# Parameters Studies on Surface Initiated Rolling Contact Fatigue of Turnout Rails by Three-Level Unreplicated Saturated Factorial Design

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulation of Rail Surface Initiated RCF

#### 2.1. Surface Initiated Rolling Contact Fatigue

_{z}is the normal wheel-rail contact stress, k is the shear yield strength of the material, and f

_{t}is the traction coefficient calculated by Equation (2).

_{x}and p

_{y}are the longitudinal and lateral wheel-rail tangential contact stresses respectively. Surface initiated RCF is predicted to occur if FI

_{surf}> 0. The fatigue index values could be higher if partial slip is considered. Therefore, this index is only used to simulate and compare the surface initiated RCF of turnout rails but not as the observation parameter in factorial design. In addition, when the state of wheel-rail contact is full-slip conditions, the traction coefficient is equal to the wheel-rail friction coefficient, and when the state of wheel-rail contact is partial slip conditions, the traction coefficient is less than the wheel-rail friction coefficient. Therefore, this method considering the partial slip conditions is also matched to the full-slip shakedown diagram shown in Figure 2.

#### 2.2. Wheel-Rail Rolling Contact Model

#### 2.3. Surface Initiated RCF of Turnout Rail

^{11}N/m

^{2}, 0.3 and 0.3 [29], respectively, and the ultimate shear strength of the rail material is 350 MPa. When the lateral wheel-set displacement y

_{w}results are 0 mm, 3 mm, 6 mm and 9 mm, respectively, the wheel-rail contact point positions of the selected section will be those as shown in Figure 4. In this figure, the coordinate zero point refers to the center of railway track, y and z refers to the lateral and vertical direction of railway line, respectively.

#### 2.3.1. Small Creep Conditions

_{x}, lateral creep f

_{y}and spin creep fin are −5 × 10

^{−4}, −5 × 10

^{−4}and 0, respectively. In this figure, the upper part refers to the wheel-stock rail contact region, and the lower part refers to the wheel-switch rail contact region. The black solid lines refer to the boundaries of the wheel-rail contact patch regions, and the colored region refers to the value of surface initiated RCF index if it greater than 0. The blank region means that the surface initiated RCF index of turnout rails is less than 0, namely no surface initiated RCF occurs. The coordinate zero point refers to the center of first contact patch, x and y refers to the longitudinal and lateral direction of railway line, respectively. Under the small wheel-rail creep conditions, the surface initiated RCF regions are small and located mainly at the trailing edge of the contact patches at the stock rail side along the rolling direction. The surface initiated RCF index within the wheel-switch rail contact patches are less than 0. The regions with surface initiated RCF account for 2.9% of the total contact patch area.

#### 2.3.2. Large Creep Conditions

_{x}, lateral creep f

_{y}and spin creep fin are −3 × 10

^{−3}, −3 × 10

^{−3}and 0, respectively. Under the large wheel-rail creep conditions, the surface initiated RCF regions are big and located mainly at the center of the contact patches at the wheel-stock rail contact patch side. The surface initiated RCF index within the wheel-switch rail contact patches are less than 0. The regions with surface initiated RCF account for 25.4% of the total contact patch area.

#### 2.3.3. Pure Spin Creep Conditions

_{x}, lateral creep f

_{y}and spin creep fin are 0, 0 and −0.5 m

^{−1}, respectively. Under the pure spin wheel-rail creep conditions, the surface initiated RCF regions are big and located mainly at the center of the contact patches at the wheel-stock rail contact patch side. The surface initiated RCF index within the wheel-switch rail contact patches are less than 0. The regions with surface initiated RCF account for 18.7% of the total contact patch area.

## 3. Parameters Studies and Discussions

#### 3.1. Three-Level Unreplicated Saturated Factorial Design Method

_{1}, y

_{2}, …, y

_{n})

^{T}is the observation vector, μ is the overall average, α

_{i}is the main influence of the first factor, β

_{j}is the main influence of the second factor, α

_{i}and β

_{j}shall satisfy the following constraint condition:

_{ij}is the mutual influence when the first factor is i level and the second factor is j level, the satisfying the following constraint condition:

_{ij}is the experimental error, ε

_{ij}is the independent normal distribution variable, and its mathematical expectation of normal distribution is 0, the variance is σ

^{2}, which can be expressed by Equation (6).

_{0}is false, it is necessary to confirm which of α

_{i}, β

_{j}or αβ

_{ij}is a non-zero value. When the deviation square and SS

_{i}of the corresponding factors are considered in this method, the following can be adopted instead of the test assumptions of (7):

_{i}is the mathematical expectation of the mean square error. The equation for the mean square error MS

_{i}is shown below [33,34]:

#### 3.1.1. H Statistical Value

_{(1)}≤ MS

_{(2)}≤ … ≤ MS

_{(m)}is the permutation of MS

_{(i)}statistical values from low to high. l is the maximum value of s and j. s is the minimum number of columns in the orthogonal table for evaluating the error variance, j is the maximum positive integer satisfying Equation (11).

_{0}will be false, and those columns satisfying Equation (12) are considered to have a non-zero effect.

^{(1)}(α; s, m)) and 1 − α quantiles (c

_{pool}(α; j, m)) were listed in the appended Table A1.

#### 3.1.2. P Statistical Value

_{0}will be false, and those columns satisfying Equation (15) are considered to have a non-zero effect.

_{1}

^{(2)}(α, m) is the 1-α quantile of the left part of Equation (14), c

_{2}

^{(2)}(α, m) is the 1-α quantile of the left part of Equation (14). During the testing process, some necessary critical values (c

_{1}

^{(2)}(α,m)) and (c

_{2}

^{(2)}(α, m)) were listed in the appended Table A2.

#### 3.1.3. B Statistical Value

_{0}should be considered as a true value and the calculation should be ended;

_{0}for being false and consider that column m-k with the maximum squared error has a non-zero effect.

^{(3)}(α, k)) were listed in the appended Table A3.

#### 3.2. Orthogonal Design Test

#### 3.2.1. Analysis Factor and Observation Parameters

_{z}and contact patch area A

_{c}are selected for reflecting the normal wheel-rail contact stress level. Similarly, the tangential wheel-rail contact stress is mainly related to the tangential wheel-rail force and contact patch area. Therefore, in this paper the longitudinal wheel-rail creep force F

_{x}, lateral creep force F

_{y}and contact patch area A

_{c}are selected for reflecting the tangential wheel-rail contact stress level. In conclusion, F

_{z}, F

_{x}, F

_{y}and A

_{c}are adopted as the observation parameters in the factorial design. In fact, these parameters are not independent markers for surface initiated RCF. The main purpose of this is to find out what is missing in the factorial design as the single parameter FI

_{surf}cannot be used.

#### 3.2.2. Orthogonal Test Plan

_{27}(3

^{13}) orthogonal table, see Table 2 for the detailed orthogonal test plan. In this Table, 1, 2 and 3 refer to the three levels. The simulation calculation for the tests of Table 2 is conducted according to the vehicle-turnout dynamic model. During the calculation, a vehicle passing the turnout in the diverging line with facing move is simulated. The dynamic response results of the obtained four observation parameters, i.e., F

_{z}, F

_{x}, F

_{y}and A

_{c}, are the observation vectors for the statistical analysis model, which, together with the 13 analysis factors selected according to H, P and B statistical values, are adopted to obtain the factors significantly influencing turnout rail RCF. It should be noted that the maximum dynamic wheel-rail response of the first wheel-set of vehicle at the switch rail sides is selected as the observation value. The calculated observation values are shown in Table 2.

#### 3.2.3. Selection of Active Factors

_{z}, l = 12 for F

_{x}, l = 11 for F

_{y}and l = 6 for A

_{c}. Table 4 shows the H statistic values of all analysis factors calculated according to Equation (10).

_{z}, the H statistic values of 3 analysis factors are higher than the critical value, namely x

_{1}, x

_{2}and x

_{13}are the significant influence factors of F

_{z}. Similarly, x

_{13}is the significant influence factor of F

_{x}and F

_{y}, x

_{4}, x

_{10}and x

_{12}are the significant influence factors of A

_{c}.

_{z}, the P statistic values of 3 analysis factors are higher than the critical value, namely x

_{1}, x

_{2}and x

_{13}are the significant influence factors of F

_{z}. Similarly, x

_{13}is the significant influence factor of F

_{x}and F

_{y}, x

_{4}, x

_{5}, x

_{10}and x

_{12}are the significant influence factors of A

_{c}.

_{z}, the B statistic values of 3 analysis factors are higher than the critical value, namely x

_{1}, x

_{2}and x

_{13}are the significant influence factors of F

_{z}. Similarly, x

_{13}is the significant influence factor of F

_{x}and F

_{y}, x

_{4}, x

_{10}and x

_{12}are the significant influence factors of A

_{c}.

_{1}, x

_{2}, x

_{4}, x

_{10}, x

_{12}and x

_{13}, referring to train speed, axle load of vehicle, wheel profile, stock rail cant, integral vertical track stiffness and wheel-rail friction coefficient, respectively. In the previous work [1], Kassa and Nielsen presented axle load, wheel–rail friction coefficient, and wheel/rail profiles as the parameters which could affect the performance of railway turnout significantly. The results of this paper are basically the same with previous conclusions, and the previous results could be corroborated by this paper. In addition, the previous work is investigated based on an ordinary railway turnout which has a significant difference with the high-speed railway turnout, such as the structure, wheel/rail contact relationship and so on. Therefore, there are some different results, such as the integral vertical track stiffness. The outcome of this work could provide a theoretical guidance for the simulation and prediction of surface initiated RCF of high-speed railway turnouts.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

RCF | Rolling Contact Fatigue |

3D | Three Dimensional |

## Appendix A

m | j or s | c_{pool} (0.25; j, m) | c^{(1)} (0.10; s, m) | c^{(1)} (0.05; s, m) | c^{(1)} (0.01; s, m) |
---|---|---|---|---|---|

13 | 1 | 1.618 | 4.905 | 5.382 | 6.250 |

2 | 1.806 | 5.081 | 5.505 | 6.322 | |

3 | 1.946 | 5.201 | 5.604 | 6.411 | |

4 | 2.049 | 5.303 | 5.689 | 6.484 | |

5 | 2.140 | 5.440 | 5.793 | 6.539 | |

6 | 2.228 | 5.627 | 5.969 | 6.631 | |

7 | 2.320 | 5.655 | 6.059 | 6.756 | |

8 | 2.428 | 5.576 | 6.048 | 6.876 | |

9 | 2.556 | 5.346 | 5.867 | 6.821 | |

10 | 2.738 | 5.080 | 5.618 | 6.655 | |

11 | 3.016 | 4.733 | 5.248 | 6.334 | |

12 | 3.663 | 4.337 | 4.838 | 5.863 |

m | A | |||||
---|---|---|---|---|---|---|

0.1 | 0.05 | 0.01 | ||||

c_{1}^{(2)} (α, m) | c_{2}^{(2)} (α, m) | c_{1}^{(2)} (α, m) | c_{2}^{(2)} (α, m) | c_{1}^{(2)} (α, m) | c_{2}^{(2)} (α, m) | |

13 | 2.346 | 0.940 | 2.888 | 1.310 | 4.370 | 2.307 |

k | A | ||
---|---|---|---|

0.1 | 0.05 | 0.01 | |

2 | 0.839 | 1.180 | 1.970 |

3 | 0.930 | 1.200 | 1.810 |

4 | 0.935 | 1.160 | 1.660 |

5 | 0.924 | 1.130 | 1.550 |

6 | 0.914 | 1.100 | 1.470 |

7 | 0.900 | 1.065 | 1.410 |

8 | 0.886 | 1.030 | 1.350 |

9 | 0.874 | 1.011 | 1.305 |

10 | 0.862 | 0.991 | 1.260 |

11 | 0.854 | 0.975 | 1.230 |

12 | 0.845 | 0.959 | 1.200 |

13 | 0.837 | 0.947 | 1.177 |

## References

- Kassa, E.; Nielsen, J.C.O. Stochastic analysis of dynamic interaction between train and railway turnout. Veh. Syst. Dyn.
**2008**, 46, 429–449. [Google Scholar] [CrossRef] - Wang, P.; Xu, J.; Xie, K.; Chen, R. Numerical simulation of rail profiles evolution in the switch panel of a railway turnout. Wear
**2016**, 366, 105–115. [Google Scholar] [CrossRef] - Nielsen, J.C.O.; Palsson, B.A.; Torstensson, P.T. Switch panel design based on simulation of accumulated rail damage in a railway turnout. Wear
**2016**, 366, 241–248. [Google Scholar] [CrossRef] - Xin, L.; Markine, V.L.; Shevtsov, I.Y. Numerical procedure for fatigue life prediction for railway turnout crossings using explicit element approach. Wear
**2016**, 366, 167–179. [Google Scholar] [CrossRef] - Nicklisch, D.; Kassa, E.; Nielsen, J.; Ekh, M.; Iwnicki, S. Geometry and stiffness optimization for switches and crossings, and simulation of material degradation. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit
**2010**, 224, 279–292. [Google Scholar] [CrossRef] - Markine, V.; Steenbergen, M.; Shevtsov, I. Combatting RCF on switch points by tuning elastic track properties. Wear
**2011**, 271, 158–167. [Google Scholar] [CrossRef] - Wan, C.; Markine, V.; Shevtsov, I. Improvement of vehicle-turnout interaction by optimising the shape of crossing nose. Veh. Syst. Dyn.
**2014**, 52, 1517–1540. [Google Scholar] [CrossRef] - Wang, P.; Ma, X.; Wang, J.; Xu, J.; Chen, R. Optimization of rail profiles to improve vehicle running stability in switch panel of high-speed railway turnouts. Math. Probl. Eng.
**2017**, 2017, 1–13. [Google Scholar] [CrossRef] - Pletz, M.; Daves, W.; Ossbergec, H. A wheel set/crossing model regarding impact, sliding and deformation-Explicit finite element approach. Wear
**2012**, 294, 446–456. [Google Scholar] [CrossRef] - Pletz, M.; Daves, W.; Yao, W.; Ossberger, H. Rolling contact fatigue of three crossing nose materials-Multiscale FE approach. Wear
**2014**, 314, 69–77. [Google Scholar] [CrossRef] - Xiao, J.; Zhang, F.; Qian, L. Contact stress and residual stress in the nose rail of a High Manganese steel crossing due to wheel contact loading. Fatigue Fract. Eng. Mater. Struct.
**2014**, 37, 219–226. [Google Scholar] [CrossRef] - Guo, S.; Sun, D.; Zhang, F.; Feng, X.; Qian, L. Damage of a Hadfield steel crossing due to wheel rolling impact passages. Wear
**2013**, 305, 267–273. [Google Scholar] [CrossRef] - Zhang, F.; Lv, B.; Zheng, C.; Zou, Q.; Zhang, M.; Li, M.; Wang, T. Microstructure of the worn surfaces of a bainitic steel railway crossing. Wear
**2010**, 268, 1243–1249. [Google Scholar] [CrossRef] - Meng, T. The Comparison Research about Plackett-Burman Data Analysis Method for Supersaturated Designs. Master’s Thesis, East China Normal University, Shanghai, China, 2008. (In Chinese). [Google Scholar]
- Chen, Y.; Kunert, J. A new quantitative method for analysing unreplicated factorial designs. Biom. J.
**2004**, 46, 125–140. [Google Scholar] [CrossRef] - Daniel, C. Applications of Statistics to Industrial Experimentation; Wiley: New York, NY, USA, 1976; ISBN 9780471194699. [Google Scholar]
- Dong, F. On the identification of active contrasts in unreplicated fractional factorials. Stat. Sin.
**1993**, 3, 209–217. [Google Scholar] - Zhang, X.; Zhang, Y.; Mao, S. Statistical analysis of 2-level orthogonal saturated design: The procedure of searching zero effects. J. East China Norm. Univ.
**2008**, 31, 51–59. (In Chinese) [Google Scholar] - Xu, J.; Wang, P.; Ma, X.; Qian, Y.; Chen, R. Parameters studies for rail wear in high-speed railway turnouts by unreplicated saturated factorial design. J. Cent. South. Univ.
**2017**, 24, 988–1001. [Google Scholar] [CrossRef] - Ekberg, A.; Sotkovszki, P. Anisotropy and rolling contact fatigue of railway wheels. Int. J. Fatigue
**2001**, 23, 29–43. [Google Scholar] [CrossRef] - Ekberg, A.; Kabo, E.; Andersson, H. An engineering model for prediction of rolling contactfatigue of railway wheels. Fatigue Fract. Eng. Mater. Struct.
**2002**, 25, 899–909. [Google Scholar] [CrossRef] - Ekberg, A.; Akesson, B.; Kabo, E. Wheel/rail rolling contact fatigue—Probe, predict, prevent. Wear
**2014**, 314, 2–12. [Google Scholar] [CrossRef] - Dirks, B.; Enblom, R. Prediction model for wheel profile wear and rolling contact fatigue. Wear
**2011**, 271, 210–217. [Google Scholar] [CrossRef] - Sichani, M.; Enblom, R.; Berg, M. A novel method to model wheel-rail normal contact in vehicle dynamics simulation. Veh. Syst. Dyn.
**2014**, 52, 1752–1764. [Google Scholar] [CrossRef] - Sichani, M.; Enblom, R.; Berg, M. An alternative to FASTSIM for tangential solution of the wheel-rail contact. Veh. Syst. Dyn.
**2016**, 54, 748–764. [Google Scholar] [CrossRef] - Sichani, M.; Enblom, R.; Berg, M. A fast wheel-rail contact model for application to damage analysis in vehicle dynamic simulation. Wear
**2016**, 366, 123–130. [Google Scholar] [CrossRef] - Ma, X.; Xu, J.; Wang, P. Study on Impact of Relative Motion of Switch/Stock Rail on Wheel Load Transfer and Distribution in Railway Turnout. J. China Railw. Soc.
**2017**, 39, 75–81. (In Chinese) [Google Scholar] - Wang, P.; Ma, X.; Xu, J.; Wang, J.; Chen, R. Numerical investigation on effect of the relative motion of stock/switch rails on the load transfer distribution along the switch panel in high-speed railway turnout. Veh. Syst. Dyn
**2017**. under review. [Google Scholar] - Johansson, A.; Palsson, B.; Ekh, M.; Nielsen, J.C.O.; Ander, M.K.A.; Brouzoulis, J.; Kassa, E. Simulation of wheel-rail contact and damage in switches & crossings. Wear
**2011**, 271, 472–481. [Google Scholar] - Xu, J.; Wang, P.; Wang, L.; Chen, R. Effects of profile wear on wheel-rail contact conditions and dynamic interaction of vehicle and turnout. Adv. Mech. Eng.
**2016**, 8, 1–14. [Google Scholar] [CrossRef] - Chen, Y. On the analysis of three-level factorial designs with only one replicate. J. Appl. Stat. Manag.
**2010**, 29, 819–829. (In Chinese) [Google Scholar] - Chen, Y.; Chan, C.K.; Leung, B.P.K. An analysis of three-level orthogonal saturated designs. Comput. Stat. Data. Anal.
**2010**, 54, 1952–1961. [Google Scholar] [CrossRef] - Hou, S.; Liu, T.; Dong, D.; Han, X. Factor screening and multivariable crashworthiness optimization for vehicle side impact by factorial design. Struct. Multidiscip. Optim.
**2014**, 49, 147–167. [Google Scholar] [CrossRef] - Hou, S.; Tan, W.; Zheng, Y.; Han, X.; Li, Q. Optimization design of corrugated beam guardrail based on RBF-MQ surrogate model and collision safety consideration. Adv. Eng. Softw.
**2014**, 78, 28–40. [Google Scholar] [CrossRef] - Holms, A.G.; Berrettoni, J.N. Chain-Pooling ANOVA for Two-Level Factorial Replication-Free Experiments. Technometrics
**1969**, 11, 725–746. [Google Scholar] [CrossRef] - Berk, K.N.; Picard, R.R. Significance tests for saturated orthogonal arrays. J. Qual. Technol.
**1991**, 23, 79–89. [Google Scholar] [CrossRef] - Bartlett, M.S. Properties of Sufficiency and Statistical Tests. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci.
**1937**, 160, 268–282. [Google Scholar] [CrossRef] - An, B.; Wang, P.; Xu, J.; Chen, R.; Cui, D. Observation and simulation of axle box acceleration in the presence of rail weld in high-speed railway. Appl. Sci.
**2017**, 7, 1259. [Google Scholar] [CrossRef] - Li, S.; Li, Z.; Nunez, A.; Dollevoet, R. New insights into the short pitch corrugation enigma based on 3D-FE coupled dynamic vehicle-track modeling of frictional rolling contact. Appl. Sci.
**2017**, 7, 807. [Google Scholar] [CrossRef]

**Figure 4.**Wheel-rail contact point positions. (

**a**) y

_{w}= 0 mm; (

**b**) y

_{w}= 3 mm; (

**c**) y

_{w}= 6 mm; (

**d**) y

_{w}= 9 mm.

**Figure 5.**Small creep condition, f

_{x}= −5 × 10

^{−4}, f

_{y}= −5 × 10

^{−4}, fin = 0. (

**a**) surface initiated rolling contact fatigue (RCF) index; (

**b**) Shakedown map.

**Figure 6.**Large creep condition, f

_{x}= −3 × 10

^{−3}, f

_{y}=−3 × 10

^{−3}, fin = 0. (

**a**) surface initiated RCF index; (

**b**) Shakedown map.

**Figure 7.**Pure spin creep condition, f

_{x}= 0, f

_{y}= 0, fin = −0.5 m

^{−1}. (

**a**) surface initiated RCF index; (

**b**) Shakedown map.

No. | Analysis Factor | Analysis Factor Levels | ||
---|---|---|---|---|

Level 1 | Level 2 | Level 3 | ||

x_{1} | Train speed/(km/h) | 40 | 80 | 120 |

x_{2} | Train axle load/(t) | 12 | 14 | 16 |

x_{3} | Nominal wheel rolling radius/(mm) | 420 | 460 | 500 |

x_{4} | Wheel profile | LMA | S1002CN | XP55 |

x_{5} | Lateral stiffness of primary suspension/(kN/mm) | 5.0 | 6.5 | 8.0 |

x_{6} | Vertical stiffness of primary suspension/(kN/mm) | 1.0 | 1.2 | 1.4 |

x_{7} | Lateral stiffness of secondary suspension/(kN/mm) | 0.10 | 0.15 | 0.20 |

x_{8} | Vertical stiffness of secondary suspension/(kN/mm) | 0.15 | 0.18 | 0.21 |

x_{9} | Rail gauge/(mm) | 1432 | 1435 | 1438 |

x_{10} | Stock rail cant | 0 | 1/40 | 1/20 |

x_{11} | Integral lateral track stiffness/(kN/mm) | 30 | 50 | 70 |

x_{12} | Integral vertical track stiffness/(kN/mm) | 80 | 100 | 120 |

x_{13} | Wheel-rail friction coefficient | 0.1 | 0.3 | 0.5 |

No. | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | x_{7} | x_{8} | x_{9} | x_{10} | x_{11} | x_{12} | x_{13} | F_{z} (kN) | F_{x} (kN) | F_{y} (kN) | A_{c} (mm^{2}) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 61.7 | 0.2 | 3.0 | 90.1 |

2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 61.6 | 4.2 | 6.6 | 126.3 |

3 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 60.0 | 7.8 | 9.2 | 67.1 |

4 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 68.9 | 10.2 | 12.7 | 70.6 |

5 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 1 | 1 | 1 | 71.7 | 1.1 | 3.8 | 64.7 |

6 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 1 | 1 | 1 | 2 | 2 | 2 | 71.8 | 4.8 | 9.9 | 67.8 |

7 | 1 | 3 | 3 | 3 | 1 | 1 | 1 | 3 | 3 | 3 | 2 | 2 | 2 | 81.4 | 5.2 | 9.9 | 77.9 |

8 | 1 | 3 | 3 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 3 | 3 | 3 | 79.0 | 7.7 | 17.2 | 55.7 |

9 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 82.3 | 0.5 | 4.7 | 72.0 |

10 | 2 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 58.4 | 2.5 | 11.7 | 60.8 |

11 | 2 | 1 | 2 | 3 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 64.4 | -0.2 | 3.8 | 50.3 |

12 | 2 | 1 | 2 | 3 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 65.0 | 3.9 | 9.0 | 53.3 |

13 | 2 | 2 | 3 | 1 | 1 | 2 | 3 | 2 | 3 | 1 | 3 | 1 | 2 | 74.9 | 4.1 | 8.7 | 120.1 |

14 | 2 | 2 | 3 | 1 | 2 | 3 | 1 | 3 | 1 | 2 | 1 | 2 | 3 | 72.9 | 9.7 | 13.1 | 163.6 |

15 | 2 | 2 | 3 | 1 | 3 | 1 | 2 | 1 | 2 | 3 | 2 | 3 | 1 | 74.3 | 0.7 | 3.9 | 83.0 |

16 | 2 | 3 | 1 | 2 | 1 | 2 | 3 | 3 | 1 | 2 | 2 | 3 | 1 | 84.4 | 0.9 | 4.1 | 75.5 |

17 | 2 | 3 | 1 | 2 | 2 | 3 | 1 | 1 | 2 | 3 | 3 | 1 | 2 | 83.0 | 7.3 | 9.4 | 69.5 |

18 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 2 | 3 | 1 | 1 | 2 | 3 | 81.1 | 11.4 | 14.0 | 78.6 |

19 | 3 | 1 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 2 | 66.9 | 4.0 | 8.6 | 78.6 |

20 | 3 | 1 | 3 | 2 | 2 | 1 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 65.3 | 9.1 | 13.5 | 72.5 |

21 | 3 | 1 | 3 | 2 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 2 | 1 | 67.6 | 0.2 | 3.4 | 68.8 |

22 | 3 | 2 | 1 | 3 | 1 | 3 | 2 | 2 | 1 | 3 | 3 | 2 | 1 | 77.3 | 0.5 | 4.4 | 68.5 |

23 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 2 | 77.7 | 3.7 | 10.8 | 51.6 |

24 | 3 | 2 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 2 | 2 | 1 | 3 | 75.8 | 7.8 | 15.6 | 59.2 |

25 | 3 | 3 | 2 | 1 | 1 | 3 | 2 | 3 | 2 | 1 | 2 | 1 | 3 | 84.8 | 10.0 | 15.1 | 129.8 |

26 | 3 | 3 | 2 | 1 | 2 | 1 | 3 | 1 | 3 | 2 | 3 | 2 | 1 | 86.1 | 1.0 | 3.8 | 146.0 |

27 | 3 | 3 | 2 | 1 | 3 | 2 | 1 | 2 | 1 | 3 | 1 | 3 | 2 | 87.2 | 6.0 | 10.7 | 98.6 |

Factors | The Mean Square Errors | |||
---|---|---|---|---|

F_{z} | F_{x} | F_{y} | A_{c} | |

x_{1} | 15.70742 | 0.02138 | 0.60160 | 44.53319 |

x_{2} | 196.36442 | 2.11052 | 2.69865 | 115.39593 |

x_{3} | 0.24615 | 0.13753 | 0.21154 | 69.21850 |

x_{4} | 0.05153 | 1.91301 | 1.04544 | 1557.16874 |

x_{5} | 0.24876 | 0.27282 | 0.08978 | 161.02762 |

x_{6} | 0.05817 | 0.88098 | 0.08544 | 13.81811 |

x_{7} | 0.02298 | 0.98206 | 0.25764 | 1.50246 |

x_{8} | 0.46019 | 0.61019 | 0.15663 | 16.25891 |

x_{9} | 0.22571 | 0.08459 | 0.41872 | 1.14816 |

x_{10} | 0.20981 | 0.02352 | 0.62467 | 219.01455 |

x_{11} | 0.09366 | 0.11422 | 0.12611 | 9.55418 |

x_{12} | 0.26766 | 0.12900 | 0.25309 | 320.33627 |

x_{13} | 4.48885 | 31.47385 | 47.28643 | 9.62576 |

H Statistic Values | Critical Value | |||||||
---|---|---|---|---|---|---|---|---|

Factors | F_{z} | Factors | F_{x} | Factors | F_{y} | Factors | A_{c} | |

x_{7} | 0.06469 | x_{1} | 0.02988 | x_{6} | 0.17214 | x_{9} | 0.14598 | 5.505 |

x_{4} | 0.14401 | x_{10} | 0.03287 | x_{5} | 0.18075 | x_{7} | 0.18981 | 5.505 |

x_{6} | 0.16229 | x_{9} | 0.11743 | x_{11} | 0.25236 | x_{11} | 1.05385 | 5.505 |

x_{11} | 0.25897 | x_{11} | 0.15806 | x_{8} | 0.31184 | x_{13} | 1.06055 | 5.505 |

x_{10} | 0.56367 | x_{12} | 0.17824 | x_{3} | 0.41736 | x_{6} | 1.42821 | 5.505 |

x_{9} | 0.60404 | x_{3} | 0.18985 | x_{12} | 0.49595 | x_{8} | 1.62203 | 5.505 |

x_{3} | 0.65548 | x_{5} | 0.37128 | x_{7} | 0.50449 | x_{1} | 3.16670 | 5.505 |

x_{5} | 0.66202 | x_{8} | 0.80207 | x_{9} | 0.79891 | x_{3} | 3.93523 | 5.505 |

x_{12} | 0.70907 | x_{6} | 1.12715 | x_{1} | 1.11541 | x_{2} | 4.77114 | 5.505 |

x_{8} | 1.16509 | x_{7} | 1.24410 | x_{10} | 1.15407 | x_{5} | 5.24434 | 5.505 |

x_{13} | 5.89688 | x_{4} | 2.22188 | x_{4} | 1.81392 | x_{10} | 5.61736 | 5.505 |

x_{1} | 8.81897 | x_{2} | 2.40877 | x_{2} | 3.77904 | x_{12} | 5.99169 | 5.505 |

x_{2} | 10.78661 | x_{13} | 10.03983 | x_{13} | 10.67472 | x_{4} | 6.76578 | 5.505 |

P statistic Values | Critical Value | |||||||
---|---|---|---|---|---|---|---|---|

Factors | F_{z} | Factors | F_{x} | Factors | F_{y} | Factors | A_{c} | |

x_{7} | 0.00790 | x_{1} | 0.00768 | x_{6} | 0.02687 | x_{9} | 0.01166 | 1.310 |

x_{4} | 0.01772 | x_{10} | 0.00845 | x_{5} | 0.02823 | x_{7} | 0.01526 | 1.310 |

x_{6} | 0.02000 | x_{9} | 0.03039 | x_{11} | 0.03965 | x_{11} | 0.09706 | 1.310 |

x_{11} | 0.03221 | x_{11} | 0.04104 | x_{8} | 0.04925 | x_{13} | 0.09778 | 1.310 |

x_{10} | 0.07215 | x_{12} | 0.04635 | x_{3} | 0.06652 | x_{6} | 0.14037 | 1.310 |

x_{9} | 0.07762 | x_{3} | 0.04942 | x_{12} | 0.07958 | x_{8} | 0.16516 | 1.310 |

x_{3} | 0.08465 | x_{5} | 0.09803 | x_{7} | 0.08101 | x_{1} | 0.45239 | 1.310 |

x_{5} | 0.08554 | x_{8} | 0.21925 | x_{9} | 0.13166 | x_{3} | 0.70315 | 1.310 |

x_{12} | 0.09204 | x_{6} | 0.31655 | x_{1} | 0.18917 | x_{2} | 1.17224 | 1.310 |

x_{8} | 0.15825 | x_{7} | 0.35287 | x_{10} | 0.19642 | x_{5} | 1.63578 | 1.310 |

x_{13} | 1.54362 | x_{4} | 0.68738 | x_{4} | 0.32873 | x_{10} | 2.22484 | 1.310 |

x_{1} | 5.40143 | x_{2} | 0.75835 | x_{2} | 0.84857 | x_{12} | 3.2541 | 1.310 |

x_{2} | 67.52536 | x_{13} | 11.30908 | x_{13} | 14.86887 | x_{4} | 15.81833 | 1.310 |

B Statistic Values | Critical Value | |||||||
---|---|---|---|---|---|---|---|---|

Factors | F_{z} | Factors | F_{x} | Factors | F_{y} | Factors | A_{c} | |

x_{7} | - | x_{1} | - | x_{6} | - | x_{9} | - | - |

x_{4} | - | x_{10} | - | x_{5} | - | x_{7} | - | 1.180 |

x_{6} | - | x_{9} | - | x_{11} | - | x_{11} | - | 1.200 |

x_{11} | - | x_{11} | - | x_{8} | - | x_{13} | - | 1.160 |

x_{10} | - | x_{12} | - | x_{3} | - | x_{6} | - | 1.130 |

x_{9} | - | x_{3} | - | x_{12} | - | x_{8} | - | 1.100 |

x_{3} | - | x_{5} | - | x_{7} | - | x_{1} | - | 1.065 |

x_{5} | - | x_{8} | - | x_{9} | - | x_{3} | - | 1.030 |

x_{12} | - | x_{6} | - | x_{1} | - | x_{2} | - | 1.011 |

x_{8} | 0.3200 | x_{7} | - | x_{10} | - | x_{5} | 0.9544 | 0.991 |

x_{13} | 1.1258 | x_{4} | - | x_{4} | - | x_{10} | 1.0293 | 0.975 |

x_{1} | 1.9125 | x_{2} | 0.9009 | x_{2} | 0.5550 | x_{12} | 1.1122 | 0.959 |

x_{2} | 3.6180 | x_{13} | 2.1199 | x_{13} | 2.1932 | x_{4} | 1.6703 | 0.947 |

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## Share and Cite

**MDPI and ACS Style**

Ma, X.; Wang, P.; Xu, J.; Chen, R.
Parameters Studies on Surface Initiated Rolling Contact Fatigue of Turnout Rails by Three-Level Unreplicated Saturated Factorial Design. *Appl. Sci.* **2018**, *8*, 461.
https://doi.org/10.3390/app8030461

**AMA Style**

Ma X, Wang P, Xu J, Chen R.
Parameters Studies on Surface Initiated Rolling Contact Fatigue of Turnout Rails by Three-Level Unreplicated Saturated Factorial Design. *Applied Sciences*. 2018; 8(3):461.
https://doi.org/10.3390/app8030461

**Chicago/Turabian Style**

Ma, Xiaochuan, Ping Wang, Jingmang Xu, and Rong Chen.
2018. "Parameters Studies on Surface Initiated Rolling Contact Fatigue of Turnout Rails by Three-Level Unreplicated Saturated Factorial Design" *Applied Sciences* 8, no. 3: 461.
https://doi.org/10.3390/app8030461