Vehicle–bridge coupling dynamic response of sea-crossing railway bridge under correlated wind and wave conditions

Abstract

The sea-crossing railway bridge is exposed to a high risk of wind and wave, which threatens the safety of the bridge and railway. A wind–wave–vehicle–bridge dynamic analysis model for sea-crossing railway bridge under wind and wave loadings is developed by extending the previous wind–vehicle–bridge model. The developed wind–wave–vehicle–bridge model involves multipoint fluctuating wind field, irregular wave field, finite element model of the bridge, and mass–spring–damper model of the vehicle. The correlation between wind and wave is considered by an empirical curve derived based on field measurement. Static, buffeting, and self-excited wind forces on the bridge and vehicle are considered with coefficients obtained from wind tunnel tests. The wave forces on the bridge are calculated by Morison equation including stretching modification. The governing equations of the wind–wave–vehicle–bridge model are solved in time domain by Newmark-β method to compute the dynamic response of bridge and vehicle. The dynamic response of bridge and vehicle is compared and discussed in both wind–wave–vehicle–bridge and wind–vehicle–bridge model. The performance of bridge and vehicle are finally evaluated. Studies of dynamic response under correlated wind and wave are found to be imperative for assessment of structural and vehicle safety and driving comfort of sea-crossing railway bridge.

Keywords

Introduction

Development of the high-speed railway network is being pursued along the east coast of China because of proximity to major population and economic centers. Sea-crossing railway bridge is one of the common infrastructure types to extend the railway from mainland to the islands. The sea-crossing bridges located in the coastal areas, as shown in Figure 1, are exposed to a high risk of strong wind and huge wave (Kennedy et al., 2011; Padgett et al., 2008). Many structural failures and vehicle accidents due to extreme wind and wave have been reported in previous studies (Ataei et al., 2010; Baker and Reynolds, 1992; Kitada, 2006; Robertson et al., 2007). Unlike traditional dynamic analyses of bridge where the vehicles are not included, dynamic analyses of a railway bridge are required to treat structure and the train running on the bridge deck as a coupled vibration system. Many previous studies have shown that the dynamic responses of the system are significantly affected by the wind and wave loadings (Guo et al., 2016; Zhu and Zhang, 2017). The wind and wave loadings may have frequencies close to structural fundamental frequency, and the lateral force components and vibration may make troubles to the vehicles crawling on the deck. Moreover, since the wind and wave in the coastal area are site-specific and correlated events, the correlation between wind and wave should not be ignored in the assessment of structures and vehicles. To this end, a wind–wave dynamic analysis framework of the vehicle–bridge coupling system is presented, where the system is decomposed into four components (e.g. wind filed, wave filed, bridge, and vehicle). The correlation of wind and wave is derived by fitting the measured wind and wave data. Dynamic response of trains and bridges are then calculated and discussed through an example study of the sea-crossing railway bridge and China Railway High Speed train (CRH2).

Figure 1.

Schematic illustration of sea-crossing railway bridge under wind and wave.

Structural dynamic analyses under wind and wave loadings are originated from the research of offshore platforms (Zaheer and Islam, 2012, 2017) and wind turbines (Wei et al., 2014). Earlier efforts made on the dynamic response of the vehicle–bridge system under environmental loads are to study the response of bridge and vehicle under wind. Many studies have been carried out to resolve the vibration problems and ensure the vehicle–bridge coupling system’s performance (Cai and Chen, 2004; Li et al., 2005; Zhang et al., 2013). In order to understand well the effect of wind and wave loadings on the structure, Guo et al. (2016) studied the effect of coupled wind and wave actions on the dynamic responses of a bridge tower experimentally in a wind tunnel and wave flume laboratory. Zhu and Zhang (2017) included the effect of wave loading in the numerical analysis framework and studied the dynamic response of the bridge under wind and wave. However, few of current research studied the vehicle–bridge vibration under wind and wave conditions, especially for railway bridge. Therefore, this article presents a wind–wave–vehicle–bridge (WWVB) model by extending the authors’ previous wind–vehicle–bridge (WVB) model for railway bridges (Li et al., 2005).

Another question addressed in this study is how to consider the effect of correlation between wind and wave. Earlier studies always ignored their correlation or only considered ideal wind–wave relationships (Li et al., 2007; Remya et al., 2014). The ideal assumption is possible for wind and wave conditions in open-ocean environment. However, they cannot be used in the island area near the coast, where the sea-crossing bridges are always locating (Ti et al., 2017). Empirical wind–wave correlation equation provides a necessary tool to tackle the problem, especially when the simultaneously measured wind and wave data at the bridge site are available. The correlation between wind and wave conditions used in this study is derived based on the field measured data in the example site.

The reminder of the article is organized as follows: the WWVB analysis framework is introduced as a general approach; the coupling relationships and motion equation of the vehicle–bridge system is explained; specific methods for calculating wind and wave field on the bridge and rail vehicles are summarized; the formula of correlation curve of wind and wave conditions is obtained by fitting the measured data of wind speed and wave height; finally, dynamic response of bridge and vehicle of the example are calculated and discussed.

Model definition and analysis procedure

General configuration of WWVB model

WWVB model provides an efficient approach to calculate the dynamic response of sea-crossing railway bridge under wind and wave conditions. It was developed by extending the WVB model proposed by Li et al. (2005) to include the wave loads that act on the bridge. Considering a sea-crossing railway bridge subjected to wind and wave loads shown in Figure 1, there are five components in the proposed model: wind field, wave field, vehicle, and bridge. The following coupling effects among these five components are considered: (1) the coupling effect between bridge and vehicle that meets Newton’s third law and the displacement coordination condition at the contact points between the tires and the bridge deck; (2) the coupling effect between wind and bridge that accounts for the wind induced aerodynamic loading acting on the bridge girders and towers; (3) the coupling effect between wind and vehicle that accounts for the wind induced aerodynamic loading acting on the vehicles; (4) the coupling effect between wave and bridge that accounts for the wave-induced hydrodynamic loading acting on the submerged elements of the bridge (e.g. pile and cap); and (5) the correlated wind and wave conditions.

To simplify the model, structural self-weight, wave forces and wind forces are assumed to act at the centroids of the corresponding structural elements. The wind and wave are assumed to be unidirectional and load the structure from a deterministic direction that is perpendicular to longitudinal bridge axis. Waves are assumed to act on the piers without any phase difference. The spatial characteristics of wave field are neglected. The effect of structure on the wave kinematics is assuming to be tiny, and hence, the wave fields do not change with the deformation of structure and vehicle. The governing equations of motion of the WWVB model can be given in a matrix form by

M_{v} {\overset{\cdot\cdot}{u}}_{v} + C_{v} {\overset{\cdot}{u}}_{v} + K_{v} u_{v} = F_{bv} + F_{stv} + F_{buv}

(1)

M_{b} {\overset{\cdot\cdot}{u}}_{b} + C_{b} {\overset{\cdot}{u}}_{b} + K_{b} u_{b} = F_{vb} + F_{wb} + F_{stb} + F_{bub} + F_{seb}

(2)

in which the subscripts b and v indicate bridge and vehicle, respectively; $M, C, and K$ are mass matrix, damping matrix, and stiffness matrix, respectively; $\overset{\cdot\cdot}{u}, \overset{\cdot}{u}, and u$ are acceleration matrix, velocity matrix, and displacement matrix, respectively; $F_{stv} and F_{buv}$ are static wind force and buffeting force acting on the vehicle, respectively; $F_{stb}, F_{bub}, F_{seb}, and F_{wb}$ are static wind force, buffeting force, self-excited force, and wave force acting on the bridge, respectively; and $F_{bv} and F_{vb}$ are the interaction forces between bridge and vehicle.

The equations of bridge subsystem and vehicle subsystem are calculated independently because bridge or vehicle is constant in the process of vehicles running through the bridge. The solution of the equations of motion can be obtained at each step through equilibrium iterations based on the Newmark-β method.

Aerodynamic loading

According to the load patterns of WWVB model shown in Figure 1, the wind induced response of bridge or vehicle is mainly caused by static wind forces, self-excited forces, and buffeting forces. Wind field is required to be generated prior to calculation of aerodynamic loadings.

Mean wind profile

Log-law profile (Zaheer and Islam, 2017) is used to describe the vertical distribution of mean wind speed within the atmospheric boundary layer and can be expressed as

\frac{U (z)}{U (z_{10})} = \frac{\ln (z / z_{0})}{\ln (10 / z_{0})}

(3)

where $U (z_{10})$ is the reference mean wind speed at 10 m height; $U (z)$ is the mean wind speed at the height of z m above the design water level (DWL); and $z_{0}$ is the sea surface roughness which is set to be 0.005 m for a rough sea surface.

Fluctuating wind

Previous studies (Chen and Kareem, 2002; Simiu and Scanlan, 1996) presented several power spectral density functions, which can be used to simulate fluctuating wind. Simiu spectrum given by equations (4) and (5) is select as the target wind power spectra to simulate horizontal fluctuating wind along the bridge deck and cross the bridge deck

\frac{n S_{u} (f)}{u_{*}^{2}} = \frac{200 f}{{(1 + 50 f)}^{5 / 3}}

(4)

\frac{n S_{v} (f)}{u_{*}^{2}} = \frac{15 f}{{(1 + 9.5 f)}^{5 / 3}}

(5)

Lumley–Panofsky spectrum is taken as the target wind power spectra for vertical fluctuation along bridge tower. Power spectral density function of Lumley–Panofsky spectrum can be expressed as

\frac{n S_{w} (f)}{u_{*}^{2}} = \frac{3.36 f}{{(1 + 10 f)}^{5 / 3}}

(6)

In equations (4) to (6), $f = nz / U (z)$ is the dimensionless frequency, n is frequency, and $u_{*} = KU (z) / \ln (z / z_{0})$ is the friction velocity, where K is von Kármán constant equals 0.4.

Multipoint wind field

Sea-crossing bridge is always a long-span structure. Traditional single-point wind field technique is not yet suitable to the long-span bridge due to excessive simulation points and strong coherence at a distance less than 100 m (Li et al., 2004). Therefore, multipoint wind field has been used in the proposed framework. The typical wind field of a long-span bridge can be described by a spatio-temporal function in a Cartesian coordinate system as follows

U = U (z) + u (y, z, t)

(7)

v = v (y, z, t)

(8)

w = w (y, z, t)

(9)

in which y represents the transversal direction that is perpendicular to the bridge axis, z represents the vertical direction that is aligned with the tower axis; $u, v, and w$ denote the three-dimensional (3D) fluctuating wind velocities of along-wind, lateral wind, and vertical wind at time t. The approach developed by Chen et al. (2014) is adopted in the developed WWVB model to generate the stationary Gaussian process of wind.

Calculation of wind loading

Static wind force is calculated by three parts: drag force $F_{H}$ , lift force $F_{V}$ , and moment $F_{M}$ . The expressions for the components of the static wind force on the unit length of vehicle or bridge are given as follows

F_{H} = \frac{1}{2} ρ U_{d}^{2} C_{d} h

(10)

F_{V} = \frac{1}{2} ρ U_{d}^{2} C_{l} b

(11)

F_{M} = \frac{1}{2} ρ U_{d}^{2} C_{m} b^{2}

(12)

where $U_{d}$ is mean wind velocity cross the bridge deck; $ρ$ is air density; h and b are the height and the width of vehicle or bridge; $C_{d}, C_{l}, and C_{m}$ are the drag coefficient, the lift coefficient, and the moment coefficient for vehicle or bridge, respectively.

Buffeting forces induced by stochastic wind and self-excited force induced by fluid–solid coupled motion can also be divided into three components—drag, lift, and moment—for both bridge and vehicle. Because the low natural vibration frequency of long-span bridge easily leads to wind vibration, the self-excited forces on the deck are considered. Details of the formulations for buffeting and self-excited forces can be found in the authors’ previous work (Li et al., 2005) and are not given hereafter for the sake of conciseness.

Hydrodynamic loading

Irregular wave kinematics

The linear irregular wave kinematic (velocity and acceleration) is used to generate the wave loading. In order to consider the influence of free surface, Chakrabarti (1971) modified this theory based on equivalent water depth method. The formula of horizontal water quality velocity and acceleration are as follows

{\overset{\cdot}{u}}_{hj} (z, t) = \frac{H}{2} w \frac{\cosh k_{i} z}{\sinh (k_{i} d + φ)} \cos (k_{i} x_{j} - wt)

(13)

{\overset{\cdot\cdot}{u}}_{hj} (z, t) = \frac{H}{2} w^{2} \frac{\cosh k_{i} z}{\sinh (k_{i} d + φ)} \sin (k_{i} x_{j} - wt)

(14)

where $x_{j}$ is the horizontal distance between the initial position and the jth pile foundation in the direction of wave travel; z is the vertical distance between the calculation point and the seabed; H is wave height; $w = 2 π / T$ is the wave frequency, T is wave period; d is the DWL; $k_{i}$ is the ith component of wave number satisfying the linear dispersion relationship such that $w^{2} = gk \tanh (kd)$ ; $φ (t)$ is the elevation of the water surface at time t, which can be calculated according to equation (15) by superposition of multiple cosine waves with individual period and random initial phase

φ (t) = \sum_{1}^{M} a_{i} \cos (k_{i} x - w_{i} t + ε_{i})

(15)

a_{i} = \sqrt{2 S_{φ} (w_{i}) Δ w}

(16)

where $a_{i}, w_{i}, and ε_{i}$ is the ith component of wave amplitude, wave frequency, and random initial phase; $Δ w = (w_{\max} - w_{\min}) / M$ is frequency resolution, M is number of frequency, $w_{\max} and w_{\min}$ are the lower and upper cutoff frequencies, respectively; $S_{φ} (w_{i})$ is wave spectral density of one side at wave frequency $w_{i}$ .

As the representative value of the ith frequency interval, $w_{i}$ is selected randomly within subinterval, this is to avoid inconsistent with actual wave conditions such that waves appear repeatedly in accordance with the cycle of $2 π / Δ w$ . JONSWAP spectrum (Goda, 1999) is adopted to simulate time history of water particle velocity and acceleration, the function is expressed as

S (w) = \frac{1}{2 π} β_{j} H_{s}^{2} T_{p}^{- 4} {(\frac{w}{2 π})}^{- 5} \exp [- \frac{5}{4} {(T_{p} \frac{w}{2 π})}^{- 4}] \cdot γ^{\exp [- {(\frac{w}{w_{p}} - 1)}^{2} / 2 σ^{2}]}

(17)

β_{j} = \frac{0.06238}{0.230 + 0.0336 γ - 0.185 {(1.9 + γ)}^{- 1}} \cdot [1.094 - 0.01915 \ln γ]

(18)

T_{p} = \frac{T_{s}}{1 - 0.132 {(γ + 0.2)}^{- 0.559}}

(19)

where $H_{s} and T_{s}$ are significant wave height and the corresponding period, respectively; $T_{p} and w_{p}$ are peak circular period and peak circular frequency, respectively; $σ = 0.07$ when $w \leq w_{p}$ , or $σ = 0.09$ when $w > w_{p}$ ; g is acceleration of gravity; $γ$ is peak enhancement factor taken as 3.3. The wave superposition method is adopted in this work which has been widely used for wave numerical simulations.

Calculation of wave loading

Morison equation is adopted to calculate the wave forces on the piles and piers, of which the structure diameter D is smaller than 0.2 times wave length L. The wave loading on the pile is given as follows

F_{p i l e} (z, t) = \frac{1}{2} ρ_{w} C_{D} D_{i j} {\overset{\cdot}{u}}_{h i j} (z, t) | {\overset{\cdot}{u}}_{h i j} (z, t) | + ρ_{w} C_{M} A_{i j} {\overset{\cdot\cdot}{u}}_{h i j} (z, t)

(20)

where $ρ_{w}$ is water density; $A_{ij}$ is cross-sectional area of the jth pier foundation for the ith pile; $D_{ij}$ is pile diameter of the jth pier foundation for the ith pile; ${\overset{\cdot}{u}}_{hij} and {\overset{\cdot\cdot}{u}}_{hij}$ are water particle velocity and acceleration normal to the displaced jth pier foundation at the ith pile; $C_{D} and C_{M}$ are drag coefficient and inertia coefficient for pile foundation, and they are set to be 1.2 and 2.0 according to Chinese code of hydrology for harbor and waterway (JTS 145-2015, 2015).

Because the pile cap is like a transition piece between pile group and bridge tower, the possibility of waves that reach and interact with the cap of the bridge substructure (Figure 1) must be considered. The wave forces acted on the pile cap are calculated according to the calculation method of wave-in-deck force, which is commonly used in the analyses of offshore wind turbine support structures (Wei et al., 2014). Considering the weight of bridge superstructure is super heavy, the effect of vertical component of wave-in-deck force on the cap is hence neglected. Only horizontal component of the force is considered for simplicity. The equation of horizontal component of the wave-in-deck loads is given as follows

F_{cap} (z_{0}, t) = \frac{1}{2} C_{D} ρ_{w} B_{j} (z_{0}, t) {\overset{\cdot}{u}}_{j} {(z_{0}, t)}^{2}

(21)

where $C_{D}$ is the drag coefficient of the cap and is taken as 2.0 according to Chinese code (JTS 145-2015), $z_{0}$ is the height above the bottom of the cap, $B_{j} (z_{0}, t)$ is the cap width of the submerged area at $z_{0}$ in time t for the jth pier foundation, and ${\overset{\cdot}{u}}_{j} (z_{0}, t)$ is the horizontal particle velocity at $z_{0}$ in t time for the jth pier foundation.

Combined method of wind and wave

The geographical environment of bridge site is always complex, the seabed in this region is rugged with valleys along the bridge, which caused that the assumption of open-ocean and parallel straight isobaths is not suitable in practice and difficult to meet the needs of bridge engineering. An empirical correlation based on the measured wind and wave data is therefore required (Wei et al., 2017). The measurement should include the following data: hourly measured significant wave height $H_{s}$ , the associated 10-min mean wind speed $U (z_{10})$ measured simultaneously at 10 m above the DWL, and the mean zero up-crossing periods $T_{s}$ .

For correlation between wind and wave, the following formula form of significant wave height and mean wind speed is suggested

H_{s} = aU {(z_{10})}^{b} + c

(22)

in which the coefficients a, b, and c can be fitted through the scatters between wave height and mean wind speed. The calculations of corresponding mean zero up-crossing periods are proposed by Valamanesh et al. (2015), and the range of peak spectral period is defined as

11.7 \sqrt{H_{s} / g} \leq T_{s} \leq 17.2 \sqrt{H_{s} / g}

(23)

WWVB analysis procedure

Once the structure and the environmental loadings have been defined, the procedure illustrated in Figure 2 is performed to complete the dynamic analyses of WWVB model:

Select several sea states and calculate wind and wave metocean conditions corresponding to each sea state, for example, significant wave height, mean wind speed, and mean zero up-crossing wave periods. The correlation between wind and wave condition are included.

Build the finite element model of bridge and vehicle by Kalker’s creep theory and Hertz’s contact theory.

Generate fluctuating wind field and irregular wave field and calculate the aerodynamic and hydrodynamic loadings.

For a given sea state, perform dynamic analysis of the sea-crossing railway bridge under in time domain to calculate the dynamic response of the sea-crossing railway bridge and the vehicle.

Repeat Step 4, changing sea states so that the responses of structure and vehicle corresponding to all selected sea states are obtained.

Assess the response of bridge and vehicle to ensure the vehicle–bridge system’s performance.

Figure 2.

Flowchart of WWVB analysis procedure.

Example study

A numerical case study of a sea-crossing railway bridge example is given in this section following the procedure described in section “Model definition and analysis procedure.” The WWVB model of the example is built, and the dynamic responses of bridge and vehicle are analyzed.

Site description and example structure

The sea-crossing railway bridge used in the study is main bridge of the Pingtan Strait Bridge Project, which has a full length of 16.32 km and connects Changle City with Pingtan Island in China East Sea. The example bridge crosses over four islands, which include Changyu Island, Renyu Island, Xiaolian Island, and Dalian Island (Figure 3(c)). The terrain of the site is very complex and the water depth of varies greatly due to submarine relief and numerous reefs. There is a high risk of typhoon in the area, which threatens the structural and driving safety.

Figure 3.

Layout of the example bridge and site: (a) layout of the bridge (unit: m), (b) cross section of main girder (unit: cm), and (c) site of the example bridge and layout of the bridge pile foundations.

As shown in Figure 3(a), the example sea-crossing railway bridge is a 1190-m long double-tower cable stayed bridge with five spans (133 m + 196 m + 532 m + 196 m + 133 m). The main towers of the bridge (N01 and N03 illustrated in Figure 3) are 222 m high and made from variable cross section H-shaped reinforced concrete. Steel truss composite girder with low damping ratio is used and cross section of the main girder is shown in Figure 3(b).

The substructures of sea-crossing railway bridge consist of two tower pier foundations for N03, N04, and four auxiliary pier foundations for N01, N02, N05, and N06. Each pier foundation is composed of three parts: (a) pier body made from reinforced concrete, (b) cap with circular end, and (c) group pile included several drilling piles and the layout is shown in Figure 3(c).

Correlated wind and wave conditions

Measurement setup

As shown in Figure 4, field monitors were set up on a platform at the bridge site to measure the marine meteorological conditions during typhoon period in 2015. SBY2-1 ultrasonic wave gauge was used to measure wave height and wave period, and a two-dimensional ultrasonic anemometer was used to monitor the wind speed simultaneously. The monitoring equipment of wave and wind fixed in a stable environment is 13 and 20.25 m away from the DWL, respectively. The sampling frequency of SBY2-1 ultrasonic wave gauge and ultrasonic anemometer are 2 and 1 Hz, respectively.

Figure 4.

Wind and wave monitoring equipment: (a) wave and (b) wind.

Correlation of wind and wave

The scatter between mean wind speed $U (z_{10})$ and significant wave height $H_{s}$ are shown in Figure 5 during typhoon period. Fitting curve is estimated by equation (22) using the least square method and can be expressed as follows

H_{s} = 0.0457 \times U {(z_{10})}^{1.5145} + 0.2974

(24)

Figure 5.

Correlation of mean wind speed and significant wave height.

It can be seen that the significant wave height and the mean wind speed have a strong correlation with R² at 0.805. The wind–wave correlation curve suggested in Chinese design code (JTS 145-2015) and previous studies (e.g. American SMB method (USACE, 1984)), Japanese design code (Ministry of Land, Infrastructure, Transport and Tourism (MLIT), 2009) are also plotted in Figure 5. It is obvious that the field measured results are far larger than the values predicted by the other methods. Therefore, it is better to determine the correlation of wind and waves by fitting method against by other traditional methods. Although this fitting curve ignores the randomness between wind and waves, it is at least an effective way to estimate the combination of wind and wave conditions in the deterministic analyses.

The empirical joint probability density distribution of mean zero up-crossing periods $T_{s}$ and the associated significant wave height $H_{s}$ are shown in Figure 6. The lower bound and upper bound of the wave period range suggested by Valamanesh et al. (2015) are given as follows

T_{s} = 11.7 \sqrt{H_{s} / g}

(25)

Figure 6.

Joint distributions of H_s and T_s based on field measurement.

It can be seen from Figure 6 that the variability of the measurements above the upper bound is much larger than the variability of measurements below the lower bound. It is a common sense that the same wave with shorter peak spectral period induces more significant impact on the structure than that with longer period. In this work the lower bound is hence adopted to describe the relationship between mean zero up-crossing periods and significant wave height.

Selected sea states

Three sea states, sea states I, II, and III, are considered based on three different mean wind speeds $U_{d}$ of 15, 20, and 30 m/s at the height of bridge deck. The wave height associated with the given mean wind speed is obtained according to equation (24), and the corresponding wave period is calculated according the lower bound equation (25). The characteristics of sea states are defined in Table 1.

Table 1.

Characteristics of sea states.

Sea state designation	$U_{d} (m / s)$	$H_{s} (m)$	$T_{s} (s)$	$U (z_{10}), (m / s)$
Sea state I	15	2.27	5.63	12.00
Sea state II	20	3.34	6.83	16.00
Sea state III	30	5.92	9.10	24.00

Model of vehicle and bridge

Vehicle model

The CRH2 high-speed train model is used in the example study. The train is consists of eight vehicles, each vehicle includes one train body, two bogies, and four wheels, as shown in Figure 7. The train model is regarded as a mass–spring–damper system in all 23 degrees of freedom (DOFs), which are composed of five DOFs on each train body and bogie and two DOFs for each wheel. American level 6 spectrums are used to simulate the rail irregularities with the 0.2-m spatial step applying in dynamic time history analysis. The speed of the train in the whole process of passing the bridge is equal to the design speed of 200 km/h.

Figure 7.

Mass–spring–damper model of a vehicle: (a) side view, (b) bottom view, and (c) elevation view.

Bridge model

The finite element model of full bridge is established (Figure 8) with 3D beam element for deck, pier foundations and bridge towers, and 3D bar element for cables. The Rayleigh damping ratio of the bridge is set to be 0.5% according to Chinese design code (TB 10621-2014, 2014).

Figure 8.

The full-bridge finite element model.

Aerodynamic parameters of bridge and vehicle

The aerodynamic parameters of section model for this bridge were obtained by wind tunnel experiment in the XNJD-1 wind tunnel (Figure 9). The wind characteristics obtained by wind tunnel tests are presented in Table 2.

Figure 9.

Aerodynamic experiment of section model.

Table 2.

Characteristics of wind.

Bridge			Vehicle			Air density
$C_{d}$	$C_{l}$	$C_{m}$	$C_{d}$	$C_{l}$	$C_{m}$	Air density
1.026	0.167	0.107	1.478	0.231	0.239	1.27 kg/m³

Results and discussion

This section investigates the dynamic response of both vehicle and bridge during the train passes through the bridge under correlated wind and wave by time history analyses. Structural dynamic response discussed in this study includes the lateral displacement, vertical displacement, torsion and lateral, vertical acceleration response at the middle point of span (hereinafter referred to as mid-span point), as illustrated in Figure 3. And, results of vehicle consist of lateral and vertical acceleration response of the windward vehicle (i.e. the first vehicle of the train) and the index of safety and comfort of the train. The performance of bridge structure and vehicle under different sea states is carefully discussed.

In order to evaluate the performance of vehicles and bridges in design situation, a list of evaluation indicators are required to be defined at first. The performance of railway bridge structure depends on the deflection and vertical acceleration of the deck. The performance of vehicle consists of driving safety and driving comfort. The former one is commonly evaluated by the following factors: maximum wheel lateral force, wheel unloading rate, and derailment coefficient. And, the latter one is evaluated by lateral and vertical maximum accelerations of vehicle. The limit values of all above index are given in Table 3 according to Chinese design code (TB 10621-2014, 2014).

Table 3.

Evaluation criteria for vehicles and bridges.

Evaluation index of bridge and vehicle			Standard limit value
Bridge	Ratio of deflection to span		1/4000
Bridge	Vertical acceleration $a_{v} (m / s^{2})$		3.5
Vehicle	Safety	Derailment coefficient, Q/P	0.8
		Wheel unloading rate, ΔP/P	0.6
		Wheel lateral force, T (kN)	80
	Comfort	Lateral acceleration, $a_{h max} (m / s^{2})$	1.0
	Comfort	Vertical acceleration, $a_{v max} (m / s^{2})$	1.3

Evaluation of bridge performance

The effect of wave on structural and vehicle response is first evaluated by numerical analyses of the example using both WVB and WWVB models. The results of WVB model are calculated under the wind conditions associated with three sea states given in Table 1 only, while the results of WWVB model are obtained considering both wind and wave conditions. Figure 10 presents the time history of lateral, vertical, and torsional displacement at the mid-span point when the train passes through the bridge under different sea states. It is clear that the structural responses calculated by both models increase with the change of wind and wave conditions from sea state I to sea state III.

Figure 10.

Structural response of the bridge at the mid-span point: (a) time history of lateral displacement of deck $y_{h}$ at the mid-span point, (b) time history of vertical displacement of deck $y_{v}$ at the mid-span point, and (c) time history of torsion of deck $y_{t}$ at the mid-span point.

It can be seen from Figure 10(a) that the lateral displacement $y_{h}$ varies periodically with the movement of vehicle on the bridge. The maximum lateral displacement $y_{h}$ of WVB under sea state III is 49.91 mm, while the maximum lateral displacement of WWVB model reaches to 143.22 mm, approximately three times of the WVB results. The maximum lateral displacement of WWVB model has exceeded the limit value of bridge deflection 133 mm, which equals the maximum ratio of deflection to span 1/4000 (Table 3) times the length of span 532 m (Figure 3(a)). It indicates that the deflection exceeds the limit of bridge, which is not acceptable for the operation of railway. In addition, the peak of lateral displacement y_h occurs when the train arrives at special positions, such as left tower, right tower, and the mid-point of main span or side span. All these positions of the main girder are worthy of attention in the design and evaluation of railway bridge performance.

The maximum torsion $y_{t}$ of WWVB model is about three times that of WVB model under sea state III, as shown in Figure 10(c). The torsional response exhibits stronger fluctuation induced by wind and wave loads in compared to that induced by wind loads only. It is worth noting that the vertical displacement of the deck changes slightly under three sea states regardless of whether the effect of wave is considered, as presented in Figure 10(b). This is because the vertical deformation of the deck is mainly caused by the component of lift force of the fluctuating wind. WVB and WWVB model use the same fluctuating wind field and hence arrive in a good agreement in structural vertical response.

The lateral and vertical acceleration responses of bridge deck at the mid-span point are given in Figure 11, when the vehicle moves in a distance from −200 to 200 m to the mid-span point. The lateral acceleration increases significantly with the change of sea state from I to III, while the vertical acceleration changes slightly regardless of the sea states. It should be noted that the vertical acceleration results are much smaller than the limit value of 3.5 m/s².

Figure 11.

Response of acceleration at mid-span point: (a) lateral acceleration response and (b) vertical acceleration response.

In order to better understand the mechanism of bridge lateral response under correlated wind and wave, the power spectral density function of lateral displacement response at the mid-span point under three sea states is shown in Figure 12.

Figure 12.

Power spectral density (PSD) for lateral displacement at mid-span point.

The response energy is mainly concentrated around two peaks, according to the frequency-domain analyses. The first peak occurs at lateral bending fundamental frequency 0.218 Hz of the bridge and is mainly caused by the first-order lateral natural vibration of the bridge. The frequency of the second peak shows significant difference under three sea states and equals 0.14, 0.11, and 0.09 Hz under sea states I, II, III, respectively. It is worth noting that the frequency of the second peak is close to frequency associated with the significant wave period given in Table 1. This phenomenon has also been reported by Zaheer and Islam (2017). It explains the reason why the WWVB lateral responses of bridge are larger than that of WVB model and hence the nature of the lateral responses is predominantly governed by irregular wave instead of fluctuating wind.

Moreover, the magnitude difference between the peaks increases with the increase in wave and wind conditions from I to III. It is evident from the fact that the lateral dynamic response is controlled by both bridge fundamental mode and wave spectrum under sea state I, while it is mainly governed by the lateral bending fundamental frequency for higher sea states.

Evaluation of vehicle performance

Figure 13 shows the WVB and WWVB time histories of vertical and lateral acceleration of the windward vehicle (i.e. the first vehicle of the train) under different sea states. And, the maximum values of the vehicle dynamic response are given in Table 4.

Figure 13.

Response of acceleration of the first vehicle of CRH2 train: (a) time history of the lateral acceleration $a_{h}$ and (b) time history of the vertical acceleration $a_{v}$ .

Table 4.

Maximum responses of vehicles.

Sea state	$a_{h max} (m / s^{2})$	$a_{v max} (m / s^{2})$	T (kN)	ΔP/P	Q/P
I	0.69	0.63	22	0.17	0.23
II	0.88	0.76	24.18	0.19	0.28
III	1.22	0.82	32.17	0.25	0.46

Note that T is wheel lateral force; ΔP is the difference between the actual vertical force and the static wheel weight; Q and P are the lateral force and the vertical force at the wheel–rail contact point.

It is clear from Figure 13 that the maximum lateral acceleration of the first vehicle of the train in the WWVB model is larger than that of WVB model regardless of sea states. The lateral and the vertical acceleration of the first vehicle in the WWVB model increases with the increase in the wind speed and wave height from sea state I to III, according to the data given in Table 4. The maximum lateral acceleration under sea states I, II, and III occurs at the time when the first vehicle moves to the mid-span point. The difference between the vertical acceleration response of the first vehicle of WVB and WWVB is very slight. In other words, the effect of wave has little influence on the vertical acceleration of the vehicle as well.

For driving safety, the wheel lateral force T, wheel unloading rate ΔP/P and derailment coefficient Q/P are all increasing with the change of sea state from I to III. Their maximum values are still smaller than the standard limit value given in Table 3. It means the high-speed railway is safe during all the given sea states. For driving comfort, the maximum lateral acceleration under sea state III equals 1.22, which exceeds the limit of lateral acceleration $a_{h max} = 1.0 m / s^{2}$ and may bring serious discomfort to passengers.

Conclusion

This article developed the WWVB model and carried out dynamic analyses of a sea-crossing railway bridge under correlated wind and wave conditions. A sea-crossing railway cable stayed bridge is taken as the example and the wind and wave measurements at the site are used to build the correlation between wind and wave. Dynamic response of the bridge and train response are calculated, and the performance of bridge and vehicle is carefully evaluated. The fluctuating wind, irregular wave, finite element model of full bridge, and mass–spring–damper model of the vehicle are considered. Correlated wind and wave conditions are obtained by taking account of correlation between significant wave height and mean wind speed and between mean zero up-crossing periods and significant wave height based on the field measurements.

According the comparison between WVB model and WWVB model, the lateral response of bridge and vehicle is dominated by waves. The PSD of structural lateral response is controlled by both bridge fundamental mode and wave spectrum for sea state I, while it is mainly governed by the lateral bending fundamental frequency for higher sea states. Dynamic responses of bridge and vehicle increase with the increase in the wind speed and wave height. Higher wind speed and larger wave height may lead to over-limited bridge’s lateral displacement at mid-span point and vehicle’s lateral acceleration, which threatens structural safety and driving comfort.

Although the developed WWVB model allows multiple coupling relationships among wind, wave, vehicle, and bridge, and provides an effective method to evaluate the dynamic response of vehicle and bridge, many assumptions adopted in the model are still need to be verified, when the field measurement is available. Some issues should be addressed and discussed in the future studies, such as the impact of train speed, direction of wind and wave, and the joint probability between wind and wave.

Footnotes

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported financially by the National Natural Science Foundation of China (Grant Nos 51525804 and 51708455) and the Fundamental Research Fund for Central Universities (A0920502051707-2-027).

ORCID iD

Yongle Li

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