*4.2. Validation of the Numerical Sleeper Models*

#### 4.2.1. Fiber Orientation Reduction Factor

The fiber orientation effect was considered, herein, when calibrating the experimentally obtained tensile constitutive relationships of UHPC depicted in Figure 1. It is well known that the tensile capacities of fiber reinforced concrete including UHPC principally depends on the fiber properties including the distribution and volume fraction [26,27]. It should be pointed out that due to the relatively thin specimen used in the tensile test (30 mm) [24], 19.5 mm long steel fibers tend to be aligned in a two-dimensional manner. On the other hand, the discontinuous steel fibers can be assumed to be three-dimensionally oriented in the 140 mm thick rail-seat section of the sleepers. The fiber orientation factors, α, are known to be 2/π and 0.5 for two-dimensional and three-dimensional fiber orientations, respectively [27,28]. Therefore, it is logical to adopt 0.785 (= 0.5 2/*<sup>π</sup>* ) as the fiber orientation reduction factor in this numerical study. Figure 5 shows the adopted stress-strain curves in the numerical analysis after considering 0.785 of the reduction factor, α. After the proportional limit of the obtained stress-strain relationships, the strength is reduced by 21.5% of the original strengths. Therefore, the constitutive relationships with α of 0.785 were used in the numerical simulations.

**Figure 5.** Stress and strain curves after applying the reduction factor of 0.785 for tension.

4.2.2. Validation of the Numerical Models

Three numerical sleeper models were prepared: (1) with 0.5% steel fiber contents, (2) with 1.0% steel fiber contents, and (3) with 1.5% steel fiber contents. A point load

is applied at the center of the rail-seat section of the models. The applied load and the crack widths were monitored and compared with the experimental tests. Figures 6–8 show the comparisons of the force and crack-width curves between the numerical simulations and the experimental tests. Overall, the numerical models agree well with the experimental test results. The figures demonstrate that the sleeper models incorporated in this study are capable of capturing the initial stiffness, yielding of the steel tendons, cracking of the concrete, and the capacity of the sleepers due to the different level of steel fiber contents. It is also worthwhile to note that the fiber orientation factor, α, of 0.785 is able to describe the strength change in the UHPC of the sleepers from the coupon tests. In addition, the 1% steel fiber UHPC sleeper tends to overestimate the strength, while 0.5% and 1.5% steel fiber UHPC sleepers underestimate the ultimate strengths as compared to the experimental results.

**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.

#### **5. Parametric Study**

#### *5.1. Design of Input Parameters*

A parametric study was conducted with respect to the cross-sectional dimensions of sleepers and different types of steel tendons and steel fiber contents in the UHPC using the developed numerical sleeper models. In this parametric study, the structural performance of the UHPC sleepers were explored in terms of the crack width, the load capacities, the safety factor, and an economical design factor.

The important mechanical and geometrical parameters of the UHPC sleepers considered, herein, are as follows: (1) the cross-sectional dimensions, (2) the diameter of the steel tendons, (3) the yielding strength of the PS tendons, and (4) the steel fiber content of the UHPC. Table 2 summarizes the input values of each parameter. When the height of the cross-section at the rail-seat (hr) changes, the height of the cross-section (hc) changes accordingly. In addition, the locations of the steel tendons on top (P1) and bottom (P2) 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.


**Table 2.** Summary of the input parameters.

**Figure 9.** The schematics of the UHPC sleeper sections.



#### *5.2. Analysis Results*

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

In order to discuss the effect of the cross-section sizes (L, M, and H), three simulation results were presented in Table 4 and Figure 10 as examples: (1) L/9.2/1275/1.0%, (2) M/9.2/1275/1.0%, and (3) H/9.2/1275/1.0%. In this discussion, the only variable is the size of the cross-section, when other parameters are kept constant: the diameter of the tendons is 9.2 mm, the yielding strength of the tendon is 1275 MPa, and the steel fiber content is 1.0%. In general, the larger the cross-section is, the greater the loading capacity of the sleepers becomes. In the figure, the simulation result of the sleeper with 140 mm of hr, 9.2 mm of the diameter, 1275 MPa of fy (steel), and 1% of the steel fiber content is represented by the black square line (L/9.2/1275/1%). ΔF1 means the change in the applied load required between the force (Frr) when the crack width is about 0.01 mm and the corresponding force (Fr0.05) when the crack width reaches about 0.05 mm. Higher

ΔF1 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 (FrB) 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.

**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.


**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.

The ratio of the cross-sectional area of the M-type sleeper to the L-type sleeper, and the ratio of the cross-sectional area of the H-type sleeper to the L sleeper are 1.19 and 1.41, respectively, while the ΔF1 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 FrB 2.5Fro , where FrB and Fro is the force at the failure and the design reference force; Fro 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 100FrB/Area. When this index is close to 1, the sleeper is structurally sound and economical. The 100FrB/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

Table 5 and Figure 11 shows the simulation results with respect to the diameter and the yielding strength of the PS tendons when the steel fiber content was kept at 1.0%. Two different diameters of the PS tendons are explored: (1) 9.2 mm (smaller diameter), and (2) 11.0 mm (larger diameter). In addition, 1080 MPa and 1275 MPa of the yielding strength, fy 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 ΔF2 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 Frr, Fr0.05, and FrB 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 100FrB/Area index is close to 1.0.

**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).


**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).

## 5.2.3. Steel Fiber Contents

This section presents the simulation results with respect to the steel fiber contents (i.e., 0.5%, 1.0%, and 1.5%). Table 6 and Figure 12 show the summary of the results of the six simulations used as examples: (1) L/9.2/1275/0.5%, (2) L/9.2/1275/1.0%, (3) L/9.2/1275/1.5%, (4) H/9.2/1275/0.5%, (5) H/9.2/1275/1.0%, (6) H/9.2/1275/1.5%. In general, as the steel fiber content increases, the load capacities, the safety factor, and the economic design factor increase. For the smaller cross-section sleepers (L-type cases), the use of 1.0% and 1.5% steel fiber contents results in the significant increase in the performance compared to that of the sleeper with 0.5% steel fiber content. The performance of L/9.2/1275/1.0% and L/9.2/1275/1.5% are similar to each other, and the increase in the load capacities are only 3~6%; furthermore, the safety factor only increases to 1.56 (1.5% of the steel fiber) from 1.51 (1.0% of the steel fiber). The performance of the L-type-0.5% steel fiber sleeper is significantly lower than that of the sleepers with the higher steel fiber contents. As observed, FrB 2.5Fro is only 1.18 and 100FrB/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; 100FrB/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

Frr, Fr0.05, FrB, the economical design factor, and the safety factor (see Tables 4 and 6). The economical design factor, 100FrB/Area of the M-type was found to be 1.12, while that of the H-type was 0.94.


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

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

Figures 13–15 show the force and crack-width relationship of all 21 numerical simulations. Regardless of the steel fiber contents, the increase in size would have the most

significant impacts on the performance of the UHPC sleepers. The improvement can be further enhanced with the combination of the higher steel fiber contents, the larger diameter PS tendons, and the higher strength PS tendons. However, the use of a larger cross-section with 1.5% steel content, 11.0 mm diameter PS tendons of 1275 MPa yielding strength is a clearly over-designed sleeper. This study also indicates that the improvement due to the higher strength PS tendons would be minimum among all the design parameters considered. It is also interesting to note that there are cases that would show similar performance even though the cross-section sizes are different. For example, M/9.2/1080/0.5% and L/11.0/1275/0.5% show similar performance, as well as M/9.2/1275/1.0% and H/9.2/1275/0.5%.

**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.

#### **6. Discussion**

In this numerical study, a 2D prestressed concrete sleeper model with a brittle cracking constitutive model was developed and validated with experimental data. In general, the numerical results were compatible with the experimental results in terms of the force and crack-width relationship at the rail-seat section. The obtained tensile stress-strain relationships of UHPC with different steel fiber contents [24] were directly used to define the cracking stress and cracking strain of the brittle cracking model. In this process, an orientation reduction factor of 0.785 was applied to the post-cracking behavior of all three UHPCs. As it has been known to the community, 2D and 3D orientation reduction factors are 2/π, and 0.5. Therefore, a single orientation reduction factor (0.785) was adopted in this study for the simulations. However, the fiber reduction factor could be dependent upon the distribution, orientation, and volume fraction of the fibers in a concrete mix. Therefore, it is challenging to use a just deterministic approach for the factor while the concrete properties including the reduction factor has the stochastic characters. With the 0.785 reduction factor, 1.0% steel fiber sleepers predict the ultimate strength higher than the experimental test. This shows that there is room for improvement. A stochastic approach can be applied to evaluate the range in the performance of the sleepers using UHPC. It should be, however, pointed out that additional experimental research should be conducted to achieve a statistically meaningful dataset to adopt the stochastic approach, especially for the fiber orientation effect. In addition, the usage of a single orientation reduction factor would be beneficial for sleeper engineers and structural engineers to practically design concrete structures with fiber reinforcements.

#### **7. Conclusions**

This numerical study focuses on investigating the performance of UHPC sleepers with respect to various design/mechanical/geometrical parameters. The parameters include the steel fiber contents, the size of the cross-section, the diameter and yielding strength of the PS tendons. The key observations and findings of this research can be summarized as follows:


This numerical study was able to provide insights on the effects of the design parameters for developing concrete sleepers using UHPC. Additional research needs to be conducted to investigate the overall behavior of UHPC sleepers, including bending at the center-section of the sleepers and the effect of the variability of concrete properties including an orientation reduction factor.

**Author Contributions:** Conceptualization, M.S., Y.B. and S.P.; methodology, M.S., Y.B. and S.P.; software, M.S.; validation, M.S. and Y.B.; formal analysis, M.S. and Y.B.; investigation, M.S., Y.B. and S.P.; resources, M.S., Y.B. and S.P.; data curation, M.S. and Y.B.; writing—original draft preparation, M.S.; writing—review and editing, M.S., Y.B. and S.P.; visualization, M.S. and Y.B.; supervision, S.P.; project administration, Y.B. and S.P.; funding acquisition, Y.B. and S.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research described herein was sponsored by a grant from R&D Program of the Korea Railroad Research Institute, Republic of Korea. This work is also supported by the Korea Agency for Infrastructure Technology Advancement (KAIA) grant funded by the Ministry of Land, Infrastructure and Transport (Grant 21CTAP-C152046-03). The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of the sponsors.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding authors.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

