Evaluation of Steel Slag as a Sustainable Alternative Aggregate for Railway Ballast: A Shakedown Theory-Based Approach

Abstract

Recent advancements in railway construction have emphasized environmental sustainability, integrating considerations of environmental impact into the planning and execution of infrastructure projects to reduce costs and mitigate adverse effects. This study investigates the use of steel slag as a sustainable alternative for railway ballast, grounded in shakedown theory. The characterization of the aggregates was performed in accordance with NBR 5564 and AREMA standards, confirming that the material meets most requirements. The mechanical behavior of the ballast was analyzed under cyclic loading conditions, assessing permanent deformation and the material’s ability to achieve stability (shakedown). Triaxial tests with repeated loading simulated real railway conditions, applying vertical stresses up to 600 kPa and confining pressures ranging from 35 to 200 kPa. The results indicate that steel slag aggregates exhibited promising performance, with seven specimens achieving stable deformation levels, characterized by residual deformations of less than 2.5 mm. Notably, these specimens approached deformations on the order of ${10}^{-7}$, indicating stability under cyclic loading. Furthermore, a comparative analysis of shakedown criteria proposed by various authors revealed variations in limits for granular materials, enhancing the understanding of steel slag aggregate behavior. The experimental results were validated through numerical simulations conducted with Systrain software 2.0, which simulated a loading condition of 32.5 tons per axle, confirming the observations with maximum principal stresses ranging from 166 to 184 kPa in the ballast. The analysis showed that steel slag aggregates can withstand stress levels higher than those of granodiorite, reinforcing their viability as a sustainable alternative for railway ballast.

1. Introduction

The development of freight rail transport in Brazil is directly tied to the need for faster and higher-capacity transportation, given the country’s fertile soil and vast mineral wealth across its continental dimensions. Since the inception of railways, the design of Brazilian railroads has remained largely unchanged, despite the significant increase in both load capacity and speed of passenger and freight trains.

Railway construction consists of a layered foundation, including a compacted sub-ballast or capping layer positioned above the formation soil. Additionally, a thick layer of granular material usually crushed rock ballast is placed above the sub-ballast. The rails are made of steel and are installed on either wooden or concrete sleepers, which transfer loads to the ballast, playing the primary role as the load-bearing layer [1].

The growing environmental concern has highlighted the importance of efficient waste management and the conservation of finite natural resources. The European Union (EU) recommends prioritizing the reuse of these wastes as byproducts, provided they possess the necessary technical characteristics for their intended applications [2,3,4,5]. The global production of steel, which has been increasing in countries such as China, the USA, Brazil, Australia, and several EU nations, generates large quantities of steel slag [5], motivating research into its utilization in various applications [6,7,8,9,10,11,12,13,14].

In the railway sector, the dependence on natural resources is evident, particularly in the large-scale use of crushed materials for railway ballast. These materials, typically sourced from quarries, are extensively used to ensure the structural integrity of the railway, supporting repeated train loads and providing a sturdy base that facilitates track geometry maintenance. However, the extraction of these materials has significant environmental impacts, emphasizing the search for more sustainable alternatives [15].

The production of natural ballast is a capital-intensive activity, and this material often needs to be sourced from distant locations, resulting in significant transportation costs that can represent a considerable portion of the final product price [16]. In Brazil, where steel production is substantial, there is a significant generation of steel slag waste that can be repurposed in civil construction, including railway ballast. This practice helps mitigate environmental impact through recycling and reuse, while also alleviating pressure on natural resources [4]. The use of recycled materials in civil construction reflects a growing trend toward sustainability [17]. Furthermore, the utilization of steel slag reduces the volume of waste destined for landfills, converting it into valuable resources for railway paving [18,19]. Investing in studies that explore the performance of steel slag as railway ballast is crucial, considering that companies frequently face high transportation costs and significant expenses related to ballast maintenance [20]. This underscores the importance of considering the environmental and economic advantages of using steel slag as an alternative to natural ballast materials in the introduction of this study.

Nascimento [21] and Silva et al. [22] highlight the use of steel slags in various countries including England, Germany, Poland, France, and Japan, among others since the early 20th century for multiple purposes. In Brazil, the use of this material in paving has been recorded since the 1970s [23]. Steel slag has been the focus of studies for railway ballast applications, particularly due to the competitive advantage provided by the proximity of steel mills to rail networks in producing countries. However, until the early 2010s, most studies focused more on the environmental performance of slag, such as its electrical resistivity and leaching processes [7], as well as on empirical parameters correlated with mechanical behavior, such as the Los Angeles and Micro-Deval tests reported by Morata and Saborido [24], Kuo and Lin [25], and Paixão and Fortunato [26]). Only recently have studies in the field, such as those by Delgado [4], Gomes [6], Rosa [27], and Cescon [28], begun to investigate the geomechanical behavior of this material under stress conditions similar.

Sahay et al. [29] explored various applications for steel aggregates, highlighting their potential use as railway ballast based on physical parameters. Kaya [30] conducted a comparison of the mechanical behavior of steel slag with a natural crushed rock aggregate, using parallel grading techniques in laboratory tests. The promising results indicate the feasibility of using slag in low-load contexts. Koh et al. [31] focused on the long-term mechanical behavior of steel slag ballast in an operational railway. The authors concluded that the slag performed favorably, supporting its use in railway applications.

Building on this research, Delgado et al. [32] presented results from monotonic triaxial tests conducted on scaled ballast samples, aiming to compare the shear strength of Inert Steel Aggregate for Construction (ISAC) materials with granite aggregate, evaluating both peak and critical state strength. In a subsequent study, Delgado et al. [4] further analyzed this by performing a series of monotonic triaxial tests, focusing specifically on the geomechanical behavior of ISAC compared with granite. The results indicate the superior performance of ISAC in terms of strength, stabilization of permanent deformation, and resilient modulus.

Computational modeling has proven to be an essential tool in the railway context. The discrete element method (DEM) has been effectively employed in various studies to simulate the behavior of granular materials under dynamic loading conditions. For instance, recent research has demonstrated the utility of DEM in assessing railway ballast performance, providing insights into particle interactions and the mechanical behavior of aggregates under load [33,34,35]. In addition to DEM, the finite element method (FEM) has also been widely utilized in railway studies. For example, Rose et al. [36] employed Kentrack 4.0 software to analyze the structural behavior of railway pavements, highlighting the importance of these tools in accurately simulating stress–strain behavior and identifying pathologies. Prakoso [37] discussed two-dimensional and three-dimensional modeling of railway superstructures using FEM in ANSYS software version 14.5, while Rangel [38] applied ABAQUS to estimate deflections in railway pavements. In this study, FEM was utilized to validate experimental results, ensuring a comprehensive analysis of the material’s performance.

Furthermore, Silva and Guimarães [39] simulated railway platforms under static loading using ANSYS and Ferrovia software version 3.0, achieving consistent results. Silva Filho et al. [40] compared the structural evaluation of railway tracks between Ferrovia 3.0 and ANSYS v15 software, highlighting limitations in traditional modeling approaches. Other studies have also utilized SysTrain software to assess the structural behavior of railway pavements. Ribeiro [41] applied SysTrain to conduct a stress–strain analysis through finite element modeling, iteratively processing data between pavement elements and determining the effective stresses acting on the structure. Cruz [42] conducted field tests and modeling to evaluate the integrity of a railway platform, while Delgado et al. [4] analyzed the structural behavior of platforms using static loading models and nonlinear elastic materials.

This study aims to evaluate the technical feasibility of using Linz–Donawitz (LD) steel slag, a byproduct generated at the Ternium facility located in the Santa Cruz district of Rio de Janeiro, Brazil, as a sustainable alternative to conventional railway ballast. The evaluation will be conducted through repeated load triaxial tests and numerical simulations using the finite element method (FEM) with Systrain software. This approach will not only validate the experimental results and ensure a comprehensive analysis of the material’s performance but also assess the ballast’s ability to achieve the shakedown regime under cyclic loading conditions. Furthermore, it will enable the determination of the shakedown limits of the ballast, ensuring it can withstand the applied stresses without exhibiting significant permanent deformations, as well as allowing for a comparison between small-scale tests and simulations.

2. Materials and Methods

2.1. Material

The steel slag was provided by Ternium, located in Santa Cruz, Rio de Janeiro, Brazil. The characterization of the slag was carried out in accordance with the guidelines of the Brazilian standard NBR 5564 [43], assessing particle shape, specific gravity, porosity, water absorption, weathering resistance, Los Angeles abrasion, and Treton toughness.

The grain size distribution applied to the ballast followed the specifications of AREMA standard No. 4 [44], as illustrated in Figure 1.

This enables laboratory tests to be conducted with a grain size suitable for main railway lines without requiring additional scaling. Although the Brazilian standard DNIT 179 [45] allows for a less strict relationship between the specimen diameter and the maximum particle size $(D/d_{\max }\ge 4)$, this can be applied to graded aggregates and other alternative materials, provided they are not chemically stabilized. The sieving and granulometric separation process is presented in Figure 2.

2.2. Preparation and Testing of Specimens

The ballast samples were prepared in accordance with international standards, specifically the British BS EN 13286-7 [46] and the Brazilian DNIT 179 [45], using cylindrical specimens with a diameter of 15 cm and a height of 30 cm. The particle size distribution followed the specifications of AREMA No. 4, ensuring a well-graded aggregate suitable for railway ballast. The shape of the particles was predominantly cubic, with approximately 7% of the particles being noncubic, which meets the standards for ballast materials. The samples were tested in a dry condition, without the addition of moisture, and exhibited an apparent density of approximately 1730 kg/m³ and a porosity of 11%. This porosity is slightly above the typical range for natural aggregates but remains within acceptable limits for steel slag aggregates.

The preparation process involved placing four layers of uniform thickness, compacted by vibration in a tripartite mold. Prior to compaction, a latex membrane with a thickness of 2.0 mm was carefully fitted to the inner walls of the mold to ensure the correct positioning of the samples.

The specimen compaction process involved applying vibration to each layer for one minute, followed by a fifth cycle of two minutes using a vibrating plate. After completing the compaction process, the specimen was manually adjusted to ensure that the particles were ideally positioned on the upper surface. To smooth the top surface of the specimens, a layer of plaster was applied, as shown in Figure 3. After 24 h, the specimen was placed in the dynamic triaxial equipment.

This procedure, previously detailed in earlier studies, was employed by several authors, such as Andrade et al. [47], Gomes et al. [6], Cescon [28], Rosa et al. [27], and Silva [48], who reported densities and void ratios similar to those obtained by other specimen preparation methods [1,49,50].

According to international standards, such as the British BS EN 13286-7 [46] and the Brazilian DNIT 179 [45], the specimens were subjected to cyclic axial loads on the top (${\sigma }_1$) and static axisymmetric loads (${\sigma }_3$). This procedure allows for a detailed analysis of the material’s response to cyclic loading, which is widely used to characterize permanent deformation (PD) resistance and resilient modulus, both fundamental parameters in pavement design. Permanent deformation tests were performed at a frequency of 5 Hz, using a servo-driven piston to apply the load. This frequency reflects typical operational conditions of freight railway transportation, with speeds below 100 km/h.

The triaxial tests were conducted using dynamic triaxial testing equipment (Owntec-MS-151), as shown in Figure 4. The press allows tests to be carried out at frequencies of 1, 2, and 5 Hz. The choice of 5 Hz was based on previous studies, which indicate that this range is representative of typical operational conditions for low-to-moderate-speed trains, as indicated by Nalsund [51] and Trinh et al. [52].

Additionally, the application of 150,000 cycles was chosen to ensure representativeness of the ballast behavior over a typical service life [45]. According to Guimarães [53], for PD studies, it is more appropriate to use repeated loading with a number of applications exceeding 100,000 cycles, as the shape of the curve relating total PD to the number of load applications is just as important as the total value obtained.

A total of 12 PD tests were performed, varying the stress ratios to establish the shakedown limit of the materials, with the values of ${\sigma }_1$ and ${\sigma }_3$ listed in

Table 1. Triaxial Test Data for Specimens.

Test Specimen (CP)	${\mathit{\sigma }}_3$ (kPa)	${\mathit{\sigma }}_{\mathit{d}}$ (kPa)	${\mathit{\sigma }}_1$ (kPa)	Stress Ratio $\left(\frac{{\mathit{\sigma }}_1}{{\mathit{\sigma }}_3}\right)$
CP1	35	70	105	3
CP2	75	150	225	3
CP3	150	300	450	3
CP4	200	400	600	3
CP5	50	200	250	5
CP6	100	300	400	4
CP7	133	400	533	4
CP8	37	150	187	5
CP9	75	300	375	5
CP10	100	400	500	5
CP11	75	375	450	6
CP12	62	313	375	6

. The stress combinations in the table were selected with the objective of identifying the occurrence of shakedown, according to the DNIT 179 standard [45], based on the studies of Guimarães et al. [54] and Dawson et al. [55], in addition to determining the yield limit of the material.

2.3. Simulations Conducted

Numerical simulations were performed using the FEM with the SysTrain software [56]. To validate the experimental results, two scenarios were simulated to evaluate the performance of ballast and sub-ballast layers under different load and thickness conditions, in accordance with railway pavement engineering practices:

• Scenario 1: Application of a 32.5 t/axle load with a 42 cm ballast layer and a 10 cm sub-ballast layer.
• Scenario 2: Application of a 32.5 t/axle load with a 30 cm ballast layer and a 20 cm sub-ballast layer, following the modeling standard used by the SysTrain software.

These scenarios were designed to analyze the influence of ballast and sub-ballast thickness combinations and load variations on the structural behavior of the railway track, comparing experimental practices with modeling standards. Figure 5a presents the three-dimensional (3D) FEM mesh used in the analyses, symmetric in the transverse direction between the tracks, while Figure 5b shows the pavement with the load distribution for wagons with a 32.5 t/axle.

For all simulations, the same geometric configurations of the superstructure elements were adopted. The infrastructure was composed of several elements, whose characteristics are described in

Table 2. Input Parameters for the Railway Infrastructure Simulation.

Element	Description
Rails	• Gauge: 1.6 m	• Modulus of elasticity: 210 GPa
	• Section: TR-68	• Poisson’s ratio: 0.3
	• Specific weight: 7867.01 kg/m3
Sleepers	• Pre-stressed concrete monoblock	• Trapezoidal section
	• Length: 2.8 m	• Height: 20 cm
	• Bottom width: 30 cm	• Top width: 28 cm
	• Specific weight: 2400 kg/m3	• Modulus of elasticity: 33 GPa
	• Poisson’s ratio: 0.2	• Spacing: 60 cm
Fastener	• Spring element
	• Stiffness coefficient: 17,000 kN/m (x/y axes)
	• Stiffness coefficient 170,000 kN/m (z axis)
Ballast	• Steel aggregate (grade no. 4) [44]	• Thicknesses: 30 cm and 42 cm
	• Shoulder: width of 40 cm	• Slope (H:V) 1:1
	• Gradient: 3% towards both sides	• Specific weight: 1730 kg/m3
Sub-ballast	• Thicknesses: 10 cm and 20 cm	• Resilient clay material
	• Nonlinear elastic behavior	• Shoulder: width of 50 cm
	• Slope: 1:1.2 at the edges	• Gradient: 1% transverse slope
Subgrade	• Thickness: 300 cm	• Behavior: linear elastic
	• Resilient modulus: 100 MPa	• Shoulder: width of 2 m
	• Slope: 1:1.5 at the edges	• Gradient: 1% transverse slope
Loading	• Hopper cars	• Distance between coupler and axle: 1.21 m
	• 2 bogies	• Axle spacing: 1.7 m
	• Spacing between bogies: 13.945 m	• Load per axle: 32.5 t/axle
	• Reference position: midpoint of the span between sleepers
	• Reference axle: first axle

. The variations in parameters reflect the two distinct conditions analyzed during the simulations.

3. Results and Discussions

3.1. Physical Characterization

In this study, the slag used is the same as that in the work of Gomes [6], and therefore, its characterization results were adopted. The values were compared with the limits specified by the NBR 5564 [43] and AREMA [44] standards, as presented in

Table 3. Characterization Parameters and Compliance with NBR 5564 and AREMA Standards.

Parameter	Obtained Value	Units	NBR 5564 Limit [43]	AREMA Limit [44]	Check
			Other Lithologies	Steel Slag
Mean Particle Shape	Cubic	-	Cubic	Cubic	Ok
Noncubic Particles	7	%	15	5	Ok
Apparent Specific Gravity	3153	kg/m3	2500	2900	Ok
Water Absorption	3.90	%	2.0	2.00	Nok
Apparent Porosity	11.00	%	2.0	-	Nok
Los Angeles Abrasion	10.60	%	30	30	Ok
Treton Toughness Index	5.20	%	25	-	Ok
Dust Material	0.10	%	1	1	Ok
Clay Clods	0.00	%	0.50	0.50	Ok
Loose Bulk Density	1.585	kg/m3	1250	-	Ok

.

The slag met normative criteria regarding shape, specific gravity, and resistance to wear and shock. However, its porosity and water absorption exceeded the limits, likely due to the material’s metallurgical origin and cooling rate [57]. While elevated porosity and absorption may impair ballast performance in regions subject to freeze-thaw cycles, their impact in tropical climates is less significant.

The increase in saturation degree adversely affects the ballast’s response to external forces [1]. The presence of water within the ballast influences its mechanical behavior, particularly in temperate climates where water in the pores can freeze, inducing stresses in the particles and leading to degradation. In tropical climates, where freezing is rare, absorbed water is unlikely to expand, making saturation and drying cycles more critical for the long-term performance of the ballast.

Although the slag is not specifically produced for use as railway ballast, its characteristics, such as high density and resistance to wear, closely resemble those of high-quality aggregates, as noted by Indraratna [1] and Selig and Waters [58].

3.2. Data Analysis of Triaxial Tests

Table 4. Properties of Molded Test Specimens.

Test Specimen (CP)	${\mathit{e}}_{\mathbf{mold}}$	${\mathit{e}}_{\mathbf{min}}$	Density (kg/m3)
CP-01	0.87	0.87	1712
CP-02	0.87	0.87	1712
CP-03	0.87	0.86	1712
CP-04	0.87	0.86	1714
CP-05	0.87	0.83	1713
CP-06	0.85	0.84	1730
CP-07	0.85	0.83	1730
CP-08	0.85	0.85	1730
CP-09	0.85	0.84	1733
CP-10	0.85	0.83	1713
CP-11	0.85	0.83	1713
CP-12	0.85	0.83	1713

presents the void ratios at the time of specimen molding ($e_{\mathrm{mold}}=e_0$) and the minimum void ratio ($e_{\min }$) after the PD tests. The specimen preparation was considered appropriate, with void ratio values consistent with the recommendations of Indraratna [1] and Delgado [32]. Additionally, the density of the materials used exceeded 1600 kg/m³, as recommended by the CRPH [59].

3.3. Permanent Deformation and Shakedown

The plastic deformations were compared with the maximum recommended limits for pavements, taking into account the contribution of the ballast to the railway deformation. Guimarães et al. [15] suggest a limit of 4.4 mm, based on studies in Brazil, while Hay [60] proposes a limit of 5.0 mm for heavy-haul railways, which was adopted by Werkmeister et al. [55] in the investigation of shakedown. Additionally, [44] establishes a limit of 6.35 mm for PDs in railway ballast, particularly under heavy-load conditions.

Figure 6 presents the graph of PD as a function of the number of cycles. Figure 7 shows the analysis of shakedown occurrence, according to the model proposed by Dawson and Wellner [61].

The analysis of PD graphs is essential for understanding the behavior of materials under repeated loading. Based on the model from the standard [45], which is grounded in [55,62], the behavior of the specimens (CPs) was classified regarding their tendency to accommodate, i.e., shakedown.

As shown in Figure 6, after a certain number of cycles, most CPs exhibit stabilization behavior, indicating the occurrence of shakedown. This behavior is similar to the findings of Guimarães et al. [63], who investigated PD in lateritic mixtures. As observed in this study, the PD curves showed significant growth during the first 10,000 cycles. After this point, the PDs converged asymptotically to constant values, reinforcing the stabilized behavior of the materials after an initial phase of deformation. This comparison highlights the importance of PD stabilization under cyclic loading, regardless of the type of granular aggregate used.

However, CP7, CP10, and CP11 did not exhibit this behavior. The observations can be organized as follows:

• CPs 1 to 6 and CP8: quickly stabilize deformation, indicating Type I behavior—Plastic Accommodation (Shakedown).
• CP9 and CP12: stabilize at higher levels, representing Type II—Accommodation with High Initial Deformation.
• CP7, CP10, and CP11: show continuous deformation increase, characterizing Type III—Continuous Deformation Accumulation.

Table 5. Summary Table of Results and Classifications.

Test Specimen (CP)	Type of Behavior	Description
CP1, CP2, CP3, CP4, CP5, CP6, and CP8	Type I—Plastic Accommodation (Shakedown)	Deformation stabilizes quickly; suitable for railway ballast.
CP9 and CP12	Type II—High Initial Deformation (Shakedown)	Deformation stabilizes after high initial deformation; acceptable with reservations.
CP7, CP10, and CP11	Type III—Continuous Accumulation of Plastic Deformation (Plastic creep)	Deformation does not stabilize, significantly increasing; unsuitable.

summarizes the results and classifications.

According to Table 5, the results indicate that the material generally tends towards shakedown under different stress levels, with most of the specimens (CPs) stabilizing permanent deformation after an initial number of cycles. Guimarães and Motta [54] emphasize that granular materials tend to accumulate high permanent deformations during the first 10,000 cycles, followed by stabilization.

In the shakedown analysis, following Werkmeister et al. [55] and Guimarães and Motta [54], as shown in Figure 7, CPs 1 to 6 and 8 exhibit Type A behavior, with a deformation increment rate on the order of ${10}^{-7}$, indicating deformation accommodation.

CPs 9 and 12 show Type AB behavior [62], with initial plastic flow (Type B) followed by deformation stabilization (Type A). Although they accumulate high deformations initially, these deformations stabilize as the number of cycles increases.

On the other hand, CPs 7, 10, and 11 exhibited Type B behavior, with continuous deformation increase and no tendency towards shakedown. The applied stresses (${\sigma }_1=533$, ${\sigma }_1=500$ and ${\sigma }_1=450$) resulted in significant deformations, as observed in CP10 (5.66 mm) and CP11 (4.14 mm). Although CP7 showed a smaller deformation (2.72 mm), it also did not demonstrate any tendency towards shakedown.

The analysis of Figure 6 and Figure 7 shows that these CPs did not reach deformation rates parallel to the Y-axis. Comparing the results with the limits of Hay [60], Lundgren et al. [64], Guimarães et al. [15], and the Arema standard [44], the CPs that exhibited shakedown presented PDs lower than 2.5 mm, within the recommended limits. The rate of deformation increment was below ${10}^{-5}$ mm/cycle after stabilization, indicating stable accommodation behavior.

Table 6. Classification of Test Specimens Based on Shakedown Behavior and Permanent Deformation.

Test Specimen (CP)	Type of Behavior	Range	Permanent Deformation (mm)	Rate of Increment (mm/cycle)	Classification
CP1	Type I—Plastic Accommodation	A	0.15	≈${10}^{-7}$	Suitable
CP2	Type I—Plastic Accommodation	A	0.03	≈${10}^{-7}$	Suitable
CP3	Type I—Plastic Accommodation	A	0.75	≈${10}^{-6}$	Suitable
CP4	Type I—Plastic Accommodation	A	1.49	≈${10}^{-6}$	Suitable
CP5	Type I—Plastic Accommodation	A	1.14	≈${10}^{-6}$	Suitable
CP6	Type I—Plastic Accommodation	A	1.19	≈${10}^{-6}$	Suitable
CP7	Type III—Continuous Accumulation of Deformation	B	2.72	≈${10}^{-5}$	Unsuitable
CP8	Type I—Plastic Accommodation	A	0.24	≈${10}^{-7}$	Suitable
CP9	Type II—High Initial Deformation	AB	2.22	≈${10}^{-6}$	Acceptable
CP10	Type III—Continuous Accumulation of Deformation	B	4.14	≈${10}^{-5}$	Unsuitable
CP11	Type III—Continuous Accumulation of Deformation	B	5.66	≈${10}^{-5}$	Unsuitable
CP12	Type II—High Initial Deformation	AB	2.46	≈${10}^{-6}$	Acceptable

summarizes the classification of the CPs in relation to shakedown and PD, based on the concepts of Werkmeister et al. [55] and Guimarães and Motta [54].

The specimens that reached shakedown (CP1, CP2, CP5, CP6, CP8, CP9, and CP12) were subjected to maximum vertical stresses (${\sigma }_1$) ranging between 105 kPa and 400 kPa, values comparable to those found in the literature. Indraratna et al. [65] reported average cyclic stresses of 230 kPa for freight trains, while Paiva et al. [66] indicated 278 kPa for passenger trains. Additionally, Indraratna et al. [1] presented values of 350 to 400 kPa. These results show that the CPs classified as suitable can accommodate typical railway stresses, ensuring their long-term stability.

On the other hand, CP7, CP10, and CP11, subjected to maximum vertical stresses between 450 kPa and 533 kPa, did not achieve shakedown, accumulating continuous deformations. These CPs exhibited a stress ratio (${\sigma }_1/{\sigma }_3$) of 4 to 6, with an increment rate of approximately ${10}^{-5}$ mm/cycle.

Notably, the behavior of CP4, CP7, and CP10, despite being subjected to the same deviatoric stress of 400 kPa, differed due to the varying confining stresses. CP4, subjected to a higher confining stress of 200 kPa, remained in the shakedown regime, while CP7 (133 kPa) and CP10 (100 kPa) experienced material flow due to the lower confining stresses.

Additionally, the shakedown analysis was conducted following the method proposed by Alnedawi et al. [67]. This method involves assessing the angle of the deformation curve, establishing mathematical criteria for determining shakedown. The curves representing specific PD (%) in relation to the number of cycles were analyzed, and the angles of these curves were measured using AutoCAD software version 2024, as illustrated in Figure 8.

The analysis conducted indicated that all the specimens exhibited Level A behavior, with angles below 22.5°. Specimen 7 (CP 7) was the closest to transitioning to the B regime, displaying an angle of 22°.

Although the shakedown methodology proposed by Alnedawi et al. [67] classifies all specimens as operating in the shakedown regime, this approach may be less accurate when compared with the models of Werkmeister et al. [55], as well as Guimarães and Motta [54], which identify behaviors that indicate material flow. These models suggest the need for more rigorous and detailed analysis, particularly for materials that may exhibit transitions between regimes A and B. It is important to note that the analysis by Alnedawi et al. [67] does not consider materials in the intermediate regime (AB), a critical aspect for a more comprehensive evaluation of material behavior.

Figure 9 illustrates the shakedown limit, determined from experimental tests conducted at various stress ratios (${\sigma }_1/{\sigma }_3$). It shows the relationship between the applied stresses and the stress ratio (${\sigma }_1/{\sigma }_3$), allowing the identification of conditions under which the material stabilizes without accumulating significant plastic deformations. Additionally, the graph presents the shakedown limit of granodiorite as described by Dawson [55] and Werkmeister [68], enabling a comparison between the materials.

The figure compares the shakedown limit of steel slag and granodiorite, highlighting the differences in the mechanical behavior of these materials. The dashed red line represents the shakedown limit for steel slag, while the green line illustrates the same limit for granodiorite, the latter being used solely as a reference for comparison. All blue points correspond to the results from tests performed on steel slag. It is observed that steel slag exhibits significantly superior performance compared with granodiorite, with considerably higher shakedown limits, reflecting its greater capacity to withstand repeated loads before entering a regime of permanent deformation.

As the stress ratio (${\sigma }_1/{\sigma }_3$) increases, a reduction in the maximum stress supported by the material before yielding is noted. The shakedown limit curve clearly illustrates this transition between stable behavior and the onset of yielding, distinguishing the regions where the material remains in shakedown from those where plastic flow occurs. Even under higher stress conditions, steel slag is less prone to yielding compared with granodiorite, reinforcing its suitability for applications involving cyclic loads and elevated stress conditions.

Figure 10 illustrates the shakedown limit as a function of confining stress. It highlights that increasing the confining stress ${\sigma }_3$ allows the material to withstand higher maximum stresses without entering a yielding regime. The blue points in the figure, representing specimens that achieved shakedown, are located below the line that defines this limit, while the red points, indicating yielding, are above this line. This clearly demonstrates the importance of confining stress in ensuring the material’s stability under cyclic loading.

The final shape of the shakedown limit was described by different types of curves, depending on the variable considered.

• In Figure 9, where the limit is expressed as a function of the stress ratio ${\sigma }_1/{\sigma }_3$, the curve that best describes the data is a decreasing exponential function, with an excellent fit ($R^2=0.96$), varying between 3 and 6.
• In Figure 10, where the limit is a function of the confining stress ${\sigma }_3$, the data is best described by an increasing exponential curve ($R^2=0.98$), with confining stress ranges between 35 and 200 kPa.

It is important to emphasize that the curve fits and variable ranges differ for each representation of the shakedown limit. Therefore, the most accurate description of the material’s behavior under the tested conditions is provided by the equations presented in the figures, which precisely reflect the shakedown limit within the analyzed stress ranges and stress ratios.

3.4. Simulation Results

The shakedown verification implemented in SysTrain is based on the equation proposed by Werkmeister [68]:

$${\sigma }_{1\max }=\alpha {\left({\displaystyle \frac{{\sigma }_{1\max }}{{\sigma }_3}}\right)}^{\beta }$$

where ${\sigma }_{1\max }$ is the peak axial stress from the repeated load triaxial test, ${\sigma }_3$ is the confining pressure, and $\alpha$ and $\beta$ are material parameters.

The index S₁/S_1max represents the ratio between the vertical stress, calculated as the sum of ${\sigma }_d$ and ${\sigma }_c$, and the maximum admissible axial stress, $S1\max$. The value of $S1\max$ is determined using the equation presented above, which specifies ${\sigma }_{1\max }$ calculated based on the confining pressure and the material parameters $\alpha$ and $\beta$.

For the condition to be deemed acceptable, indicating that the soil is in a shakedown regime, the result of this ratio must be less than 1. Figure 11 and Figure 12 illustrate this relationship, demonstrating that the ballast layer exhibits shakedown behavior.

The shakedown behavior was verified:

• Scenario 1: S₁/S_1max = 0.54
• Scenario 2: S₁/S_1max = 0.51

The ballast layer remained within the shakedown regime, ensuring that PD were minimal and that the material responded adequately to repeated loads. The results indicate the feasibility of using steel aggregates for this application, clearly establishing the shakedown limit based on the experimental data.

Moreover, the comparison with numerical modeling reinforced the observed behavior, confirming the accommodation of deformations over time. The simulations showed that the stresses applied to the ballast were within safety parameters, ensuring the durability and stability of the structure under real loading conditions.

The maximum principal stresses (S₁) obtained in the ballast simulation ranged from 166 to 184 kPa. These values align with the results of Cruz [42], who reported similar stresses in simulations using equivalent modeling in the Systrain software, as well as with Delgado et al. [4], who obtained similar results for another type of steel aggregate. Additionally, these values can be compared with those of specimen CP8, which was subjected to a stress of ${\sigma }_1$ of 187 kPa in triaxial tests. Specimen CP8 not only entered the shakedown regime but also showed minimal residual deformation, less than 0.5 mm, indicating stable behavior under the applied loading conditions.

This finding confirms the relationship and validation of the simulations in relation to the experimental tests. In both cases, with comparable stress levels, it was possible to verify the shakedown regime of the material. This further substantiates its suitability for railway ballast applications, given that the simulations were conducted in the context of a railway system with a load of 32.5 tons per axle. The congruence of experimental and simulated data underscores the reliability of the numerical model, ensuring that the material can effectively accommodate repeated loading while maintaining structural integrity.

4. Summary and Conclusions

In order to evaluate the technical feasibility of Linz–Donawitz (LD) steel slag as a sustainable alternative for railway ballast, this research was based on repeated load triaxial tests, complemented by numerical simulations using the finite element method, with SysTrain software employed to validate the experimental results.

• The characterization tests indicated that, despite some porosity and absorption indices exceeding the limits established by NBR 5564, the overall mechanical behavior of the aggregates was satisfactory, with permanent deformations within the standards accepted by AREMA.
• Based on shakedown theory criteria, most of the specimens were classified in A and AB behavior regimes, showing a low permanent deformation increase rate after stabilization. Although the model proposed by a new approach suggested that all specimens were in shakedown, the traditional model identified flow in some cases, demonstrating greater precision in detecting failures. Additionally, the traditional model allows the analysis of specimens according to Guimarães, which predicts an AB-type regime, an aspect that the new model does not cover.Alnedawi’s model does not cover.
• The analysis of the shakedown limits and the quantification of permanent deformations were complemented by computer simulations using SysTrain software, which corroborated the experimental data. The simulations confirmed that permanent deformations remained below the established limits, demonstrating the occurrence of shakedown in the ballast layer. The ratio $S_1/S_{\max }$ remained below 1, indicating the material’s stability under the analyzed load conditions.
• Moreover, the research highlighted that increasing the confining stress ${\sigma }_3$ is a crucial factor for the material’s strength, enabling it to withstand higher maximum stresses without exhibiting flow. This robust behavior underscores the importance of confining stress in ballast stability under cyclic loading conditions. Thus, the analysis of this parameter provides a significant contribution to understanding the performance of steel aggregates, suggesting that optimizing confining stresses could lead to substantial improvements in railway ballast durability and effectiveness.

In summary, the results of this research not only establish steel aggregates as a sustainable and effective alternative for railway ballast but also pave the way for future investigations in the field, contributing to innovations and improvements in transportation infrastructure. The conclusions presented here highlight the importance of a comprehensive and detailed assessment of material behavior, considering multiple factors that influence their performance under real operational conditions.