Performance Analysis of Prefabricated Steel-Spring Floating-Slab Track and Its Application to Urban Express Rail Transit

)e aim of the research is the design of prefabricated steel-spring floating-slab track to be applied in urban express rail transit systems. Using a developed vehicle-track dynamic-coupling equation for steel-spring floating-slab track, the effects of length, thickness, vertical damping, and use of side-mounted isolators on the floating-slab track were investigated experimentally using full-scale model and under different working conditions. )e finding of the study revealed the following: (1) )e prefabricated steel-spring floating-slab track can be applied to urban express rail transit, because it meets the requirements of high-speed transit while efficiently reducing noise. (2))e floating-slab track’s stability slightly increases with the increase of its length and thickness. As thickness increases, vertical displacement of the rail increases slightly, and lateral stability increases, thereby slightly improving the vehicle’s running stability. (3) When the intercity electric multiple-unit train travels along the prefabricated steel-spring floating-slab-track bed at different speeds, the wheel-axle lateral force, wheel-rail vertical force, the derailment coefficient, the wheel-weight reduction rate, and the lateral acceleration of the vehicle body are all less than the specified limits of Chinese code, thus fully meeting the safety requirements of train operation. (4) Appropriately increasing the vertical-support damping of the floating slab can improve the vehicle’s vertical dynamic performance, reduce the vertical displacement of the rail, and lower the vibration response of the floating slab. (5) Adding side-mounted vibration isolators at the joint of the floating slab could greatly improve the stability of the floating slab itself and appropriately reduce the vehicle’s vertical vibration response. Due to the optimization and establishment of relevant factors influencing the performance of prefabricated steel-string floating-slab track achieved in the study, the results obtained are particularly useful for setting safety, comfort, and stability requirements of the floating slab.


Introduction
In recent years, the rapid development of urban express rail transit has brought great convenience to people's travel and transportation [1]. Urban express rail transit is an emerging category of rail transit, which is a cross between railway and urban rail transit, and is mainly used to solve intercity traffic problems. e development of urban express rail transit provides a new model for urban residents to live in one city and work in a neighboring city, which is of great significance for optimizing urban patterns and alleviating traffic problems in dense urban areas. us, the urban express rail transit system came into being to optimize urban patterns, ease the traffic pressure in dense urban areas, and better address publictransportation challenges between city centers and suburbs or satellite communities and between key towns.
Owing to the tight use of urban land, urban express rail transit lines are commonly located very close to residence areas or even in tunnels below residents' gathering places or on viaduct structures, which are gradually adopted in large areas [2]. Because of this close proximity, a serious problem to be addressed is environmental vibration due to the high speed of urban express rail transit (such as 160 km/h, the design speed of Guangzhou Rail Transits nos. 18 and 22) [3,4]. e floating-slab track is a structure that can effectively reduce vibration and noise caused by vehicles on rails. e proposed steel-spring floating-slab track uses a prefabricated reinforced-concrete structure to form an integral track. A steel-spring isolator is used to elastically isolate the track slab from the foundation to form a mass-spring vibration-isolation system. Furthermore, a side-mounted vibration isolator is incorporated to constrain lateral displacement and vibration of the track slab. e advantages are that the steelspring floating-slab track has better three-dimensional elasticity, less lateral displacement, good vibration-isolation performance, and easiness to maintain and replace [5,6].
Many scholars have investigated the dynamic characteristics of floating-slab track under vehicle loads [7][8][9][10][11][12][13]. In 2009, Zhai et al. proposed a method to investigate the dynamics of vehicle-track systems with emphasis on theoretical modeling, numerical simulation, and experimental validation. ey modeled a traditional ballasted track as two continuous parallel beams supported by a discrete elastic foundation consisting of three layers, including sleepers and ballasts. ey also modeled the nonballasted slab track as two continuous parallel beams supported by a series of elastic rectangular slabs on a viscoelastic foundation [7]. On the basis of the above, a coupleddynamics computation model for metro vehicles, and steelspring floating-slab track, was developed as the influence of factors on the coupled system was explored such as the floating-slab dimensions (thickness and length) and mass, spring rate and spatial arrangement, and running speed [8]. MFM Hussein et al. proposed a new modeling method for discontinuous floating-slab track applied to subway. e coupling relationship between two submodels of track and tunnel under train load was expressed by Fourier series. e floating-slab track studied was found to display good vibration response [9]. Lombaert established a vehicle-track-foundation coupled dynamics model through three-dimensional numerical modeling method and evaluated the vibration reduction effect of the floating-slab track under different working conditions [10]. Hunt combined a vehicle with a track model to create a method that can be used to calculate the vibration transfer between the track and building. e results obtained in the study are valuable in evaluating the vibration response between rails, fasteners, floating track, and foundation [11]. Xu et al. proposed and applied a probabilistic model to simulate the characteristic of track irregularities by employing vehicletrack and vehicle-slab coupled system [12,13]. Wang et al. established a spatial dynamic model of train steel-spring floating-slab-track interaction and analyzed the vibration characteristics of train passing through the steel-spring floating-slab track [14]. Huang et al. built a vehicle-track coupled dynamic system and investigated how the parameters of the floating-slab track and the train's speed influence the vehicletrack coupling system [15]. Liang et al. investigated the vibration characteristics of the damping-pad floating slab on the long-span steel-truss cable-stayed bridge in urban rail transit by developing a theoretical model of the train-track-bridge coupling interaction in the frequency domain [16]. Lu et al. established a vehicle-track dynamic interaction model to investigate the wheel-rail interaction characteristics of the steelspring floating-slab track and carried out the dynamic analysis between the subway vehicle and the steel-spring floating-slab track under emergency braking conditions [17].
At the same time, in the research of the new floating-slab track structure that has side-mounted isolator to reduce lateral vibration, Park et al. proposed a new type of vibration isolator to overcome the shortcomings of the conventional floating-slab track and achieved good results [18]. Zhu et al. effectively suppressed the low-frequency vibration of the steel-spring floating-slab track by using a dynamic vibration absorber [19]. Ding et al. obtained the low-frequency vibration performance of the floating slab by vibration testing of the floating-slab track and optimized the parameters of the floating slab [20]. e above researches have achieved certain results in the optimization and improvement of the floating-slab structural parameters, but there is no research on the effect of lateral displacement limit effect of the floating slab. e existing researches considered the interaction principles between the subway train and the steel-spring floating-slab track under normal circumstances and lower speed [21,22]. Currently, there is still limited experience in applying steel-spring floating-slab track in urban express rail transit for higher speed worldwide. erefore, this study focuses on the use of prefabricated steel-spring floating-slab track in urban express rail transit for higher speed. e wheel-rail dynamic performance under high-speed conditions was investigated by applying vehicle-track coupled dynamics theory and simulation technology. e analysis and evaluation of the vehicle-track dynamics, the vehicle operational safety, ride comfort, and track structural stability, were conducted according to Chinese railway dynamic performance evaluation standards. Ultimately, the study aimed to demonstrate the feasibility of using prefabricated steel-spring floating-slab track for urban express rail transit at very high running speed. In the developed model, optimization of the key structural parameters of the floating-slab track was performed to provide the theoretical basis and technical support for its engineering design and application to urban express rail transit.

Vehicle-Track Coupled Dynamics Model of Steel-Spring
Floating-Slab Track. e study proposes a model to simulate the dynamic interaction between the vehicle and the floating-slab track in order to effectively evaluate the running safety and stability of an intercity electric multiple-unit (EMU) train under conditions of different running speeds and floating-slab-track bearing stiffness. In addition, floating-slab-track vibration characteristics were investigated. e vehicle-track coupled dynamics model of the intercity EMU train was based on vehicletrack coupled-dynamics theory [23] (Figure 1). In the model, the vehicle is simulated as a rigid multicomponent system consisting of a car body, a frame, and a wheelset. e lateral, vertical, side roll, shake, and nod movements of each part were considered. e rail is simulated as a Bernoulli-Euler beam supported on a base of elastic points. e rail-support points are arranged according to the actual fastener nodes' spacing by considering lateral, vertical, and rotational degrees of freedom. e vertical direction of the floating slab is simulated as a bidirectional curved elastic thin slab on an elastic foundation; the lateral direction is simulated as a rigid body, considering translational and rotational degrees of freedom. e concrete foundation is also simulated as a bidirectional curved elastic thin slab on the elastic foundation. e wheel-rail normal force was determined by the Hertz nonlinear elastic-contact theory, and the tangential force was determined by the nonlinear creep theory [24].

Vehicle Dynamic Equations.
According to the multibody system dynamics, the vehicle subsystem is built by considering seven rigid parts involving a car body, two bogies, and four wheelsets with the primary and the secondary suspensions. Each component is, respectively, assigned with 5 degrees of freedom (DOFs) involving the vertical displacement Z, the lateral displacement Y, the roll angle Φ, the yaw angle Ψ, and the pitch angle β. erefore, the vehicle subsystem has a total of 35 DOFs. For more details about the vehicle dynamic equations, monograph [25] can be consulted for readers.

Track Dynamic Equations.
According to the method mentioned in [26], the track model consists of rail and floating slab. e vibrations of both were considered at the same time. e equations of motion are shown in (1)-(5).
(1) Rail dynamic equation e rail is treated as a Bernoulli-Euler beam resting on the rail pads, and the lateral, vertical, and torsional vibrations are simultaneously taken into account. By adopting the modal superposition method, the second-order ordinary differential equations of the rail vibration can be obtained: where E r and G r are Young's modulus and shear modulus of the rail, respectively; A r and ρ r are the cross-sectional area and mass density of the rail, respectively; J ry , and J rz are, respectively, the moments of inertia of the rail section to the lateral and vertical axes; J r0 and J rt are the polar moments of inertia and torsional moment of inertia of the rail section, respectively; N s and N w are the numbers of sleepers and the number of axles in the rail section; F rVi , F rHi , and F rTi are the vertical reaction force, lateral reaction force, and torsion reaction force of the i-th fulcrum of the rail, respectively; P Vj , P Hj , and P Tj are the vertical force, lateral force, and torque of the rail acted by j-th wheel, respectively; and x Fj and x Pj are the x-coordinate of the i-th fulcrum of the rail and the x-coordinate of the j-th wheel set, respectively. (2) Floating-slab dynamic equation e floating slab is regarded as an elastic thin plate, whose governing equation is given by where P rVi is the vertical force of the i-th rail fastener point on the track slab; F sVj is the vertical reaction force of the j-th steel spring isolator under the track slab; F cVk is the vertical shear force of the k-th force hinge between the floating slabs; z s (x, y, t) is the vertical displacement or deflection of the floating slab; x Pi and y Pi are the positions of the i-th rail fasten points on the floating slab; x Fj and y Fj are the positions of the j-th steel spring isolators under the floating slab; x Ck and y Ck are the positions of the k-th shear joints between the floating slabs; and h s , ρ s , C s , E s , v s , and D s are slab thickness, density, damping coefficient, modulus of elasticity, Poisson's ratio, and bending stiffness, respectively.

Advances in Civil Engineering
Floating slab

Steel spring
Side-mounted isolators  e generalized coordinate T mn (t) of the floating slab is introduced, and the above partial differential equation is converted into a second-order ordinary differential equation by the Ritz method, as shown in equation (7): where m � 1, 2, . . . , N x and n � 1, 2, . . . , N y .
At moment t, the vertical displacement at point (x, y) of the track is where N x and N y are the cutoff mode orders of directions for length and width of the floating slab, respectively, and X m (x) and Y n (y) are the beam-mode functions of directions for length and width of the floating slab, respectively.

e Wheel and Rail Interaction
Principle. e vehiclefloating-slab track (FST) is a dynamic interaction system, and the wheel-rail relationship is the link between the vehicle subsystem and the track subsystem. In previous vehicletrack dynamics equations, if the wheel-rail interaction force was determined, the numerical simulation method was applied, and the dynamics simulation analysis of the vehicletrack system could be performed. In this research, the wheelrail contact geometry was determined according to the principle of wheel-rail contact mentioned in [27]. e wheel and rail normal force and wheel and rail creep force were calculated according to the method mentioned in [26,28]. On obtaining the wheel-rail force, the values can be substituted into the dynamic equations of the vehicle and the track as the reaction force of the wheel and the external load of the track.

Track Irregularities.
Because the vehicle-track coupled dynamic system is very complicated and extensive, a fast explicit-integration method should be used to solve its dynamic response problem [29]. At present, China has no reliable track-irregularity data for urban express rail transit. For the purpose of analysis, considering the characteristics of the floating-slab track and the deterioration of railway-line smoothness after long-term operation, the excitation input of the vehicle-track dynamic system was based on the U.S. six-grade track spectrum that closely matches the Chinese urban express rail transit [30]. Accordingly, the powerspectral-density expressions of the track-vertical-profile, track-alignment, rail-gauge, and track-cross-level irregularities can be expressed as shown in equations (10) and (11): (1) Track-vertical-profile irregularity: where S v (Ω) is the power-spectral density of trackvertical-profile irregularity [cm 2 /(rad/m)], A v is the roughness constant (cm 2 ·rad/m), Ω c is the cutoff frequency (rad/m), and k is the safety coefficient. (2) Track-alignment irregularity: where S v (Ω) is the power-spectral density of trackalignment irregularity [cm 2 /(rad/m)], A a is the roughness constant (cm 2 ·rad/m), and Ω c is the cutoff frequency (rad/m). (3) Rail-gauge and track-cross-level irregularities: where S c (Ω) and S g (Ω) are the power-spectral densities of rail-gauge and track-cross-level irregularities [cm 2 /(rad/m)], respectively, and Ω s is the cutoff frequency (rad/m). According to the track power spectrum density expression, a new algorithm based on the frequency domain power spectrum equivalent was used to obtain the amplitude and stochastic phase of the spectrum. e inverse Fourier transform was used to obtain time-domain samples of the stochastic irregularity of the track (Figures 2∼5), which were used as the excitation input of the vehicle-track dynamics system.

Numerical Integration Method.
It can be seen that the developed dynamics model has large DOFs involving many nonlinear factors and time-varying parameters. Consequently, an efficient numerical integration algorithm is essential for this problem. In this paper, the Zhai method [31] Advances in Civil Engineering is adopted to solve such a large-scale dynamic model, which has the integration form as follows: where X, _ X, and € X are the generalized displacement, velocity, and acceleration of the system, respectively; Δt is the time step for numerical integration; the independent parameters φ and ψ are used for controlling the stability of the algorithm; the subscript n indicates the integration at the time of nΔt.

Vehicle Parameters.
e fully loaded CRH6 intercity EMU train was considered as model vehicle when setting the vehicle parameters required for the vehicle-track-coupled dynamics simulation while keeping the overall parameters within safety limits (Table 1).

Rail Parameters.
To ensure overall vehicle safety, the full-load parameters for the CRH6 intercity EMU train were considered. e basic calculation parameters for the rail are shown in Table 2.

Floating-Slab-Track Parameters and Layout Scheme.
We studied and analyzed mainly 3.6 m and 4.8 m long floating-slab tracks. In order to reduce the vertical displacement at the joint of adjacent floating slabs, an increase in the stiffness transition of the floating slab by sidemounted vibration isolator was proposed. e side-mounted isolator was added to each end of the floating slab, as shown in Figures 6∼8. e basic parameters of the 3.6 m long prefabricated floating slab are shown in Table 3, and layout schematics are shown in Figures 9-11. e basic parameters of the prefabricated 4.8 m long floating slab are shown in Table 4, and layout schematics are shown in Figures 6, 12, and 13.

Analysis of Driving Safety and Stability
Running on Prefabricated Steel-Spring Floating Slab    Advances in Civil Engineering investigated. Also, the performance and the stability of the track itself were analyzed. e working conditions (Tables 5-13) considered in this study are straight line-140, straight line-160, and straight line-200, which indicates that the intercity EMU train passes through a straight line at the speeds of 140, 160, and 200 km/h, respectively, and Curve-140 and Curve-160, which indicates that the intercity EMU train travels the curve sections of R � 1100 m and R � 1500 m at the speeds of 140 and 160 km/h, respectively. L values of 3.6 and 4.8 m indicate the two lengths of the prefabricated steel-spring floating-slab track. e calculation results of wheel-railsystem dynamic response, vehicle stability and comfort, rail dynamic response, and floating-slab dynamic response under various conditions are shown in Tables 5-8. ese tabulated results suggest the following: (i) When the intercity EMU train runs on either the 3.6 m or 4.8 m long prefabricated floating track beds, its dynamic performance is basically the same, although the stability of the 3.6 m floating slab is slightly lower than that of the 4.8 m slab. (ii) Whether the intercity EMU train runs at 140, 160, or 200 km/h on the straight sections and 140 or 160 km/ h in the curved sections (R � 1100 m or R � 1500 m) of either the 3.6 m or 4.8 m prefabricated floatingslab-track bed, the wheel-axle lateral force, the wheel-rail vertical force, the derailment coefficient, and the wheel-load shedding rate are all less than the specified code limit. e lateral acceleration of the body is lower than the specified code limit, the stability index is "excellent," and the comfort rating is "comfortable."               thickness increases, the vertical displacement of the track bed increases, and lateral stability improves. (iii) e thickness variation of the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab tracks has little effect on the operational-safety and ridecomfort indexes of the intercity EMU trains. (iv) With the increase of the thickness of the 3.6 m and 4.8 m long prefabricated steel-spring floating-slabtrack beds, the wheel and rail safety index slightly increased, and the vehicle-stability and ride-comfort indicators were slightly reduced. e reason for the increase in the wheel-rail safety index is that the increase in the thickness of the floating slab increases the wheel-rail impact at the transition joint of the track slab.

Analysis of the Influence of Vertical-Support Damping of the Floating Slab on Vehicle-Track Dynamics Characteristics.
To clearly investigate the influence of vertical-support damping of the floating slab (vertical damping of the vibration isolator) on the vehicle-track dynamics, no sidemounted isolator was provided during the analysis, the    ( Figure 16) was added at each end of the floating slab (Tables 14 and 15). We considered the situations without (Figures 17(a) and 18(a)) and with (Figures 17(b) and 18(b)) side-mounted isolators. For the five working conditions (straight lines and curves), the dynamic response of the intercity EMU train running on the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab-track beds was calculated.
Results of the analysis revealed the following: (i) For the 3.6 m and 4.8 m long prefabricated steelspring floating-slab-track beds, whether or not the side-mounted isolators are installed, the runningsafety index and the ride-comfort index are basically similar during the intercity EMU train operation; it can be concluded that the impact of the sidemounted isolator on vehicle dynamic performance is not significant when the device is installed. Specifically, according to the analysis results, the addition of the side-mounted isolator can notably reduce the vertical force between the wheel and rail and the vertical-vibration acceleration of the vehicle body while slightly reducing the vertical stability of the vehicle body (Table 16). Accordingly, the sidemounted isolator can improve the vertical dynamic performance of the vehicle. (ii) For the 3.6 m and 4.8 m long steel-spring floatingslab-track beds, after adding the side-mounted vibration isolator, the track deformation and vibration-response index are significantly reduced. at is, side-mounted isolator improves the stability of the floating-slab bed significantly.

Conclusions
Research shows that prefabricated steel-spring floating-slab track, which is traditionally used for low-speed lines, can actually be used for higher-speed lines and can also achieve significant results for the vibration and noise reduction function under conditions of driving safety and operational stability. e study is an exploratory study of the application of the new prefabricated steel-spring floating-slab track in the field of high-speed rail transportation. From the above analysis, we can draw the following conclusions: (1) A prefabricated steel-spring floating-slab track can be applied in urban express rail transit systems and can meet the requirements of safety, comfort, and stability of high-speed vehicles while efficiently reducing noise. is is a great guiding principle for the popularization of prefabricated steel-spring floatingslab tracks on high-speed railway lines. (2) e dynamic performance of the 3.6 m and 4.8 m long prefabricated steel-spring floating slabs is comparable, although the stability of the former is slightly lower than that of the latter. For the shorter length of the prefabricated steel-spring floating-slab track is often used in curved sections, it can be seen that when the short section of the steel-spring floating-slab track is used in curved sections, the stability of the vehicle-track coupled dynamic system will be reduced but will still remain within the acceptable range of the engineering project. (3) For same-length prefabricated steel-spring floatingslab tracks, as slab thickness increases, the vertical displacement of the rail increases slightly, and lateral stability improves. A change of slab thickness has a little effect on the running-safety and ride-comfort indexes of the intercity EMU train, but the running stability of the vehicle can be slightly improved with increasing thickness of the floating slab. It is shown that the thickness of the track slab has a great effect on the vehicle-track coupled dynamic system; the limit of the thickness of the track slab can achieve good economic results within the acceptable range of the project. (4) When the intercity EMU train runs at 140, 160, or 200 km/h in the straight section and at 140 or 160 km/ h in the curved section (R � 1100 m and R � 1500 m) on the 3.6 m or 4.8 m long prefabricated steel-spring floating-slab track, the wheel-axle lateral force, the wheel-rail vertical force, the derailment coefficient, and the wheel-weight reduction rate are each less than the specified code limit value. e lateral and vertical accelerations of the vehicle body are each lower than the specified code limit value. In addition, the stability index is "excellent," and the comfort rating is "comfortable." e analysis results have subverted the previous perception that steel-spring floating-slab track can only be used for low-speed lines. e steelspring floating-slab track in this project research has achieved very good results and can be promoted as a theoretical basis for the application of floating-slab track for high-speed railway lines. (5) Appropriately increasing the vertical-support damping of the floating-slab track can improve the vertical dynamic performance of the vehicle, reduce the vertical displacement of the rail, and lower the vibration response of the floating-slab track. We comprehensively considered the dynamic performance of the vehicle and the stability of the track and found that the optimal ranges of vertical damping for the 3.6 m and 4.8 m long prefabricated floating-slabtrack isolators are 10-30 and 40-60 kN·s/m, respectively. Choosing the vibration isolator damping in this range can obtain good vibration isolation effect and can save engineering investment. (6) Adding side-mounted vibration isolators at the joint of the floating slabs can greatly improve the stability of the floating slab and appropriately reduce the vertical vibration response of the vehicle. e invention of the side-mounted vibration isolator is a new exploration to improve the stability of the floating-slab track. Compared with the past, simply increasing the thickness of the track slab to improve the overall quality of the track slab to achieve improved stability, the side-mounted vibrator is undoubtedly economical and effective.

Data Availability
e experimental data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.