# A Numerical Study on Structural Performance of Railway Sleepers Using Ultra High-Performance Concrete (UHPC)

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{f}. Instead of performing a direct tension test on concrete, they conducted splitting, three-point bending, and four-point bending tests. In their study, they reported that the ratio of the maximum to minimum loads is not necessarily corresponding to the limit of the input parameters. Valikhani et al. [21] studied numerical modeling of bonding of regular concrete and UHPC since UHPC can be used for the repair of concrete structures. In order to model the interface between concrete and UHPC, they used a zero-thickness volume element with post-failure tension-separation laws. They demonstrated the importance of the interface between two different materials. Shin and Yu [22] presented a numerical study on the splitting performance of prestressed concrete prisms by incorporating bond-slip behavior of prestressed concrete using a cohesive element. They used a user-defined material model to describe the bond-slip behavior at the interface.

## 2. Mix Design and Fabrication of UHPC Sleeper

## 3. Finite Element Modeling

^{2}= 2 × 66.48 mm

^{2}) at the specific heights. The prestressing tendons were fully embedded into the concrete body. Figure 3 shows the 2D numerical model developed in ABAQUS (ABAQUS 6.14, Dassault Systèmes Simulia Corp, Providence, RI, USA). The total number of the elements and the nodes were 1840 and 1985, respectively. Then, 69,000 N (1038 MPa) of the prestressing force was assigned to each tendon. A pin and roller boundary conditions were assigned at 197 mm and 697 mm nodes from the free end. A point load was applied at 447 mm from the free end on the top surface at the rail-seat section, similar to the experimental test. An explicit dynamics analysis was performed for a quasi-static process [23].

## 4. Comparisons with Experimental Data

#### 4.1. Summary of the Testing at the Rail-Seat Section

_{o}of 126.8 kN, was computed [17]. The force and crack-width relationship of each sleeper was obtained and compared to each other. Overall, the higher the steel fiber contents are, the higher load capacities become. The 1.5% UHPC sleepers showed the highest failure forces and were able to mitigate the crack propagation. In Section 4.2.2, the experimental force and crack-width relationships together with numerical results are presented. The detail of the experimental tests and results can be found in the previous study [8].

#### 4.2. Validation of the Numerical Sleeper Models

#### 4.2.1. Fiber Orientation Reduction Factor

#### 4.2.2. Validation of the Numerical Models

## 5. Parametric Study

#### 5.1. Design of Input Parameters

_{r}) changes, the height of the cross-section (h

_{c}) changes accordingly. In addition, the locations of the steel tendons on top (P

_{1}) and bottom (P

_{2}) have to be adjusted (see Figure 9). Three different heights at the rail-seat section have been explored: 140 mm (L-type), 165 mm (M-type), and 195 mm (H-type). L, M, and H stands for lower, medium, and high height of the cross-sections, respectively. In the railway industry in South Korea, a 9.2 mm diameter tendon with 1080 MPa of the yielding strength has been commonly used. However, there is a growing interest in adopting larger diameter tendons and/or high strength steel such as 11.0 mm and 1275 MPa of the yielding strength when manufacturing prestressed concrete sleepers. Three different levels of steel fiber contents (0.5%, 1.0%, and 1.5%) are also explored. The total number of the simulation cases is 21, and Table 3 summarizes the 21 different simulation cases. Specimen numbers 1~7 were designed to have 0.5% of steel fiber of the UHPC, specimen numbers 8~14, 1.0%, and specimen numbers 15~21, 1.5%, respectively.

#### 5.2. Analysis Results

#### 5.2.1. Cross-Sectional Dimensions: L, M and H

_{r}, 9.2 mm of the diameter, 1275 MPa of f

_{y}(steel), and 1% of the steel fiber content is represented by the black square line (L/9.2/1275/1%). ΔF

_{1}means the change in the applied load required between the force (Fr

_{r}) when the crack width is about 0.01 mm and the corresponding force (Fr

_{0.05}) when the crack width reaches about 0.05 mm. Higher ΔF

_{1}is observed from the larger section sleepers. This means that the larger cross-section sleepers are capable of delaying crack propagations. In other words, when the cross-section of the sleeper gets larger, the moment of inertia becomes greater, which results in increased flexural rigidity and sustains higher moments without significant damages. After the crack width reached 0.05 mm, the secant and tangent modulus of the force-crack width diagram were gradually reduced. At approximately 0.12 mm crack width, the PS tendons reached the yielding. Soon after the yielding of the prestressing tendons, the sleepers reached the failure (Fr

_{B}) of the rail-seat section due to the significantly reduced flexural rigidity. Similar trends were observed when the steel fibers were 0.5% and 1.5% as well. The force and crack-width graphs of other cases are presented in Section 5.2.3.

_{1}ratios of the M to L sleeper and H to L sleeper were 1.30 and 1.74, respectively. This means that the increased capacity ratios of the sleepers were higher than the increased area ratios. The safety factor of each sleeper can be computed by $\frac{{\mathrm{F}}_{\mathrm{r}\mathrm{B}}}{2.5{\mathrm{F}}_{\mathrm{r}\mathrm{o}}}$, where Fr

_{B}and Fr

_{o}is the force at the failure and the design reference force; Fr

_{o}is 126.8 kN and 2.5 is the dynamic factor [17]. L, M, and H’s safety index was found to be 1.51, 1.97, and 2.64. Too large a safety index means the sleeper is over-designed. This study suggests an economical design factor, which can be computed by 100Fr

_{B}/Area. When this index is close to 1, the sleeper is structurally sound and economical. The 100Fr

_{B}/Area index value of the L, M, and H sleepers were found to be 1.02, 1.12, and 1.25, which indicate that the L-type sleeper is the most economical design.

#### 5.2.2. The Diameter and the Yielding Strength of PS Tendons

_{y}are considered. As examples, five simulations are presented in Table 5 and Figure 11: (1) L/9.2/1275/1.0%, (2) L/11.0/1275/1.0%, (3) L/11.0/1080/1.0%, (4) H/9.2/1275/1.0%, and (5) H/9.2/1080/1.0%. Given that the cross-sections and the steel fiber contents are kept constant, about 20% higher yielding strength PS tendons results in only 4.4% and 9.5% increase in ΔF

_{2}for H/9.2 types, and L/11.0 types, respectively. This is due to the area of the PS tendons to the area of the cross-sectional area of concrete being relatively low for the H/9.2 type. When using the larger diameter PS tendons, the load capacities of the sleepers increase accordingly. When comparing the results between L/9.2/1275/1.0% and L/11.0/1275/1.0%, the area of the larger diameter PS tendons is 1.43 times to that of the smaller diameter tendons; and the increase in Fr

_{r}, Fr

_{0.05}, and Fr

_{B}is 20%, 11%, and 20%, respectively. These results indicate that the use of the larger diameter tendons would be more efficient than the use of the higher strength PS tendons in terms of the load increase capacities. In addition, these simulation results give some insights on whether sleeper (or crosstie) engineers would like to use a combination of (1) smaller diameter with higher strength PS tendons or (2) larger diameter with lower strength PS tendons. L/11.0/1080/1.0% case shows higher load capacities and safety factors than those from L/9.2/1275/1.0%. However, when engineers prefer an economical design, L/9.2/1275/1.0% can also be adopted since the safety factor is 1.51 and the 100Fr

_{B}/Area index is close to 1.0.

#### 5.2.3. Steel Fiber Contents

_{B}/Area is 0.79 for L/9.2/1275/0.5%. For the larger cross-section sleepers (H-type cases), the trends are similar to those from the L-type cases. The steel fiber 1.0% and 1.5% sleepers show good performance while the difference between two cases is not as great as the L-types. The H-type-0.5% steel fiber sleeper shows lower load capacities and safety factors when compared to those of the higher steel fiber content sleepers; 100Fr

_{B}/Area is 0.94, which is still less than 1.0. When casting a smaller cross-section UHPC sleeper (L-type case), the use of 0.5% steel fiber content is not adequate. In addition, the difference in the performance of the sleepers between 1.0 and 1.5% steel fiber UHPC in terms of the force and crack-width at the rail-seat is not much different. Therefore, 1.0% steel fiber UHPC can be an economical design choice. For the larger cross-section sleepers, all three steel fiber contents would be acceptable. However, instead of H-type 0.5% sleeper, M-type sleepers could be a good alternative since M-type sleepers shows the similar performance while they are more economical. As an example, the performance of M/9.2/1275/1.0% is similar to that of H/9.2/1275/0.5% in term of Fr

_{r}, Fr

_{0.05,}Fr

_{B,}the economical design factor, and the safety factor (see Table 4 and Table 6). The economical design factor, 100Fr

_{B}/Area of the M-type was found to be 1.12, while that of the H-type was 0.94.

## 6. Discussion

## 7. Conclusions

- The developed numerical 2D-UHPC sleeper model was capable of representing the force and crack-width relationships. Three UHPC direct tension tests with the 0.5%, 1.0%, and 1.5% steel fiber contents were used for the UHPC tensile constitutive models.
- The fiber orientation factor, $\mathsf{\alpha}$, of 0.785 is used to represent the realistic stress-strain behavior of the UHPC in 3D as opposed to the thin coupon test where the fibers are well aligned in a 2D manner.
- The numerical analysis results indicate that the bigger the cross-section is, the higher the load capacities and the safety factor become. However, using a too large cross-section can result in uneconomical design sleepers. The economical design factor, 100Fr
_{B}/Area is computed to evaluate the economical factor of the UHPC sleeper. When 100Fr_{B}/Area is close to 1.0, the UHPC sleeper is economical. - There are growing interests in using a larger diameter tendon and/or a higher strength tendon. This study recommends using a larger diameter tendon with a lower strength for an economical design.
- A steel fiber content of 0.5% tends to yield to lower strengths UHPC sleepers relative to the 1.0% and 1.5% steel fiber content sleepers.
- Some M-type sleepers with 1.0% steel fiber UHPC show similar performance to H-type sleepers with 0.5% steel fiber UHPC.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Sadeghi, J.; Kian, A.R.T.; Khabbazi, A.S. Improvement of mechanical properties of railway track concrete sleepers using steel fibers. J. Mater. Civil. Eng.
**2016**, 28, 04016131. [Google Scholar] [CrossRef] - Ferdous, W.; Manalo, A. Failures of mainline railway sleepers and suggested remedies–review of current practice. Eng. Fail. Anal.
**2014**, 44, 17–35. [Google Scholar] [CrossRef] - Kaewunruen, S.; Remennikov, A. Dynamic properties of railway track and its components: A state-of-the-art review. In New Research on Acoustics; Weiss, B.N., Ed.; Hauppauge, Nova Science: New York, NY, USA, 2008; Volume 28, pp. 197–220. [Google Scholar]
- Pyo, S.; Alkaysi, M.; El-Tawil, S. Crack propagation speed in ultra high performance concrete (UHPC). Constr. Build. Mater.
**2016**, 114, 109–118. [Google Scholar] [CrossRef][Green Version] - Ramezanianpour, A.A.; Esmaeili, M.; Ghahari, S.A.; Najafi, M.H. Laboratory study on the effect of polypropylene fiber on durability, and physical and mechanical characteristic of concrete for application in sleepers. Constr. Build. Mater.
**2013**, 44, 411–418. [Google Scholar] [CrossRef] - Shin, H.O.; Yang, J.M.; Yoon, Y.S.; Mitchell, D. Mix design of concrete for prestressed concrete sleepers using blast furnace slag and steel fibers. Cem. Concr. Compos.
**2016**, 74, 39–53. [Google Scholar] [CrossRef] - Yang, J.M.; Shin, H.O.; Yoon, Y.S.; Mitchell, D. Benefits of blast furnace slag and steel fibers on the static and fatigue performance of prestressed concrete sleepers. Eng. Struct.
**2017**, 134, 317–333. [Google Scholar] [CrossRef] - Bae, Y.; Pyo, S. Effect of steel fiber content on structural and electrical properties of ultra high performance concrete (UHPC) sleepers. Eng. Struct.
**2020**, 222, 111131. [Google Scholar] [CrossRef] - EN 13230-2. Railway Applications-Track-Concrete Sleepers and Bearers-Part 2: Prestressed Monoblock Sleepers; European Committee for Standardization (CEN): Brussles, Belgium, 2016. [Google Scholar]
- AS-1085.14. Railway Track Material Part 14: Prestressed Concrete Sleepers; Standard Australia: Sydney, Australia, 2012. [Google Scholar]
- UIC. 713R. Design of Monoblock Concrete Sleepers; UIC Leaflet, International Union of Railways: Paris, France, 2004. [Google Scholar]
- Hashim, D.T.; Hejazi, F.; Lei, V.Y. Simplified Constitutive and Damage Plasticity Models for UHPFRC with Different Types of Fiber. Int. J. Concr. Struct. Mater.
**2020**, 14, 1–21. [Google Scholar] [CrossRef] - Pyo, S.; El-Tawil, S.; Naaman, A.E. Direct tensile behavior of ultra high performance fiber reinforced concrete (UHP-FRC) at high strain rates. Cem. Concr. Res.
**2016**, 88, 144–156. [Google Scholar] [CrossRef][Green Version] - Park, S.; Wu, S.; Liu, Z.; Pyo, S. The role of supplementary cementitious materials (SCMs) in ultra high performance concrete (UHPC): A review. Mater.
**2021**, 14, 1472. [Google Scholar] [CrossRef] [PubMed] - Pyo, S.; Abate, S.Y.; Kim, H.K. Abrasion resistance of ultra high performance concrete incorporating coarser aggregate. Constr. Build. Mater.
**2018**, 165, 11–16. [Google Scholar] [CrossRef] - Pyo, S.; El-Tawil, S. Capturing the strain hardening and softening responses of cementitious composites subjected to impact loading. Constr. Build. Mater.
**2015**, 81, 276–283. [Google Scholar] [CrossRef] - Bae, Y.; Pyo, S. Ultra high performance concrete (UHPC) sleeper: Structural design and performances. Eng. Struct.
**2020**, 210, 110374. [Google Scholar] [CrossRef] - Auersch, L.; Said, S.; Knothe, E.; Rücker, W. The dynamic behavior of railway tracks with under sleeper pads, finite-element boundary-element model calculations, laboratory tests and field measurements. In Proceedings of the 9th European Conference on Structural Dynamics (EURODYN 2014), Porto, Portugal, 30 June–2 July 2014; pp. 805–812. [Google Scholar]
- Chandra, S.; Shukla, D. Sustainability of Railway Tracks. In Sustainability Issues in Civil Engineering; Springer: Singapore, 2017; pp. 91–104. [Google Scholar]
- Sucharda, O.; Mateckova, P.; Bilek, V. Non-Linear Analysis of an RC Beam Without Shear Reinforcement with a Sensitivity Study of the Material Properties of Concrete. Slovak J. Civ. Eng.
**2020**, 28, 33–43. [Google Scholar] [CrossRef] - Valikhani, A.; Jahromi, A.J.; Mantawy, I.M.; Azizinamini, A. Numerical Modeling of Concrete-to-UHPC Bond Strength. Materials
**2020**, 13, 1379. [Google Scholar] [CrossRef] [PubMed][Green Version] - Shin, M.; Yu, H. Numerical Evaluation of Splitting Performance of Prestressed Concrete Prisms With Larger Diameter Prestressing Wires. In Proceedings of the 2019 ASME Joint Rail Conference, Snowbird, UT, USA, 9–12 April 2019. [Google Scholar] [CrossRef]
- ABAQUS. ABAQUS Documentation; Dassault Systemes: Providence, RI, USA, 2012. [Google Scholar]
- Pyo, S.; Kim, H.K.; Lee, B.Y. Effects of coarser fine aggregate on tensile properties of ultra high performance concrete. Cem. Concr. Compos.
**2017**, 84, 28–35. [Google Scholar] [CrossRef] - Japan Society of Civil Engineers (JSCE). Recommendations for Design and Construction of High Performance Fiber Reinforced Cement Composites with Multiple Fine Cracks (HPFRCC); Concrete Engineering Series; Springer: Tokyo, Japan, 2008. [Google Scholar]
- Naaman, A.E. Engineered steel fibers with optimal properties for reinforcement of cement composites. J. Adv. Concr. Technol.
**2003**, 1, 241–252. [Google Scholar] [CrossRef][Green Version] - Pyo, S.; Wille, K.; El-Tawil, S.; Naaman, A.E. Strain rate dependent properties of ultra high performance fiber reinforced concrete (UHP-FRC) under tension. Cem. Concr. Compos.
**2015**, 56, 15–24. [Google Scholar] [CrossRef] - Wille, K.; Kim, D.J.; Naaman, A.E. Strain-hardening UHP-FRC with low fiber contents. Mater. Struct.
**2011**, 44, 583–598. [Google Scholar] [CrossRef]

**Figure 1.**Experimentally obtained averaged stress-strain relationships under the direct tensile test on the UHPC with various fiber volume contents and the corresponding numerical constitutive models.

**Figure 2.**Geometrical dimension of L-150 series sleeper (unit: mm): (

**a**) top view; (

**b**) front view; (

**c**) rail-seat section; (

**d**) center section.

**Figure 6.**Comparison of the force and crack-width curves with 0.5% steel fiber UHPC at the rail-seat section.

**Figure 7.**Comparison of the force and crack-width curves with 1.0% steel fiber UHPC at the rail-seat section.

**Figure 8.**Comparison of the force and crack-width curves with 1.5% steel fiber UHPC at the rail-seat section.

**Figure 10.**The force-crack width diagram of the L, M, and L-type sleepers with 9.2 mm diameter, 1% steel fiber, and fy of 1275 MPa.

**Figure 11.**Force and crack width diagrams with respect to the diameter and the yielding strength of the PS tendons (the steel fiber content was kept at a 1.0% constant).

**Figure 12.**Force and crack-width relationships of the L- and H-type sleepers with respect to the three different steel fiber contents.

**Figure 13.**Simulation results of the force and crack-width curves with 0.5% steel fiber UHPC at the rail-seat section.

**Figure 14.**Simulation results of the force and crack-width curves with 1.0% steel fiber UHPC at the rail-seat section.

**Figure 15.**Simulation results of the force and crack-width curves with 1.5% steel fiber UHPC at the rail-seat section.

Concrete | Young’s modulus | 51.0 GPa |

Compressive strength | 150 MPa | |

Poisson’s ratio | 0.2 | |

Tensile strength (steel fiber 0.5%) | 8.82 MPa | |

Tensile strength (steel fiber 1.0%) | 15.6 MPa | |

Tensile strength (steel fiber 1.5%) | 18.4 MPa | |

Direct stress onset of cracking (steel fiber 0.5%) | 3.17 MPa | |

Direct stress onset of cracking (steel fiber 0.5%) | 6.52 MPa | |

Direct stress onset of cracking (steel fiber 0.5%) | 5.58 MPa | |

Steel tendon | Young’s modulus | 200 GPa |

Yielding strength | 1275 MPa | |

Poisson’s ratio | 0.3 |

Steel fiber contents (%) | 0.5 | 1.0 | 1.5 |

Yielding stress of prestressing tendon (f_{y}) (MPa) | 1 080 | 1 275 | |

Diameter of prestressing tendon (φ) (mm) | 9.2 | 11.0 | |

Cross-sectional parameters | L-Type | M-Type | H-Type |

Height of the rail-seat section, h_{r} (mm)(h _{r1}, mm) | 140 (125) | 165 (150) | 195 (180) |

Height of the center section, h_{c} (mm) | 125 | 150 | 180 |

Location of the prestressing tendon, P_{1} (mm) | 32.5 | 35 | 50 |

Location of the prestressing tendon, P_{2} (mm) | 60 | 75 | 80 |

Sp. No. | Steel Fiber 0.5% | Sp. No. | Steel Fiber 1.0% | Sp. No. | Steel Fiber 1.5% |
---|---|---|---|---|---|

No.1 | L-φ9.2-f_{y}1 275 | No.8 | L-φ9.2-f_{y}1 275 | No.15 | L-φ9.2-f_{y}1 275 |

No.2 | L-φ11.0-f_{y}1 275 | No9 | L-φ11.0-f_{y}1 275 | No.16 | L-φ11.0-f_{y}1 275 |

No.3 | L-φ11.0-f_{y}1 080 | No.10 | L-φ11.0-f_{y}1 080 | No.17 | L-φ11.0-f_{y}1 080 |

No.4 | M-φ09.2-f_{y}1 275 | No.11 | M-φ9.2-f_{y}1 275 | No.18 | M-φ9.2-f_{y}1 275 |

No.5 | M-φ09.2-f_{y}1 080 | No.12 | M-φ9.2-f_{y}1 080 | No.19 | M-φ9.2-f_{y}1 080 |

No.6 | H-φ09.2-f_{y}1 275 | No.13 | H-φ9.2-f_{y}1 275 | No.20 | H-φ9.2-f_{y}1 275 |

No.7 | H-φ09.2-f_{y}1 080 | No.14 | H-φ9.2-f_{y}1 080 | No.21 | H-φ9.2-f_{y}1 080 |

**Table 4.**Summary of the L, M, and H-type sleepers with the following parameters: 9.2 mm diameter, fy of 12,175 MPa and 1% steel fiber.

Simulation Case | Rail-Seat Section Area (mm ^{2}) | Force (kN) | Crack Width (mm) | 100Fr_{B}/Area | ΔF_{1} (kN) = (Fr_{0.05} − Fr_{r}) | ΔF_{2} (kN) = (Fr_{B} − Fr_{0.05}) | Fr_{B}/2.5Fr_{0} | |
---|---|---|---|---|---|---|---|---|

L/9.2/1 275/1.0% | 47 200 | Fr_{r} | 230.4 | 0.008 6 | 1.02 | 158.4 | 91.2 | 1.51 |

Fr_{0.05} | 388.8 | 0.056 4 | ||||||

Fr_{B} | 480.0 | 1.28 | ||||||

M/9.2/1 275/1.0% | 56 075 | Fr_{r} | 300.0 | 0.009 | 1.12 | 206.2 | 118.8 | 1.97 |

Fr_{0.05} | 506.2 | 0.051 | ||||||

Fr_{B} | 625.0 | 2.94 | ||||||

H/9.2/1 275/1.0% | 66 725 | Fr_{r} | 400.8 | 0.009 4 | 1.25 | 276.5 | 158.7 | 2.63 |

Fr_{0.05} | 676.3 | 0.052 8 | ||||||

Fr_{B} | 835.0 | 1.988 |

**Table 5.**Summary of the simulation results with respect to the diameter and the yielding strength of the PS tendons (the steel fiber content was kept at a 1.0% constant).

Analysis Case | Rail-Seat Section Area (mm ^{2}) | Force (kN) | Crack Width (mm) | 100Fr_{B}/Area | ΔF_{1} (kN) = (Fr_{0.05} − Fr_{r}) | ΔF_{2} (kN) = (Fr_{B} − Fr_{0.05}) | Fr_{B}/2.5Fr_{0} | |
---|---|---|---|---|---|---|---|---|

L/9.2/1 275/1.0% | 47 200 | Fr_{r} | 230.4 | 0.0086 | 1.02 | 158.4 | 91.2 | 1.51 |

Fr_{0.05} | 388.8 | 0.0564 | ||||||

Fr_{B} | 480.0 | 1.28 | ||||||

L/11.0/1 275/1.0% | 47 200 | Fr_{r} | 276.0 | 0.0098 | 1.22 | 155.3 | 142.7 | 1.81 |

Fr_{0.05} | 431.3 | 0.0452 | ||||||

Fr_{B} | 575.0 | 7.292 | ||||||

L/11.0/1 080/1.0% | 47 200 | Fr_{r} | 249.6 | 0.0115 | 1.10 | 171.6 | 98.8 | 1.64 |

Fr_{0.05} | 421.2 | 0.0532 | ||||||

Fr_{B} | 520.0 | 2.104 | ||||||

H/9.2/1 275/1.0% | 66 725 | Fr_{r} | 400.8 | 0.0094 | 1.25 | 276.5 | 158.7 | 2.63 |

Fr_{0.05} | 676.3 | 0.0528 | ||||||

Fr_{B} | 835.0 | 1.988 | ||||||

H/9.2/1 080/1.0% | 66 725 | Fr_{r} | 384.0 | 0.0084 | 1.12 | 264.0 | 152.0 | 2.52 |

Fr_{0.05} | 648.0 | 0.0539 | ||||||

Fr_{B} | 800.0 | 1.922 |

**Table 6.**Summary of the simulation results with respect to the 0.5%, 1.0%, and 1.5% steel fiber contents.

Analysis Case | Rail-Seat Section Area (mm ^{2}) | Force (kN) | Crack Width (mm) | 100Fr_{B}/Area | ΔF_{1} (kN) = (Fr_{0.05} − Fr_{r}) | ΔF_{2} (kN) = (Fr_{B} − Fr_{0.05}) | Fr_{B}/2.5Fr_{0} | |
---|---|---|---|---|---|---|---|---|

L/9.2/1 275/0.5% | 47 200 | Fr_{r} | 180.0 | 0.0103 | 0.79 | 101.3 | 93.7 | 1.18 |

Fr_{0.05} | 281.3 | 0.0487 | ||||||

Fr_{B} | 375.0 | 2.282 | ||||||

L/9.2/1 275/1.0% | 47 200 | Fr_{r} | 230.4 | 0.0086 | 1.02 | 158.4 | 91.2 | 1.51 |

Fr_{0.05} | 388.8 | 0.0564 | ||||||

Fr_{B} | 480.0 | 1.28 | ||||||

L/9.2/1 275/1.5% | 47 200 | Fr_{r} | 217.8 | 0.0108 | 1.05 | 183.1 | 94.1 | 1.56 |

Fr_{0.05} | 400.9 | 0.0532 | ||||||

Fr_{B} | 495.0 | 1.724 | ||||||

H/9.2/1 275/0.5% | 66 725 | Fr_{r} | 300.0 | 0.0105 | 0.94 | 168.7 | 156.3 | 1.97 |

Fr_{0.05} | 468.7 | 0.0454 | ||||||

Fr_{B} | 625.0 | 3.137 | ||||||

H/9.2/1 275/1.0% | 66 725 | Fr_{r} | 400.8 | 0.0094 | 1.25 | 276.5 | 158.7 | 2.63 |

Fr_{0.05} | 676.3 | 0.0528 | ||||||

Fr_{B} | 835.0 | 1.988 | ||||||

H/9.2/1 275/1.5% | 66 725 | Fr_{r} | 388.5 | 0.0109 | 1.32 | 273.8 | 220.7 | 2.79 |

Fr_{0.05} | 662.3 | 0.0452 | ||||||

Fr_{B} | 883.0 | 2.224 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shin, M.; Bae, Y.; Pyo, S.
A Numerical Study on Structural Performance of Railway Sleepers Using Ultra High-Performance Concrete (UHPC). *Materials* **2021**, *14*, 2979.
https://doi.org/10.3390/ma14112979

**AMA Style**

Shin M, Bae Y, Pyo S.
A Numerical Study on Structural Performance of Railway Sleepers Using Ultra High-Performance Concrete (UHPC). *Materials*. 2021; 14(11):2979.
https://doi.org/10.3390/ma14112979

**Chicago/Turabian Style**

Shin, Moochul, Younghoon Bae, and Sukhoon Pyo.
2021. "A Numerical Study on Structural Performance of Railway Sleepers Using Ultra High-Performance Concrete (UHPC)" *Materials* 14, no. 11: 2979.
https://doi.org/10.3390/ma14112979