#### 1. Introduction

The rail seat load (RSL) is the load transferred from the rail to the underneath slabs via fastenings, rail seat plates, and sleepers. The load sharing ratio (LSR) is the ratio of the RSL to the axle load, reflecting the axle-load transmission law among wheelsets, rail, and fastenings. The maximum RSL, acting as the main parameter in the design and construction of railway tracks [1], needs to be calculated from the maximum axle load and the LSR. Moreover, recently increasing investigations have been devoted to reducing the model scale (for example, using 2.5D numerical methods) of the dynamic response prediction of railway tunnels [2] and the environment [37]. However, these models still include rail and fastenings, partly repeating with train-track coupling models. Omitting rail and fastenings and imposing the RSL time history calculated from the dynamic axle load and the LSR time history as the excitation can further simplify 2.5D numerical models.

Current practices in the analysis and design of railway track systems assume the axle load to be static [8]. A dynamic coefficient is generally adopted worldwide to modify the axle load to reflect the effect of the load dynamic properties [9]. However, the current dynamic coefficient formulas recommended are different both in form and main parameters [1019]. Early methods for determining LSR are also based on the static assumption using the static load or the dynamic modified static load, mainly including the three adjacent sleepers method [11, 14, 20, 21], Australian formula [22], AREA method [23], ORE method [24], Shinkansen method, and Chinese method [25]. Numerous studies have applied such approaches [9]; however, these methods are too rough to satisfy the requirement of railway track design [26] or environmental dynamic response prediction. Besides, there is also a lack of research on the matter of whether the static assumption is reasonable and how to determine the LSR time history function.

The LSR must be determined in light of the actual conditions because it is affected by many parameters of the train-track coupled dynamic system [9]. Jiang et al. [27, 28] developed a simple LSR formula based on the results of a high-speed railway model test, bringing a more accurate method for determining LSR than before. The only parameter in the function is the fastening stiffness; however, Zhou and Chen [29] hold that some other factors should not be ignored, such as the fastening spacing. Moreover, the wheelset is often located in any position between the adjacent fastenings during the train running, while all current studies merely focus on the point above one fastening or the middle point between two adjacent fastenings.

In this paper, a “moving loading method” is proposed to obtain the LSR under moving train excitation, verified to be reasonable by comparing with the literature [27]. At the same time, a procedure for establishing the LSR multi-factor prediction model is brought up. Using the method and the program, an LSR multi-factor prediction model is developed. The parameters in the model are determined according to Shijiazhuang Metro Line 1, in which the A-type vehicle and the monolithic trackbed are utilized. It is worth noting that establishing an accurate LSR multi-factor prediction model requires a lot of working conditions and data. Hence, the train-track coupled numerical model based on multi-body (MB) and finite element (FE) methods instead of model tests is used to acquire the main results in order to avoid considerable workload and high cost.

#### 2. Methodology for Obtaining the LSR under Moving Train Loads

##### 2.1. The LSR under the Excitation of a Single Wheel

Existing studies generally use the “static loading method,” that is, applying a static load at a fixed position on the rail to calculate the LSR of surrounding fastening. When the axle load acts on a specific point, the load is distributed mainly by a total of six fastenings on both sides (five fastenings when the axle is directly above one fastening), as shown in Figure 1(a). This method is only useful when the wheel axle is located in a particular situation. Therefore, this article proposes a “moving loading method” as follows.

For a sure fastening, record the distance between the wheel axle and denote it as D. As shown in Figure 1(b), the LSR is not zero only when D is less than 3a (a is the distance between adjacent fastenings). When the train runs at a constant speed, for a specific train-rail system, the LSR of a single fastening is merely related to D. Regarding the metro track as a periodic structure, the LSR of each fastening varies with D in the same way. Therefore, as shown in Figure 2, the LSR of a single fastening time history includes all the static loading method results, and the complete LSR curve is obtained only by transforming the x-axis from time to D.

##### 2.2. The LSR under the Excitation of a Bogie

The “moving loading method” under a single wheel excitation regards the wheel as a point load and does not consider the real structure of the train and the complicated dynamic interaction between the wheel and the rail. The following assumptions are made to simplify the problem [28]: (1) when a single wheelset passes by, the LSR time-history curve is smooth and symmetrical; (2) in a certain train-track system, the LSR stimulated by each wheelset is exactly the same as the LSR stimulated by a single wheelset; and (3) all the LSR functions are superimposed assuming a linear superposition.

When loads from two wheels in a bogie are superimposed on a fastening, the corresponding M-shaped LSR time-history curve is shown in Figure 3(a). Due to the superposition of adjacent axle loads, the LSR curve of a single wheel is incomplete, but considering the symmetry, a complete curve can be obtained by half of the curve, as illustrated in Figure 3(b). The LSR under a whole train can be obtained by adding the LSR under the excitation of each wheel.

#### 3. Methodology for Establishing an LSR Prediction Model

A procedure for establishing the LSR multi-factor prediction model is put forward, which includes the following:(1)Determination of the LSR function form and fitting algorithm.(2)Parameter sensitivity analysis for determining the main parameters of the LSR function.(3)Quadratic regression orthogonal tests for obtaining the prediction formula of the LSR function coefficients.

##### 3.1. The LSR Function

The LSR curve is approximately a Gaussian curve [27]. Therefore, a Gaussian-like function is used to fit the LSR curve as follows:where is the LSR; x is the ratio of D to a (a is fastening spacing); and A and B are constants determined by actual conditions. The Levenberg–Marquardt optimization algorithm is used for fitting.

##### 3.2. Parameter Sensitivity Analysis

The LSR is affected by many factors, including train speed, axle load, fastening stiffness, fastening damping, fastening spacing, trackbed elastic modulus, and subsoil stiffness [30]. To achieve quantitative analysis, a parameter sensitivity analysis method is employed to find the main parameters. The parameter sensitivity in the form of the ratio of the objective function to the relative change rate of the influencing parameter is adopted as follows:where is the sensitivity of the LSR to the jth parameter at the ith position of the load; is the parameter variation; is the LSR variation with parameter ; is the initial value of the LSR at the ith position; is the initial value of the jth parameter ; ; ; and n and m are the maximum numbers of positions and parameters, respectively.

The LSR function sensitivity to different parameters is still a function related to variable D, bringing difficulties in selecting the main parameters. Regarding the coefficients A and B as target parameters, equation (2) is transformed into the following form:where and are separately the sensitivity factors of A and B to the jth parameter ; and are separately the variations of A and B with ; and and are individually the initial values of A and B.

##### 3.3. Determining the Coefficient Formulas

To generalize equation (1) into a universal empirical formula, it is necessary to further explore the valuing method for A and B. After determining the main parameters of the LSR function, a quadratic regression orthogonal test is designed, and the value functions of A and B under multiple parameters are established, respectively. To simplify the treatment process of data, the actual value of each test factor is firstly linearly transformed into the factor level code as follows:where is the average of the upper and lower bounds of the ith experimental factor ; is the upper bound of ; is the variation range of ; and γ is the asterisk arm which can be calculated as follows:where m is the number of test factors; is the number of zero-level tests; and is the number of two-level trials.

Quadratic regression orthogonal analysis needs to consider the influence of a single factor and the impact between two factors ; hence, the quadratic regression equations of parameters A and B are shown as follows:where and are the estimated values of parameters A and B; n is the number of test factors; and , , , , , , , and are all regression coefficients.

#### 4. Case Study

The concrete monolithic trackbed has the characteristics of good integrity and high rigidity and is a commonly used track form in metro underground lines [31]. Shijiazhuang Metro Line 1 using the A-type train and the monolithic trackbed was taken as an example. The LSR data were obtained employing a numerical model based on the moving loading method, and a multi-factor prediction model of LSR was developed.

##### 4.1. Numerical Model

The instance is an A-type metro train with six cars including four motor cars and two trailer cars. The train size is shown in Figure 5, and the main parameters are shown in Table 1. Each vehicle subsystem had several rigid bodies including one carbody, two frames, four wheelsets, and eight axle boxes, and each body had six degrees of freedom in the longitudinal, lateral (transverse), vertical (up and down), side roll, pitch, and yaw. Therefore, each vehicle subsystem had 90 degrees of freedom. The train model established by the MB software named Universal Mechanism (UM) is shown in Figure 6(a). The axle load of the fully loaded train was about 14.8 t, and the vehicle runs at a constant speed of 80 km/h.

The rails were modeled as Timoshenko beams, and the fastenings were modeled as a series of spring-damper pairs [32] with a fastening spacing of . The monolithic trackbed and the circular cross section tunnel lining were modeled as a whole FE system without creating separate sleeper models. The geometry and properties of the track-tunnel-soil system are separately shown in Figure 7(a) and Table 2. Based on the linear viscoelastic constitutive model, a FE model of the trackbed-tunnel bottom was established using the element type of SOLID 45 in the software ANSYS, as shown in Figure 7(b). The longitudinal length of the model was 120 m, and the element size was 0.11–0.3 m. The Craig–Bampton method was adopted to couple the FE model and the MB model in the software UM. Spring-damper pairs were used to simulate the subsoil, whose stiffness and damping coefficients were determined separately according to the literatures [33, 34].

There are many wheel-rail interaction models. In this paper, the Kik–Piotrowski multi-point contact algorithm [35] was used to simulate the contact between wheels and rail. Figure 7(b) shows the train-track coupling model.

Track irregularity is the main reason for the dynamic excitation of trains [36]. The environmental vibration response caused by the metro operation is mainly vertical [37]. Hence, it is generally considered that the vertical wheel-rail force is the primary excitation source, so only vertical track irregularities were considered [38]. At present, the American Class 6 track irregularity spectrum and the Sato track irregularity spectrum are commonly used to simulate medium and long wave and short wave irregularities in metro tracks, respectively [36]. However, some studies have shown that the American Class 6 track irregularity spectrum differs significantly from the actual metro track irregularities [3941]. This research used the Shanghai Metro track spectrum [39] and Sato track irregularity spectrum [42] to simulate vertical track irregularities. Figure 8 shows the sample data.

##### 4.2. Results

and obtained from equation (3) do not change with D, facilitating an intuitive comparison of the influence of various parameters. The reference values of parameters were determined according to the actual situation of Shijiazhuang Metro Line 1, and the change rate of each parameter took the same small amount (10%). The calculation conditions are shown in Table 3, where condition 3 increases the mass of each rigid body in the train model by 10%. In fact, the mass of carbody, bogie, and wheelset has different influence on LSR, but here the vehicle is taken as a whole, and only the influence of total vehicle mass on LSR is considered to reduce the number of influencing factors. In the process of changing the mass of each body, the change of vehicle structure is not expected. Therefore, assuming that the influence of mass change as small as 10% on vehicle structural characteristics can be ignored, the mass of each rigid body is increased by 10%.

According to the conditions in Table 3, the vehicle-rail coupling calculation is performed to obtain the corresponding LSR data. The calculated discrete data cannot fully reflect the LSR when the wheel is at any position, so the Levenberg–Marquardt optimization algorithm was used to fit each data group according to equation (1) (see Figure 9).

Figures 10(a) and 10(b) show the sensitivity curves of LSR and the coefficients A and B to each parameter calculated by equations (2) and (3) individually. As shown in Figure 10(a), factors have different effects on the LSR of different locations, resulting in different parameter sensitivity distribution laws. Besides, the impact of axle load varies with D in the opposite trend with other factors. Figure 10(b) implies that the coefficients A and B have considerable absolute values of the parameter sensitivity of the fastening stiffness and the fastening spacing, so they were regarded as the main influencing parameters.

The fastening stiffness change range was taken to be 20–50 MN m−1, and the fastening spacing change range was assumed to be 0.5–0.65 m. A two-factor quadratic regression orthogonal test was designed, and the factor level coding is shown in Table 4. The number of test factors was , and the length of the asterisk arm was . The number of two-level tests was ; the number of asterisks was ; and the number of zero-level tests was . Therefore, the total number of test groups was . Table 5 shows the values of test factor levels and the corresponding coefficients of fitting functions.

By carrying out regression analysis according to the orthogonal experiment results, the regression equations were obtained as follows:

According to equations (4) and (6), equations (7a) and (7b) are transformed into functions represented by the actual levels as follows:where kf is the fastening stiffness in MN·m−1 and a is the fastening spacing in m.

Equations (1), (8a), and (8b) are the final LSR formulas. Through this group of functions, the LSR can be predicted from the fastening stiffness and the fastening spacing.

##### 4.3. Error Analysis

The estimated values of coefficients A and B were compared with the calculated values, and the relative errors δ were calculated individually, as shown in Figure 11(a). The maximum prediction errors of coefficients A and B appear in the seventh and the sixth groups separately, and the errors are 5.72% and 5.21% individually, both less than 6%. Hence, using equations (8a) and (8b) for prediction is reliable.

Under the sixth group of test parameters, the LSR curve was generated utilizing the prediction model composed of equations (1), (8a), and (8b). The LSRs obtained by the prediction model and numerical calculation are compared in Figure 11(b). The absolute error is always less than 3%, which is acceptable.

The LSR curve under the excitation of a bogie was obtained by superimposing two LSR curves under the excitation of a single wheel. Figure 11(c) shows the curves of the prediction result, numerical result, and the absolute error. The prediction error is within 5%, indicating that the prediction effect is satisfactory.

#### 5. Conclusions

The load sharing ratio (LSR) is a significant factor for designing railway tracks, but current LSR formulas or calculation methods are too rough to satisfy the requirements, and the widely used “static loading method” for obtaining the LSR is inconsistent with reality.

For similar trains and track types, only one prediction model needs to be established according to the proposed method; then, the LSR prediction results can be obtained by adjusting the main parameters. Replacing current LSR calculation approaches with those recommended in this research will considerably improve the accuracy of the LSR prediction model.

It is worth noting that the LSR prediction model can only consider the influence of quantifiable factors, while non-quantitative factors (including train type and track type) need to be reflected by establishing a new LSR prediction model. For example, cars of type A, type B, or type C may be used in metros, and passenger cars or freight cars may be used in railways. Track types include ballastless tracks and ballasted tracks, and track vibration reduction measures such as vibration damping fasteners may be adopted. The influence of different train and track types on the main parameters of LSR and the distribution of LSR requires to be further investigated in future research.

#### Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

#### Disclosure

The opinions expressed in this paper are those of the authors.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This study was supported by the National Key R&D Program of China (grant no. 2018YFC0808704) and Shijiazhuang Rail Transit Group Co. Ltd. (grant no. SJZM01B-ZGCSBGS005-FW-2019). The authors are very grateful for the support. The authors also gratefully acknowledge Dr. Liu Wei for assistance in software operation and Prof. Wang Jianxi for suggestions on writing.