The dynamic amplification factors for continuous beam bridges along high-speed railways

Abstract

In this paper, the dynamic amplification factors (DAFs) of high-speed railway continuous girder bridges are studied. The vehicle-bridge interactions (VBIs) of 13 concrete continuous girder bridges with spans ranging from 48 to 130 m are analyzed, the influences of the train speed, the train marshalling and the bridge fundamental frequency on the DAF are investigated, and the DAF design standard for high-speed railway bridges is discussed. The results indicate that for the continuous beam bridge whose fundamental frequency is less than 3.0 Hz, the maximum DAF is no more than 1.15; while for the bridge examples with a fundamental frequency larger than 3.0 Hz, the maximum DAF reaches 1.25 because the resonance occurs at high train speed. The empirical formulas of the DAFs in the Japan Railway Technical Research Institute (JRTRI) code could provide a conservative estimation of the DAFs of high-speed railway continuous bridges.

Keywords

continuous beam bridges dynamic amplification factor high-speed railway bridges impact coefficient vehicle-bridge interaction

Introduction

With the rapid development of high-speed railway construction, the dynamic response of bridges becomes stronger due to the increase in train speed and axle load; thus, the dynamic vehicle-bridge interactions (VBIs) of high-speed railway bridges become more prominent (Zhai and Xia, 2011). In bridge design, the dynamic amplification factor (DAF) is generally used to consider the dynamic effect of a train on a bridge structure, and it is defined as the ratio of the maximum dynamic response to the maximum static response. The DAF of bridges depends on a number of factors, including the bridge span length or fundamental frequency, the train speed and marshalling, and the track irregularity (Li et al., 2015). Due to the complexity of the problem, there are significant differences between the empirical DAF formulas in the current bridge design codes of different countries and districts. Therefore, the dynamic VBI and the DAF of high-speed railway bridges are still important topics of research for bridge scholars and engineers.

Since the 1960s, high-speed railways have been built throughout the world, and a large number of tests and theoretical studies have been carried out regarding the dynamic VBI of high-speed trains. Matsuura (1976), Matsuura and Zhou (1980) established a four-axle vehicle model with upper and lower suspensions totalling 10 degrees of freedom (DOFs) and included the influence of the rail irregularities in their studies. Diana and Cheli (1989) established vehicle-bridge dynamic analysis frameworks that included the influence of the elasticity of the rail and the creep forces between the tire set and the rail, respectively. Yang and Wu (2001) proposed the dynamic condensation method that condensed the DOFs of the sprung mass of a vehicle model into the bridge element in contact. Frýba (1999), Yang et al. (2004) reviewed the early studies on the VBI of railway bridges. Zhai and Xia (2011) implemented a systematic investigation on the running stability and smoothness of high-speed railways to promote the technical standard of Chinese railway lines.

Considering the complexity of the dynamic VBI, Eurocodes tend to adopt moving load models to investigate the dynamic response of railway bridges and the additional damping ratios to include the damping effect of the vehicle suspension system on the bridge. The European Rail Research Institute (ERRI) (1999) implemented a systematic study on the DAFs of bridges in railways with running speeds higher than 200 km/h and proposed the additional damping model (ADM). Their suggestions were then adopted by the Eurocode (EN 1991-2, 2003). Other studies (Doménech et al., 2014; Yau et al., 2019) further investigated the equivalent additional damping ratios from the VBI benefits.

The DAF of bridges is closely related to the VBI (Deng and Cai, 2010). In the 1970s, Matsuura (1976), Matsuura and Zhou (1980) investigated the effect of the train formation, axle distances, vehicle speed, vehicle suspension parameters, rail irregularity, and bridge span on the DAFs of railway bridges. They proposed that regular axle alignment could cause a significant resonance phenomenon when the load period is equal to the integral multiple of the natural period of the bridge. Yau et al. (2001) studied the dynamic response of an elastically supported beam bridge under train loads and showed that under most resonance conditions, the DAFs of the elastic-supported girder bridge are higher than those of a simply supported girder bridge. Frýba (2001, 2004) investigated the dynamic responses of various kinds of concrete bridges and steel bridges with spans of 5–50 m with three kinds of high-speed trains, that is, the German ICE, French Train à Grande Vitesse (TGV)/Euro-Star, and Spanish Talgo. The influence of the running speed (5–500 km/h) and the initial state of the running train before arriving at the bridge, etc. on the VBI were analyzed.

In recent years, Hamidi and Danshjoo (2010) studied the effects of the velocity, train axle distance, and number of axles on the DAFs of railway steel bridges. The results showed that the DAFs clearly increase with increasing speed, and an increase in the ratio of the axle distance to the span length leads to a decrease in the DAFs. Li et al. (2015) studied the VBI and the influencing factors of the DAFs of a cable-stayed railway bridge. The study showed that different DAFs occur to different components of the long-span railway cable-stayed bridges, and the parameters such as vehicle speed, marshalling, and driving direction affect the DAFs. Gunmo (2015) applied an implicit integration approach to investigate vehicle-bridge resonance in a 100 m beam-type railway bridge carrying a TGV. The solution revealed that at speeds approaching 100 km/h, the car body deflection is excited by resonance despite the suspension system. Martínez-Rodrigo et al. (2018) investigated the influence of soil-structure interaction on the vibrations of a simply supported bridge. The results showed that at resonant speeds, the soil-structure interaction reduces the DAF of the bridge.

The above studies show that the DAFs of railway bridges are affected by many factors such as the vehicle speed, rail irregularity, and train formation. However, previous studies have mostly focused on simply supported beam bridges. Owing to different boundary conditions, the structural forms and dynamic characteristics of continuous beam bridges may be quite different from those of simply supported bridges (Deng et al., 2015), and the DAFs of continuous bridges needs to be further discussed. Moreover, the results of the formulas recommended in the various codes are quite different. Therefore, it is necessary to conduct a systematic analysis of the DAF for continuous beam bridges in high-speed railways.

In this study, 13 continuous beam bridges are chosen as bridge examples. The main span lengths of these bridges range from 48 to 130 m. Based on VBI analyses, the present study investigates the DAFs of these bridges under the action of high-speed trains and discusses the DAFs of railway continuous bridges.

Bridge and vehicle analysis models

Bridge models

The bridge examples selected for this study are listed in Table 1, among which the first eight are existing bridges. To increase the span range, another five bridges, Design Bri1 to Design-Bri5, are designed. All of these bridge examples are continuous concrete box girder bridges with variable cross-sections.

Table 1.

Primary parameters of the bridges studied in this paper.

Bridge name	Span lengths (m)	Bridge width/cantilever length (m)	Position	Girder depth (m)	Top plate/bottom plate/web plate thickness (m)
JHGT-Bri1	32 + 48+ 32	12.00/3.25	Midspan	2.978	0.40/0.48/0.40
JHGT-Bri1	32 + 48+ 32	12.00/3.25	Support	4.078	0.40/0.80/0.80
JHGT-Bri2	36 + 54 + 36	12.00/3.25	Midspan	3.000	0.40/0.48/0.40
JHGT-Bri2	36 + 54 + 36	12.00/3.25	Support	4.750	0.40/0.80/0.80
JHGT-Bri3	40 + 64 + 40	13.00/3.60	Midspan	3.560	0.40/0.48/0.40
JHGT-Bri3	40 + 64 + 40	13.00/3.60	Support	5.450	0.40/0.80/0.86
XJ-Bri	48 + 80 + 48	11.00/2.65	Midspan	3.800	0.40/0.48/0.40
XJ-Bri	48 + 80 + 48	11.00/2.65	Support	5.936	0.40/0.90/1.00
LS-Bri	48 + 80 + 48	13.40/3.35	Midspan	3.850	0.40/0.48/0.40
LS-Bri	48 + 80 + 48	13.40/3.35	Support	6.650	0.40/0.90/1.00
YN-Bri	40 + 64 × 2 + 40	13.00/3.60	Midspan	3.400	0.40/0.40/0.40
YN-Bri	40 + 64 × 2 + 40	13.00/3.60	Support	5.200	0.40/0.80/0.76
WSJ-Bri	60 + 100 + 60	12.00/2.65	Midspan	4.850	0.40/0.80/0.40
WSJ-Bri	60 + 100 + 60	12.00/2.65	Support	7.850	0.40/1.00/1.12
GXH-Bri	60 + 100 × 2 + 60	12.00/2.65	Midspan	4.535	0.40/0.60/0.40
GXH-Bri	60 + 100 × 2 + 60	12.00/2.65	Support	7.035	0.40/1.00/1.18
Design-Bri1	42 + 75 + 42	13.70/3.45	Midspan	3.800	0.40/0.40/0.40
Design-Bri1	42 + 75 + 42	13.70/3.45	Support	6.600	0.40/0.75/0.80
Design-Bri2	60 + 108 + 60	12.00/2.60	Midspan	5.000	0.40/0.40/0.40
Design-Bri2	60 + 108 + 60	12.00/2.60	Support	9.000	0.40/0.80/1.20
Design-Bri3	54 + 90 × 2 + 54	13.70/3.45	Midspan	4.500	0.40/0.60/0.40
Design-Bri3	54 + 90 × 2 + 54	13.70/3.45	Support	7.000	0.40/0.90/1.08
Design-Bri4	75 + 120 + 75	13.70/3.45	Midspan	5.800	0.40/0.60/0.46
Design-Bri4	75 + 120 + 75	13.70/3.45	Support	9.400	0.40/1.00/1.44
Design-Bri5	80 + 130 + 80	13.00/3.10	Midspan	6.300	0.40/0.60/0.46
Design-Bri5	80 + 130 + 80	13.00/3.10	Support	10.200	0.40/1.00/1.56

For continuous bridges, the piers and beams are typically not rigidly connected to form a whole structure, and actually piers support the main beams by using bearings. So, the main beams’ vibrations of continuous bridges are almost independent on the piers. Thus, the piers are not included in the finite element bridge models in this paper. Finite element bridge models (FBMs), which are all 3D models, are established using ANSYS 15.0. The Beam44 element is adopted, and the modulus of elasticity and mass density are set to be 3.55E + 10 Pa and 2600 kg/m³, respectively. Rayleigh damping is used (Deng and Cai, 2011), and the damping ratios of the first two flexural modes are set to be 2%. The frequencies of the first two vertical bending modes are given in Table 2.

Table 2.

Natural frequencies and DAFs of the bridges from empirical formulas.

Bridge name	L_max (m)	$L_{ϕ}$ / $L_{φ}$ (m)	L_c (m)	n (Hz)	JRTRI code (2004)	UIC code (2006)	CNRA code (2014)
JHGT-Bri1	48	48.53/48.53	33	4.28	1.2471	1.0492/1.1196	1.0328/1.0797
JHGT-Bri2	54	54.60/54.60	37	3.53	1.2475	1.0305/1.0972	1.0203/1.0648
JHGT-Bri3	64	64.00/64.00	41	3.16	1.2162	1.0069/1.0782	1.0046/1.0521
XJ-bri	80	80.00/80.00	51	2.29	1.2006	1.0000/1.0412	1.0000/1.0275
LS-Bri	80	80.00/80.00	49	2.43	1.1914	1.0000/1.0476	1.0000/1.0318
YN-Bri	64	72.80/72.80	49	2.18	1.2896	1.0000/1.0476	1.0000/1.0318
WSJ-Bri	100	100.00/100.00	65	1.91	1.1891	1.0000/1.0047	1.0000/1.0032
GXH-Bri	100	112.00/112.00	75	1.41	1.2432	1.0000/1.0000	1.0000/1.0000
Design-Bri1	75	75.00/75.00	46	2.73	1.1930	1.0000/1.0582	1.0000/1.0388
Design-Bri2	108	108.00/108.00	61	1.84	1.1815	1.0000/1.0138	1.0000/1.0092
Design-Bri3	90	100.80/100.80	67	1.75	1.2239	1.0000/1.0005	1.0000/1.0003
Design-Bri4	120	120.00/120.00	79	1.53	1.1913	1.0000/1.0000	1.0000/1.0000
Design-Bri5	130	130.00/130.00	85	1.43	1.1877	1.0000/1.0000	1.0000/1.0000

(1) L_max denotes the maximum span length; $L_{ϕ}$ is the determinant length in UIC code, $L_{φ}$ is the load length in CNRA code, and the two lengths are calculated in the same way; L_c denotes the characteristic length according to the literature (Yang et al., 1995); n denotes the basic frequency corresponding to the flexural mode; (2) when the formula in UIC code is used, the tracks are assumed to be standardly maintained and thus equation (1b) is adopted; (3) when the formula in the JRTRI code is used, the railways are assumed to be new and double-line, and thus K_a is equal to 1.0 and the reduction coefficient α is used; the design speed V is 350 km/h in equation (2).

According to the span length and frequencies of these continuous beam bridge models, the empirical DAFs can be obtained using the formulas by the Japan Railway Technical Research Institute (JRTRI) (2004) code, the International Union of Railways (UIC) (2006) code 776-1 R, and the China National Railroad Administration (CNRA) code (TB10621, 2014). Here, the empirical formulas recommended by these codes are briefly introduced.

The International Union of Railways (French: Union Internationale Des Chemins De Fer, UIC) takes different DAF values of railway bridges according to the determinant length and maintenance condition:

ϕ_{2} = \frac{1.44}{\sqrt{L_{ϕ}} - 0.2} + 0.82, (1.00 < ϕ_{2} < 1.67)

(1a)

ϕ_{3} = \frac{2.16}{\sqrt{L_{ϕ}} - 0.2} + 0.73, (1.00 < ϕ_{3} < 2.00)

(1b)

where $ϕ$ denotes the dynamic factor (DAF) and is taken as $ϕ_{2}$ or $ϕ_{3}$ for carefully maintained and standardly maintained tracks, respectively. $L_{ϕ}$ denotes the determinant length (m). For main girders of a continuous beam, the determinant length is taken as k × L_m and not less than the maximum span length (L_max). L_m denotes the average span length of the continuous beam; k is the amending coefficient. The coefficient k is used to include the DAF difference between a continuous beam and a simply supported beam with the same span length. When the span number of the continuous beam is equal to 2, 3, 4, or ≥5, k is taken to be 1.2, 1.3, 1.4, or 1.5, respectively.

For single-line railway bridges, the DAF formula recommended by the JRTRI code (2004) is as follows:

1 + μ = 1 + \frac{K_{a} V}{7.2 nL} + \frac{10}{65 + L}

(2)

where $1 + μ$ denotes the DAF with $μ$ being the impact coefficient; K_a is a constant, which is 2.0 and 1.0 for old and new bridges, respectively; V denotes the design speed (km/h); $n$ denotes the basic frequency of the bridge (Hz); and L is the span length of the bridge (m). For double-line railway bridges, the calculated impact coefficient $μ$ should also be multiplied by a reduction coefficient α, which is equal to 0.6 when the span length L of the bridge is larger than 80 m, whereas a value of 1 − L/200 is used for bridges with a span length less than 80 m.

The DAF formula specified in the CNRA code (TB10621, 2014) is similar to that of UIC code:

1 + μ = 1 + (\frac{1.44}{\sqrt{L_{φ}} - 0.2} - 0.18)

(3)

where $1 + μ$ denotes the DAF, which is set to be 1.0 if its calculated value is less than 1.0; and L_φ represents the load length (unit: m). The load length is calculated in the same way as the determinant length in UIC code. When the calculated load length is less than the maximum span length L_max, it should be set to be L_max. Some studies (Ma et al., 2019; Yang et al., 1995) have recommended that span lengths or load lengths should be replaced by the characteristic lengths (i.e. the distances between zero nodes of curvature modes) in the empirical formulas for DAFs.

The calculated determinant/load lengths and characteristic lengths of the above bridge examples 1 are listed in Table 2. The DAFs from the JRTRI code (2004), the UIC code (2006) and the CNRA code (2014) are also given in Table 2. Here, the maximum span lengths and basic flexural frequencies are the fundamental parameters for the JRTRI code (2004). The determinant or the characteristic lengths are the fundamental parameters for the UIC code (2006) while the load lengths or the characteristic lengths are the fundamental parameters for the CNRA code (2014). Thus, there are two DAF values for each bridge example when the UIC code (2006) code or the CNRA code (2014) is applied. For example, when the UIC code (2006) code is applied, there are two DAF values, that is, 1.0492 and 1.1196, for the JHGT-Bri1 bridge, in which the former value is calculated using the determinant length while the latter is calculated using the characteristic length. Clearly, the DAFs from the empirical formulas specified in the different bridge codes are quite different, and the DAFs from the JRTRI (2004) are much larger than those from the UIC code (2006) and CNRA code (2014). Moreover, the characteristic length is much smaller than the load length of each bridge example, and thus the DAFs using the characteristic length are evidently larger than those using the load length. However, the DAFs from the UIC code (2006) or the CNRA code (2014) are still less than those from the JRTRI code (2004), even if the characteristic lengths are used.

Vehicle model

In the present numerical simulation, a typical vehicle model with four wheelsets is adopted, which consists of seven rigid bodies, namely, one body, two bogies, and four wheelsets. These parts are connected by suspension springs and dampers as shown in Figure 1. The ith body has five DOFs, including lateral displacements Y_ci, vertical displacements Z_ci, rolling rotations θ_ci, pitching rotations φ_ci, and yawing rotations Ψ_ci. The jth bogie also has five DOFs, including lateral displacements Y_tij, vertical displacements Z_tij, rolling rotations θ_tij, pitching rotations φ_tij, and yawing rotations Ψ_tij. Finally, the lth wheelset has three DOFs, including lateral displacements Y_wijl, vertical displacements Z_wijl, and rolling rotations θ_wijl. Thus, a single vehicle model with four wheelsets has a total of 27 DOFs. For the jth bogie of the ith body, the coefficients of lateral stiffness and damping of the primary suspension spring are designated as k^h_1ij and c^h_1ij, respectively; the coefficients of vertical stiffness and damping are designated as k^v_1ij and c^v_1ij, respectively. Similarly, the four parameters of the secondary suspension spring are correspondingly k^h_2ij and c^h_2ij, k^v_2ij and c^v_2ij. The CRH3 train composed of 2 × (1 trailer + 1 motor + 1 motor + 1 trailer) is employed in this study, in which a single vehicle is 24.775 m long. The primary parameters refer to the literature (Shen, 2011), which are shown in Table 3. Among the 27 DOFs of the vehicle model, 15 DOFs of the vehicle body and bogies are independent, while 12 DOFs of the four wheel sets are dependent on the bridge motion. The independent displacement vector of the vehicle is as follows:

\begin{array}{l} V_{v} = {Y_{c i}, θ_{c i}, ψ_{c i}, Z_{c i}, φ_{c i}, Y_{t i 1}, θ_{t i 1}, ψ_{t i 1}, Z_{t i 1}, φ_{t i 1}, \\ Y_{t i 2}, θ_{t i 2}, ψ_{t i 2}, Z_{t i 2}, φ_{t i 2}}^{T} \end{array}

(4)

Figure 1.

Analytical model and DOFs of the CRH3 train.

Table 3.

Primary parameters of the CRH3.

Parameters	Units	Motor	Trailer
Mass of the body	t	48.000	44.000
Rolling moment of inertia of the body	t·m²	115.000	100.000
Pitching moment of inertia of the body	t·m²	2700.000	2700.000
Yawing moment of inertia of the body	t·m²	2700.000	2700.000
Mass of the bogie	t	3.200	2.400
Rolling moment of inertia of the bogie	t·m²	3.200	1.800
Pitching moment of inertia of the bogie	t·m²	7.200	2.200
Yawing moment of inertia of the bogie	t·m²	6.800	2.200
Mass of the wheelset	t	2.400	2.400
Rolling moment of inertia of the wheelset	t·m²	1.200	1.100
Coefficient of lateral stiffness of the primary suspension spring	MN/m	3.000	5.000
Coefficient of vertical stiffness of the primary suspension spring	MN/m	1.040	0.700
Coefficient of lateral damping of the primary suspension spring	KN·s/m	0	0.000
Coefficient of vertical damping of the primary suspension spring	KN·s/m	40.000	40.000
Coefficient of lateral stiffness of the secondary suspension spring	MN/m	0.240	0.280
Coefficient of vertical stiffness of the secondary suspension spring	MN/m	0.400	0.300
Coefficient of lateral damping of the secondary suspension spring	KN·s/m	30.000	25.000
Coefficient of vertical damping of the secondary suspension spring	KN·s/m	60.000	60.000
Half of the distance between the wheelbases	m	2.500/2	2.500/2
Half of the distance between the bogies	m	17.375/2	17.375/2
Distance between the bodies (hook to hook)	m	24.775	24.775
Half of the lateral distance between the nominal rolling circle of the wheel	m	1.496/2	1.496/2
Height of the center of gravity of the body from the rail surface	m	1.700	1.700
Height of the center of gravity of the bogie from the rail surface	m	0.700	0.700
Half of the lateral distance between the primary suspension springs	m	2.000/2	2.000/2
Half of the lateral distance between the secondary suspension springs	m	1.900/2	1.900/2
Height of the center of the primary suspension spring from the rail surface	m	0.600	0.600
Height of the center of the secondary suspension spring from the rail surface	m	0.900	0.900

The calculated frequencies and modes of the CRH3 are listed in Table 4. There are only subtle differences between the first five frequencies of the motor and those of the trailer, which correspond to the movement of the motor’s body and the trailer’s body. However, the frequencies corresponding to the movement of the motor’s bogies are evidently smaller than those of the trailer’s bogies. The reason is mainly that the moments of inertia of the motor’s bogies are larger than those of the trailer’s ones.

Table 4.

Frequencies and modes of the CRH3.

Mode	Motor		Trailer
Mode	Frequency (Hz)	Mode description	Frequency (Hz)	Mode description
1st mode	0.1540	Lateral movement of body	0.1609	Lateral movement of body
2nd mode	0.1840	Yawing movement of body	0.1840	Yawing movement of body
3rd mode	0.3835	Rolling movement of body	0.4112	Rolling movement of body
4th mode	0.6007	Vertical movement of body	0.6276	Vertical movement of body
5th mode	0.6949	Pitching movement of body	0.6952	Pitching movement of body
6th mode	2.1735	Pitching movement of bogies	4.3470	Pitching movement of bogies
7th mode	2.1735	Pitching movement of bogies	4.3470	Pitching movement of bogies
8th mode	4.6768	Rolling movement of bogies	5.4423	Vertical movement of bogies
9th mode	4.6790	Rolling movement of bogies	5.4437	Vertical movement of bogies
10th mode	4.7144	Vertical movement of bogies	5.6404	Rolling movement of bogies
11th mode	4.7166	Vertical movement of bogies	5.6425	Rolling movement of bogies
12th mode	5.5913	Yawing movement of bogies	9.8301	Yawing movement of bogies
13th mode	5.5913	Yawing movement of bogies	9.8301	Yawing movement of bogies
14th mode	10.9053	Lateral movement of bogies	12.5924	Lateral movement of bogies
15th mode	10.9053	Lateral movement of bogies	12.5924	Lateral movement of bogies

The body represents the vehicle body, and the bogies denote the vehicle bogies.

Simulation of the rail irregularity

The rail irregularity of the actual line is a random wave that is formed by the superposition of random irregularity with different wavelengths, different amplitudes, and different phases. The rail irregularity is generally assumed to be a Gaussian stationary random process, described by the power spectral density (PSD) function. The rail irregularity samples adopted in this study are generated from a German low-interference (GLI) spectrum (Schiehlen, 1982). The German high-speed railway interference spectrum is the spectrum function widely used in Europe and is also adopted in Chinese high-speed railway design, in which the low interference spectrum is a high-speed spectrum and is applicable to railways with a traveling velocity up to 300 km/h. The PSD functions of GLI are as follows:

{\begin{matrix} S_{v} (Ω) = A_{v} Ω_{c}^{2} / [(Ω^{2} + Ω_{r}^{2}) (Ω^{2} + Ω_{c}^{2})] \\ S_{a} (Ω) = A_{a} Ω_{c}^{2} / [(Ω^{2} + Ω_{r}^{2}) (Ω^{2} + Ω_{c}^{2})] \\ S_{c} (Ω) = A_{v} b^{- 2} Ω_{c}^{2} Ω^{2} / [(Ω^{2} + Ω_{r}^{2}) (Ω^{2} + Ω_{c}^{2}) (Ω^{2} + Ω_{s}^{2})] \end{matrix}

(5)

where S_v(Ω) (m²/(rad/m)), S_a(Ω) (m²/(rad/m)), and S_c(Ω) (1/(rad/m)) denote the PSDs of the vertical, alignment, and horizontal irregularities; Ω (rad/m) is the spatial frequency; Ω_c, Ω_r, and Ω_s denote the cut-up spatial frequencies; A_v and A_a (m²·rad/m) denote the irregularity coefficients; and b denotes half of the lateral distance between the rolling circle of the wheel. For the GLI spectrum, Ω_c, Ω_r, and Ω_s are 0.8246, 0.0206, and 0.4380, respectively; and A_v and A_a are 4.032e⁻⁷ and 2.119e⁻⁷, respectively. The produced sample rail irregularities are shown in Figure 2.

Figure 2.

Samples from the GLI spectrum: (a) vertical irregularity, (b) horizontal irregularity, and (c) alignment irregularity.

Equation of motion of the vehicle-bridge coupled system and numerical methods

For the discrete dynamic model such as the vehicle model, the Lagrange principle is a general method for establishing its governing differential equation of motion. By using this general method, the equation of motion for the vehicle model can be obtained as follows:

M_{v} {\overset{\cdot\cdot}{V}}_{v} + C_{v} {\overset{\cdot}{V}}_{v} + K_{v} V_{v} = F_{v}

(6)

where M_v, C_v, and K_v represent the mass matrix, the damping matrix, and the stiffness matrix, respectively; $V_{v}$ , ${\overset{\cdot}{V}}_{v}$ , and ${\overset{\cdot\cdot}{V}}_{v}$ denote the displacement, velocity, and acceleration vectors of the vehicle; and F_v denotes the load produced by the rail irregularities.

The bridge structure is discretized into a series of beam elements using finite element method and its nodal equation of motion can be expressed as:

M_{b} {\overset{\cdot\cdot}{V}}_{b} + C_{b} {\overset{\cdot}{V}}_{b} + K_{b} V_{b} = F_{bg} + F_{bv}

(7)

where M_b, C_b, and K_b denote the mass matrix, the damping matrix, and the stiffness matrix of the bridge; $V_{b}$ , ${\overset{\cdot}{V}}_{b}$ , and ${\overset{\cdot\cdot}{V}}_{b}$ denote the displacement, velocity, and acceleration vectors of the nodes of the bridge; F_bg denotes the dead load acting on the bridge (e.g. the gravity of the structure); and F_bv denotes the forces acting on the bridge produced by the moving vehicles.

The equation of motion of the vehicle-bridge coupled system could be obtained by combining the motion equations of the vehicle and the bridge as:

{\begin{matrix} M_{v} {\overset{\cdot\cdot}{V}}_{v} + C_{v} {\overset{\cdot}{V}}_{v} + K_{v} V_{v} = F_{v} \\ M_{b} {\overset{\cdot\cdot}{V}}_{b} + C_{b} {\overset{\cdot}{V}}_{b} + K_{b} V_{b} = F_{bg} + F_{bv} \end{matrix}

(8)

In the above equation set, the excitation of the vehicle sub-system consists of the rail irregularity and bridge vibration, while the dynamic loads of the bridge sub-system are also correlated with the motion of the vehicle. Thus, the equations of motion of the vehicle and the bridge are not independent but rather a coupled one. Moreover, the right side of equation (7) includes some items that are dependent on the bridge’s displacement, velocity, and acceleration at the vehicle-bridge contacting points. Due to the movement of the vehicle, these items at these contacting points cause the mass, stiffness, and damping matrices of the bridge sub-system in equation (7) to be time-varying; thus, the coupled motion equation set, equation (8), is a coupled equation set with time-varying coefficients.

Numerical analysis is typically used to solve a coupled system of differential equations with time-varying coefficients. For the vehicle-bridge coupled system, there are three categories of widely used approaches for conducting numerical analyses: the coupled degrees of freedom method (CDFM) (Deng et al., 2014, 2015; Frýba, 1999; Zhai and Xia, 2011) and the separated iteration method (SIM) (Ma et al., 2019; Zhu and Law, 2002). In the CDFM, the coupled equation set including the vehicle and bridge sub-systems is solved directly, and the coefficient matrices are updated at any time step. In the SIM, the vehicle and bridge sub-systems are independently solved, the coefficient matrices remain unchanged throughout the whole time series, and an iteration solution method is used to satisfy the deformation compatibility between the two sub-systems. Another method for solving the vehicle-bridge coupled system is the dynamic condensation method (Yang et al., 2001, 2004), in which Newmark’s finite difference formulations are used to discretize the equation of motion of a sprung mass with suspension units and the DOFs of the sprung mass are condensed into the bridge element in contact.

In this paper, the first two solution strategies are used to validate each other. In one scheme, the mode superposition method is applied to solve the bridge sub-system, and the CDFM is implemented to solve the coupled system. In the other scheme, the direct integration method is applied to solve the bridge motion, while the SIM method is used to solve the coupled system. In both methods, the Newmark-β method is used to solve the equations of motion of the vehicle, the bridge and the coupled system, which is unconditionally steady when it uses the constants of γ = 0.5 and β = 0.25. After the displacement and acceleration of vehicle-bridge system are obtained, the bending moment and shearing forces of the bridge structure could be further obtained by using the shape function of beam elements. According to the Newmark-β method and the above procedures, the vehicle-bridge coupled vibration analysis system is developed using Visual Fortran 6.5. Self-complied calculation programs are applicable to different train formations and bridge structural systems. The time step length Δt should be less than one-tenth of the period of the highest-order mode considered. Here, Δt = 0.005 s.

Taking the Wusong Jiang Bridge (WSJ-Bri) as an example, the FEM model is established in ANSYS 15.0. Then, all information regarding the finite element bridge models, including the node coordinates, the element formations, the material data, the cross-sectional data, and the boundary conditions, are recorded into a text file. The information in the text file is then read by the self-compiled programs using Fortran to implement the VBI analyses. The displacement, acceleration, and internal force responses under the action of a single moving vehicle at a speed of 170 km/h are calculated using the CDFM and SIM, respectively, as shown in Figure 3. The results from the SIM are highly consistent with those from the CDFM when the first 50 modes are included, and the relative errors between the two methods are less than 0.5%. In the CDFM, the stiffness and damping matrices of the coupled system need to be repeatedly reproduced, which greatly affects the efficiency of calculation. The SIM is shown to be more efficient and is therefore used to conduct the subsequent analyses.

Figure 3.

Dynamic responses of the VBI system of the WSJ-Bri: (a) vertical deflection (VD) at midspan of the third span, (b) vertical moment (VM) at midspan of the third span, (c) vertical acceleration (VA) at midspan of the third span, and (d) lateral displacement (LD) at midspan of the third span.

In addition, two damping definition methods are compared in this paper: one is Rayleigh damping, in which the damping coefficients are calculated by assuming the damping ratios of the first two flexural modes to be 2%; the other is to use a constant damping ratio, in which all damping ratios for all modes considered are kept constant (2%). Although different damping definitions could produce some difference in the damping coefficients for the high-order modes of the bridge, there is hardly any difference in the calculated dynamic responses of the bridge using the two damping definitions because the low-order modes dominate the dynamic response of the bridge.

Case study

Influence of vehicle speed

To investigate the influence of vehicle speed on the DAF, the WSJ-Bri is taken as an example, and a CRH3 with an 8-vehicle formation is used for the VBI analyses. The rail irregularity samples are generated by the GLI spectrum. The DAFs of the vertical deflection (VD) and vertical moment (VM) at the mid-spans and the VM and vertical shear force (VSF) at the interior supports of the WSJ-Bri at various speeds are shown in Figure 4. Here, the five irregularity samples are produced using Monte Carlo simulation, and the mean DAFs from different profile samples are obtained. Vehicle speeds ranging from 100 to 500 km/h at an interval of 20 km/h are used. In the peak area of the DAF, the interval of the vehicle speed is reduced.

Figure 4.

DAFs of the WSJ-Bri at various speeds (100–500 km/h).

The results reveal that as the vehicle speed increases from 100 to 500 km/h, the DAF increases on the whole. In the low-speed region, the DAF peaks at 170 km/h speed; in the high-speed region, the DAF peaks at 320 km/h speed. These peaks are caused by the periodic loading of vehicles. When a train passes through a simply supported beam bridge, the regular axle load will periodically load the bridge, which will cause resonance of the bridge. The resonance speed (Matsuura, 1976; Matsuura and Zhou, 1980) can be expressed as follows:

V_{res} = \frac{3.6 f_{bn} d_{v}}{i} (n, i = 1, 2, 3, \dots)

(9)

where V_res denotes the resonance speed (km/h); the multiplier i = 1, 2, 3, …; f_bn denotes the nth-order natural frequency of the bridge (Hz); and d_v denotes the length of a single vehicle (m). Equation (9) was derived from mode functions of a simply supported beam, that is, sinusoidal functions. It can be rewritten as 3.6d_v/V_res = i/f_bn = iT_bn, which means that when the loading period of the train axle load 3.6d_v/V_res is i times the nth-order natural vibration period of the bridge T_bn, the train applies a periodic load on the bridge and facilitates a continually superimposed response of the bridge, and then resonance occurs. In fact, the periodicity is only related to the speed and length of the vehicle and the frequency of the bridge, which is independent on whether the modes of the bridge are sinusoidal.

According to equation (9), the first resonance speed is calculated as 170.3 km/h. Figure 5 shows the time histories for the VD at the midspan of the third span caused by the first four vehicles passing over the bridge at a speed of 170 km/h, in which the corresponding static responses are removed from the total responses. At a speed of 170 km/h, the impact effects caused by the first four vehicles are very consistent, and the dynamic responses become superimposed, which forms a good resonance condition. Therefore, equation (9) is also applicable to continuous beam bridges and can accurately estimate the resonance speed of continuous beam bridges under periodic loading.

Figure 5.

Time histories of the VD caused by the first four vehicles (170 km/h).

Hamidi and Danshjoo (2010) also investigated the influence of vehicle speed taking a simple beam as the object of study. Only one peak occurs in the DAF curve for the simple beam with the vehicle speed owing to its high fundamental frequency. In the present study, there are two peaks in the DAF curve of a long continuous beam bridge under different operating speeds because of the relatively low frequencies.

Influence of train formation

In high-speed railway transportation, different numbers of trailers and motors are marshalled into a group in a certain order, which is usually called the train formation. To study the influence of the train formation on the DAF, the WSJ-Bri is taken as an example, and CRH3 is used for the VBI analysis. The rail irregularity samples were still generated based on the GLI spectrum. The cases of the different formations are shown in Table 5, and the maximum DAF of the VD at the mid-spans are shown in Figure 6.

Table 5.

The formations of motors and trailers.

Case number	Number of vehicles	Formation
1	4	1 × (1 trailer + 1 motor + 1 motor + 1 trailer)
2	8	2 × (1 trailer + 1 motor + 1 motor + 1 trailer)
3	16	4 × (1 trailer + 1 motor + 1 motor + 1 trailer)

Figure 6.

DAF of the VD of the WSJ-Bri under different formations.

The results reveal that the DAF varies with the vehicle speed with the same trend for the 4-vehicle, 8-vehicle, and 16-vehicle formations, with the peaks being at speeds of 170 and 320 km/h. Among these peak points, the minimum peak value occurs for the 4-vehicle formation, while the maximum peak value occurs for the 16-vehicle formation. This is because the longer the train formation is, the more sufficient the superposition of the dynamic responses of the bridge at the resonance speed and the more highlighted the resonance. However, the DAFs from the formation of 8 vehicles approach those from the formation of 16 vehicles, which indicates that the influence of the train formation is weakened when the number of vehicles increases to 8.

DAFs of 13 continuous beam bridges

To explore the variation law of the bridge DAF within a wider span range, 13 continuous concrete beam bridges are adopted here, and a CRH3 with an 8-vehicle formation and the GLI spectrum are used for dynamic analyses. Considering that the accuracy of the MLM and EADA depends on the choices of vehicle mode and bridge mode, the VBI analyses are carried out in this part. Five irregularity samples are used, and the mean values of the DAFs from the five different irregularity samples are obtained. By considering six vehicle speeds (160, 200, 240, 280, 320, and 360 km/h), the maximum DAFs of the 13 bridges versus their fundamental frequencies are shown in Figure 7.

Figure 7.

DAFs versus the fundamental frequencies of 13 continuous beam bridges: (a) 160 km/h, (b) 200 km/h, (c) 240 km/h, (d) 280 km/h, (e) 320 km/h, and (f) 360 km/h.

At different vehicle speeds, the peaks of the DAFs of the VD and VM at the mid-spans occur in bridges with different fundamental frequencies. At a speed of 160 km/h, the DAF reaches peaks in the range of 1.7–1.9 and 3.5 Hz. The former occurs in the Design-Bri3, Design-Bri2, and WSJ-Bri (i = 1), and the latter occurs in the JHGT-Bri2 (i = 2). These peaks are caused by the resonance condition (the periodic loading of the moving force). In the speed range of 200–320 km/h, the DAFs arrive at their peaks at different fundamental frequencies, and the peak frequencies increase with increasing vehicle speed. The second DAF peak does not occur at these speeds because the fundamental frequencies of the bridge satisfying the resonance condition are higher than 4.5 Hz at the speed range when the multiplier i is equal to 2, which is beyond the frequency range of the bridge examples discussed here. The peak DAFs at the resonance frequencies also increase with the speed, which indicates that the combination of the resonance and high speed could further amplify the bridge vibration when resonance occurs in high-frequency bridges at high train speeds.

The DAF of the VM and VSF at the interior supports presents neither an obvious resonance regularity nor a clear increasing or decreasing trend, which is different from those of the VD and VM at the mid-spans.

Discussion of the empirical formulas for the DAF in bridge design codes

In this part, the maximum DAF values at different train speeds are obtained in the speed range of 140–360 km/h from both the numerical simulations and codes as presented in Figure 8. The results show that the DAFs of the bending moments are less than those of the displacements at the mid-spans. When the fundamental frequency of a bridge is less than 3.0 Hz, the DAFs of both the displacements and internal forces are less than 1.15, although they fluctuate irregularly. When the fundamental frequency is higher than 3.0 Hz, the DAFs of the bridge increase with the increasing fundamental frequency, and the maximum DAF reaches 1.25. The DAFs of the VM and VSF at the interior supports are much less than those of the VD and VM at the mid-spans and are less than 1.1 in most cases.

Figure 8.

Comparison of the calculated DAFs to those from empirical formulas.

In calculating DAFs, the maximum span lengths are used in the formulas in the JRTRI code (2004), while the characteristic lengths are used in the formulas in the UIC code (2006) and the CNRA code (2014). The DAF values from the UIC code and the CNRA code are less than 1.05 for the bridge examples with basic flexural frequencies of less than 3.0 Hz. However, according to the results from the present VBI analyses, even when the traveling velocities of high-speed railways are 160 or 200 km/h, the maximum DAF values can still reach 1.1, as shown in Figure 7(a) and (b). Thus, in viewpoint of the results from the numerical simulations of this paper, it is deemed that these two codes may underestimate the dynamic effect of train loads on a bridge.

The empirical formula of the DAF provided by the JRTRI code (2004) includes more factors, such as vehicle speed, bridge fundamental frequency, and rail irregularity. The results in Figure 8 indicate that for bridges whose fundamental frequency is larger than 3.0 Hz, the JRTRI code provides a reasonable estimation of the DAFs. However, for bridges with fundamental frequencies smaller than 3.0 Hz, JRTRI code tends to overestimate their DAFs. Overall, based on the results of the studied 13 bridges, the JRTRI code is more reasonable than the other two HSR codes, although it tends to be conservative for bridges with low fundamental frequencies.

Conclusions

Based on the VBI analyses, the influencing factors of the DAFs of 13 continuous concrete beam bridges along high-speed railways are studied. The main conclusions are as follows:

Both the coupled degrees of freedom method (CDFM) and the separated iteration method (SIM) are applied to solve the train-bridge coupled vibration system in this paper. The results show that the calculated displacement, acceleration, bending moment responses of the bridge in vertical and lateral directions from the SIM are consistent with the corresponding responses from the CDFM.

For the bridge examples whose basic frequency is less than 3.0 Hz discussed in this paper, the maximum DAF is no more than 1.15, and for the bridge examples with a basic frequency larger than 3.0 Hz, the maximum DAF reaches 1.25. The reason is that for the high-frequency bridge, the resonance speed is enhanced, and the resonance and high train speed work together to produce a larger DAF.

The DAFs of the VM and VSF at the interior supports are less than 1.1 in most cases.

Regarding the 13 continuous beam bridge examples, the empirical formula of the DAF provided by the JRTRI code (2004) is generally more reasonable than that of the other HSR codes. For bridges whose fundamental frequency is larger than 3.0 Hz, the empirical formulas for the DAF in the JRTRI code provide a reasonable estimation of the DAFs; for bridges with fundamental frequencies smaller than 3.0 Hz, the empirical formulas in the JRTRI code tend to be conservative. In viewpoint of the results from the numerical simulations, the empirical formulas of the DAFs in the UIC code and CNRA code may underestimate the DAFs of high-speed railway continuous beam bridges to some degree. However, many factors, such as the chosen rail irregularity type, two-way running of rail vehicles, affect the calculation results, which should be discussed in further studies.

It should be noted that a relatively lower value of DAF does not mean an unsafe bridge design. In fact, the internal force calculation for structural design is based on the elastic theory in the CNRA code, and thus there are enough safety reserves. This manuscript investigates the DAFs of high speed railway continuous bridges purely by numerical simulations. As such, it is necessary to further investigate field experiment data before a more convincing conclusion could be made, which will be paid more attention to in further study.

Footnotes

Acknowledgements

The authors acknowledge the financial support from China Scholarship Council (No. 201906715009) and Fundamental Research Funds for the Central Universities (No. 2019B13014).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Data availability

All data, models, and codes that support the findings of this study are available from the corresponding author and the first author upon reasonable request.

ORCID iDs

Lin Ma

Steve C.S. Cai

Shaofan Li

References

China National Railway Administration (2014) Code for Design of High Speed Railway. TB 10621, China National Railway Administration: Beijing.

Deng

Cai

(2010) Development of dynamic impact factor for performance evaluation of existing multi-girder concrete bridges. Engineering Structures 32(1): 21–31.

Deng

Cai

(2011) Identification of dynamic vehicular axle loads: Demonstration by a field study. Journal of Vibration and Control 17(2): 183–195.

Deng

Zou

, et al. (2014) State-of-the-art review of dynamic impact factors of highway bridges. Journal of Bridge Engineering 20(5): 04014080.

Deng

Shao

(2015) Dynamic impact factors for shear and bending moment of simply supported and continuous concrete girder bridges. Journal of Bridge Engineering 20(11): 04015005.

Diana

Cheli

(1989) Dynamic interaction of railway systems with large bridges. Vehicle System Dynamics 18(1–3): 71–106.

Doménech

Museros

Martínez-Rodrigo

(2014) Influence of the vehicle model on the prediction of the maximum bending response of simply-supported bridges under high-speed railway traffic. Engineering Structures 72: 123–139.

European Rail Research Institute (1999) Rail bridges for speeds>200km/h. Technical Report, D 214/RP 9, European Rail Research Institute, Paris.

European Standard (2003) Eurocode 1: Actions on structures – Part 2: Traffic loads on bridges. EN 1991-2, Paris.

10.

Frýba

(1999) Vibration of Solids and Structures Under Moving Loads. London: Thomas Telford.

11.

Frýba

(2001) A rough assessment of railway bridges for high speed trains. Engineering Structures 23: 548–556.

12.

Frýba

(2004) Dynamic behaviour of bridges due to high speed trains. Bridges for High-Speed Railways 6(3–4): 137–158.

13.

Gunmo

(2015) Resonance in long-span railway bridges carrying TGV trains. Computers and Structures 152: 185–199.

14.

Hamidi

Danshjoo

(2010) Determination of impact factor for steel railway bridges considering simultaneous effects of vehicle speed and axle distance to span length ratio. Engineering Structures 32: 1369–1376.

15.

International Union of Railways (2006) Load to be considered in railway bridge design. UIC CODE 776-1 R, Paris, French: Union International Des Chemins De Fer, UIC.

16.

Japan Railway Technical Research Institute (2004) Design Standards for Railway Structures and Other Structures-Concrete Structures. Japan Railway Technical Research Institute: Tokyo.

17.

Dong

, et al. (2015) Impact coefficient analysis of long-span railway cable-stayed bridge based on coupled vehicle-bridge vibration. Shock and Vibration 2015: 641731.

18.

Zhang

Han

, et al. (2019) Determining the dynamic amplification factor of multi-span continuous box girder bridges in highways using vehicle-bridge interaction analyses. Engineering Structures 181: 47–59.

19.

Martínez-Rodrigo

Galvín

Doménech

, et al. (2018) Effect of soil properties on the dynamic response of simply-supported bridges under railway traffic through coupled boundary element-finite element analyses. Engineering Structures 170: 78–90.

20.

Matsuura

(1976) A study on the dynamic problems of high-speed railway bridges. Proceed Japan Civil Society 12: 35–47.

21.

Matsuura

Zhou

(1980) Study on dynamic performance of bridges on high speed railway. World Bridge 1980: 46–63.

22.

Schiehlen

(1982) Dynamics of High-Speed Vehicles Design and Evaluation of Trucks for High-Speed Wheel/Rail Application. New York: Spring-Verlag Wien.

23.

Shen

(2011) Dynamic response of continuous girder bridge due to high-speed trains. Master degree Thesis of Wuhan University of Technology. (in Chinese).

24.

Yang

Liao

Lin

(1995) Impact formulas for vehicles moving over simple and continuous beams. Journal of Structural Engineering 21: 1644–1650.

25.

Yang

(2001) A versatile element for analysing vehicle-bridge interaction response. Engineering Structures 23(5): 452–469.

26.

Yang

Yau

(2004) Vehicle-Bridge Interaction Dynamics – with Applications to High Speed Railways. Taiwan: World Science Publishing.

27.

Yau

Martínez-Rodrigo

Doménech

(2019) An equivalent additional damping approach to assess vehicle-bridge interaction for train-induced vibration of short-span railway bridges. Engineering Structures 188: 469–479.

28.

Yau

Yang

(2001) Impact response of bridges with elastic bearings to moving loads. Journal of Sound and Vibration 248(1): 9–30.

29.

Zhai

Xia

(2011) Train-Track-Bridge Dynamic Interaction: Theory and Engineering Application. Beijing: Science Publishing House.

30.

Zhu

Law

(2002) Dynamic load on continuous multi-lane bridge deck from moving vehicles. Journal of Sound and Vibration 251(4): 697–716.