superposition in time domain method Velocity 40 km/h 50 km/h 60 km/h Loads I II III I II III I II III Node (m) 2 0.248 0.324 0.372 0.261 0.342 0.389 0.271 0.356 0.399 6 0.380 0.497 0.564 0.435 0.578 0.648 0.450 0.600 0.660 7 0.339 0.444 0.505 0.393 0.523 0.586 0.412 0.551 0.604 108 0.228 0.299 0.343 0.269 0.359 0.402 0.288 0.385 0.420 11 0.058 0.076 0.087 0.072 0.097 0.108 0.078 0.105 0.113 453   JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 9 (2022) 447–454  The reason can be explained based on the resonance as well as the position of the axial loads during the moving process of vehicles. For example, although the vehicles move with lower speed, they can cause a greater displacement for the structure if the axial puts on the mid-span and the resonance phenomenon occurs. Table 7 – Dynamic displacement according to direct time integration method Velocity 40 km/h 50 km/h 60 km/h Loads I II III I II III I II III Node (m) 2 0.187 0.243 0.278 0.200 0.262 0.294 0.205 0.269 0.295 6 0.380 0.497 0.561 0.425 0.563 0.634 0.434 0.576 0.633 7 0.357 0.467 0.530 0.400 0.531 0.598 0.410 0.543 0.596 108 0.264 0.346 0.396 0.292 0.388 0.441 0.294 0.390 0.428 11 0.090 0.118 0.137 0.101 0.134 0.152 0.099 0.131 0.145  Table 8 – Dynamic displacement according to direct solution in the frequency domain method Velocity 40 km/h 50 km/h 60 km/h Loads I II III I II III I II III Node (m) 2 0.188 0.244 0.279 0.198 0.261 0.293 0.205 0.269 0.294 6 0.382 0.500 0.563 0.423 0.561 0.633 0.438 0.582 0.636 7 0.358 0.469 0.531 0.398 0.528 0.597 0.413 0.549 0.600 108 0.265 0.348 0.397 0.291 0.386 0.439 0.297 0.394 0.431 11 0.090 0.118 0.137 0.100 0.133 0.151 0.100 0.133 0.145  Table 9 shows that modal superposition in the frequency domain and modal superposition in the time domain require lower time to finish the dynamic analysis process of the considered structures than direct time integration and direct solution in the frequency domain. Table 9 – Computational time. Velocity 40 km/h 50 km/h 60 km/h Loads I II III I II III I II III Modal superposition in frequency domain 0.42 0.30 0.31 0.30 0.27 0.31 0.24 0.22 0.21 Modal superposition in time domain 0.41 0.37 0.41 0.36 0.32 0.30 0.26 0.27 0.27 Direct time integration 13.29 13.37 13.36 10.72 10.58 10.84 8.96 8.86 8.88 Direct solution in the frequency domain 13.24 13.249 13.29 10.76 10.78 10.65 8.96 8.87 8.93 4 Conclusions This paper investigates the dynamic response of a bridge under the effect of moving loads. The considered main parameters consist of moving mass, moving velocity, and type of moving loads. To analyze the structural dynamic response, the time history method (Wilson-𝜃 method) is employed. From the obtained results, some main conclusions are drawn: The displacement along the vertical axis is proportional to the load of the vehicles. The obtained results from the four aforementioned methods are almost the same. This demonstrates the reliability of the used methods.   454   JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 9 (2022) 447–454  Modal superposition in the frequency domain and modal superposition in the time domain outperforms direct time integration and direct solution in the frequency domain in terms of reducing the computational time. The displacement at the mid-span is the largest in all cases. The displacement of the bridge is not completely dependent on the speed of the moving load. In other words, the bridge appears maximum displacement when the resonance phenomenon occurs.  Acknowledgements This research is funded by University of Transport and Communications (UTC) under the grant number T2022-CT-006TĐ. Nguyen Huu Quyet was funded by the Master, PhD scholarship Programme of Vingroup Innovation Foundation (VINIF), code VINIF.2022.ThS.075. REFERENCES [1]-  L. Ngoc-Nguyen, H. Ngoc-Tran, S. Khatir, T. Le-Xuan, Q. Huu-Nguyen, G. De Roeck, T. Bui-Tien, M. 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Dynamic response analysis of truss bridges under the effect of moving vehicles

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Dynamic response analysis of truss bridges under the effect of moving vehicles

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Université Mouloud Mammeri de Tizi Ouzou (UMMTO): Research Review of Sciences and Technologies

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