Analysis on running safety of train on the bridge considering sudden change of wind load caused by wind barriers

Abstract

The calculation formulae for change of wind load acting on the car-body are derived when a train moves into or out of the wind barrier structure, the dynamic analysis model of wind-vehicle-bridge system with wind barrier is established, and the influence of sudden change of wind load on the running safety of the train is analyzed. For a 10-span simply-supported U-shaped girder bridge with 100 m long double-side 3.5 m barrier, the response and the running safety indices of the train are calculated. The results are compared with those of the case with wind barrier on the whole bridge. It is shown that the sudden change of wind load caused by wind barrier has significant influence on the lateral acceleration of the car-body, but no distinct on the vertical acceleration. The running safety indices of train vehicle with sectional wind barriers are worse than those with full wind barriers, and the difference increases rapidly with wind velocity.

Keywords

wind barrier / sudden change of wind load / dynamic response / running safety / comfort

Introduction

The strong wind gives rise to a serious threat to the safety of rail transport, and brings significant economic losses and casualties [¹,²]. Experience of Japan’s Shinkansen has shown that setting wind barrier is an effective measure to reduce the impact of wind on the train and stopping operation times, as the running safety of the train is greatly improved [³,⁴]. Therefore, the wind barrier is very important to ensure the operation safety and ride comfort of the train on the bridge or subgrade. However, the sudden change of the wind load on the train after installing the wind barrier on the bridge when the train driving into and out of the wind barrier will affect the running safety and ride comfort of the train.

There have been studies on the vibration response of the vehicle-bridge coupling system [⁵,⁶], the aerodynamic properties and running safety of trains in wind field [⁷,⁸], the wind-induced vibration of the long span bridges [⁹,¹⁰], and identification of the aerodynamic parameters of the bridge according to the wind tunnel test or the field measurement [¹¹,¹²], and the vehicle-bridge dynamic analysis under wind load including the automobile and the train passing on the bridge [⁷,¹³,¹⁴]. Moreover, with the large-scale construction of the railway in the windy zones currently the study on the windbreak measures of the railway, such as windbreak barrier on the bridge and windbreak wall on the roadbed, has become a hot spot gradually. For the wind barrier design on the bridge, some results are obtained by wind tunnel tests regarding the possibilities of protecting the vehicles on highway viaducts and the effects of wind barrier porosity on a flow field [¹⁵]. In addition, Kwon and Kim et al. [¹⁶,¹⁷] present the design reference criteria for wind barriers to protect vehicles running on an expressway under a high side wind and the decision process for installation of wind barrier. And Zhang et al. [¹⁸] has progressed some research work for the wind barrier design of the high-speed railway bridge and the running safety of the train in strong wind field. But the research on the running safety of the train into and out of the wind barriers, which will lead to the sudden change on the train in wind field, is given little attention. Some researchers have carried out the analysis and testing study for the impact of the wind barrier around the pylon on the automotive operation [¹⁹], and the impact of the bridge tower shielding effect on the safe operation of the train [¹⁴].

As above mentioned, wind load on the train will be greatly reduced with wind barriers to improve operating safety, but when the train runs into or out of the wind barrier on the bridge, the wind load will suddenly change due to different aerodynamic properties of the front or rear vehicle. This change will induce additional vibration of the vehicle, which may lead to safety problems of the train. However, there are few research achievements on this issue up to now.

In this paper, the running safety of the train considering the sudden change of the wind load acting on the train is discussed when the train passes into or out of the wind barrier. The calculation formulas of the wind load on the vehicle are derived, the dynamic analysis model of wind-vehicle-bridge system with wind barrier is established, and then the dynamic response and safety indices of the train are calculated for a 10-span simply-supported bridge with U-shaped girders, with sectional wind barriers and full wind barriers on the bridge at different wind speed. Finally, appropriate conclusions from the work are presented.

Sudden change of wind load on the train

When a train runs on the bridge without wind barrier, the wind loads include the side force F_S (or F_Y), the lift force F_L (or F_Z) and the rolling moment M_q (or M_X). When the wind barrier is installed on the bridge, the wind load F_S, F_L and M_X will change as the train moves into or out of the barrier, generating the additional yawing moment M_Z and pitching moment M_Y, as shown in Fig. 1.

A Cartesian coordinate system OXYZ is established with the vehicle mass center as the origin, the wind forces of the vehicle body can be shown visually. The static wind load of the i-th vehicle under crosswind is represented as

(1)F_{\mathrm{V}i}^{\mathrm{no}}={\left[\begin{array}{lllll}{F}_{\mathrm{Y}}\hfill & F_{\mathrm{Z}}\hfill & M_{\mathrm{X}}\hfill & \mathrm{0}\hfill & \mathrm{0}\hfill \end{array}\right]}^{\mathrm{T}},

(2)F_{\mathrm{V}i}^{\mathrm{ba}}={\left[\begin{array}{lllll}{F}_{\mathrm{Y}}\hfill & F_{\mathrm{Z}}\hfill & M_{\mathrm{X}}\hfill & M_{\mathrm{Y}}\hfill & M_{\mathrm{Z}}\hfill \end{array}\right]}^{\mathrm{T}},

where the superscript “no” and “ba” denote the wind forces without and with wind barriers, respectively.

As the length of wind barrier l is generally greater than a vehicle length L, namely l >L, when the i-th vehicle passes by the wind barrier at a speed V, there are five stages for the process of the train moving into, in and out of the wind barrier. Let t=t_{\mathrm{0}}indicate the vehicle head arrives at the beginning of the wind barrier. The process of the train into and out of the wind barrier is shown in Fig. 2.

Let the car-body height H, length L, the windward area of the car-body is A=H\times L. The aerodynamic coefficients of the car-body are C_{\mathrm{S}}^{\mathrm{I}} for the side force, C_{\mathrm{L}}^I for the lift force and C_{\mathrm{M}}^{\mathrm{I}} for the rolling moment without wind barrier, and C_{\mathrm{S}}^{\mathrm{II}}, C_L^{\mathrm{II}}andC_{\mathrm{M}}^{\mathrm{II}} are the corresponding aerodynamic coefficients with wind barrier, respectively; \bar{u}represents the mean wind velocity;\rhois the density of air. Therefore, the wind forces on the car-body in the five stages can be expressed as follow.

Stage 1: t\le t_{\mathrm{0}}. When the vehicle moves outside the wind barrier, the wind force acting on the car-body include the side force F_S, lift force F_L and rolling moment M_q that do not change with time t, so the wind forces can be calculated by the three-component coefficients of the vehicle without wind barriers.

(3)F_{\mathrm{V}i}^I={\left[\begin{array}{lllll}{F}_{\mathrm{Y}}^I\hfill & F_{\mathrm{Z}}^I\hfill & M_{\mathrm{X}}^I\hfill & \mathrm{0}\hfill & \mathrm{0}\hfill \end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{lllll}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}A{C}_S^I\hfill & {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}A{C}_L^I\hfill & {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}AH{C}_M^I\hfill & \mathrm{0}\hfill & \mathrm{0}\hfill \end{array}\right]}^{\mathrm{T}}.

Stage 2: . During the vehicle moves into the wind barrier (from arrival of the vehicle head to arrival of the rear at the beginning of the barrier), the wind forces F_S, F_L and M_q acting on the car-body change with time linearly, while M_Y and M_Z quadratically with time. They can be expressed as:

(4)F_{\mathrm{V}i}^{\mathrm{II}}=\left[\begin{array}{l}{F}_{\mathrm{Y}}^{\mathrm{II}}\\ F_{\mathrm{Z}}^{\mathrm{II}}\\ M_{\mathrm{X}}^{\mathrm{II}}\\ M_{\mathrm{Y}}^{\mathrm{II}}\\ M_{\mathrm{Z}}^{\mathrm{II}}\end{array}\right]=\left[\begin{array}{l}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}H\{L{C}_{\mathrm{S}}^{\mathrm{I}}-V\cdot (t-t_{\mathrm{0}})\cdot (C_{\mathrm{S}}^I-C_{\mathrm{S}}^{\mathrm{II}})\}\\ {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}H\{L{C}_{\mathrm{L}}^{\mathrm{I}}-V\cdot (t-t_{\mathrm{0}})\cdot (C_{\mathrm{L}}^I-C_{\mathrm{L}}^{\mathrm{II}})\}\\ {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}{H}^{\mathrm{2}}\{L{C}_{\mathrm{M}}^{\mathrm{I}}-V\cdot (t-t_{\mathrm{0}})\cdot (C_{\mathrm{M}}^I-C_{\mathrm{M}}^{\mathrm{II}})\}\\ {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}H\left\{{\displaystyle \frac{V\cdot (t-t_{\mathrm{0}})\cdot [L-V\cdot (t-t_{\mathrm{0}})]}{\mathrm{2}}}\cdot (C_{\mathrm{L}}^I-C_{\mathrm{L}}^{\mathrm{II}})\right\}\\ {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}H\left\{{\displaystyle \frac{V\cdot (t-t_{\mathrm{0}})\cdot [L-V\cdot (t-t_{\mathrm{0}})]}{\mathrm{2}}}\cdot (C_{\mathrm{S}}^I-C_{\mathrm{S}}^{\mathrm{II}})\right\}\end{array}\right].

Stage 3: . During the vehicle moves in the wind barrier (from arrival of the vehicle rear at the beginning of the barrier to the head at the end), the car-body is sheltered by the barrier, but the wind forces F_S, F_L and M_q still exist owing to the porosity of the barrier. In this stage, the wind forces do not change with time, which can be calculated with three-component coefficients of the car-body with wind barriers.

(5)F_{\mathrm{V}i}^{\mathrm{III}}={\left[\begin{array}{lllll}{F}_Y^{\mathrm{III}}\hfill & F_Z^{\mathrm{III}}\hfill & M_X^{\mathrm{III}}\hfill & \mathrm{0}\hfill & \mathrm{0}\hfill \end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{lllll}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}A{C}_{\mathrm{S}}^{\mathrm{II}}\hfill & {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}A{C}_{\mathrm{L}}^{\mathrm{II}}\hfill & {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}AH{C}_{\mathrm{M}}^{\mathrm{II}}\hfill & \mathrm{0}\hfill & \mathrm{0}\hfill \end{array}\right]}^T.

Stage 4: . During the vehicle moves out of the wind barrier (from arrival of the vehicle head to arrival of the rear at the end of the barrier), the wind forces F_S, F_L and M_q acting on the car-body change with time linearly, while M_Y and M_Z quadratically with time.

(6)F_{\mathrm{V}i}^{\mathrm{IV}}=\left[\begin{array}{l}{F}_Y^{\mathrm{IV}}\\ F_Z^{\mathrm{IV}}\\ M_X^{\mathrm{IV}}\\ M_Y^{\mathrm{IV}}\\ M_Z^{\mathrm{IV}}\end{array}\right]=\left[\begin{array}{l}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}H\{[V\cdot (t-t_{\mathrm{0}})-l]\cdot C_{\mathrm{S}}^I+[L-V\cdot (t-t_{\mathrm{0}})+l]\cdot C_{\mathrm{S}}^{\mathrm{II}}\}\\ {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}H\{[V\cdot (t-t_{\mathrm{0}})-l]\cdot C_{\mathrm{L}}^I+[L-V\cdot (t-t_{\mathrm{0}})+l]\cdot C_{\mathrm{L}}^{\mathrm{II}}\}\\ {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}{H}^{\mathrm{2}}\{[V\cdot (t-t_{\mathrm{0}})-l]\cdot C_{\mathrm{M}}^I+[L-V\cdot (t-t_{\mathrm{0}})+l]\cdot C_{\mathrm{M}}^{\mathrm{II}}\}\\ {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}H\left\{{\displaystyle \frac{[V\cdot (t-t_{\mathrm{0}})-l]\cdot [L-V\cdot (t-t_{\mathrm{0}})+l]}{\mathrm{2}}}\cdot (C_{\mathrm{L}}^{\mathrm{II}}-C_{\mathrm{L}}^I)\right\}\\ {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}H\left\{{\displaystyle \frac{[V\cdot (t-t_{\mathrm{0}})-l]\cdot [L-V\cdot (t-t_{\mathrm{0}})+l]}{\mathrm{2}}}\cdot (C_{\mathrm{S}}^{\mathrm{II}}-C_{\mathrm{S}}^I)\right\}\end{array}\right].

Stage 5: . After the vehicle leaves away from the wind barrier, that is, when the vehicle moves outside the wind barrier, the wind forces on the car body are the same as in stage 1.

(7)F_{\mathrm{V}i}^{\mathrm{V}}={\left[\begin{array}{lllll}{F}_Y^{\mathrm{V}}\hfill & F_Z^{\mathrm{V}}\hfill & M_X^{\mathrm{V}}\hfill & \mathrm{0}\hfill & \mathrm{0}\hfill \end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{lllll}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}A{C}_{\mathrm{S}}^{\mathrm{I}}\hfill & {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}A{C}_{\mathrm{L}}^{\mathrm{I}}\hfill & {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\overline{u}}^{\mathrm{2}}AH{C}_{\mathrm{M}}^{\mathrm{I}}\hfill & \mathrm{0}\hfill & \mathrm{0}\hfill \end{array}\right]}^{\mathrm{T}}.

So far, the static wind forces acting on the car-body during the vehicle moves into, in and out of the wind barrier have been given. On the basis of the buffeting theory, the buffeting wind forces on the car-body can be calculated similarly to the static wind forces [²⁰]. It can be seen from the above equations that the wind forces on the car-body change suddenly, which will affect the safe operation of the train. Further analysis and discussion are performed hereinafter for the train running safety when the train passing into and out of the wind barrier.

Dynamic interaction model of wind-vehicle-bridge-wind barrier system

The analysis model of wind-vehicle-bridge-wind barrier coupling system is formed in accordance with the existed vehicle model, bridge model and wind load model in Xia et al. [²⁰]. Especially, the bridge is discretized as a three-dimensional FE model. By applying the modal superposition method, the generalized coordinates of bridge vibration modes are solved rather than the motion equations of the bridge directly. Because the stiffness of the wind barrier is much less than that of the bridge and only the stand columns of the wind barrier connect with the bridge, the influence of wind barrier on the bridge modes is neglected. The influence of the wind barrier on the wind load on the bridge and the train is considered in the coupling system. The wind loads on the bridge include the static wind forces by the mean wind speed, the buffeting forces by the turbulent wind speed and the self-exciting forces by the dynamic interaction of the wind and the bridge. The wind loads on the train only have the static wind force and unsteady wind force on the car-body, while those on the bogies and the wheel-sets are neglected because of their small windward area.

The motion equations of the vehicle-bridge coupling system in the wind field can be stated in matrix form:

(8)\left[\begin{array}{ll}{M}_{\mathrm{vv}}& \mathrm{0}\\ \mathrm{0}& M_{\mathrm{bb}}\end{array}\right]\left\{\begin{array}{l}{\ddot{X}}_{\mathrm{v}}\\ {\ddot{X}}_{\mathrm{b}}\end{array}\right\}+\left[\begin{array}{ll}{C}_{\mathrm{vv}}& C_{\mathrm{vb}}\\ C_{\mathrm{bv}}& C_{\mathrm{bb}}\end{array}\right]\left\{\begin{array}{l}{\dot{X}}_{\mathrm{v}}\\ {\dot{X}}_{\mathrm{b}}\end{array}\right\}+\left[\begin{array}{ll}{K}_{\mathrm{vv}}& K_{\mathrm{vb}}\\ K_{\mathrm{bv}}& K_{\mathrm{bb}}\end{array}\right]\left\{\begin{array}{l}{X}_{\mathrm{v}}\\ X_{\mathrm{b}}\end{array}\right\}=\left\{\begin{array}{l}{F}_{\mathrm{v0}}\\ F_{\mathrm{b0}}\end{array}\right\}+\left\{\begin{array}{l}{F}_{\mathrm{v}}^{\mathrm{st}}+F_{\mathrm{v}}^{\mathrm{ust}}\\ F_{\mathrm{b}}^{\mathrm{st}}+F_{\mathrm{b}}^{\mathrm{bf}}+F_{\mathrm{b}}^{\mathrm{se}}\end{array}\right\},

where M, C and K are the mass, damping and stiffness matrices, X,\dot{X} and \ddot{X}are the displacement, velocity and acceleration vectors, with the subscripts vv and bb representing the train and bridge, bv and vb the interaction between the train and the bridge, respectively; F_v0 and F_b0 are the force vectors due to the train-bridge interaction through the track and wheels under the moving train, respectively; F_{\mathrm{b}}^{\mathrm{s}\mathrm{t}}, F_{\mathrm{b}}^{\mathrm{b}\mathrm{f}}, and F_{\mathrm{b}}^{\mathrm{s}\mathrm{e}}are the static wind force vector, buffeting force vector and the self-excited force vector of the bridge, respectively; F_{\mathrm{v}}^{\mathrm{s}\mathrm{t}} and F_{\mathrm{v}}^{\mathrm{ust}}are the steady force vector and unsteady force vector of the vehicle; the subscripts st, bf, se and ust represent the static force, buffeting force, self-excited force and the unsteady aerodynamic force, respectively. It is important to note that the corresponding aerodynamic coefficients should be adopted to calculate the wind load on the bridge and train when the train moving in different place on the bridge with or without the wind barrier.

Case study

Based on the wind-vehicle-bridge system analysis model, and considering the wind load change on the train when the train moves into and out of the wind barrier, the calculation program is compiled to achieve the dynamic response and running safety indices of the train.

Calculation conditions and parameters

A 169.6 m long bridge with ten-span simply-supported U-shaped girders is used in the analysis. The girder is 16 m in span, 13.4 m in width and 2.0 m in height, which are supported on three-column piers with 12 m height. The bridge is simulated by the finite element method, as shown in Fig. 3, and the first 60 order vibration frequencies of the bridge range from 2.55 Hz to 21.46 Hz.

The calculation includes three cases: without wind barrier on the bridge (Case A), with wind barrier on the whole bridge (Case B) and with a section of wind barrier on the bridge (Case C, a 100 m wind barrier is installed at 40 m from the bridge head). The wind barrier is the double-side vertical column structure with 3.5 m height, where 10% porosity is adopted at the bottom 1.5 m, 20 % at the other part, as shown in Fig. 4.

The train concerned is the ICE3 train composed of 4×(3M+1T), where M represents the motor-car and T the trailer-car. Each vehicle includes seven rigid parts, i.e., one car-body, two bogies and four wheel-sets, which are connected by springs and dashpots. The height and width of the car-body are 3.5 m and 2.7 m, respectively. The average static axle load is 160 kN for a motor-car and 146 kN for a trailer car. The other parameters of the ICE3 train can be found in Xia et al. [²¹]. The train runs on the bridge at the constant speed of 250 km/h. In the wind-train-bridge vibration analysis model the initial conditions such as the initial vibration displacement, speed and acceleration are the important input parameters of the model. For overcoming this problem, let the train run a certain distance on the same line conditions with the bridge before passing the bridge, until the vibration of the car reaches steady state, then the train derives into the bridge. The distance before the bridge is taken as 500 m in this research.

The track irregularities are generated with the harmony superposition method from the German Track PSD functions for high-speed railway, which are recommended by the technical condition for high-speed railways in China [²¹]. The length of the simulated data is 2000 m with the maximum amplitudes being 4.33 mm in the lateral direction, 5.51 mm in the vertical direction and 0.003 rad in the torsional direction.

The wind velocity time histories at the bridge site are simulated by AR method that is described in detail and applied in Zhang et al. [²²] to calculate the buffeting wind forces on the bridge and the unsteady wind forces on the train. The following lateral and vertical wind auto-spectra are adopted in the code of China (JTG/T D60-01-2004) [²³]:

(9){\displaystyle \frac{f{S}_u(f)}{u_{*}^{\mathrm{2}}}}={\displaystyle \frac{\mathrm{200}{f}_{*}}{{(\mathrm{1}+\mathrm{50}{f}_{*})}^{\mathrm{5}/\mathrm{3}}}},

(10){\displaystyle \frac{f{S}_w(f)}{u_{*}^{\mathrm{2}}}}={\displaystyle \frac{\mathrm{6}{f}_{*}}{{(\mathrm{1}+\mathrm{4}{f}_{*})}^{\mathrm{2}}}},

where f_{\ast } stands for the reduced frequency,f_{\ast }={\displaystyle \frac{fz}{\bar{u}(z)}},u_{*}={\displaystyle \frac{K\bar{u}(z)}{\mathrm{In[(}z-z_d)/z_0]}},z_d=\bar{H}-z_{\mathrm{0}}/K; S_u(f)andS_w(f) are the power spectral density function on the lateral and vertical direction, respectively; u_{*}is the friction velocity of the air flow , in m/s; K is a non-dimensional const, K ≈ 0.4; z is the height from ground or water surface, in m; \bar{u}(z)is the mean wind velocity at the height z, in m/s; z₀ is the ground roughness height, in m; \bar{H} is the mean height of surrounding buildings.

The time histories of vertical and lateral fluctuating wind velocity at a point of the bridge are shown in Fig. 5, corresponding to the mean wind velocity 25 m/s.

For calculation of dynamic response of the train-bridge system under cross-winds, the aerodynamic data, including both static force coefficients and turbulent characteristics, for the bridge and the moving train are required. C_{\mathrm{D}},C_{\mathrm{L}},C_{\mathrm{M}},{{{\displaystyle C}}_{\mathrm{D}}}^{\text{'}},{{{\displaystyle C}}_L}^{\text{'}}and{{{\displaystyle C}}_{\mathrm{M}}}^{\text{'}} represent the drag, lift, moment coefficient and their first order derivatives, respectively. These values are acquired from a wind tunnel experiment, as listed in Table 1 for the bridge, where there are two cases—a vehicle on the bridge and no vehicle on the bridge. While in the experiment the train is stationary on the bridge and the wind is perpendicular to the train running direction, in fact the aerodynamic force of the vehicle is related to both the running speed of the train and wind velocity, the aerodynamic coefficient of the train should be the function of the yaw angle β (the angle between the train direction of travel and the wind vector relative to the train). In order to calculate the dynamic response of the train at any train speed and wind velocity, based on the References [²⁴,²⁵], assume that the aerodynamic coefficients can be represented by sine curves, and thus fully specified by a representative value at 90°, so C_n(β) = C_n(90°)·sinβ (n = S, L, M).

Listed in Table 2 are the aerodynamic coefficients of the vehicle at zero wind attack angle and 90° yaw angle, with and without wind barriers on the bridge.

Results analysis

According to the given condition and parameters, the wind forces on the bridge and vehicle are calculated considering the different cases of the wind barriers on the bridge, based on which the whole histories of the ICE train running on the bridge in the wind field with or without wind barriers are simulated. The dynamic responses of the vehicle and the running safety indices of the train are obtained under different wind velocities to evaluate the performance of the train, especially when the train runs into and out of the wind barriers.

The dynamic response of the train mainly includes the acceleration of the car-body and Sperling index W to evaluate the riding comfort of the train. The running safety of train currently is evaluated with the derailment factor Q/P₁, the offload factor\Delta P/\bar{P}, the lateral wheel-rail force Q, and the overturn factor P_d/P_st. When these evaluation indices are used, the bigger index value, the worse the running safety state for the train vehicle. The expressions and allowable values of these indices given in the Chinese codes are as follows in Xia et al. [²⁰]:

(11)

where Q is the lateral wheel-rail force, P₁ is the vertical force of the wheel at the climbing-up-rail side; DP is the offload vertical wheel-rail force, \bar{P} is the average vertical wheel-rail force of the two wheels on a wheel-set; P_d is the change value of the vertical wheel-rail force acted by lateral load, and P_st the vertical static wheel-rail force without lateral load. The allowable lateral wheel-rail forces for the motor-car and trailer-car of the ICE3 high-speed train are 52.97 and 49.08 kN, corresponding to their static loads of 156.96 and 143.22 kN, respectively.

Shown in Fig. 6 are the distributions of maximum lateral acceleration of the car-body versus the mean wind velocity. It can be seen that for Case C (only 100 m wind barrier on the bridge), the maximum lateral acceleration of the car-body significantly increase with the wind velocity, and the higher the wind velocity, the greater the growth. For Case B (wind barrier on the whole bridge), the maximum lateral acceleration of the car-body changes very little with the wind velocity. While for Case A (without wind barrier), when the mean wind velocity is less than 30 m/s, the maximum lateral acceleration slowly increase with the mean wind velocity. The lateral accelerations of the trailer car are smaller than those of the motor-car, with maximum being 0.578 m/s² for Case A, 0.501 m/s² for Case B and 2.123 m/s² for Case C at the mean wind velocity of 30 m/s.

Shown in Fig. 7 are the distributions of the maximum vertical acceleration of the car-body versus the mean wind velocity. It can be found that the maximum vertical acceleration of the motor car is usually greater than that of the trailer car at the same case. The vertical acceleration has an increasing trend with the mean wind velocity, but the growth for Case B is less than that for Case A and Case C. For Case C the growth of the lateral acceleration is much more than that of the vertical acceleration. These results indicate that the sudden change of wind load induced by sectional wind barriers affects the lateral acceleration of the car-body significantly, but the vertical acceleration relatively less.

The effect of the wind load sudden change on the train is more clearly shown with the time history curves. Figures 8 and 9 illustrate them for the same vehicle in Case A, Case B and Case C at the train speed of 250 km/h and the mean wind velocity of 25 m/s, in which the thick dotted lines mark the position of train moving into and out of wind barrier. It can be seen that the lateral acceleration of the car-body changes sharply for when the train runs into and out of the wind barrier (Case C), but not obviously for Case A and Case B, indicating that sectional wind barrier affects the lateral acceleration of the car-body strongly, while the vertical acceleration not obviously.

Only the distributions of maximum offload factor and overturn factor versus the mean wind velocity are given in Figs. 10 to 11 because of limited space. It can be seen that these indices change with similar same rules for motor-car and trailer-car, and the indices of the trailer-car are bigger than that of the motor-car because of lighter axle load. These indices increase with the wind velocity for Case A and C, and the higher the wind velocity, the faster the indices, but in Case B, they are smaller and change little. So the running safety of the train can be obviously improved when the wind barrier is installed on the whole bridge. At the same mean wind velocity, the maximum offload factors and overturn factors in Case C is much bigger than in Case A without wind barriers, especially at high wind velocity.

The time histories of offload factors and overturn factors of the train are plotted in Figs. 12 and 13, to visually illustrate the effect of wind barrier on the running safety of the train. It can be seen that offload factors for Case C changes more drastically than for other cases and the maximum value appears after the train moves out of the wind barrier. The differences of overturn factors are more obvious for different cases. In Case C, the overturn factor changes rapidly when the train moves into and out of the wind barrier, which is marked with the vertical thick dotted lines, but when the train runs in the wind barrier, it less than without wind barrier.

Sperling index is commonly used to evaluate the riding comfort of the train, and the smaller the Sperling value, the better the riding comfort. The calculated Sperling indices for different cases are listed in Table 3.

It is demonstrated that the sudden change of the wind load greatly reduces the lateral comfort, but it little affects the vertical comfort. When sectional wind barrier is used, the lateral Sperling index increases with wind velocity, while the vertical ones change little, the growth of the Sperling value is much bigger than without wind barrier for the lateral, but opposite for the vertical. The lateral and vertical comfort is the best when the wind barrier is installed on the whole bridge, and they slowly increase with the mean wind velocity.

Conclusions

In this paper the equations for wind loads on car-body are derived when the train moves into and out of the wind barrier. The dynamic analysis model of wind-vehicle-bridge system with wind barrier is established, to calculate the dynamic response of the train. A case study is performed to evaluate the running safety and riding comfort of the ICE3 train running on a ten-span simply-supported bridge with U-shaped girder. By analyzing the calculated results such conclusions have been drawn as follows:

(1) The wind load on car-body is different due to the different aerodynamic coefficients for different conditions, including no wind barrier, full wind barriers, and sectional barriers on the bridge. In the case with sectional wind barriers, the wind loads on the car-body suddenly change when the train runs on different position.

(2) For the bridge with sectional wind barriers, the maximum lateral acceleration of car-body increases significantly with the wind velocity and the higher the wind velocity, the faster the increase. And it is bigger than the value without wind barrier when the mean wind velocity is less than 30 m/s. The influence of wind load sudden change induced by sectional wind barriers is relatively small on the vertical acceleration of car-body.

(3) The lateral comfort of the train greatly decreases when the train moving into and out of the wind barrier, and the higher the mean wind velocity, the greater the effect, but the influence is relatively small on the vertical comfort.

(4) The running safety indices of the train rapidly increase with the mean wind velocity for the bridge without wind barrier or with sectional wind barriers, and the higher the mean wind velocity, the greater the growth. The indices change slowly in the case with wind barrier, which are much smaller than the values in the case with sectional wind barriers. In the case with sectional wind barriers, the offload factor and the overturn factor are greater than without wind barrier, but it has a negative impact on the running safety of the train, even less favorably than without wind barriers.

Consequently, it is better to install wind barrier on the whole bridge, and some appropriate measures should be adopted to reduce such adverse effects when sectional wind barrier is used.

References

[1] Qian Z Y.Strong wind disaster and control countermeasure for northwest China railways, China Railway, 2009, 51(3), 1–4

[2] Liu Q K, Du Y L, Qiao F G. Train crosswind and strong wind countermeasure research in Japan. Journal of the China Railway Society, 2008, 30(1): 82–88

[3] Noguchi T, Fujii T. Minimizing the effect of natural disasters. Japan Railway & Transport Review, 2000, 23: 52–59

[4] Fujii T, Maeda T, Ishida H, Imai T, Tanemoto K, Suzuki M. Wind-induced accidents of train/vehicles and their measures in Japan. Quarterly Report of Railway Technical Research Institute, 1999, 40(1): 50–55

[5] Li X Z, Zhu Y. Stochastic space vibration analysis of a train-bridge coupling system. Interaction and Multiscale Mechanics, 2010, 3(4): 333–342

[6] Zhang N, Xia H. Dynamic analysis of coupled vehicle-bridge system based on inter-system iteration method. Computers & Structures, 2013, 114-115(1): 26–34

[7] Li Y L, Qiang S Z, Liao H L, Xu Y L. Dynamics of wind-rail vehicle-bridge systems. Journal of Wind Engineering and Industrial Aerodynamics, 2005, 93(6): 483–507

[8] Chen R L, Zeng Q Y, Huang Y Q, Xiang J, Wen Y, Guo X G, Yin C J, Dong H, Zhao G. Analysis theory of random energy of train derailment in wind. Science China. Physics, Mechanics & Astronomy, 2010, 53(4): 751–757

[9] Xiang H F, Ge Y J, Zhu L D, Chen A R, Gu M, Xiao R C. Modern Theory and Practice on Bridge Wind Resistance. Beijing: China Communications Press, 2005

[10] Zhang W M, Ge Y J, Levitan M L. Aerodynamic flutter analysis of a new suspension bridge with double main spans. Wind and Structures, 2011, 14(3): 187–208

[11] Han Y, Chen Z Q, Hua X G. New estimation methodology of six complex aerodynamic admittance functions. Wind and Structures, 2010, 13(3): 293–307

[12] Nikitas N, Macdonald J H G, Jakobsen J B. Identification of flutter derivatives from full-scale ambient vibration measurements of the clifton suspension bridge. Wind and Structures, 2011, 14(3): 221–238

[13] Cai C S, Chen S R. Framework of vehicle-bridge-wind dynamic analysis. Journal of Wind Engineering and Industrial Aerodynamics, 2004, 92(1): 579–607

[14] Li Y L, Chen N, Cai X T. Wake effect of bridge tower on coupling vibration of wind-vehicle-bridge System. Journal of Southwest Jiaotong University, 2010, 45(6), 875–887

[15] Kozmar H, Procino L, Borsani A, Bartoli G. Sheltering efficiency of wind barriers on bridges. Journal of Wind Engineering and Industrial Aerodynamics, 2012, 274–284

[16] Kim D H, Kwon S D, Lee I K, Jo B W. Design criteria of wind barriers for traffic. Part 2: Decision making process. Wind and Structures, 2011, 14(1): 71–80

[17] Kwon S D, Kim D H, Lee S H, Song H S. Design criteria of wind barriers for traffic. Part 1: Wind barrier performance. Wind and Structures, 2011, 14(1): 55–70

[18] Zhang T, Xia H, Guo W W. Analysis on running safety of train on bridge with wind barriers subjected to cross wind. Wind and Structures, 2013, 17(2): 203–225

[19] Smith B W, Barker C P. Design of wind screens to bridges: Experience and applications on major bridges. Bridge Aerodynamics, 1998, 3: 289–298

[20] Xia H, Zhang N. Dynamic interaction of vehicles and structures. 2nd ed. Beijing: Science Press, 2005

[21] Xia H, Roeck G D, Goicolea J M. Bridge vibration and controls: New Research. New York: Nova Science Publishers, 2011

[22] Zhang T, Xia H, Guo W W. Simulation of bridge stochastic wind field using multi-variate auto-regressive model. Journal of Central South University, 2012, 43(3): 1114–1121

[23] Ministry of Communications of PRC. Wind Resistant Design Specification for Highway Bridges JTG/T D60-01-2004. Beijing: China Communications Press, 2004

[24] Baker C J. The simulation of unsteady aerodynamic cross wind forces on trains. Journal of Wind Engineering and Industrial Aerodynamics, 2010, 98(2): 88–99

[25] Baker C J, Hemida H, Iwnicki S, Xie G, Ongaro D. Integration of crosswind forces into train dynamic modelling. Journal of Rail and Rapid Transit, 2011, 225(2): 154–164

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature