Study on the threshold for superposed deformation of simply supported bridge creep and track slab upwarp in high-speed railway

Abstract

Based on long-term inspection data about track geometric irregularity of 32 m-span simply supported girder bridge in China Railway High-Speed for nearly 10 years, it is found that there is continuous periodic track longitudinal level (TLL) with different wavelengths, thereinto, simply supported girder bridges creep and ballastless track slab upwarp are the primary reasons. Moreover, there are two types of variation trends for bridge creep of 32 m simply-supported bridges with ballastless track, that is, stable type and increasing type, and the maximum $V_{\mathrm{pp}}^{\mathrm{ll}}$ (the peak-to-peak value of track longitudinal level) is 7.1 mm for track longitudinal level caused by bridge creep, and the maximum $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by upwarp deformation of ballastless track slab is 3.2 mm. Hence, this paper investigated the effects on running performance and structure service performance of random track irregularity and additional track irregularity caused by bridge creep, track slab upwarp deformation or superposition of both, a three-dimensional finite element model of vehicle-track-bridge for dynamic simulation calculation was established, the threshold values for bridge creep and track slab deformation under 350 km/h were proposed based on running performance and structure service performance. Finally, research results show that bridge creep influences on the ride comfort significantly, but running safety barely. Due to the short wavelength of TLL, the track slab upwarp deformation affects both the ride comfort and running safety heavily. The car bodies are more sensitive to bridge creep and produce mainly low-frequency vibration, but the rate of wheel load reduction is more sensitive to the track deformation and shows the high-frequency vibration. For 32 m simply supported bridges without track slab deformation, the $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by bridge creep should be limited to 7 mm, and 9 mm in difficult situations. When the $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by bridge creep is small (less than 5 mm), the $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by track slab upwarp deformation must be less than 4 mm. When both deformations exist simultaneously, the $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by superposition deformations of both should not exceed 10 mm, with track slab upwarp deformation not exceeding 4 mm. In the presence of both structural deficiencies, it is suggested that the track slab deformation must be eliminated when it is difficult to eliminate the bridge creep.

Introduction

It has been 10 years for the operation of China high-speed railway. Various deficiencies of infrastructure and track structure occur, which cause obvious track geometric irregularity, thereby affecting ride comfort and running safety. For simply supported girder bridge with continuous equal span, bridge creep will cause continuous periodic TLL ¹ (track longitudinal level, also referred to as longitudinal irregularity or profile). Under various actions of temperature^2,3 creep or train vibration load, ⁴ track slab deformations would also occur, such as arching or upwarp. Wang ⁵ studied the influence of bridge pier settlement and girder bridge creep, which are two typical types of long-term bridge deformation, on the vibration of high-speed trains. Based on the vehicle-bridge coupling dynamics theory, Yang et al., ⁶ Gou et al. ⁷ investigated the dynamic response of vehicle-track-bridge structure and running performance of vehicle when there is bridge creep. Xia et al. ⁸ studied the influence of PC beam deformation induced by creep and temperature effect on the dynamic responses and running safety of HSR train-bridge system. Zhao et al. ⁹ developed two finite element models of CRTS II ballastless track superstructure were to simulate temperature deformations of the precast concrete track slab. Based on dynamic inspection data of TLL, Tian et al. ¹ analyzed the causes of periodic TLL of simply supported girder bridge, and proposed the control standard for bridge creep via the vehicle-track-bridge vertical coupling vibration analysis model. Ren et al. ¹⁰ studied influence of cement asphalt mortar debonding on the damage distribution and mechanical responses of CRTS I prefabricated slab, Li et al. ¹¹ studied the mechanical behavior characteristics of CRTS II track slab on bridge and the mean value, standard deviation, and variation coefficient of periodic track irregularity. Zhu ¹² revealed the effect of train–track interaction and environment loads on the mechanical characteristic variation of ballastless tracks in high-speed railways. To capture the mechanical characteristic variation of ballastless tracks under environment loads such as the temperature load, Zhong ¹³ established a three-dimensional finite element model of slab track to study the deformation and interface stress of a slab track under daily changing temperature. Wang et al. ¹⁴ evaluated the mechanical performance of the CA mortar of the slab track interlayer under extreme temperatures, by conducting the flexural strength and fatigue tests and a transient thermal simulation. Cai et al. ¹⁵ investigated the arching mechanism of slab joints under high temperature conditions considering the buckling instability of track structure, and the damage process of joint concrete and its effects on the arching instability were analyzed with the help of the field investigation.

TLL caused by bridge creep and slab upwarp deformation is continuous and periodic, which has a great influence on ride comfort and running safety. However, there are few researches on the track state influenced by bridge creep and slab upwarp deformation simultaneously. The variation trend of TLL caused by bridge creep and slab upwarp deformation for 32 m simply supported bridge obtained using 10 years’ high-speed comprehensive inspection train data. To make the track management more targeted and ensure the safety and comfort for high-speed railway operation, the threshold values are proposed based on the simulation results of 3D vehicle-track-bridge coupling dynamic model considering current operation and maintenance situation. ¹⁶

Analysis of TLL on the bridge

For 32 m-span simply supported girder bridge with ballastless track slab in high-speed railway, the bridge creep and slab upwarp deformation are two main factors affecting the TLL. Figure 1 presents the schematic of ballastless track slab on 32 m-span simply supported girder bridge and common slab deficiencies. Due to significant difference between structures, TLL with different wavelength would be caused by both deficiencies. ¹⁷ For example, the TLL in C line (represents a high-speed railway line that has been in operation for several years, A line and B line are the same) of high-speed railway with CRTS I caused by bridge creep and track slab deformation is shown by the black line in Figure 2(a). As shown in Figure 2(b) of the analysis results by power spectral density, ¹⁸ the wavelength of TLL caused by bridge creep is 32.7 m, and the values caused by track slab deformation are mainly 4.67 and 5.45 m, which is basically the same as the length of bridge and CRTS I. Thus the individual influence of bridge creep and track slab deformation on TLL can be identified by low-pass, high-pass, or band-pass filtering methods because of the different wavelengths, as shown in Figure 2(a).

Figure 1.

Ballastless track slab and deficiencies of 32 m-span simply supported girder bridge in high-speed railway: (a) layout drawing of ballastless track slab on 32 m-span simply supported girder bridge, (b) mud pumping and rising soil of CRTS I track slab, and (c) the broken wide and narrow joints of CRTS II track slab.

Figure 2.

The TLL caused by bridge creep of 32 m-span simply supported girder bridge and track slab upwarp deformation in C line of high-speed railway and power spectral density: (a) the TLL of in C line of high-speed railway and (b) the PSD of TLL of left and right rail.

Analysis of TLL caused by bridge creep

Thirty-two metre simply supported PC girders with box sections are widely used in China Railway High-Speed. Under the interaction of prestressed steel and dead load, each section is in an eccentric compression state. In a section, the compressive stress varies along the height linearly, with larger compressive stress in the upper flange, and less stress in the lower flange, even tensile stress. Due to the creep characteristics of concrete, under the reverse bending moment caused by prestressed steel, the creep occurs, ¹⁹ as shown in Figure 3. The main factors affecting the creep include the arrangement of prestressed tendons, elastic modulus, strength of concrete, the age at prestressing and time between prestressing and application, and the age at of non-structural loads. Moreover, the actual train load is lower than the design value, which also intensifies the amplitude of bridge arching.

Figure 3.

Diagram for creep of simply supported girder bridge.

Slab ballastless track slab and rail are laid on 32 m-span simply supported girder bridges. Slab ballastless track is consolidated with bridge deck. The vertical stiffness of bridge is much larger than the bending stiffness of slab ballastless track and rail. The deformation caused by bridge creep can be transmitted to the rail surface through ballastless track. As a continuous elastic body, there is limited mitigation effect for rails on the beam end angle of simply supported girder bridge. ⁶ However, due to the poor bending stiffness of rail, the difference between the vertical deformation of rail and bridge creep deformation is small. ¹ Therefore, the creep characteristics of simply supported girder bridge can be obtained through the TLL. Bridge creep will cause continuous periodic TLL, as shown in Figure 4, when the TLL is large, the standard deviation of TLL data will exceed the threshold value 11.

Figure 4.

Influence on TLL of continuous 32 m simply supported girder bridge creep.

Ballastless track slab is laid on the simply supported bridges of high-speed railway, with the increase of service time, slab upwarp deformation might occur. Because two structural deformations cause different wavelength components, band-pass filtering of track dynamic longitudinal level was carried out, and the components with wavelength at 10–42 m were refined. The wavelength of angle irregularity caused by creep at beam end is relatively short, and band-pass filtering will change the real situation of TLL of beam joint. Meanwhile, considering the influence of different settlement of piers, the median of the filtered peak to the two valleys of the original waveform is taken as the $V_{\mathrm{pp}}^{\mathrm{ll}}$ which caused by the bridge creep. (Same below) See formula (1) for specific calculation method and Figure 5 shows the $V_{\mathrm{pp}}^{\mathrm{ll}}$ diagram.

$$V_{\mathrm{ppi}}^{\mathrm{ll}}=A_i-\frac{B_{i1}-B_{i2}}{2}\begin{array}{c}i=1,2,\cdots \hfill \end{array}N$$

(1)

Where $V_{\mathrm{ppi}}^{\mathrm{ll}}$ is the peak to peak value of track longitudinal level which caused by bridge creep, $A_i$ is the peak value of TLL caused by bridge creep, $B_{i1}\mathrm{and}{B}_{i2}$ represent the two valley values of the TLL caused by bridge creep respectively, and i represents the span number.

Figure 5.

Schematic diagram of TLL and $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by 32 m simply supported girder bridge creep in high-speed railway.

High-speed comprehensive inspection trains are used to regular detect track geometric irregularity in China.^20,21 By nearly 10 years, massive track geometric irregularity data have been obtained. There is certain temperature effect for simply supported girder bridge, that is, the TLL is larger in summer and smaller in winter. To better ensure running safety, the maximum value envelope curve was built to capture the development trend of bridge creep. As seen in Figures 6 and 7, the $V_{\mathrm{pp}}^{\mathrm{ll}}$ in A line and B line of high-speed railway were fitted and analyzed statistically. And the CEB-FIP model ²² is adopted by concrete creep coefficient in Code for Design of Concrete Structures. ²³ Referring to the CEB-FIP model, the $V_{\mathrm{pp}}^{\mathrm{ll}}$ is fitted by a function as follow:

$$z(t)=a•{\left(\frac{t+b}{t+b+c}\right)}^d$$

(2)

Where z(t) represents the variation of $V_{\mathrm{pp}}^{\mathrm{ll}}$ with time caused by 32 m simply supported girder bridge creep, t is the operation time of railway (d), and symbols a, b, c, d are various related coefficients, respectively.

Figure 6.

Development of a 32 m-span simply supported girder bridge creep in A line of high speed railway.

Figure 7.

Development of a 32 m-span simply supported bridge creep in B line of high-speed railway.

Figures 6 and 7 show the trend of $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by creep on most bridges, by analyzing the $V_{\mathrm{pp}}^{\mathrm{ll}}$ on 32 m simply supported girder bridges in the A line and B line of high-speed railway, it is found that there are two main trends for bridge creep. First, for most simply supported girder bridges, the creep increases faster in former years after railway operation, and then increases slowly, finally, turns to be stable, and this situation (stable) accounts for 97.26% and 97.89% of all 32 m simply supported bridges on A line and B line. Second, for some simply supported girder bridges, the creep presents continuous increase, and this situation (increasing) accounts for 2.74% and 2.11% of all 32 m simply supported bridges on A line and B line. Therefore, it is necessary to continue monitoring the development of bridge creep. In addition, the growth trend is non-monotonic because of the different temperatures over 1 year.

Tian et al. ¹ pointed out that the theoretical calculation of the maximum bridge creep deformation is about 6.2–7.3 mm for 32 m-span simply supported girder bridges. Based on the statistics result of $V_{\mathrm{pp}}^{\mathrm{ll}}$ , the value reaches 7.1 mm, as shown in Figure 7. For prestressed concrete girder bridge with design speed large than 200 km/h and span less than 50 m, it is specified that the vertical creep deformation shall not be greater than 10 mm in Code for Design of High-Speed Railway. ²⁴ At present, the creep deformation meets the requirements of design code for 32 m-span simply supported girder bridges in China Railway High-speed.

Analysis of upwarp deformation characteristics of track slab on the bridge

There are various forms of slab ballastless track in China Railway High-Speed, ²⁵ such as CRTS I,^26–28 CRTS II,^29,30 CRTS III, ³¹ and other forms. Under the action of train traffic load and external environment effect, ballastless track slab is prone to various deficiencies, such as mud-pumping, mortar seam opening, joint damage, and upper arch of track slab end. ³² According to track dynamic detection data, the slab arching has become one of the most prominent deficiencies, and the superposition of slab arching and bridge creep would intensify the dynamic response of vehicle.

CRTS II is laid on the simply supported girder bridge in B line of high-speed railway. The track geometric irregularity from October 2011 to August 2020 are shown in Figure 8(a). The TLL of three-span simply supported girder bridge near K532+500 (equal to 532.500 km, same bellow) not only includes the 32.7 m wavelength component caused by bridge creep, but also includes the 6.5 m wavelength component, which is equal to the length of CRTS II. This point can be proved by the power spectral density of TLL around K532+400–K532+600, as shown in Figure 8(b). Meanwhile, it can be seen from Figure 8(c) that the slab upper arching at K532+496 occurs in July 2017, and the $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by slab deformation reaches 3.2 mm in July 2020, and the maximum difference of TLL between bridge midspan and bearing position is 6.2 mm.

Figure 8.

(a) The superposition curve of TLL of B line of high-speed railway. (b) Power spectral density of TLL in K532+400–K532+600. (c) The variation of the $V_{\mathrm{pp}}^{\mathrm{ll}}$ with time in K532+496.

Model of vehicle-track-substructure interaction

The accurate three-dimensional finite element model of vehicle-track-slab-bridge is established, as shown in Figure 9. And the dynamic wheel-rail coupling relationship, the reasonable equations and solving methods are established. The influence of slab and bridge deficiencies on TLL is taken as the input excitation of the whole model, and the dynamic simulation calculation is carried out to provide the premise for evaluating the driving performance and structural service performance.

Figure 9.

Vehicle-track-bridge dynamic interaction model.

Vehicle Model

There are multiple vehicles for a train, each of which contains one body, two bogies, and four wheelsets, and each of the seven parts is considered as a rigid body. The analysis of two suspension train is carried out, and 35 independent degrees of freedom are considered, including lateral motion, vertical motion, roll motion, pitch motion, and yaw motion, respectively, as shown in Table 1. The vehicle dynamic analysis model is presented in Figure 10. The physical meaning of each variable in Figure 10 is shown in reference. ³³ The CRH380A is used as the dynamic simulation model. To improve calculation efficiency, one motor car and one trail car are selected, and they are different in quality, size and the suspension parameter.

Table 1.

Degrees of freedom of passenger vehicle dynamics model.

Vehicle component	Lateral motion	Vertical motion	Roll motion	Pitch motion	Yaw motion
Car body	$Y_{\mathrm{c}}$	$Z_{\mathrm{c}}$	${\theta }_{\mathrm{c}}$	${\beta }_{\mathrm{c}}$	${\phi }_{\mathrm{c}}$
Bogie (i = 1, 2)	$Y_{\mathrm{t}i}$	$Z_{\mathrm{t}i}$	${\theta }_{\mathrm{t}i}$	${\beta }_{\mathrm{t}i}$	${\phi }_{\mathrm{t}i}$
Wheelset (i = 1, 4)	$Y_{\mathrm{w}i}$	$Z_{\mathrm{w}i}$	${\theta }_{\mathrm{w}i}$	${\beta }_{\mathrm{w}i}$	${\phi }_{\mathrm{w}i}$

Figure 10.

Vehicle dynamic analysis model: (a) side view of vehicle model and (b) front view of vehicle model.

Dynamic model of Substructure

Compared with other slab types, the length of CRTS I track slab is the shortest, and the wavelength of TLL caused by slab deformation is also the shortest, which is more unfavorable to running safety. Therefore, this paper establishes a three-dimensional finite element model of 10-span 32 m simply supported girder bridge-CRTS I-seamless rail based on a commonly used finite element software, and the model is called substructure including rail, fastening system, track slab, base slab, and bridge structure components, as shown in Figure 11(a). The detailed model is shown in Figure 11(b) and (c). Some material characteristic parameters of the bridge can refer to this paper. ³⁴ Furthermore, the length of every bridge span is 32.6 m, the length between the cross sections of the two bearings is 32 m, and the gap width between the center line of every bridge span is 0.01 m, finally the overall length of rail is 900 m considering the influence of train entering and leaving the bridge, one about 300 m track is added on both sides of bridges, then the train will go from one end of the track to the other. And the height of every pier is 20 m (a average value of high-speed railway pier, and pier height has little influence on TLL and driving performance, so the average value is used for calculation.). Moreover, one end of the track fastener is connected to the rail and the other is consolidated on the ground. In the model, the beam element is used to simulate the rail of CHN60, and spring-damping element simulates the constraint of fastener and rubber on rail, the constraint between sleeper and track slab, and the connection between track slab and base slab instead of the CA mortar mixture, with the fastener spacing being 0.629 m, in addition, the rigid connection is used to restrict motion between the base slab and bridge. According to structure and stress characteristics, the track slab, base slab, and simply supported girder bridge are modeled by beam element with 12 freedoms for two nodes. ³⁵ The section parameters of simply supported girder bridge are introduced in literature. ⁸ The main-slave relation simulates the bridge bearing without considering the friction of the active bearing, and the piers bottom are fixed on the ground, leaving out the friction of the active bearing. Consequently, the whole rail-track-bridge model has 9042 nodes and 11,375 elements. In addition, the structural parameters of track slab on bridge are listed in Table 2.

Figure 11.

Three-dimensional finite element model of rail-track-bridge: (a) Overall model drawing, (b) detail drawing of the bridge and (c) detail drawing of the track.

Table 2.

The parameters of CRTS I track slab.

Type	Name	Value	Units
Track slab	Size (length × width × thickness)	4.962 (3.685) × 2.4 × 0.19	m × m × m
	Concrete grade	C60	–
	Density	2500	kg/m3
	Spacing	0.07	m
	Elastic modulus	34	GPa
	Poisson ratio	0.2	–
	Slab joint	0.07	m
CA mortar mixture	Size (length × width × thickness)	4.962 (3.685) × 2.4 × 0.05	m × m × m
	Density	1900	kg/m3
	Stiffness	1.25	GPa/m
Base slab	Size (length × width × thickness)	20.108 × 3.0 × 0.3	m × m × m
	Concrete grade	C40	–
	Density	2500	kg/m3
	Spacing	0.02	m
	Elastic modulus	33	GPa
	Poisson ratio	0.2	–
Convex retaining platform	Concrete grade	C40	–
	Height	0.25	m
	Radius	0.26	m

The values in brackets express the track slab size of the bridge end.

Rayleigh damping is adopted for the damping matrix of track system and substructure. Meanwhile, the influence of fasteners on the stiffness matrix and damping matrix is considered according to the principle “seat by number.” ³⁶ The Rayleigh damping calculation can be expressed as:

$$C_{\mathrm{r}}^{\mathrm{e}}=\alpha M_{\mathrm{r}}^{\mathrm{e}}+\beta K_{\mathrm{r}}^{\mathrm{e}}$$

(3)

where $M_{\mathrm{r}}^{\mathrm{e}}$ and $K_{\mathrm{r}}^{\mathrm{e}}$ are the overall mass matrix and stiffness matrix individually. Then, α and β can be obtained as:

$$\alpha =2{\omega }_1{\omega }_2\frac{({\xi }_2{\omega }_1-{\xi }_1{\omega }_2)}{{\omega }_1^2-{\omega }_2^2},\beta =2\frac{{\xi }_1{\omega }_1-{\xi }_2{\omega }_2}{{\omega }_1^2-{\omega }_2^2}$$

(4)

where ${\omega }_1,{\omega }_2$ are the first and second natural frequencies, ${\xi }_1,{\xi }_2$ are the corresponding damping coefficients respectively. For rail, ${\xi }_1,{\xi }_2$ are usually about 1%, however, for concrete track slabs and bridges, ${\xi }_1,{\xi }_2$ are usually about 2%.

The vertical static stiffness of rail pad is 25 MN/m, and the vertical damping is 75 kNs/m. The lateral static stiffness of rail pad is 20 MN/m, and the lateral damping is 60 kNs/m. ³³

Wheel-rail contact principle

The Hertz nonlinear elastic contact theory is adopted in the normal direction of wheel-rail, the vertical force between wheel and rail can be expressed as:

$$N_{\mathrm{Z}}(t)={[\frac{1}{G}\delta Z(t)]}^{3/2}$$

(5)

Where G is the wheel rail contact constant, the units is m/N^2/3. $\delta Z(t)$ is the elastic compression between wheel and rail, the units is m.

First, the longitudinal creep force $F_x$ , the lateral creep force $F_y$ , and spin creep torque $M_z$ between wheel and rail are solved based on the Kalker’s linear creep theory, expressed as:

$$\{\begin{array}{c}{F}_x=-f_{11}{\xi }_x\hfill \\ \multicolumn{1}{c}{F_y=-f_{22}{\xi }_y-f_{23}{\xi }_{\varphi }}\\ \multicolumn{1}{c}{M_z=f_{23}{\xi }_y-f_{33}{\xi }_{\varphi }}\end{array}$$

(6)

where $f_{\mathrm{ij}}$ represents the creep coefficients, and the ${\xi }_x$ , ${\xi }_y$ , and ${\xi }_{\varphi }$ represent the longitudinal, lateral, and spin creepages at the wheel–rail contact points respectively.

Then, the Shen-Hedrick-Elkins theory is used to make nonlinear corrections for the longitudinal and lateral creep slip forces between wheel and rail, and the calculation details and parameters description are presented in the literature. ³³

Selection of TLL

Creep of prestressed concrete simply supported girder bridge and slab upwarp deformation will cause additional rail deformation (also referred to as additional TLL). To obtain the mapping relationship between two deformations and TLL, based on the finite element model of vehicle-track-bridge shown in Section “Dynamic model of substructure,” the specified forced displacement is arranged, accordingly, the rail deformation curve is the additional TLL. The bridge creep deformations of 5, 6, 7, 8, 9, and 10 mm are applied to the mid-span of each bridge, and the rail deformation curve caused by the bridge creep is obtained, as shown in Figure 12. Similarly, the slab upwarp deformations of 1, 2, 3, 4, and 5 mm are applied to the middle of each track slab, and the rail deformation curve caused by the slab upwarp deformation is obtained, as seen in Figure 13. Two kinds of rail deformations are input as the additional track irregularity for the vehicle-track-substructure dynamic interaction analysis.

Figure 12.

Rail vertical deformation caused by creep of simply supported girder bridge.

Figure 13.

Rail vertical deformation caused by slab upwarp deformation.

According to the ballastless track spectrum of China Railway High-Speed, ²⁴ the calculation formula is shown as:

$$S(f)=\frac{A}{f^n}$$

(7)

where $S(f)$ is the spectrum, units is $\mathrm{m}{\mathrm{m}}^2/(1/\mathrm{m})$ , $f$ is the spatial frequency, units is 1/m, A and n are fitting parameters.

The frequency-time domain conversion method is used to generate random track irregularity, and the generated random track irregularities of longitudinal level and alignment are shown in Figure 14. Then the additional TLL caused by the bridge creep and track slab upwarp are superimposed with the random irregularity of longitudinal level as wheel-rail excitation. Results of bridge creep of 7 mm superimposed random irregularity of longitudinal level are shown in Figure 15. Results of track slab upwarp of 4 mm superimposed random irregularity of longitudinal level are shown in Figure 16. Results of bridge creep of 7 mm superimposed track slab upwarp of 4 mm and random irregularity of longitudinal level are shown in Figure 17.

Figure 14.

Random track irregularity.

Figure 15.

Superposition of bridge creep of 7 mm and random track irregularity.

Figure 16.

Superposition of slab upwarp deformation of 4 mm and random track irregularity.

Figure 17.

Superposition of bridge creep of 7 mm, slab upwarp deformation of 4 mm, and random track irregularity.

Coupled equation and numerical solution method for vehicle-track-substructure

Vehicle, track, and bridge constitute the dynamic simulation analysis system, which is coupled by wheel-rail force. The system vibration should obey the force equilibrium and deformation compatibility at any time. The motion equation of the system can be expressed as:

$$[M]\{\stackrel{··}{x}\}+[C]\{\stackrel{·}{x}\}+[K]\{x\}=\{P\}$$

(8)

Where $\{x\}$ , $\{\stackrel{·}{x}\}$ , $\{\stackrel{··}{x}\}$ , and $\{P\}$ represent the displacement, velocity, acceleration, and load vector of the whole system individually. The array $\{x\}$ and $\{P\}$ are expressed as:

$$\{x\}={\left\{\begin{array}{ccc}{x}_{\mathrm{c}}\hfill & x_{\mathrm{r}}\hfill & x_{\mathrm{b}}\hfill \end{array}\right\}}^{\mathrm{T}}$$

(9)

$$\{P\}={\left\{\begin{array}{ccc}{P}_{\mathrm{c}}\hfill & P_{\mathrm{r}}\hfill & P_{\mathrm{b}}\hfill \end{array}\right\}}^{\mathrm{T}}$$

(10)

where $[M]$ , $[C]$ , $[K]$ are the mass matrix, damping matrix, and stiffness matrix of the whole system, individually. These parameters are expressed as:

$$\begin{array}{c}[M]=\left[\begin{array}{ccc}{M}_c\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& M_r\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & M_{\mathrm{b}}\hfill \end{array}\right][C]=\left[\begin{array}{ccc}{C}_{\mathrm{cc}}\hfill & C_{\mathrm{cr}}\hfill & 0\hfill \\ \multicolumn{1}{c}{C_{\mathrm{rc}}}& C_{\mathrm{rr}}\hfill & C_{\mathrm{rb}}\hfill \\ \multicolumn{1}{c}{0}& C_{\mathrm{br}}\hfill & C_{\mathrm{bb}}\hfill \end{array}\right]\hfill \\ [K]=\left[\begin{array}{ccc}{K}_{\mathrm{cc}}\hfill & K_{\mathrm{cr}}\hfill & 0\hfill \\ \multicolumn{1}{c}{K_{\mathrm{rc}}}& K_{\mathrm{rr}}\hfill & K_{\mathrm{rb}}\hfill \\ \multicolumn{1}{c}{0}& K_{\mathrm{br}}\hfill & K_{\mathrm{bb}}\hfill \end{array}\right]\hfill \end{array}$$

(11)

Where $x_{\mathrm{c}},x_{\mathrm{r}},x_{\mathrm{b}}$ represent the displacement vectors of each degree of freedom for vehicle, track, and substructure, individually. $P_{\mathrm{c}},P_{\mathrm{r}},P_{\mathrm{b}}$ represent the load vectors of each degree of freedom for vehicle, track, and substructure, individually. $M_{\mathrm{c}},M_{\mathrm{r}},M_{\mathrm{b}}$ represent the mass matrixes of each degree of freedom for vehicle, track, and substructure, individually. $K_{\mathrm{cc}},K_{\mathrm{rr}},K_{\mathrm{bb}}$ represent the stiffness matrix of each degree of freedom for vehicle, track, and substructure, individually. The damping matrixes are similar. $K_{\mathrm{cr}}$ represents the additional stiffness matrix of vehicle caused by wheel-rail contact. $K_{\mathrm{br}}$ represent the additional stiffness matrix of substructure caused by track stiffness. $K_{\mathrm{rc}},K_{\mathrm{rb}}$ are equal to $K_{\mathrm{cr}},K_{\mathrm{br}}$ for the absolute value, but the meaning is opposite. $C_{\mathrm{rc}},C_{\mathrm{rb}},C_{\mathrm{cr}},C_{\mathrm{br}}$ are similar to the stiffness matrixes.

The Newmark- $\beta$ method is an unconditionally stable implicit integral scheme, belonging to self-starting method. The value of time step $\mathrm{\Delta }t$ does not affect the stability of the solution, 5 × 10⁻⁵ second is adopted in the current study.

Determination threshold of bridge creep and slab upwarp deformation based on running performance and structural service performance

Based on vehicle-track-substructure dynamic model, track longitudinal level caused by bridge creep, slab upwarp deformation and superposition of both are determined to be various calculation cases, with the train speeds at 200, 230, 250, 280, 300, 330, and 350 km/h for each case. The car body vertical vibration acceleration, Sperling index, rate of wheel load reduction (ΔP/P), derailment coefficient, and wheel-rail vertical force are used to evaluate the running performance, and the detailed reference standard mentioned above is presented in the literature. ³⁷ Finally, the vertical displacement at midspan and vertical accelerations at midspan and beam end for tracks, slabs and bridges are employed to investigate the structure service performance. Based on running safety and ride comfort, the deformation thresholds are proposed.

Case study of bridge creep

According to current situation of bridge creep for 32 m-span simply supported girder bridge for high-speed railway and track geometric irregularity presented in Section “Selection of TLL,” the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of bridge creep are adjusted to 5, 6, 7, 8, 9, and 10 mm individually, and then dynamic simulation calculation is carried out. The threshold value for bridge creep is determined according to the running and structure performance. From Figure 18(a), the vehicle vibrates periodically due to the bridge creep, and the vertical acceleration of car body are larger at mid-span and beam joint, moreover, from the FFT curve at the Figure 18(b), it can be seen that the vehicles produce mainly low frequency vibration, and the vehicles are motivated to vibrate obviously at 2.93 Hz by bridge creep. On the other hand, the rate of wheel load reduction time curve and its FFT when the bridge creep is 7 mm are shown in Figure 19, it can be seen, the rate of wheel load reduction includes low and high frequency characteristics, and the 2.93 Hz caused by bridge creep is also obvious. It should be noted that the maximum cut-off frequency is 40 Hz according to the literature. ³⁷

Figure 18.

Vertical acceleration time curve of car body and its FFT: (a) time curve and (b) FFT.

Figure 19.

Rate of wheel load reduction time curve and its FFT when the bridge creep is 7 mm: (a) time curve and (b) FFT.

From Figure 20(a) to (e), it can be seen that the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of bridge creep shows a significant effect on ride comfort and little influence on running safety. When the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of bridge creep reaches 10 mm and the train speed is 350 km/h, the maximum ΔP/P is 0.16. Meanwhile, the derailment coefficient and wheel-rail vertical force are small. The vertical vibration acceleration and ride comfort present a linear growth trend with the increase of $V_{\mathrm{pp}}^{\mathrm{ll}}$ of bridge creep and train speed. When the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of bridge creep reaches 10 mm and the train speed is 350 km/h, the vertical vibration acceleration of car body is 1.02 m/s², just beyond the class I-regular maintenance index of track dynamic management threshold. When the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of bridge creep reaches 9 mm and the train speed is 350 km/h, the ride comfort has exceeded the qualified threshold of 3.0. ³⁷ It can be seen from Table 3 (just listing certain maximum results under different speeds) that the vertical displacement and vertical vibration acceleration for rail, slab, and bridge increase with the increase of speed. Bridge creep has little influence on vertical displacement and vertical vibration acceleration. Moreover, the vertical vibration accelerations at beam end are larger than those at midspan for tracks, slabs, and bridge, but do not exceed the structure dynamic acceptance standard. ³⁸ Considering structure deterioration, creep and maintenance capacity for high-speed railway under long-term service conditions, without deficiencies of track slabs on bridge, the $V_{\mathrm{pp}}^{\mathrm{ll}}$ threshold caused by creep for 32 m simply supported girder bridge with continuous equal span arrangement is recommended to be 7 and 9 mm in difficult cases, which is consistent with the reference. ¹

Figure 20.

Vehicle dynamic response: (a) Vertical acceleration of car body, (b) Sperling/vertical, (c) rate of wheel load reduction, (d) derailment coefficient, and (e) wheel-rail vertical force.

Table 3.

Maximum dynamic response results of structure.

Amplitude of bridge creep (mm)	Speed (km/h)	Vertical displacement at midspan			Vertical vibration acceleration at midspan			Vertical vibration acceleration at beam end
		Rail (mm)	Track slab (mm)	Bridge (mm)	Rail (m/s2)	Track slab(m/s2)	Bridge (m/s2)	Rail (m/s2)	Track slab (m/s2)	Bridge (m/s2)
5	300	−1.244	−0.757	−0.688	40.181	1.606	0.257	44.381	1.855	0.342
	330	−1.248	−0.761	−0.692	39.162	1.795	0.281	51.479	3.009	0.358
	350	−1.250	−0.759	−0.695	50.049	1.906	0.272	102.168	3.377	0.438
6	300	−1.239	−0.753	−0.684	41.419	1.610	0.262	53.328	2.231	0.378
	330	−1.240	−0.755	−0.687	39.097	1.739	0.274	61.445	3.175	0.411
	350	−1.248	−0.761	−0.696	51.261	1.895	0.277	116.262	3.481	0.502
7	300	−1.233	−0.755	−0.686	39.958	1.555	0.253	62.419	2.841	0.390
	330	−1.234	−0.758	−0.690	38.878	1.726	0.270	71.272	3.304	0.439
	350	−1.248	−0.760	−0.695	52.745	1.930	0.284	127.781	3.585	0.523
8	300	−1.230	−0.751	−0.682	39.935	1.623	0.261	70.792	2.710	0.429
	330	−1.229	−0.753	−0.684	39.179	1.771	0.275	79.911	3.630	0.482
	350	−1.250	−0.762	−0.698	53.377	1.920	0.284	141.364	3.700	0.577
9	300	−1.230	−0.749	−0.680	40.937	1.552	0.245	80.111	2.692	0.458
	330	−1.227	−0.750	−0.682	40.470	1.734	0.275	89.617	3.637	0.532
	350	−1.252	−0.763	−0.699	54.827	1.932	0.302	153.536	3.810	0.621
10	300	−1.223	−0.747	−0.678	48.328	1.646	0.233	88.330	2.909	0.483
	330	−1.222	−0.747	−0.679	53.179	1.720	0.284	96.719	3.815	0.570
	350	−1.250	−0.763	−0.701	55.692	2.209	0.330	166.315	3.932	0.650

Analysis of track slab deformation

According to current situation for slab ballastless track deficiencies for high-speed railway and track geometric irregularity presented in Section “Selection of TLL,” the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of track slab upwarp deformation are adjusted to 1, 2, 3, 4, and 5 mm, and then dynamic simulation calculation is carried out. The wavelength of track geometric irregularity caused by track slab deformation is about 5–6 m, which is shorter than that caused by bridge creep. Track slab deformation has a great impact on running performance, especially running safety. From Figure 21(a), there is no influence of bridge creep, but the vertical acceleration of car body are larger when the train passes the positions of mid-span and beam joint, in particular, the vertical acceleration of car body reaches the maximum value because the train will be impacted at the beam joint. At the same time, the vehicle gets some high-frequency shocks when the train passes the tracks, and the FFT curve of the vertical acceleration shows in the Figure 21(b), it can be seen that the vehicles produce high frequency vibration by track slab deformation, which the frequencies are larger than 17 Hz, for instance 17.77 and 20.90 Hz or greater, but the low frequency vibrate at 2.93 Hz by bridge structure is still most pronounced. On the other hand, the rate of wheel load reduction time curve and its FFT when the track slab deformation is 4 mm are shown in Figure 22, it can be seen, the rate of wheel load reduction main includes high frequency characteristics, which are 17.97 and 20.90 Hz or greater, but the 2.93 Hz caused by bridge creep is not obvious, and the rate of wheel load reduction will be the greatest when the train passes beam joints.

Figure 21.

Vertical acceleration time curve of car body and its FFT: (a) time curve and (b) FFT.

Figure 22.

Rate of wheel load reduction time curve and its FFT: (a) time curve and (b) FFT.

It can be obtained from Figure 23, When the amplitude of track slab deformation reaches 5 mm and train speed is 350 km/h, the ΔP/P is 0.82, beyond the safety threshold of 0.8. With the same track slab deformation, the derailment coefficient is 0.81 and 0.82 at train speed of 330 and 350 km/h, respectively, exceeding the safety threshold of 0.8. The wheel-rail vertical force increases rapidly with the increase of $V_{\mathrm{pp}}^{\mathrm{ll}}$ of track slab deformation and train speed. The maximum value is 113.25 kN, which does not exceed the safety threshold of 170 kN . ³⁷ Both the vertical vibration acceleration and ride comfort index increase with the increase of slab deformation and train speed. When the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of track slab deformation reaches 5 mm and train speed is 350 km/h, the vertical vibration acceleration is 1.03 m/s², just beyond the class I-regular maintenance index of track dynamic management threshold. The vertical Sperling index is 2.29, corresponding to the best level. It can be seen from Table 4 (just listing certain maximum results under different speeds) that the vertical displacements for rail, slabs, and bridge barely vary. The vertical vibration accelerations at the beam end increase rapidly with the increasing train speed and slab deformation. Among them the rail is the most significant, however, they are all within the dynamic acceptance standard. Therefore, considering slab ballastless track, deficiencies and maintenance capacity for 350 km/h high-speed railway, without bridge creep, the $V_{\mathrm{pp}}^{\mathrm{ll}}$ threshold caused by slab upwarp deformation is proposed to be less than 4 mm (Figure 23).

Figure 23.

Vehicle dynamic response: (a) vertical acceleration of car body, (b) Sperling/vertical, (c) rate of wheel load reduction, (d) derailment coefficient, and (e) wheel-rail vertical force.

Table 4.

Maximum dynamic response results of structure.

Amplitude of slab deformation (mm)	Speed (km/h)	Vertical displacement at midspan			Vertical vibration acceleration at midspan			Vertical vibration acceleration at beam end
		Rail (mm)	Track slab (mm)	Bridge (mm)	Rail (m/s2)	Track slab (m/s2)	Bridge (m/s2)	Rail (m/s2)	Track slab (m/s2)	Bridge (m/s2)
1	300	−1.249	−0.767	−0.698	38.222	0.747	0.182	38.222	0.747	0.365
	330	−1.238	−0.757	−0.687	37.387	0.810	0.196	37.387	0.810	0.392
	350	−1.231	−0.740	−0.671	46.244	0.834	0.200	46.244	0.834	0.400
2	300	−1.243	−0.767	−0.699	36.665	0.764	0.184	84.934	1.275	0.226
	330	−1.226	−0.758	−0.687	32.080	0.793	0.196	116.479	2.363	0.306
	350	−1.216	−0.742	−0.671	43.188	0.935	0.220	68.615	2.506	0.304
3	300	−1.232	−0.769	−0.699	36.494	0.794	0.186	238.418	4.456	0.383
	330	−1.216	−0.759	−0.688	30.504	0.828	0.251	304.626	6.110	0.466
	350	−1.199	−0.743	−0.672	40.530	1.072	0.285	626.497	8.372	0.628
4	300	−1.221	−0.770	−0.700	31.930	0.759	0.245	786.942	9.305	0.581
	330	−1.201	−0.759	−0.689	28.536	1.073	0.434	907.817	13.191	0.785
	350	−1.266	−0.743	−0.674	33.792	1.133	0.491	1302.593	12.759	0.990
5	300	−1.211	−0.773	−0.702	32.303	0.939	0.469	1534.785	7.817	0.849
	330	−1.389	−0.761	−0.691	27.948	1.065	0.548	2605.088	23.062	1.317
	350	−1.463	−0.745	−0.675	32.848	1.220	0.564	2324.422	19.876	1.320

Analysis of superposition of bridge creep and slab upwarp deformation

Based on the above bridge creep and slab upwarp deformation, the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of bridge creep are adjusted to be 5, 6, 7, and 8 mm, with the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of slab deformation of 1, 2, 3, 4, and 5 mm superimposed, respectively. The dynamic simulation calculation is carried out, and the threshold value under multiple deficiencies are determined according to running performance and structural service performance. From Figure 24(a), there is two effects of bridge creep and track deformation, and the vehicles get more vibration when the vehicles pass the positions of mid-spans, beam joints, and tracks, so the vertical acceleration of car body are larger, in particular, and the vertical acceleration of car body excited by bridge creep are greater than that by track, furthermore, the vibration of car body caused by bridge creep is low-frequency vibration, but the vibration cause by tracks is high-frequency vibration, and which are proved by the FFT analysis shown in the Figure 24(b). On the other hand, the rate of wheel load reduction time curve and its FFT when the track slab deformation is 4 mm with the bridge creep is 5 and 7 mm are shown in Figure 25, it can be seen, the rate of wheel load reduction main includes high frequency characteristics, which are 17.87 and 20.80 Hz or greater, and the low frequency of 2.93 Hz caused by bridge creep is very little.

Figure 24.

Vertical acceleration time curve of car body and its FFT: (a) time curve and (b) FFT.

Figure 25.

Rate of wheel load reduction time curve and its FFT: (a) time curve and (b) FFT.

From Sections “Case study of bridge creep” to “Analysis of track slab deformation” and Figure 26, it can be seen that the creep and slab deformation have great influence on ride comfort. The bridge creep has little influence running safety, however, the slab deformation has great influence on running safety. When the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of slab deformation reaches 5 mm and train speed is 350 km/h or 330 km/h, no matter what bridge creep, the ΔP/P and derailment coefficient exceed the allowable threshold. When the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of slab deformation reaches 4 mm and train speed is 350 km/h, no matter what bridge creep, the derailment coefficient approaches the allowable threshold. Meanwhile, the wheel-rail vertical force increases rapidly with the increase of slab deformation, but do not exceed the threshold value.

Figure 26.

Vehicle dynamic response: (a) vertical acceleration of car body, (b) Sperling index/vertical, (c) rate of wheel load reduction, (d) derailment coefficient, and (e) wheel-rail vertical force.

The vertical vibration acceleration of car body and Sperling index increase with the increase of both deformations and train speed. When the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of bridge creep is 8 mm, and the $V_{\mathrm{pp}}^{\mathrm{ll}}$ of slab continuous deformation is 5 mm, and the train speed is 350 km/h, the vertical vibration acceleration of car body reaches 1.22 m/s², within the class II Comfort index of the dynamic management threshold. ¹⁶ However, the Sperling index is unqualified. When the slab deformation reaches 4 mm, the Sperling is close to the unqualified threshold. It can be seen from Table 5 (just listing certain maximum results under different speeds) that the bridge creep and slab deformation have little effect on the vertical displacement for tracks, slabs, and bridge. Compared to the bridge creep, the vertical vibration acceleration at the midspan and beam end is more sensitive to the slab deformation. Among them the rail is the most significant, however, they are all within the dynamic acceptance range. Therefore, considering slab ballastless track, deficiencies and maintenance capacity for 350 km/h high-speed railway, the threshold of $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by slab deformation is proposed to less than 4 mm, and the threshold of $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by bridge creep is proposed to less than 7 mm. Therefore, the superposition of both deformations is 11 mm. For the sake of safety redundancy, referring to the relevant standards in the maintenance rules of ballastless track line of high-speed railway, it is finally recommended that when both deformations presence, the threshold of $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by continuous slab deformation is less than 4 mm, and the threshold of $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by the superposition of both deformation is less than 10 mm. The engineering department should eliminate slab deformation when it is hard to eliminate bridge creep.

Table 5.

Maximum dynamic response results of structural.

Amplitude of bridge creep (mm)	Amplitude of slab deformation (mm)	Speed (km/h)	Vertical displacement at midspan			Vertical vibration acceleration at midspan			Vertical vibration acceleration at beam end
			Rail (mm)	Track slab (mm)	Bridge (mm)	Rail (m/s2)	Track slab (m/s2)	Bridge (m/s2)	Rail (m/s2)	Track slab (m/s2)	Bridge (m/s2)
5	4	300	−1.206	−0.762	−0.691	31.641	0.734	0.249	72.812	7.621	0.951
		330	−1.181	−0.744	−0.675	28.298	1.053	0.434	81.309	14.266	1.222
		350	−1.279	−0.732	−0.657	33.679	1.128	0.510	134.302	17.375	1.501
	5	300	−1.202	−0.764	−0.693	33.592	0.957	0.490	48.212	18.402	1.346
		330	−1.395	−0.746	−0.676	37.914	1.068	0.563	120.944	24.554	1.903
		350	−1.527	−0.735	−0.658	44.502	1.272	0.590	87.492	26.954	2.046
6	4	300	−1.203	−0.760	−0.689	33.029	0.783	0.257	340.424	8.014	0.941
		330	−1.177	−0.742	−0.672	31.486	1.055	0.436	689.835	15.259	1.263
		350	−1.283	−0.730	−0.657	35.956	1.194	0.513	771.808	20.400	1.514
	5	300	−1.214	−0.763	−0.685	30.159	0.903	0.486	235.234	18.338	1.337
		330	−1.396	−0.743	−0.665	43.354	1.079	0.559	424.611	25.784	1.982
		350	−1.546	−0.733	−0.655	50.399	1.264	0.587	832.177	26.546	2.141
7	4	300	−1.218	−0.758	−0.687	33.798	0.950	0.255	863.240	9.235	0.974
		330	−1.183	−0.738	−0.669	53.333	1.113	0.423	1005.458	15.412	1.264
		350	−1.284	−0.731	−0.659	46.554	1.270	0.508	1287.680	19.644	1.500
	5	300	−1.211	−0.762	−0.690	41.254	1.110	0.473	738.483	18.332	1.346
		330	−1.396	−0.740	−0.671	48.442	1.392	0.574	1008.266	26.240	2.075
		350	−1.555	−0.733	−0.660	49.628	1.597	0.604	1289.775	27.213	2.263
8	4	300	−1.184	−0.757	−0.685	38.324	1.162	0.264	819.298	10.854	0.986
		330	−1.170	−0.735	−0.667	65.692	1.238	0.421	2213.806	15.962	1.254
		350	−1.285	−0.733	−0.661	60.878	1.309	0.494	2476.532	20.647	1.487
	5	300	−1.196	−0.759	−0.688	43.790	1.179	0.504	820.578	17.364	1.361
		330	−1.396	−0.737	−0.668	60.531	1.455	0.544	2216.379	26.859	2.152
		350	−1.565	−0.736	−0.662	53.941	1.677	0.587	2566.860	28.445	2.341

Conclusions

• (1) To make the track management more targeted and ensure the safety and comfort for high-speed railway operation, 10 years’ high-speed comprehensive inspection train data are analyzed, and the variation trends and level for bridge creep and ballastless track slab upwarp are given, which can give more clear understanding for bridge creep of 32 m simply-supported bridges with ballastless track and the ballastless track slab, and can guide the design, construction, and management. The main conclusions are as follows. Simply supported girder bridge creep and ballastless track slab upwarp are two important reasons for periodic track longitudinal level in high-speed railway. And these deformation mainly affect the vertical vibration response of vehicle.

• (2) There are two types of variation trends for bridge creep of 32 m simply-supported bridges with ballastless track China Railway High-Speed, that is, stable type and increasing type. The maximum $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by bridge creep reaches 7.1 mm.

• (3) The maximum $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by ballastless track slab upwarp deformation reaches 3.2 mm for China Railway High-Speed.

• (4) The car bodies are more sensitive to bridge creep and produce mainly low-frequency vibration, but the rate of wheel load reduction is more sensitive to the track deformation and shows the high-frequency vibration.

• (5) Bridge creep affects ride comfort significantly. For 32 m-span simply supported girder bridges of 350 km/h high-speed railway without slab upwarp deformation, it is proposed that the $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by bridge creep should be limited to 7, and 9 mm in difficult situations.

• (6) Due to the short wavelength of TLL, the slab upwarp deformation has influence on both the ride comfort and running safety heavily. For 350 km/h high-speed railway with small bridge creep (less than 5 mm), it is proposed that the $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by the continuous slab deformation should not exceed 4 mm.

• (7) When the bridge creep and slab upwarp deformation exist simultaneously, the $V_{\mathrm{pp}}^{\mathrm{ll}}$ caused by continuous slab upwarp deformation should not exceed 4 mm, and that caused by superposition deformations of both should not exceed 10 mm. It is suggested that the track slab deformation should be eliminated when it is difficult to eliminate the bridge creep.