| Citation: | YU Jian, ZHOU Wang-bao, JIANG Li-zhong, FENG Yu-lin, LIU Xiang. Equivalent method for designed earthquake-induced track geometric irregularities on high-speed railway bridges[J]. Journal of Traffic and Transportation Engineering, 2024, 24(3): 110-123. doi: 10.19818/j.cnki.1671-1637.2024.03.007
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The bridge type studied in this article is a high-speed railway multi span simply supported beam bridge with a span of 32 meters. The main components include box girders, supports, and piers. The type of box girder is double track single box single chamber box girder, and the type of bridge pier is round end solid variable cross-section bridge pier. There are four pot type rubber supports arranged at the bottom of the box girder, and the directions of their movement are fixed, longitudinal sliding, transverse sliding, and bi-directional sliding. The track type studied in this article is the CRTS II slab ballastless track for high-speed railways. The main components include sliding layers, shear tooth slots, base plates, cement asphalt mortar layers, shear steel bars, track slabs, lateral stops, fasteners, and rails. The steel rail is fixed to the rail support platform of the track plate by fasteners; The track board is laid on the cement asphalt mortar layer, and a longitudinal continuous structure is formed by connecting the longitudinal steel bars extending from both ends of the track board; The cement asphalt mortar layer is laid on the base plate, and the mortar can be used to buffer train loads. The base plate is a longitudinally continuous reinforced concrete structure; The base plate and the surface of the box girder are separated by a sliding layer, which is used to reduce the deformation difference between the base plate and the box girder caused by temperature changes; Shear steel bars are arranged between the track plate and the base plate above the expansion joint of the box girder to improve the consistent deformation ability of the base plate, track plate, and steel rail at the expansion joint of the box girder; The shear tooth groove is set between the base plate and the box beam above the fixed support to limit the horizontal movement of the base plate; Lateral blocks are anchored to the surface of the box girder to limit the lateral displacement of the base plate and track plate.
Using ANSYS numerical simulation platform to establish a numerical simulation model of high-speed railway track bridge system, such asFigure 1As shown. The components modeled using BEAM189 elastic beam elements include base plates, track plates, and steel rails. The components modeled using BEAM189 nonlinear generalized beam elements are bridge piers[21-23]The components modeled using CONBIN39 nonlinear spring elements include sliding layers, shear tooth slots, cement asphalt mortar layers, shear steel bars, steel rails, lateral stops, and fasteners. The horizontal constitutive relationship of nonlinear spring elements is considered as ideal elastoplastic[24-29], such asTable 1As shown, the vertical constitutive relationship is considered as linear elasticity. Fixed constraints are applied at the bottom of the bridge pier and at both ends of the base plate, track plate, and steel rail to simulate pile-soil interactions and subgrade constraint effects, respectively. Using Rayleigh damping to establish the damping matrix of the numerical simulation model, and the mass ratio coefficient of Rayleigh dampingaProportional coefficient to stiffnessbrespectively
| a=2ξω1ω2ω1+ω2 | (1) |
| b=2ξω1+ω2 | (2) |
| 构件 | 屈服力/kN | 屈服位移/mm | ||
| 横向 | 纵向 | 横向 | 纵向 | |
| 剪力齿槽 | 1 465 | 1 465 | 0.12 | 0.12 |
| 固定支座 | 1 000 | 1 000 | 2.00 | 2.00 |
| 滑动支座 | 100 | 100 | 2.00 | 2.00 |
| 侧向挡块 | 453 | 0 | 2.00 | 0.00 |
| 水泥沥青砂浆层 | 42 | 42 | 0.50 | 0.50 |
| 钢轨扣件 | 24 | 9 | 2.00 | 2.00 |
| 剪切钢筋 | 23 | 23 | 0.08 | 0.08 |
| 滑动层 | 6 | 6 | 0.50 | 0.50 |
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In the formula:ω1andω2Two angular frequencies with the maximum mass ratio of formation participation in the direction of motivation;ξFor 2 angular frequenciesω1andω2The damping ratio is taken as 0.05 in this article.
Develop a numerical simulation model for the high-speed train track bridge system using MATLAB software. Taking the CRH2C train set as the modeling object, the body, bogie, and wheelset are considered as rigid bodies during modeling, and the suspension system is considered as a linear spring damping unit[30]Each carriage has a horizontal orientationyvVerticalzvRoll sidewaysθvShaking your headΨvNodding and noddingφvThere are a total of 5 degrees of freedom, and each bogie has a lateral orientationybVerticalzbRoll sidewaysθbShaking your headΨbNodding and noddingφbThere are a total of 5 degrees of freedom, and each wheelset has a lateral orientationywVerticalzwRoll sidewaysθwShaking your headΨwThere are a total of 4 degrees of freedom, and each carriage consists of 1 body, 2 bogies, and 4 wheelsets. Each carriage contains a total of 31 degrees of freedom. Using blade contact to simulate wheel rail contact relationship[31]Simulate the wheelset tread and rail using ideal cones and hinge points respectively; The calculation of normal force, creep force, and creep force correction generated by wheel rail contact are based on Hertz theory, respectively[32]、 Kalker's linear theory[33]The Shen Hedrick Empirical Nonlinear Theory[34]Develop a train sub model of the numerical simulation model for the high-speed train track bridge systemFigure 2As shown. This article focuses on analyzing the train response under the excitation of seismic track geometric irregularities. When modeling, the track bridge system is simplified, and elastic beam elements are used to model the main beam, bridge pier, base plate, track plate, and steel rail. Elastic spring elements are used to simulate various interlayer components including sliding layers and shear tooth slots. The linear stiffness of the elastic spring element isTable 1Same.
| Wn={exp{−2[5(nLw−1−0.5)]2}1⩽n⩽Lw0n>Lw | (3) |
In the formula:nNumber the data points;LwNon zero segment data length for window function.
| S=N−1∑n=0P2exp(−2qπknN) | (4) |
auxiliary word for ordinal numbersiGeometric irregularity of track caused by seismic activityYiEvolution of power spectral density matrixEido
| Ei=lg(S2iL−1wΔL) | (5) |
| Eu=(σi)zα+(μi) | (6) |
correspondingEuThe upper bound matrix of the spectrumSudo
| Su=√10EuLwΔL−1 | (7) |
| P2u=N−1∑k=0Suexp(2qπknN) | (8) |
supportP2uMove right to the initial position to obtain the upper bound matrix of the initial pulseP1u, willP1uWindow functionWDividing the maximum value to obtain the design earthquake induced geometric irregularity of the trackYddo
| Yd=max | (9) |
| \begin{array}{l} Y_{\mathrm{f}, x}= \begin{cases}0 & x \leqslant d_1 \\ \frac{0.05 A\left(x-d_1\right)}{d_2} & d_1<x \leqslant d_1+d_2 \\ A \sin \frac{{\rm{ \mathsf{ π}}}\left(x-d_5\right)}{L_1} & d_1+d_2<x \leqslant x_1 \\ 0.05 A-\frac{0.05 A\left(x-x_1\right)}{d_3} & x_1<x \leqslant 350-d_4 \\ 0 & 350-d_4<x \leqslant 350\end{cases} \\ x_1=350-d_3-d_4 \end{array} | (10) |
| 墩高/m | 设防地震 | 罕遇地震 | ||||
| Yd下加速度峰值/(m·s-2) | Yf下加速度峰值/(m·s-2) | 误差/% | Yd下加速度峰值/(m·s-2) | Yf下加速度峰值/(m·s-2) | 误差/% | |
| 5 | 0.022 | 0.023 | 4 | 0.143 | 0.133 | -8 |
| 6 | 0.023 | 0.024 | 4 | 0.141 | 0.140 | -1 |
| 7 | 0.024 | 0.025 | 4 | 0.140 | 0.133 | -5 |
| 8 | 0.024 | 0.026 | 8 | 0.139 | 0.132 | -5 |
| 9 | 0.024 | 0.025 | 4 | 0.140 | 0.150 | 7 |
| 10 | 0.027 | 0.028 | 4 | 0.145 | 0.152 | 5 |
| 11 | 0.027 | 0.028 | 4 | 0.153 | 0.148 | -3 |
| 12 | 0.030 | 0.031 | 3 | 0.156 | 0.144 | -8 |
| 13 | 0.030 | 0.031 | 3 | 0.171 | 0.159 | -8 |
| 14 | 0.031 | 0.032 | 3 | 0.174 | 0.165 | -5 |
| 15 | 0.037 | 0.037 | 0 | 0.178 | 0.175 | -2 |
| 16 | 0.041 | 0.042 | 2 | 0.190 | 0.197 | 4 |
| 17 | 0.046 | 0.047 | 2 | 0.232 | 0.252 | 8 |
| 18 | 0.053 | 0.055 | 4 | 0.220 | 0.221 | 0 |
| 19 | 0.061 | 0.060 | -2 | 0.212 | 0.228 | 7 |
| 20 | 0.069 | 0.073 | 5 | 0.212 | 0.228 | 7 |
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Under different earthquake intensities, the equivalent lateral acceleration spectrum of the designed earthquake induced track geometry roughness has significant differences in the envelope of the random structure. In order to enhance the rationality of the equivalent design of seismic induced track geometric irregularities, a correction factor for the equivalent amplitude response spectrum is introduced. Multiply the equivalent fitting model of seismic induced track geometry roughness with different correction coefficients to obtain multiple sets of correction fitting models. Using the first-order lateral natural period of the numerical simulation model of the high-speed railway track bridge system as the horizontal axis, and the peak lateral acceleration of the vehicle body under the excitation of the modified fitting model as the vertical axis, a modified lateral acceleration spectrum of the vehicle body is established. supportFigure 12The scatter set of acceleration spectra is denoted asp1The set of corrected lateral vehicle acceleration spectrum curve points with the same period as the scatter points of the acceleration spectrum is denoted asp2The ratio set of the two is denoted aspr.
| S_{\mathrm{p}}=\left[\frac{1}{J} \sum\limits_{j=1}^J\left(p_{\mathrm{r}j}-\bar{p}\right)^3\right]\left[\frac{1}{J} \sum\limits_{j=1}^J\left(p_{\mathrm{r}j}-\bar{p}\right)^2\right]^{-\frac{3}{2}} | (11) |
| K_{\mathrm{p}}=\left[\frac{1}{J} \sum\limits_{j=1}^J\left(p_{\mathrm{r}j}-\bar{p}\right)^4\right]\left[\frac{1}{J} \sum\limits_{j=1}^J\left(p_{\mathrm{r}j}-\bar{p}\right)^2\right]^{-2} | (12) |
| C_{\mathrm{N}}=\frac{J}{6}\left[S_{\mathrm{p}}^3+\frac{\left(K_{\mathrm{p}}-3\right)^2}{4}\right]-\chi_\beta^2(2) | (13) |
| \varepsilon(p)=\frac{1}{J} \sum\limits_{j=1}^J p_{\mathrm{r} j} | (14) |
| \varepsilon\left\{[p-\varepsilon(p)]^r\right\}=\frac{1}{J} \sum\limits_{j=1}^J\left(p_{\mathrm{r} j}-\bar{p}\right)^r | (15) |
| \varepsilon(p)=\frac{1}{J} \sum\limits_{j=1}^J p_{\mathrm{r} j}=\bar{p} | (16) |
| \varepsilon\left\{[p-\varepsilon(p)]^2\right\}=\frac{1}{J} \sum\limits_{j=1}^J\left(p_{\mathrm{r} j}-\bar{p}\right)^2=s^2 | (17) |
| p_{\mathrm{u}}=s z_\alpha+\bar{p} | (18) |
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Figures(19) / Tables(2)
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YU Jian, ZHOU Wang-bao, JIANG Li-zhong, FENG Yu-lin, LIU Xiang. Equivalent method for designed earthquake-induced track geometric irregularities on high-speed railway bridges[J]. Journal of Traffic and Transportation Engineering, 2024, 24(3): 110-123. doi: 10.19818/j.cnki.1671-1637.2024.03.007
| 构件 | 屈服力/kN | 屈服位移/mm | ||
| 横向 | 纵向 | 横向 | 纵向 | |
| 剪力齿槽 | 1 465 | 1 465 | 0.12 | 0.12 |
| 固定支座 | 1 000 | 1 000 | 2.00 | 2.00 |
| 滑动支座 | 100 | 100 | 2.00 | 2.00 |
| 侧向挡块 | 453 | 0 | 2.00 | 0.00 |
| 水泥沥青砂浆层 | 42 | 42 | 0.50 | 0.50 |
| 钢轨扣件 | 24 | 9 | 2.00 | 2.00 |
| 剪切钢筋 | 23 | 23 | 0.08 | 0.08 |
| 滑动层 | 6 | 6 | 0.50 | 0.50 |
DownLoad:
CSV
| 墩高/m | 设防地震 | 罕遇地震 | ||||
| Yd下加速度峰值/(m·s-2) | Yf下加速度峰值/(m·s-2) | 误差/% | Yd下加速度峰值/(m·s-2) | Yf下加速度峰值/(m·s-2) | 误差/% | |
| 5 | 0.022 | 0.023 | 4 | 0.143 | 0.133 | -8 |
| 6 | 0.023 | 0.024 | 4 | 0.141 | 0.140 | -1 |
| 7 | 0.024 | 0.025 | 4 | 0.140 | 0.133 | -5 |
| 8 | 0.024 | 0.026 | 8 | 0.139 | 0.132 | -5 |
| 9 | 0.024 | 0.025 | 4 | 0.140 | 0.150 | 7 |
| 10 | 0.027 | 0.028 | 4 | 0.145 | 0.152 | 5 |
| 11 | 0.027 | 0.028 | 4 | 0.153 | 0.148 | -3 |
| 12 | 0.030 | 0.031 | 3 | 0.156 | 0.144 | -8 |
| 13 | 0.030 | 0.031 | 3 | 0.171 | 0.159 | -8 |
| 14 | 0.031 | 0.032 | 3 | 0.174 | 0.165 | -5 |
| 15 | 0.037 | 0.037 | 0 | 0.178 | 0.175 | -2 |
| 16 | 0.041 | 0.042 | 2 | 0.190 | 0.197 | 4 |
| 17 | 0.046 | 0.047 | 2 | 0.232 | 0.252 | 8 |
| 18 | 0.053 | 0.055 | 4 | 0.220 | 0.221 | 0 |
| 19 | 0.061 | 0.060 | -2 | 0.212 | 0.228 | 7 |
| 20 | 0.069 | 0.073 | 5 | 0.212 | 0.228 | 7 |
DownLoad:
CSV
| 构件 | 屈服力/kN | 屈服位移/mm | ||
| 横向 | 纵向 | 横向 | 纵向 | |
| 剪力齿槽 | 1 465 | 1 465 | 0.12 | 0.12 |
| 固定支座 | 1 000 | 1 000 | 2.00 | 2.00 |
| 滑动支座 | 100 | 100 | 2.00 | 2.00 |
| 侧向挡块 | 453 | 0 | 2.00 | 0.00 |
| 水泥沥青砂浆层 | 42 | 42 | 0.50 | 0.50 |
| 钢轨扣件 | 24 | 9 | 2.00 | 2.00 |
| 剪切钢筋 | 23 | 23 | 0.08 | 0.08 |
| 滑动层 | 6 | 6 | 0.50 | 0.50 |
| 墩高/m | 设防地震 | 罕遇地震 | ||||
| Yd下加速度峰值/(m·s-2) | Yf下加速度峰值/(m·s-2) | 误差/% | Yd下加速度峰值/(m·s-2) | Yf下加速度峰值/(m·s-2) | 误差/% | |
| 5 | 0.022 | 0.023 | 4 | 0.143 | 0.133 | -8 |
| 6 | 0.023 | 0.024 | 4 | 0.141 | 0.140 | -1 |
| 7 | 0.024 | 0.025 | 4 | 0.140 | 0.133 | -5 |
| 8 | 0.024 | 0.026 | 8 | 0.139 | 0.132 | -5 |
| 9 | 0.024 | 0.025 | 4 | 0.140 | 0.150 | 7 |
| 10 | 0.027 | 0.028 | 4 | 0.145 | 0.152 | 5 |
| 11 | 0.027 | 0.028 | 4 | 0.153 | 0.148 | -3 |
| 12 | 0.030 | 0.031 | 3 | 0.156 | 0.144 | -8 |
| 13 | 0.030 | 0.031 | 3 | 0.171 | 0.159 | -8 |
| 14 | 0.031 | 0.032 | 3 | 0.174 | 0.165 | -5 |
| 15 | 0.037 | 0.037 | 0 | 0.178 | 0.175 | -2 |
| 16 | 0.041 | 0.042 | 2 | 0.190 | 0.197 | 4 |
| 17 | 0.046 | 0.047 | 2 | 0.232 | 0.252 | 8 |
| 18 | 0.053 | 0.055 | 4 | 0.220 | 0.221 | 0 |
| 19 | 0.061 | 0.060 | -2 | 0.212 | 0.228 | 7 |
| 20 | 0.069 | 0.073 | 5 | 0.212 | 0.228 | 7 |
DownLoad:
DownLoad:
