Study on Dynamic Amplification Coefficient of U-Shaped Girder Based on Vehicle-Bridge Coupling Dynamics

Bai, Tangying; Li, Hui; Tian, Yu; Chen, Yipeng; Wen, Yudong; Xia, Chaoyi

doi:https://doi.org/10.1155/2023/2676495

Shock and Vibration

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 2676495 | https://doi.org/10.1155/2023/2676495

Study on Dynamic Amplification Coefficient of U-Shaped Girder Based on Vehicle-Bridge Coupling Dynamics

Tangying Bai,¹Hui Li,²Yu Tian,¹Yipeng Chen,¹Yudong Wen,³and Chaoyi Xia³

Academic Editor: Madalina Dumitriu

Received22 Mar 2023

Revised17 Jul 2023

Accepted20 Jul 2023

Published10 Aug 2023

Abstract

In this paper, a vehicle-bridge coupling dynamic model is established considering the vertical wheel-rail tight contact and the lateral simplified wheel-rail creeping. Through the site test and the numerical calculation, the dynamic deflections and the stresses of the U-shaped girder in the Nanjing S6 Urban Rail Transit in China are obtained, and the effectiveness of the established numerical model is verified. By changing the vehicle types and speeds, the dynamic amplification coefficients (DAFs) of the vertical deflections and the biaxial stresses at the girder bottom including the key points at the plate and section intersections are calculated. The research shows that the distribution of the lateral stresses is more complex than that of the deflection and the longitudinal stresses. Based on the calculation results considering various vehicle types, it is suggested that the DAFs of the vertical deflections and the longitudinal stresses are taken as 1.30, and the DAF of the lateral stresses remains at 1.40 as stipulated by the code. The research of this paper is to provide a reference for the structural design and size optimization of the U-shaped girders for urban rail transit.

1. Introduction

As an open thin-walled structure, a U-shaped girder comprises a bottom plate carrying the track slab, two webs on both sides, and two flange plates above the webs, and all of these structural components form a “U-shape” from the cross-sectional view [1]. The structure has the advantages of low building height, good noise-reduction effect, high utilization of sectional space, fine driving safety, and beautiful appearance. As a novel type of bridge structure suitable for urban rail transit, U-shaped girder has been increasing widely used in recent years.

In the past decades, many scholars carried out research works on the static and dynamic performance of U-shaped girders. Xu et al. [2] studied the behaviour of the U-shaped thin-walled girders under the combined actions of bending, shear, and torsion and developed a method to predict the failure modes and the ultimate loads. Li et al. [3] compared the vibration and noise characteristics of the concrete bridges with different sections by using a 3D vibroacoustic finite element method and found that the U-shaped girder generally leads to slightly lower total noise levels than the box girders. Zhang et al. [4] carried out a full-scale test to examine the ultimate bearing capacity of the U-shaped girder and developed a refined numerical model to simulate the damage evolution and the failure process. Their study shows a flexural failure occurs on the structure under the vertical loads. Xu et al. [5] observed the temperature stress of a U-shaped girder for a 48-hour period during the winter, and the research found the web, and bottom plates have significant longitudinal tensile stresses which should be considered in the design.

To design a bridge structure including the U-shaped girder, it is necessary to determine the dynamic effect of the moving vehicles on the structure [6]. In the design codes [7–9], this dynamic effect is always considered by multiplying the static live load by the dynamic amplification factor (1 + μ), in which μ is termed as the dynamic coefficient. Generally, the DAF (1 + μ) is represented by using the ratio of dynamic response to static response. To meet the requirements of structural stiffness and cracking resistance, both the deformation and stress characteristics of the bridge structure should be considered. Ma et al. [10] investigated the DAFs of the continuous bridges by selecting 15 continuous bridges and conducting a vehicle-bridge interaction analysis, and the research indicates that the DAFs of the continuous bridges increase dramatically when resonance phenomena occur. Zhang et al. [11] analyzed the DAFs and the equivalent uniform distribution load of the bridges under the random traffic flows and adopted the Monte Carlo simulations to obtain the results of some given bridge spans. Mensinger et al. [12] studied the variations of the analytical and experimental observations on the dynamic responses of the steel railway bridge and compared the measured stresses of a locomotive passing through a historical steel railway bridge with the calculated stresses contemplating the dynamic factor proposed by EN1991-2.

The mechanical characteristics of a U-shaped girder are quite different from those of ordinary box girders. Under a vertical load, the bottom plate of a U-shaped girder not only carries the longitudinal bending moments but also has the transverse bending moments, thus producing a biaxial deflection. In 2011, Wu et al. [13, 14] carried out a field test and vehicle-bridge coupling vibration analysis to study the dynamic responses of a single-track simply supported U-shaped girder. By calculating the longitudinal and transverse stresses at the bottom plates, Wu pointed out that the differences between the DAFs of the longitudinal and transverse stresses should be considered in the design. Normally, it is convenient to use a single dynamic coefficient in structural strength and stiffness design. In China, according to the Code for Design of Urban Rail Transit Bridges (GBT51234-2017) [9], the DAF (1 + μ) is recommended to value 1.40 for the single-track U-shaped girder. However, due to the complex and different mechanical characteristics of U-shaped girder in spanning and transverse directions, using a single and conservative DAF value as done in the ordinary girder design may lead to a waste of building materials. Furthermore, the present research calculated the DAFs at only a few structural positions, and some key points at complex local positions were neglected. For the U-shaped girders, an accurate study is required considering the key points as more as possible, such as the ones at the intersection of the strengthened and unstrengthened sections in the spanning direction and at the intersections of the web and bottom plates in the transverse direction. In addition, various vehicle types with different vehicle lengths, axle weights, wheelbase, and bogie spacing are adopted on the U-shaped girders in reality, but the existing research did not investigate the influence of vehicle types on the DAFs. Insufficient selection of the key points at complex structural local positions and overlooking the diversity of vehicle types may cause an inaccurate calculation of the DAFs. Therefore, it is necessary to conduct a thorough research on the DAFs of the U-shaped girders considering more structural positions, more vehicle types, and more study cases.

In this paper, a vehicle-bridge coupling model is established, and its effectiveness is verified through a site test. By selecting more key points at the plate and section intersections, the dynamic responses of the bridge under different vehicle types and speeds are calculated, and the DAFs of the biaxial stresses and deflections are obtained. A comprehensive analysis in this research indicates the distribution of the DAFs considering different structural positions and vehicle parameters and discusses the rationality to separate the DAF values of the deflections and the longitudinal and lateral stresses in the U-shaped girder design.

2. Vehicle-Bridge Coupling Dynamic Model

2.1. Model Description

In recent years, many research achievements have been made in the study of vehicle-bridge coupling vibration [15–17]. Based on the classical analysis methods [18], many researchers developed a series of solution methods and calculation models [19–25].

In general, a vehicle-bridge coupling model consists of the vehicle subsystem and the bridge subsystem, and they are connected by the wheel-rail interaction force. In this paper, it assumes that the wheel and rail are in a tight contact state in the vertical direction to save the calculation time. A simplified creep theory [26] is adopted to simulate the wheel-rail contact in the lateral direction, and the creep forces in the forward and swing movements of the wheel set are ignored. Only the creep force in the yaw movement of the wheel set is considered, and it equals to the product of the relative wheel-rail deformation and the creep coefficient. In this study, the creep coefficient could be obtained using the following equation:where V is the vehicle speed, N is the static axle load of the vehicle, and the parameter S₂₂r²/³ could be determined by the wheel-set radius [26].

The dynamic equation of the vehicle-bridge coupling system iswhere the subscripts “” and “b” represent the vehicle and the bridge.

The advantage of this method is that the motions of the vehicles and the bridge are coupled directly, and the dynamic responses of the system at each time step could be solved without any iteration to guarantee the calculation convergence.

2.1.1. Matrices of the Vehicles

There are vehicles on the bridge, and their displacements could be expressed as follows:where , .

The displacements of the ith car body and its two bogies are written as follows:

The mass and stiffness matrices (subscript “”) of the vehicles arewhere

The mass matrices of the ith car body, its jth bogie, and the lth wheel set are

The stiffness matrix of the ith vehicle iswherewhere , , , and are the lateral and vertical stiffness, as well as the lateral and vertical damping ratios of the primary suspension on the lth wheel set of the jth bogie of the ith vehicle. , , , and are the lateral and vertical stiffness, as well as the lateral and vertical damping ratios of the secondary suspension of the jth bogie of the ith vehicle. h_1i, h_2i, and h_3i are the vertical distances between the center of the car body, the axle center of the secondary suspension, and the axle center of the wheel sets of the ith car, respectively; a_i and b_i are half of the transverse distances of the primary and secondary suspension systems, and d_i and s_i are half of the longitudinal distances between the bogie centers and the bogie wheelbase of the ith car.

The damping submatrices of the vehicle could be obtained by replacing “K” with “C,” and the only difference between the “” and “” is the submatrices of the wheel set:

2.1.2. Matrices of the Bridge

The number of the mode shapes considered in the calculation is N_b, and the generalized displacements can be expressed as follows:

The mass, stiffness, and damping submatrices of the bridge (subscript “bb”) arewhere where .where .

2.1.3. Matrices of Vehicle-Bridge Interaction

The stiffness matrices coupling the vehicles and the bridge (subscripts “” and “”) could be expressed as follows:where the submatrices coupling the jth bogie of the ith vehicle and the generalized displacement of the nth mode of the bridge are

Similarly, the damping submatrix of the vehicle could also be obtained from equations (19)–(22) by replacing the “K” with “C.”where i = 1, 2, …, N_v; n = 1,2, …, N_b; j = 1,2.

2.1.4. Forces on the Vehicle-Bridge System

(1)Forces on the vehicleswhere(2)Forces on the bridge

where

In the previous equation, the values of , , and represent, in the nth mode shape, the lateral, torsional, and vertical modal displacements at the location x_ijl, where the lth wheel set of the jth bogie of the ith vehicle arrives. θ_s(x_ijl) and Z_s(x_ijl) are the rotational and vertical track irregularities at the wheel-set location x_ijl. , , and are the velocities of the lateral, rotational, and vertical irregularities at x_ijl, and they are equal to the slopes of the relative irregularities divided by the vehicle speed. These modal displacements and irregularities change when the vehicle moves on the bridge, so that the dynamic equation of the vehicle-bridge system becomes linear differential equations with time-varying coefficients. In this study, the equations are solved by using the Newmark-β method.

2.2. Model Verification

2.2.1. Project Introduction

Taking the 30 m prestressed concrete (PC) simply supported U-shaped girder in Nanjing S6 Urban Rail Transit as a study case, a series of site tests and numerical simulations was carried out. The height of the girder is 1.80 m, and the bottom width is 3.91 m. The thicknesses of the bottom plates at the midspan and the girder end are 260 mm and 400 mm, respectively. The key cross sections of the girder are shown in Figure 1.

(a)

(b)

To consider the stress diffusion on the track slab as well as the spatial mechanical characteristics of the U-shaped girder, a 3D finite element model of the U-shaped girder including the track slab is established using the ANSYS software, as shown in Figure 2.

By the eigenvalue calculation, the frequencies and the mode shapes of the 1st∼5th modes are obtained and shown in Table 1.

The train formation is “motor + trailer + trailer + motor,” and the running speed during the test is 110 km/h. Due to the data lack of the track irregularities, the US six-level spectrum is used to generate the irregularities in lateral (alignment), rotational (cross level), and vertical (longitudinal level) directions, as shown in Figure 3.

(a)

(b)

(c)

The key cross sections at the 1/4, 1/2, and 3/4 spans of the girder where the maximum responses may usually appear are considered in the calculation. Moreover, the responses of the cross sections near the movable and fixed ends should also be checked. To take a more accurate analysis than the previous work, hundreds of key points are selected along the girder. A local rectangular coordinate system is created, by which all of the key points could be expressed more clearly, as shown in Figure 4(a). In the transverse direction, there are 13 sets of lateral coordinates (y₁∼y₁₃) including the intersections of the web and bottom plates, as shown in Figure 4(b). In the spanning direction, there are 17 sets of longitudinal coordinates (x₁∼x₁₇) including the intersections of the strengthened sections near midspan and unstrengthened sections near girder end, as shown in Figure 4(c).

(a)

(b)

(c)

2.2.2. Results Comparison

To verify the established vehicle-bridge analysis model, a site test was carried out on a U-shaped girder in Nanjing S6 Urban Rail Transit. Due to the limitation on the number of measuring points in the test, only a few of the key points with the x-coordinate of x₁, x₇, x₉, x₁₁, x₁₇ and with the y-coordinate of y₁, y₄, y₇, y₁₀, x₁₃ in Figure 4 are selected as the measuring points. In the site test, the vehicles are type-B metro trains with empty loaded, and the weights of the empty motor and trailer cars are 9.0t and 8.25t, respectively. To record the dynamic vertical deflections of the girder, the multipoint video detection system of HPQN-X was adopted in the test site, as shown in Figure 5(a).

(a)

(b)

(c)

At the same time, using the INV3062V high-precision acquisition instrument shown in Figure 5(b) as the dynamic strain collecting tool, combined with the INV2312N wireless static strain measuring instrument shown in Figure 5(c) to eliminate the additional static strains such as the thermal strain, the vehicle-induced strains at the girder bottom were obtained.

(1) Vertical Deflection. The time histories of the vertical deflections d_z at the transverse centers of the bottom plate as well as the intersections of the web and the bottom plates at the 1/4, 1/2, and 3/4 spans are calculated and compared with the test data, as shown in Figure 6.

(a)

(b)

(c)

(d)

(e)

(f)

By comparing the previous figures, it can be observed that the vertical deflections at the midspan are the largest, and those at the 1/4 span are the smallest. On the same cross section, the deflections at the intersections on both sides are almost equivalent to those at the transverse center. The comparison demonstrates that the calculated and measured time histories are in good agreement.

(2) Strain at Girder Bottom. On the cross sections at the 1/4, 1/2, and 3/4 spans and near the girder ends, the longitudinal strains ε_X and the lateral strains ε_Y at the measuring points are plotted with the test values, as shown in Figure 7.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

From the calculation results, it is clear that the stresses at the transverse centers of the bottom plate gradually increase from the girder ends to the midspan, and the increasing trend slows near the midspan. In most cases, the longitudinal stresses are greater than the lateral ones.

By comparing the test and calculation results, it can be found that they are close, which proves that the vehicle-bridge coupling dynamic model established in this paper could be used to predict the structural deflections and stresses under different vehicle speeds. The stress distribution on the U-shaped girder is very complex, and a detailed analysis of the DAF is always required.

3. Case Study

3.1. Dynamic Responses

(1) Vertical Deflection. For railway and metro bridges, an excessive structural deformation will cause severe vibration or even vehicle derailment on the bridge, and the structural deformation caused by the live load should be limited. The time histories of the vertical deflections at the key points are calculated, and the results when the type-B vehicle runs at the operation speed of 120 km/h are listed in Figure 8.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 8

Time histories of the vertical deflections at the bottom plate (V = 120 km/h): (a) d_Z at the locations of (x₁, y₁∼y₇), (b) d_Z at the locations of (x₁, y₇∼y₁₃), (c) d_Z at the locations of (x₇, y₁∼y₇) (d) d_Z at the locations of (x₇, y₇∼y₁₃), (e) d_Z at the locations of (x₉, y₁∼y₇), (f) d_Z at the locations of (x₉, y₇∼y₁₃), (g) d_Z at the locations of (x₁₁, y₁∼y₇), (h) d_Z at the locations of (x₁₁, y₇∼y₁₃), (i) d_Z at the locations of (x₁₇, y₁∼y₇), and (j) d_Z at the locations of (x₁₇, y₇∼y₁₃).

By comparing the time histories in Figure 8, it is clear that the closer the midspan, the greater the deflection. At the midspan, the deflections exceed 6.0 mm. For the deflections on the same cross sections, their difference is not obvious.

(2) Longitudinal Stress. For most girder structures, the longitudinal stresses at the girder bottom are normally greater than the lateral ones, and it is generally necessary to consider the longitudinal stresses when determining the DAF. The time histories of the longitudinal stresses σ_X when the type-B vehicle runs at 120 km/h are listed in Figure 9.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 9

Time histories of the longitudinal stresses at the bottom plate (V = 120 km/h): (a) σ_X at the locations of (x₁, y₁∼y₇), (b) σ_X at the locations of (x₁, y₇∼y₁₃), (c) σ_X at the locations of (x₇, y₁∼y₇), (d) σ_X at the locations of (x₇, y₇∼y₁₃), (e) σ_X at the locations of (x₉, y₁∼y₇), (f) σ_X at the locations of (x₉, y₇∼y₁₃), (g) σ_X at the locations of (x₁₁, y₁∼y₇), (h) σ_X at the locations of (x₁₁, y₇∼y₁₃), (i) σ_X at the locations of (x₁₇, y₁∼y₇), and (j) σ_X at the locations of (x₁₇, y₇∼y₁₃).

From the time histories in Figure 9, it can be found that the bottom of the girder carries longitudinal tensile stresses at the 1/4, 1/2, and 3/4 spans during the vehicle passing through. The maximum tensile stress is close to 2.0 MPa, appearing at the midspan. The girder bottom is mainly in compression near the girder ends, and the maximum stress near the fixed end is greater than that near the movable end.

(3) Lateral Stress. The lateral stresses are mainly caused by the transverse bending of the girder, and the analysis of these stresses is essential to determine the DAF of the U-shaped girder. When the speed of the type-B vehicle is 120 km/h, the time histories of the lateral stresses σ_Y are listed in Figure 10.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 10

Time histories of the lateral stresses at the bottom plate (V = 120 km/h): (a) σ_Y at the locations of (x₁, y₁∼y₇), (b) σ_Y at the locations of (x₁, y₇∼y₁₃), (c) σ_Y at the locations of (x₇, y₁∼y₇), (d) σ_Y at the locations of (x₇, y₇∼y₁₃), (e) σ_Y at the locations of (x₉, y₁∼y₇), (f) σ_Y at the locations of (x₉, y₇∼y₁₃), (g) σ_Y at the locations of (x₁₁, y₁∼y₇), (h) σ_Y at the locations of (x₁₁, y₇∼y₁₃), (i) σ_Y at the locations of (x₁₇, y₁∼y₇), and (j) σ_Y at the locations of (x₁₇, y₇∼y₁₃).

Comparing Figure 10 with Figure 9, it can be observed that the distribution of the lateral stresses is more complicated than that of the longitudinal stresses, although the amplitudes of the lateral stresses are lower than the longitudinal ones in most cases. In most cases, the girder experiences both tensile and compressive stresses. At the 1/4, 1/2, and 3/4 spans, the key points with y₃, y₄, y₁₀, and y₁₁ coordinates have the greatest lateral tensile stress, meaning that the most unfavourable positions are near the transverse midspan of the bottom plate. Near the girder ends, the key points with y₁∼y₄ and y₁₀∼y₁₃ coordinates near the intersection areas are in compression, and the ones near the transverse center of the bottom plate are mainly in tension.

3.2. The DAFs

In this study, the DAF is defined as follows:where D_dy is the structural dynamic response caused by the moving vehicle and D_st is the structural static response caused by the design live load of the vehicle.

The quasistatic loading is taken by moving the vehicle at a low speed of 1.0 km/h, and the corresponding responses are set to be D_st. By using the vehicle-bridge coupling model introduced in Section 2, the calculated dynamic responses D_dy could be obtained. To conduct a thorough study of the DAFs, the vehicles including the type-A, type-B, and type-D subway trains are considered, and all of them are widely used in China [9]. Due to the difference between the lengths, the wheelbases, the bogie spacings, and the axle weights of the vehicles, the structural dimensions of the girders for various vehicle types are different. The information on the vehicles and girders is shown in Table 2.

Where, the intervals of y3∼y11 are 8 × 368.25 mm and 8 × 393.25 mm, for the girders designed for the type-A and type-D trains, respectively.

Through a series of calculations, the DAFs of the U-shaped girder under different cases can be obtained. Table 3 shows the distribution of the DAFs of the girder designed for the type-A vehicle at speeds of 80 km/h, 100 km/h, 120 km/h, 140 km/h, and 160 km/h.

From Table 3, it is clear that the DAFs of the deflections are all lower than 1.10 in all listed cases. Similarly, most of the DAFs of the longitudinal stresses do not exceed 1.10, except for those at the strengthened section close to the fixed end. For the DAFs of the lateral stresses, the values are greater than 1.10 at many key points along the girder, and the maximum DAF of 1.320 appears at the left plate intersection of the 1/4 span when V = 120 km/h.

Table 4 shows the distribution of the DAFs of the girder designed for the type-B vehicle at the speeds of 80 km/h, 100 km/h, 120 km/h, 140 km/h, and 160 km/h.

By observing Table 4, it is found that the DAFs of the deflections and the longitudinal stresses are around 1.20 when the vehicle speed V > 120 km/h. From the distribution trend in the figures, the average DAFs of the lateral stresses increase with the vehicle speed, and the values at some positions exceed 1.20 when V ≥ 120 km/h. At the left plate intersection of the 1/4 span, the maximum DAF reaches 1.385.

Table 5 shows the distribution of the DAFs of the girder designed for the type-D vehicle at the speeds of 120 km/h, 140 km/h, 160 km/h, 180 km/h, and 200 km/h.

It is clearly seen from Table 5 that the DAFs of the deflections and the longitudinal stresses are greater than the cases of type-A and type-B because of the higher speed range for the type-D train. At the listed speeds, the DAFs greater than 1.20 are mainly at the strengthened section close to the fixed end. Although the operation speed of the type-D train is clearly higher than others, the DAFs of the lateral stresses are still less than the stipulated value of 1.40 in the code.

To follow the conservative principles in structural design, the maximum calculated speeds of the vehicles in this study are set to be 40 km/h higher than the operation speeds. In the calculation, the vehicle speed is increased gradually from half of the operation speed to the maximum calculated speed (with an increment of 10 km/h). Tables 6–8 list the maximum DAFs of all study cases.

From the summarized results in Tables 6–8, it can be concluded that the maximum DAFs of different responses occur at the girder designed for different vehicle types; thus, the vehicle types should not be neglected in the DAF calculation. For the maximum calculated speeds in this study, the maximum DAFs of the vertical deflections and the longitudinal stresses are 1.251 and 1.262, appearing in the study cases of the type-D train. The maximum DAF of the lateral stresses is 1.385, and it occurs in the study case of the type-A train. For the operation speeds, the related maximum DAFs of these three responses are 1.127, 1.164, and 1.302, respectively. As it is known, a lower DAF means a smaller live load applied on the girder, thus may lead to a reduction of the girder dimensions, especially a reduction in the thickness of the bottom plate. Considering the redundancy in vehicle speed and structural design, the DAFs of the vertical deflections and longitudinal stresses representing the longitudinal mechanical characteristics are suggested as 1.30, and the DAF of the lateral stresses is still taken as 1.40. As suggested, a separate definition of the DAFs in longitudinal and lateral directions may cause a lower live load when checking the longitudinal stresses, and it will be beneficial in reasonable dimension optimization and effective material saving.

4. Conclusions

Based on the vehicle-bridge coupling dynamics theory, the DAFs of the deflections and stresses of the U-shaped girder are analyzed by the field test and numerical calculation. Through a comprehensive study considering the local positions at the web and plate intersections and the diversity of the vehicle types, the distribution of the DAFs is studied. The results show the followings:(1)The girder bottom is in longitudinal tensile at the 1/4, 1/2, and 3/4 spans, and it is in longitudinal compression near the girder ends. For the key points along the transverse direction of the same cross sections, the difference between the vertical deflections or the longitudinal stresses is small. Although the amplitudes of the lateral stresses are lower than the longitudinal ones, the stress distribution is more complex, and there are both lateral tensile and compressive stresses at the girder bottom.(2)The maximum DAFs considering the type-A, type-B, and type-D vehicles are different, and the influence of the vehicle types could not be neglected in the DAF calculation. After incorporating the results of various vehicle types, the maximum DAFs of the vertical deflections and the longitudinal stresses appear at the left plate intersection near the fixed end, and the maximum one of the lateral stresses is at the left plate intersection of the 1/4 span. As a key position with the clear significant DAF values, the intersection of the web and bottom plates should be concerned carefully when checking cracking resistance.(3)For the operation vehicle speeds, the maximum DAFs of the vertical deflections, the longitudinal stresses, and lateral stresses are 1.127, 1.164, and 1.302. For the maximum calculated speeds 40 km/h higher than the operation ones, the maximum DAFs of the related responses are 1.251, 1.262, and 1.385, respectively. Considering the redundancy principle in structural design, it is suggested that the DAFs of the vertical deflections and the longitudinal stresses are taken as 1.30, and the DAF of the lateral stresses remains 1.40 as stipulated by the code.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This study was supported by the Fundamental Research Funds for the Central Universities of China (2023JBZY025), the Programme of Introducing Talents of Discipline to Universities (B13002) and (B20040).

References

A. R. Cusens and J. L. Rounds, “Tests of a U-beam bridge deck,” Structure Engineering, vol. 51, pp. 377–382, 1973.
View at: Google Scholar
J. C. Xu, B. Diao, Q. Q. Guo, Y. H. Ye, Y. L. Mo, and H. B. Chen, “Interaction of bending, shear and torsion in U-shaped thin-walled RC girders,” Engineering Structures, vol. 179, pp. 655–669, 2019.
View at: Publisher Site | Google Scholar
Q. Li, B. R. Dai, Z. H. Zhu, and D. J. Thompson, “Comparison of vibration and noise characteristics of urban rail transit bridges with box-girder and U-shaped sections,” Applied Acoustics, vol. 186, 2022.
View at: Google Scholar
J. F. Zhang, Y. Jing, P. D. Li, W. S. Han, N. Zhang, and Y. L. Zhou, “Experimental and numerical investigation on the ultimate vertical bearing capacity of U-shaped girder with damaged web,” Sensors, vol. 19, no. 17, p. 3735, 2019.
View at: Publisher Site | Google Scholar
D. Xu, G. F. Sun, and G. N. Xu, “Study on the influence of negative temperature gradient of U-shaped girder of rail transit in winter,” Journal of Physics: Conference Series, vol. 1549, no. 4, 2017.
View at: Google Scholar
P. Paultre, O. Chaallal, and J. Proulx, “Bridge dynamics and dynamic amplification factors a review of analytical and experimental findings,” Canadian Journal of Civil Engineering, vol. 19, no. 2, pp. 260–278, 1992.
View at: Publisher Site | Google Scholar
Aashto, The American Association of State Highway and Transportation Officials, Washington, D. C, 2020.
Code for Design on Railway Bridge and Culvert (TB 10002 2017), China railway publishing house, Beijing, China, 2017.
Code for Design of Urban Rail Transit Bridge (GBT51234 2017), China Architecture & Building Press, Beijing, China, 2017.
L. Ma, W. Zhang, W. S. Han, and J. X. Liu, “Determining the dynamic amplification factor of multi-span continuous box girder bridges in highways using vehicle-bridge interaction analyses,” Engineering Structures, vol. 181, no. 15, pp. 47–59, 2019.
View at: Publisher Site | Google Scholar
Q. L. Zhang, A. Vrouwenvelder, and J. Wardenier, “Dynamic amplification factors and EUDL of bridges under random traffic flows,” Engineering Structures, vol. 23, no. 6, pp. 663–672, 2001.
View at: Publisher Site | Google Scholar
M. Mensinger, R. R. Fard, A. Hacker, and A. Näßl, “Validation of the dynamic amplification factor in case of historic railway steel bridges with short and medium spans,” Procedia Engineering, vol. 156, pp. 233–240, 2016.
View at: Publisher Site | Google Scholar
L. Q. Wu, D. J. Wu, and Q. Li, “Vehicle-bridge dynamic response analysis of U-girder for urban rail transit,” Journal of Railway Science and Engineering, vol. 7, no. 6, pp. 6–11, 2010.
View at: Google Scholar
L. Q. Wu, D. J. Wu, and Q. Li, “Experimental study on vehicle-bridge dynamic characteristics of thin-walled trough girder bridge for urban rail transit,” China Railway Science, vol. 32, no. 4, pp. 31–37, 2011.
View at: Google Scholar
Y. B. Yang, J. D. Yau, and Y. S. Wu, Vehicle-bridge Interaction Dynamics with Application to High-Speed Railways, World Scientific Publisher, Singapore, 2004.
W. M. Zhai and H. Xia, Train-Track-Bridge Dynamic Interaction: Theory and Engineering Application, Science Press, Beijing, 2011.
H. Xia, N. Zhang, and W. W. Guo, Dynamic Interaction of Train-Bridge Systems in High-Speed Railways: Theory and Application, Springer Nature, London, 2018.
W. M. Zhai, Z. L. Han, Z. W. Chen, L. Ling, and S. Y. Zhu, “Train–track–bridge dynamic interaction: a state-of-the-art review,” Vehicle System Dynamics, vol. 57, no. 7, pp. 984–1027, 2019.
View at: Publisher Site | Google Scholar
N. Zhang and H. Xia, “Dynamic analysis of coupled vehicle-bridge system based on inter-system iteration method,” Computers & Structures, vol. 114-115, pp. 26–34, 2013.
View at: Publisher Site | Google Scholar
P. Xiang, Z. Yan, J. H. Lin, D. Kennedy, and F. W. Williams, “Random vibration analysis for coupled vehicle-track systems with uncertain parameters,” Engineering Computations, vol. 33, no. 2, pp. 443–464, 2016.
View at: Google Scholar
X. F. Zhang, X. K. Wang, M. D. Pandey, and J. D. Sørensen, “An effective approach for high-dimensional reliability analysis of train-bridge vibration systems via the fractional moment,” Mechanical Systems and Signal Processing, vol. 151, Article ID 107344, 2021.
View at: Publisher Site | Google Scholar
W. Guo, C. Zeng, H. Y. Gou et al., “Real-time hybrid simulation of high-speed train-track-bridge interactions using the moving load convolution integral method,” Engineering Structures, vol. 228, Article ID 111537, 2021.
View at: Publisher Site | Google Scholar
H. L. Li, T. Y. Wang, and G. Wu, “A Bayesian deep learning approach for random vibration analysis of bridges subjected to vehicle dynamic interaction,” Mechanical Systems and Signal Processing, vol. 170, Article ID 108799, 2022.
View at: Publisher Site | Google Scholar
X. Zhang, Y. Han, L. D. Wang, H. Y. Liu, and C. S. Cai, “An adaptive surrogate model approach for random vibration analysis of the train–bridge system,” Engineering Structures, vol. 278, Article ID 115490, 2023.
View at: Publisher Site | Google Scholar
Z. H. Zhai, C. B. Cai, and S. Y. Zhu, “Implementation of Timoshenko curved beam into train-track-bridge dynamics modelling,” International Journal of Mechanical Sciences, vol. 247, Article ID 108158, 2023.
View at: Publisher Site | Google Scholar
H. Xia and N. Zhang, “Coupling vibration of train-bridge system,” China Science Publishing & Media Ltd, Beijing, China, 2014.
View at: Google Scholar

Copyright

Copyright © 2023 Tangying Bai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

159

Downloads

208

Citations