Efficacy of the Simplex-Centroid Method for Optimization of Mixtures of Soil, Ladle Furnace Slag and Fly Ash Applied in Pavement Construction

Abstract

Integrating industrial wastes into soils to enhance their properties is a potential solution to current waste management challenges. Since the current literature lacks systematic studies on the mechanical performance of mixtures of soil, ladle furnace slag (LFS) and fly ash (FA), this research investigated the chemical stabilization of two different soils (clayey or sandy soil) using a concomitant mix of distinct types of industrial wastes: LFS and FA. A design of experiments (DoE) methodology was employed to systematically generate distinct mixtures for each soil sample, utilizing a simplex-centroid design. The mixtures were subjected to unconfined compressive strength (UCS), California Bearing Ratio (CBR) and resilient modulus (RM) tests. The industrial by-products improved the mechanical properties of the soils, providing UCS, CBR index and RM increases up to 130.5%, 324.4% and 132.6%, respectively. Synergistic and antagonistic effects related to the combination of different wastes were discussed, based on mathematical models with coefficients of determination ranging from 0.760 to 0.998, in addition to response surfaces generated for each response variable. The desirability function was applied to identify the optimal component proportions. The best mixture proportion was 80% soil, 20% LFS and 0% FA, which improved the formation of cemented compounds that contributed to the enhanced mechanical strength. The use of industrial waste for soil stabilization has therefore proven to be technically feasible and environmentally friendly.

1. Introduction

Ladle furnace slag (LFS) is a steelmaking by-product that originates during the secondary steel refinement in electric arc furnaces (EAF) [1]. Fly ash (FA) consists of fine particles produced by the combustion of pulverized coal in thermal power plants [2]. Given the large generation of these by-products and the potential environmental damage related to their improper disposal in the environment, their management has become a challenge for the industry, which is looking for sustainable and economical applications for these materials [3,4].

An alternative way to reuse LFS and FA is applying both materials as chemical soil stabilizers for highway engineering purposes. This alternative is becoming more attractive with the recent issues regarding the environmental impact of production of Portland cement and lime, which are conventional materials used as chemical soil stabilizers [5]. In this context, several studies have reported the technical suitability of adding individual LFS or FA to different soil types, aiming to enhance the engineering properties of the natural soils [4,6,7,8,9,10,11].

The combination of soils, slags and ashes in ternary mixtures has been another sustainable alternative for the road construction industry. However, most studies focused on the study of mixes between soil, ground granulated blast furnace slag and FA, which were recently reviewed by Abdila et al. [12]. In contrast, some previous works investigated the engineering properties of mixtures of soil, LFS and FA [13,14,15,16,17]. These works combined random contents of these two types of by-products and clayey soils and determined the index properties, compaction behavior, permeability, unconfined compressive strength (UCS) and/or California Bearing Ratio (CBR) of the mixtures. However, the CBR swelling (CBR-S) and the resilient modulus (RM) of mixtures of soil, LFS and FA were not investigated in the previous literature. Moreover, the mechanical performance of mixtures of LFS, FA and other types of soils (different from clayey soils) has not yet been investigated in previous papers. The synergistic use of LFS and FA is expected to combine the benefits of the high calcium content of LFS and the high silica and alumina content of FA, improving the development of pozzolanic and self-cementing reactions and enhancing the mechanical properties of the stabilized soil [14]. Moreover, the combination of these industrial by-products offers a cost-effective solution for waste management in industries producing both LFS and FA [17].

In addition, despite the existence of several studies addressing the efficiency of industrial by-products as chemical soil stabilizers, there is no consolidated method focused on the optimization of the design of different types of unconventional chemical stabilizers in mixtures with soils. The absence of systematic criteria to guide the design of waste–soil mixtures is one of the main factors limiting a large-scale application of these types of materials in the road construction industry [18].

An interesting strategy to define mixture proportions that efficiently meet specific design criteria is the design of experiments (DoE). DoE is defined as a branch of applied statistics that deals with planning, conducting, analyzing and interpreting controlled tests to evaluate the factors that control the value of a parameter or group of parameters. DoE is a powerful data collection and analysis tool that can be used in a variety of experimental situations. One of the methods of DoE is the simplex-centroid network. This method allows the elaboration of accurate mathematical models using a reduced number of tests [19]. These mathematical models have the polynomial form presented by Jiao et al. [20] (Equation (1)), which correlates response variables with input variables, allowing the assessment of the influence of an input variable on the response variable, either individually or in combination. Moreover, the optimization of the mathematical model enables the determination of an “optimal” mixture design, in which the response variable is optimized as a function of the input variables [20,21].

$$Y \left(x_1{x}_2{x}_3\right)={ \mathsf{\beta }}_1{x}_1+{\mathsf{\beta }}_2{x}_2+{\mathsf{\beta }}_3{x}_3+{\mathsf{\beta }}_{12}{x}_1{x}_2+{\mathsf{\beta }}_{13}{x}_1{x}_3+{\mathsf{\beta }}_{23}{x}_2{x}_3+{\mathsf{\beta }}_{123}{x}_1{x}_2{x}_3$$

(1)

where:

• $Y \left(x_1{x}_2{x}_3\right)$ is the response variable in function of components proportions;
• β_i is the calculated regression coefficient;
• $x_1,x_2,x_3$ are the input variables (percentage of each component in the mixture $(x_1+x_2+x_3=1$).

Although the simplex-centroid method has been used in several engineering applications such as the design of ceramic products, mortars and concrete [20,21,22,23], there are few studies exploring the efficiency of this method for roadway engineering purposes. Iwański et al. [24] and Onyelowe et al. [25] successfully used the method to optimize the design of cementing compounds used in base course pavements, enhance load-bearing capacity and minimize the expansion of different compounds. However, there is a gap in the technical literature regarding the application of the simplex-centroid method for designing mixtures of different industrial wastes and distinct types of soils. In addition, there is a notable lack of research on the effects of combining LFS and FA for improving clayey and sandy soils commonly used in pavement structural layers.

To narrow these knowledge gaps, this paper presents the following original contributions to the current state-of-the-art: (i) an evaluation of the technical feasibility of the simplex-centroid method for optimizing the UCS, CBR, CBR-S and RM of ternary soil–LFS–FA mixtures; (ii) the viability of using the simplex-centroid method to design soil–LFS–FA mixtures that meet technical criteria for their application as construction materials of pavements; (iii) the application of the RM as a dosage parameter for the design of soil–waste mixtures; (iv) a pioneering investigation of the synergistic and antagonistic effects of the combined incorporation of LFS and FA on the UCS, CBR, CBR-S and RM of road materials; and (v) a comparison between the effects of LFS and FA on the mechanical performance of soils with different particle size distributions. The consolidation of an efficient method for designing this type of mixture will optimize the reuse of these by-products and provide a technical, environmental and economic option to replace conventional soil stabilizer agents.

Therefore, this work addresses the pressing issue of waste management by proposing an innovative solution that combines LFS and FA as sustainable stabilizers for various types of soils used in pavement construction. By focusing on the beneficial reuse of these materials, this research offers an eco-friendly alternative to traditional soil stabilization methods, reducing the need for landfill disposal and tackling a significant global environmental problem. To systematically implement this waste management strategy, the present study proposes the application of the DoE methodology, utilizing a simplex-centroid design to identify synergistic and antagonistic effects from the combination of LFS and FA. This research also proposes the use of the desirability statistical function to determine the mix proportions that best meet multiple criteria related to UCS, CBR, CBR-S and RM values required for application in pavement construction as base, subbase and subgrade materials. This practical application not only improves the performance of pavement structures but also promotes the sustainable use of waste materials, offering a viable solution to current waste management challenges.

2. Materials

2.1. Soils

Two soil samples collected in Viçosa, Minas Gerais State, Brazil (Soil C and Soil V) were used in this study. One sample is predominantly clayey, referred to as Soil C. The other sample is predominantly sandy, referred to as Soil V. Figure 1 and

Table 1. Geotechnical characterization and classification of the soil samples.

Parameter	Soil		Parameter	Soil
	C	V		C	V
Clay (Φ ≤ 0.002 mm) (%)	61	5	Specific unit weight (kN/m3)	28.14	26.57
Silt (0.002 mm < Φ ≤ 0.06 mm) (%)	14	24	Liquidity limit (LL)/Plasticity limit (PL)	78/43	36/18
Fine sand (0.06 mm< Φ ≤ 0.2 mm) (%)	12	13	Plasticity index (PI) (%)	35	18
Medium sand (0.2 < Φ ≤ 0.6 mm) (%)	10	40	USC classification	CH-MH	SC
Coarse sand (0.6 mm < Φ ≤ 2 mm) (%)	3	15	TRB classification	A-5-7 (32)	A-2-6
Gravel (Φ > 2.0 mm) (%)	0	3	MCT classification	LA’	NA

Φ: particle diameter.

show the particle size distribution for both soil samples, in accordance with NBR 7181 [26]. Table 1 presents the results of soil characterization, according to NBR 6458 [27], NBR 6459 [28] and NBR 7180 [29].

2.2. Ladle Furnace Slag (LFS) and Fly Ash (FA)

The LFS used in this research was collected at the facility of a steel tube manufacturer company, headquartered in Jeceaba, State of Minas Gerais, Brazil. Before being used in this study, the raw LFS was air-dried, milled and sieved to eliminate particles with diameter larger than 0.6 mm (sieve #30) and to increase the LFS-specific surface area in order to reach similar levels considered in other studies dealing with steel slags [30,31,32]. The FA sample used in this research is classified as type C FA and meets the pozzolanic criteria prescribed in NBR 12653 [33] for pozzolanic materials. Figure 1 shows the LFS and FA particle size distribution determined by laser diffraction according to ISO 13320 [34].

Table 2. LFS characterization.

Property	LFS	FA	Standard Test Methods
Specific surface area (cm2/g)	1604.4	1788.24	NBR 16372 [35]
Fineness index (%)	55.8	23.71	NBR 11579 [36]
Specific unit weight (kN/m3)	29.1	20.8	NBR 16605 [37]

presents the results of other laboratory tests performed to physically characterize the LFS and FA.

2.3. Chemical Analysis

Table 3. Chemical composition of soils, LFS and FA.

Oxide	Soil C (%)	Soil V (%)	LFS (%)	FA (%)
CaO	0.02	0.07	43.65	1.76
MgO	1.60	1.95	2.43	1.10
SiO2	33.74	50.19	22.22	61.18
Al2O3	30.52	33.63	11.16	24.15
Fe2O3	15.52	2.66	17.41	4.26
K2O	0.08	1.80	0.35	2.84
Na2O	1.93	3.32	0.97	1.41
SO3	0.06	0.04	0.71	0.37
Cl	0.33	0.30	0.30	0.21
TiO2	1.90	0.25	0.71	1.13
LOI *	14.30	5.80	0.09	1.60

* Loss on ignition.

shows the oxide percentages identified in the chemical composition of the materials, which were determined by X-ray fluorescence (XRF) analysis in PANalytical Epsilon3x equipment (Almelo, The Netherlands). The XRF analysis indicated high amounts of SiO₂ and Al₂O₃ in both soil samples and significant Fe₂O₃ content in Soil C sample. Additionally, it was verified that the LFS sample is mainly composed of CaO, SiO₂, Fe₂O₃ and Al₂O₃, while SiO₂ and Al₂O₃ are the main oxides in FA chemical composition.

2.4. Mineralogical Analysis

Figure 2 presents the mineralogical phases identified in the samples, based on results of X-ray diffraction (XRD) analyses using a Bruker D8-Discover diffractometer (Billerica, MA, USA) (CuKα radiation, 40 kV, 30 mA, λ = 1.5418 Å, measurement θ–2θ ranging from 5° to 80°, 0.02°/step, 1 s/step). The XRD spectrum of the LFS indicated that portlandite, olivine, calcite and larnite are the main mineralogical phases of this material. The diffractogram of FA revealed mineralogical phases of mullite, wollastonite, quartz and calcite. In the diffractograms of Soil C and V, the main peaks are attributed to quartz and kaolinite, which are common mineralogical phases of soils [38]. The XRD spectrum of Soil C also revealed hematite traces, whereas Soil V also exhibited dolomite, portlandite and biotite signatures.

2.5. Microstructural Analysis

Figure 3 presents the scanning electron microscopy (SEM) images of the raw materials, obtained with a JEOL JSM-6010LA microscope (Tokyo, Japan) (resolution of 4 nm at 20 kV, magnification of ×8 to ×300,000, and accelerating voltage between 500 V and 20 kV). Figure 3a shows that the LFS has variable geometry, varying from rounded to angular-shaped particles, in addition to rough surfaces and porous aspects. In contrast, Figure 3b indicates that FA particles are mainly hollow spheres with different sizes. The Soil C SEM image [Figure 3c] reveals that this soil presents a compact structure with particle voids filled by smaller-sized minerals. The SEM image of Soil V [Figure 3d] showed a slightly less compacted structure with quartz minerals of different particle sizes.

3. Methods

The methodology used in this research is shown in Figure 4. The flowchart provides a visual representation of the sequential procedures undertaken throughout this study. Each procedure outlined in the flowchart is detailed in the following subsections.

3.1. Design of Experiments

In this study, three materials were used to formulate different experimental mixtures: natural soil (Soil C or Soil V), LFS and FA. The proportions of each mixture composition component were calibrated according to recommendations of industrial by-products contents added to different types of soils in several previous studies [1,15,30,39,40,41]. LFS contents in the mixture compositions ranged from 5% to 20%, FA contents varied from 0% to 15% and the soil content changed between 80% and 95%, all percentages related to the mixture’s total dry mass.

The simplex-centroid ternary diagram shown in Figure 5 illustrates the 7 experimental mixtures (M1 to M7) considered in this study for each soil sample. The proportions of each component in the composition of each experimental mixture are presented in

Table 4. Experimental mixture composition (% in mass).

Material	M1	M2	M3	M4	M5	M6	M7
Soil (%)	95.0	87.5	80.0	80.0	80.0	87.5	85.0
LFS (%)	5.0	12.5	20.0	12.5	5.0	5.0	10.0
FA (%)	0.0	0.0	0.0	7.5	15.0	7.5	5.0

. The lower and upper limits of the proportions of the individual components were input parameters for the statistical software. Since the simplex-centroid design was used in this work, the variation between the lower and upper limits for all components was the same. The software distributed the experimental points such that there were six equally spaced points on the edges of an equilateral triangle, forming a simplex network. In addition, a central point was included to enhance the accuracy of the final response.

The mixture proportions obtained from the DoE considered only the masses of the materials (soil, LFS and FA) in their dry state. To determine the required water content for producing specimens for mechanical tests, compaction tests were performed at standard Proctor energy, as described below. The optimum moisture content from these tests was then used to define the amount of water needed for each mixture.

3.2. Geomechanical Characterization of Natural Soils and Experimental Mixtures

The pure soil samples (Soil C and Soil V) and the experimental mixtures (M1 to M7) were subjected to laboratory tests to determine (i) UCS (NBR 12025 [42]); (ii) CBR and CBR-S (NBR 9895 [43]) and (iii) RM (ME 134 [44]). These properties were determined using specimens compacted at the optimum moisture content and maximum dry unit weight for the standard Proctor compaction energy. After compaction, all specimens were extracted, wrapped with PVC film and subjected to 7 days of curing in a moist room before the geomechanical characterization tests. All mechanical properties are directly related to the criteria for the selection of materials for the composition of structural layers of road pavements, as recommended in the technical standard NBR 12253 [45]. The UCS, CBR and CBR-S tests were carried out in triplicate and the results were calculated as the average between all determinations, considering a maximum variation between results of 10%. The RM tests were performed in duplicate.

In the RM tests, the specimens were subjected to dynamic cyclic stress and a static confining stress, using a triaxial pressure chamber and following the procedures prescribed by ME 134 [44]. The strain values were measured using two LVDTs (linear variable displacement transducers) symmetrically positioned within the triaxial cell, aligned with the specimen’s central axis. The stress states applied during all RM tests are presented in

Table 5. Stress state representative of structural layers of flexible pavements.

Layer	${\mathit{\sigma }}_3$ (MPa)	${\mathit{\sigma }}_{\mathit{d}}$ (MPa)
Base	0.030	0.160
Subbase	0.010	0.090
Subgrade reinforcement	0.009	0.037

. The confining (${\sigma }_3)$ and deviator (${\sigma }_d$) stresses presented in this table represent the stresses acting on pavement base, subbase and subgrade reinforcements. These values were calculated from the structural analysis of the typical pavement cross-section proposed by Nega and Nikraz [46], considering the stresses acting at the center of each layer and under the wheel of a standard axle [47]. The structural analysis was conducted using the AEMC software [48].

The experimental data obtained from the RM tests were used to develop composite models (Equation (2)), following the recommendations of previous papers and the IS-247 standard [7,49,50], within the context of the MeDiNa (National Pavement Design Method) methodology. Regression analyses were performed to develop a mathematical model (k₁, k₂ and k₃ were the regression coefficients) that predicts the material’s deformational characteristics, providing an RM value for design purposes.

$$RM=k_1\times {\left({\sigma }_3\right)}^{k_2}\times {\left({\sigma }_d\right)}^{k_3}$$

(2)

where:

• $RM$ is the resilient modulus (MPa);
• ${\sigma }_3$ is the confining stress (MPa);
• ${\sigma }_d$ is the deviator stress (MPa);
• $k_1, k_2$, $k_3$ are the regression model coefficients.

3.3. Mathematical Models and Response Surfaces

The UCS, CBR, E-CRB and RM responses of each experimental mixture (M1 to M7) allowed the determination of polynomial models that correlate each property to the mixture component contents. These models were estimated using statistical analysis software for development of stepwise regression and analysis of variance (ANOVA), considering a significance level of 5% and a confidence limit of 95%. These mathematical models provided response surfaces that graphically described the variation of the experimental response as a function of the mass percentage of each mixture component.

3.4. Simplex-Centroid Method Efficiency and Mixture Design Optimization

The efficiency of the simplex-centroid method in predicting the mechanical behavior of soil–LFS–FA mixtures was evaluated by comparing some experimental results with data estimated using mathematical models. The mixture design optimization was performed using the desirability function. This function provides desirability values ranging from 0 to 1 for each mixture, where 1 represents that the mixture fully met the defined design criteria, and 0 indicates that the design criteria were not met.

The soil–LFS–FA mixture design optimization was performed, aiming to reach a mix composition that better met the different criteria established for UCS, CBR, CBR-S and RM, allowing its practical application in structural layers of pavements. Therefore, the mixture with higher desirability value corresponded to the mixture whose design was optimized. To evaluate each mixture’s suitability as a base, subbase and subgrade construction material, the mechanical and swelling criteria presented in

Table 6. Criteria for design optimization of the experimental mixtures.

Base Course
Property	Soil C	Soil V	Reference
UCS (kPa)	Min. 690		AASHTO [51] and Ardah et al. [52]
CBR (%)	Min. 60		DNIT [53]
CBR-S (%)	Max. 0.5
RM (MPa)	Min. 82.74	Min. 165.47	AASHTO [54]
Subbase Course
Property	Soil C	Soil V	Reference
UCS (kPa)	Min. 690		AASHTO [51] and Ardah, Chen and Abu-Farsakh [52]
CBR (%)	Min. 20		DNIT [53]
CBR-S (%)	Max. 1.0
RM (MPa)	Min. 82.74	Min. 165.47	AASHTO [54]
Subgrade Reinforcement Course
Property	Soil C	Soil V	Reference
UCS (kPa)	Min. 345		Ardah, Chen and Abu-Farsakh [52]
CBR (%)	Min. 2		DNIT [53]
CBR-S (%)	Max. 1.0
RM (MPa)	Min. 68.95	Min. 68.95	AASHTO [54]

were considered.

4. Results and Discussion

4.1. Experimental Response, Response Surfaces and Mathematical Models

Table 7. UCS (kPa) of natural soils and experimental mixtures.

Material	M0	M1	M2	M3	M4	M5	M6	M7
Soil C	203.7	243.2	293.7	469.6	309.0	377.2	339.5	322.0
Soil V	102.1	117.2	150.6	165.5	161.7	166.4	148.5	162.1

shows the UCS values of the natural soils (M0) and each tested mixture (M1 to M7). These values were used to derive the prediction equations for UCS of mixtures produced with Soil C and Soil V (Equations (3) and (4), respectively). The UCS contour surfaces are shown in Figure 6. The parameters S, L and F correspond to the percentages of the contents of soil, LFS and FA, respectively.

$$UCS=166\times S+1573\times L+2139\times F-18733\times L\times F; {\mathrm{R}}^2=0.8752$$

(3)

$$UCS=109.6\times S+396.6\times L+382.1\times F;{ \mathrm{R}}^2=0.8897$$

(4)

The partial replacement of natural soils with by-products provided significant increases in UCS. The highest UCS increases observed in specimens produced with Soils C and V were observed in M3 and M5 mixtures, respectively. In the M3 mixture, the replacement of 20% of Soil C by LFS provided a percentage UCS increase of 130.5%. In the M4 mixture, the replacement of 5% of Soil V by LFS and 15% of Soil V by FA provided a percentage UCS increase of about 63.0%.

The maximum percentage UCS increases observed in the present study were higher than those reported in previous studies. For example, Fathonah, Intari, Mina, Kusuma and Mahfudoh [13] observed that the best UCS improvement was approximately 50% (after a curing period of 7 days), which was obtained for a mixture of 70% of clayey soil, 20% of FA and 10% of LFS. The maximum UCS increase reported by Rajakumaran [16] was about 7.3%, in a mixture of 90% of clayey soil, 4% of LFS and 6% of FA (the curing period was not mentioned in this paper).

Table 8. CBR (%) of the soil samples and experimental mixtures.

Mixture	M0	M1	M2	M3	M4	M5	M6	M7
Soil C	11.2	17.8	29.0	32.6	18.2	12.1	18.2	28.4
Soil V	8.2	14.1	23.9	34.8	18.6	17.2	14.9	18.7

presents the CBR values obtained from the laboratory tests of natural soils (M0) and experimental mixtures (M1 to M7). The mathematical descriptive models representing the CBR of mixtures containing Soil C and Soil V were presented in Equations (5) and (6), respectively. The response surfaces related to this mechanical property are presented in Figure 7.

$$CBR=17.36\times S+96.35\times L-41.44\times F;{ \mathrm{R}}^2=0.7792$$

(5)

$$CBR=7\times S+145\times L+92\times F-1583\times S\times L\times F;{ \mathrm{R}}^2=0.9987$$

(6)

The CBR index of the soils was significantly improved by the addition of both by-product types. For both soils, the greatest improvements in CBR index were verified in M3 mixtures. In this case, the replacement of 20% of Soil C and V by LFS provided percentage CBR index increases of 191.1% and 324.4%, respectively. All percentage CBR improvements observed in this study were lower than the maximum CBR increases provided by the LFS and FA analyzed by Mina, Kusuma and Ulfah [14]. After a 7-day curing period, they reported a maximum CBR increase of 642% when 20% of FA and 20% of LFS were added to clayey soils. For a 7-day curing period, Soyonar, Fırat, Yilmaz and Okur [17] observed that the CBR index of mixtures of 80% of LFS, 15% of FA and 5% of clayey soil was 51.4% higher than the CBR index of their reference clay–slag mixture.

Table 9. CBR-S (%) of the soil samples and the experimental mixtures.

Mixture	M0	M1	M2	M3	M4	M5	M6	M7
Soil C	0.7	0.2	0.3	0.3	0.8	0.5	0.5	0.4
Soil V	0.3	0.8	0.5	0.2	0.5	0.3	0.2	1.0

presents the results for CBR-S values obtained from the laboratory tests of natural soils (M0) and experimental mixtures (M1 to M7). The mathematical descriptive models representing the CBR-S of mixtures produced with Soil C and Soil V are presented in Equations (7) and (8), respectively. The response surfaces obtained from the CBR-S results are presented in Figure 8.

$$CBR-S=0.1697\times S+0.6845\times L-0.397\times F+70.3849\times S\times L\times F;{ \mathrm{R}}^2=0.9063$$

(7)

$$CBR-S=0.849\times S-1.551\times L-1.737\times F;{ \mathrm{R}}^2=0.2281$$

(8)

Significant data scatter was observed in CBR-S results, which compromised the predictive ability of the developed mathematical models, as discussed in the next subsections. The average CBR-S values of all mixtures with Soil C were lower than the average CBR-S of the natural soil, except for mixture M4. In contrast, the average CBR-S values of all mixtures with Soil V were higher than the CBR-S of the natural soil, except for mixtures M3 and M6. When Soil C was used, the highest swelling reduction was provided by the replacement of 5% of Soil C by LFS (mixture M1). When Soil V was used, the highest swelling reductions were provided by the replacement of 20% of Soil V by LFS (mixture M3) or replacement of 5% of Soil V by LFS and 7.5% of Soil V by FA (mixture M6). Despite the natural variability of the experimental data, the incorporation of by-products into both soil types did not result in significant increases in soil expansion, which is a promising result to enable their application in road construction.

Table 10. RM values of soil and experimental mixtures.

Experimental Mixture		RM Model Coefficient			Coefficient of Determination (R2)	RM Obtained from Experimental Data (MPa)
		k1	k2	k3		Base	Subbase	Subgrade Reinforcement
Soil C	M0	423.609	−0.016	−0.634	0.729	28.32	29.63	35.81
	M1	42.041	0.173	−0.139	0.685	29.57	26.49	29.43
	M2	64.314	0.237	−0.228	0.939	42.54	37.39	44.66
	M3	70.111	0.153	−0.160	0.732	54.97	50.95	57.79
	M4	45.12	0.139	−0.179	0.819	38.47	36.61	42.30
	M5	51.607	0.241	−0.160	0.728	29.72	25.01	28.10
	M6	34.038	0.096	−0.160	0.728	32.59	32.16	36.70
	M7	62.018	0.164	−0.151	0.751	46.02	41.92	47.12
Soil V	M0	146.541	0.563	−0.026	0.690	21.34	11.67	11.26
	M1	57.391	0.308	−0.249	0.731	30.76	25.31	30.57
	M2	158.408	0.305	0.129	0.921	42.92	28.50	24.61
	M3	311.022	0.401	0.246	0.972	48.57	27.14	20.90
	M4	107.273	0.356	−0.114	0.686	37.94	27.40	29.20
	M5	108.962	0.327	−0.017	0.938	35.71	25.18	24.70
	M6	89.001	0.294	−0.008	0.941	32.21	23.43	22.88
	M7	118.068	0.448	−0.253	0.729	39.02	27.59	32.95

displays the coefficients of the RM models of each experimental mixture (M1 to M7) and natural soils (M0). These values were obtained from the experimental data considering the stress states that are representative of each pavement structural layer considered in this study.

Equations (9)–(11) show the mathematical models that describe the RM of mixtures containing Soil C, considering the state stress representative of the base, subbase and subgrade layer, respectively. Equations (12)–(14) exhibit the models obtained for the RM of mixtures containing Soil V, considering the state stress representative of the base, subbase and subgrade layer, respectively. The response surfaces that graphically illustrate these equations are shown in Figure 9, Figure 10 and Figure 11.

$$RM=24.51\times S+175.66\times L+14.46\times F; { \mathrm{R}}^2=0.8604$$

(9)

$$RM=18\times S+183\times L-1191\times F+1499\times S\times F;{ \mathrm{R}}^2=0.9511$$

(10)

$$RM=20\times S+212\times L-1339\times F+1682\times S\times F;{ \mathrm{R}}^2=0.9670$$

(11)

$$RM=26.48\times S+136.75\times L+39.60\times F;{ \mathrm{R}}^2=0.8952$$

(12)

$$RM=24.51\times S+44.86\times L+20.88\times F;{ \mathrm{R}}^2=0.4980$$

(13)

$$RM=32.7\times S-25.85\times L-89.64\times F+1588.89\times L\times F;{ \mathrm{R}}^2=0.6700$$

(14)

Considering the lack of prior investigations into the RM of soil–LFS–FA mixtures, the RM dataset obtained in the present research complements the current knowledge on the resilient behavior of these mixtures, considering their application to different pavement structural layers. Regardless of the structural layer, the RM of both types of soils was improved by the addition of LFS and/or FA. Previous research [55] also verified that waste-based pavement materials can exhibit improved stiffness compared to conventional pavements, enabling them to withstand high stresses with less deformation. For both soil types, the best RM improvements were observed in M3 mixtures. In this situation, the replacement of 20% of Soil C and V by LFS provided percentage RM increases up to 94.1% and 132.6%, respectively.

According to previous research [3,6,15,56,57,58], the development of cementitious and/or pozzolanic reactions due to the incorporation of LFS or FA leads to the formation of calcium aluminates and calcium silicates that improve the filler effect and the connections between soil aggregates, which explains the improvements in mechanical performance described in the present section. Previous work [58] also reported that rod-like crystals are formed after the carbonation of steel slag, which has a binding effect and is also advantageous for strength development. The present study introduced a comparative analysis of the effects of the addition of LFS and FA on different types of soils. According to the results verified in most of the mixtures investigated in this work, it was possible to note that the percentage increases in UCS, CBR and RM due to the addition of LFS and FA were higher in Soil V (sandy soil) than in Soil C (clayey soil). This can be attributed to stronger contact bridges between the surface of particles of granular soils (Soil V) provided by LFS and FA. Such strong anchoring mechanisms between particles of Soil C were not possible due to the small size of the clay particles [7,59]. Consequently, the benefits provided by LFS and FA were more pronounced in the sandy soil than in the clayey soil.

In the present work, UCS, CBR, CBR-S and RM were determined under controlled laboratory conditions. However, it is essential to consider that different variables in real-world environments can impact the long-term performance of the soil–waste mixtures. Our research group recently performed an environmental characterization of the LFS [60] and FA [6] used in this study, based on leaching and solubilization tests [61]. These tests classified the wastes as Class II A solid wastes, indicating that they are non-hazardous and non-inert. Nevertheless, future studies should conduct in-situ tests alongside laboratory assessments, as previous research [62] has revealed several challenges in field trials, such as rutting issues and layer settling in mixtures containing high waste dosages. These in-situ evaluations can offer valuable insights into the performance of these materials under actual environmental conditions and traffic loads, helping to identify and address potential issues that may not be detected through laboratory testing alone.

4.2. Response Surfaces and Mathematical Models Evaluation

Regarding the Soil C mixtures, Figure 6a and Figure 7a suggest that the mixtures with the highest amount of LFS (M3) exhibited the greatest UCS and CBR values. This fact indicates that FA addition had less influence on UCS and CBR than the LFS addition. This hypothesis is confirmed by analyzing the model coefficients obtained for these properties (Equations (3) and (5)). Once β₁ and β₂ are positive in both models, it is expected that the UCS and CBR values increase with increases in LFS content. Regarding the effects of FA on the UCS and CBR, the negative β₂₃ coefficient (Equation (3)) indicated that the combination between LFS and FA had a negative impact on those properties, so the influence of an increasing FA content in the mixture was represented by a negative β₃ coefficient (Equation (5)).

Figure 9a indicated that the highest swelling values were verified in the lower central portion of the diagram (M4), the region in which the mixtures had the lowest soil contents and highest dosages of by-products in their composition. In addition, the coefficient β₁₂₃ (Equation (7)) was greater than the other model coefficients. These findings indicate that the combination of the three components contributed to increase the swelling of the mixtures.

Regarding the RM values of the Soil C mixtures [Figure 9a, Figure 10a and Figure 11a], it is noticed that the greatest RM values were obtained for the mixtures located at the left bottom corner of the diagram, regardless of the stress state. This behavior is an indication that LFS had a beneficial effect on the RM of the mixtures. This hypothesis is reinforced by the evaluation of Equations (9)–(11). In most equations the β₂ coefficient is positive, while β₃ is negative, indicating that the LFS addition had positive impacts, while the FA addition had detrimental effects on the RM of the mixtures, which is pretty similar to the general behavior verified in the response surfaces. In addition, the β₁₃ coefficient of Equations (10) and (11) shows that the interaction between Soil C and FA contributes positively to increasing the mixtures’ RM only for stress state levels corresponding to the load acting on subbase and subgrade reinforcement layers.

Even though some coefficients of the proposed descriptive models (Equations (3), (5), (7) and (9)–(11)) did not show statistical relevance in some statistical analyses, it is important to note that the models presented R² values ranging between 0.7792 and 0.9063, which indicates good correlation between the experimental data and the proposed models.

Regarding the Soil V mixtures, the mixtures M3, M4 and M5 located at the lower central edge of the diagram of Figure 6b presented the highest UCS values. This finding indicates that the LFS and FA addition contributed to an increase in the UCS values. This hypothesis was confirmed by the coefficients β₁, β₂ and β₃ (Equation (4)). Since these coefficients were positive and had similar magnitude, it is possible to infer that the addition of LFS and FA had a positive and quantitatively similar impact on the UCS of the experimental mixtures.

Figure 7b indicates that mixtures with higher CBR values are located at the left bottom corner of the diagram. A similar trend is observed for RM values associated with stress states acting on base and subbase layers of flexible pavements [Figure 9b and Figure 10b]. In addition, β₂ and β₃ coefficients of the respective models are positive (Equations (6), (12) and (13)). These observations indicated that LFS had greater influence on CBR and base and subbase RM values compared to the influence of FA on these mechanical properties. Regarding the CBR descriptive equation of soil V mixtures (Equation (6)), it is also noteworthy that the β₁₂₃ coefficient was negative and statistically significant, which indicates that the combination of all three components had a negative impact on the CBR of the mixtures.

Concerning the RM values associated with the stress state representative of subgrade reinforcement, it was verified that the highest RM values are located at the lower central portion of the diagram [Figure 11b]. In this case, the model coefficients presented in Equation (14) indicated that the LFS and FA addition had a negative influence on the RM values, since β₂ and β₃ were negative. However, Equation (14) suggested that the interaction between LFS and FA (coefficient β₂₃) positively influenced the RM of the mixtures, which justifies the behavior indicated in the diagram presented in Figure 11b.

Regarding the CBR-S of Soil V mixtures [Figure 8b], it was observed that the highest values were found in the upper middle portion of the diagram. According to Equation (8), both β₂ and β₃ coefficients are negative, which indicates that LFS and FA addition tended to reduce the CBR-S of the experimental mixtures when keeping the other independent variables constant.

It is important to notice that the mathematical model presented in Equation (8) did not show good data correlation, resulting in an R² equal to 0.2281, which is significantly inferior to the R² obtained for the mathematical models that represent all of the other properties evaluated in this study. Since this CBR-S equation (Equation (8)) did not show a good fit to the experimental data, it was not taken as representative for predicting the CBR-S of the mixtures. Therefore, the CBR-S criteria were not taken into consideration during the optimization design process developed for the Soil V mixtures in the next subsections.

Although some coefficients of the proposed models presented in Equations (4), (6) and (12)–(14) did not show statistical significance in some analyses, the mathematical models still presented reasonable correlation with the experimental data since R² values between 0.4980 and 0.9987 were obtained. For UCS and CBR values, the models presented an excellent fit to the experimental data, with R² values greater than 0.8897. RM models of mixtures prepared with Soil V for subbase and subgrade reinforcement layers (R² = 0.4980 and R² = 0.6700, respectively) provided a poorer fit to the experimental data compared to the model obtained for the RM of base layers (R² = 0.8952). Despite this, the models referring to the subbase and subgrade reinforcement layers were reasonably sufficient to represent the RM behavior of the mixtures and carry out the optimization process.

4.3. Simplex-Centroid Method Efficiency Evaluation

Table 11. Simplex-centroid effectiveness evaluation in predicting the UCS, CBR and CBR-S of the experimental mixtures.

UCS (kPa)
Mixture	Soil C			Soil V
	Experimental	Model	Variation	Experimental	Model	Variation
M1	243.2	236.4	2.8%	117.2	123.9	−5.8%
M2	293.7	341.9	−16.4%	150.6	145.9	3.1%
M3	469.6	447.4	4.7%	165.5	167.9	−1.4%
M4	309.0	314.2	−1.7%	161.7	167.8	−3.8%
M5	377.2	391.8	−3.9%	166.4	167.7	−0.8%
M6	339.5	314.1	7.5%	148.5	145.8	1.8%
M7	322.0	311.7	3.2%	162.1	153.2	5.5%
	Absolute mean variation		5.7%	Absolute mean variation		3.2%
CBR (%)
Mixture	Soil C			Soil V
	Experimental	Model	Variation	Experimental	Model	Variation
M1	17.8	21.3	−20.0%	14.1	13.9	1.4%
M2	29.0	27.2	6.0%	23.9	24.3	−1.5%
M3	32.6	33.2	−1.9%	34.8	34.6	0.6%
M4	18.2	22.8	−25.1%	18.6	18.8	−0.8%
M5	12.1	12.5	−3.2%	17.2	17.2	0.3%
M6	18.2	16.9	7.1%	14.9	15.1	−1.2%
M7	28.4	22.3	21.5%	18.7	18.3	2.0%
	Absolute mean variation		12.1%	Absolute mean variation		1.1%
CBR-S (%)
Mixture	Soil C			Soil V
	Experimental	Model	Variation	Experimental	Model	Variation
M1	0.2	0.2	−22.2%	0.8	0.7	2.8%
M2	0.3	0.2	16.4%	0.5	0.5	−3.6%
M3	0.3	0.3	−4.9%	0.2	0.4	−130.6%
M4	0.8	0.7	5.3%	0.5	0.4	26.0%
M5	0.5	0.5	−6.5%	0.3	0.3	−17.6%
M6	0.5	0.4	18.3%	0.2	0.5	−167.5%
M7	0.4	0.5	−23.0%	1.0	0.5	49.5%
	Absolute mean variation		13.8%	Absolute mean variation		56.8%

and

Table 12. Simplex-centroid effectiveness evaluation in predicting the resilient modulus (RM) of the experimental mixtures.

RM (MPa)—Soil C
Mixture	Base			Subbase			Subgrade reinforcement
	Experimental	Model	Variation	Experimental	Model	Variation	Experimental	Model	Variation
M1	29.57	32.07	−8.4%	26.49	26.25	0.9%	29.43	29.60	−0.6%
M2	42.54	43.40	−2.0%	37.39	38.63	−3.3%	44.66	44.00	1.5%
M3	54.97	54.74	0.4%	50.95	51.00	−0.1%	57.79	58.40	−1.1%
M4	38.47	42.65	−10.9%	36.61	37.89	−3.5%	42.30	43.00	−1.6%
M5	29.72	30.56	−2.8%	25.01	24.78	0.9%	28.10	27.59	1.8%
M6	32.59	31.31	3.9%	32.16	33.95	−5.6%	36.70	38.06	−3.7%
M7	46.02	39.12	15.0%	41.92	37.76	9.9%	47.12	42.74	9.3%
	Mean absolute variation		6.2%	Mean absolute variation		3.5%	Mean absolute variation		2.8%
RM (MPa)—Soil V
Mixture	Base			Subbase			Subgrade reinforcement
	Experimental	Model	Variation	Experimental	Model	Variation	Experimental	Model	Variation
M1	30.76	32.0	−4.0%	25.31	25.53	−0.9%	30.57	29.77	2.6%
M2	42.92	40.3	6.2%	28.50	27.05	5.1%	24.61	25.38	−3.1%
M3	48.57	48.5	0.1%	27.14	28.58	−5.3%	20.90	20.99	−0.4%
M4	37.94	41.2	−8.7%	27.40	26.78	2.2%	29.20	31.10	−6.5%
M5	35.71	34.0	4.9%	25.18	24.98	0.8%	24.70	23.34	5.5%
M6	32.21	33.0	−2.4%	23.43	25.26	−7.8%	22.88	26.56	−16.1%
M7	39.02	38.2	2.2%	27.59	26.36	4.4%	32.95	28.67	13.0%
	Mean absolute variation		4.1%	Mean absolute variation		3.8%	Mean absolute variation		6.7%

show the average variation between the experimental results and each property estimated by the developed mathematical models. Analyses of the UCS and CBR results of Soil C mixtures indicated that, except for mixtures M2, M1, M4 and M7, average variations between obtained and predicted data were lower than 7.1%. The high variations observed in some mixtures are probably related to variations inherent to the specimen’s compaction process and to the heterogeneity of the materials. Although a few high-magnitude variations (~20%) were verified, the average variation was around 12.1%, indicating that the mathematical models can provide a good estimate of the UCS and CBR values of the experimental mixtures.

Regarding the simplex-centroid efficiency in predicting the UCS and CBR of Soil V mixtures, variations lower than 6.00% were verified between estimated and obtained UCS and CBR values, which highlights the method effectiveness in predicting these properties for Soil V mixtures.

On the other hand, a high variation was observed in the CBR-S values. Therefore, it was observed that the method was not efficient in describing this property, as most mixtures prepared with Soil C presented variation results higher than 15%. For Soil V mixtures, the variation between experimental and predicted data for CBR-S was even greater (sometimes higher than 100%) than the variation verified for Soil C mixtures, which evidenced the inefficiency of this method in predicting the mixture CBR-S response. Such variations are attributed to the poor correlation between the experimental data and the predictions provided by the developed mathematical models.

The evaluation of the RM results indicated that the maximum variation between predicted and obtained RM values was 15% for the Soil C mixtures, and 16.1% for the Soil V mixtures. Despite these variations, the absolute mean variation between the RM values was around 2.8% and 6.7% for Soil C and Soil V mixtures, respectively, demonstrating the effectiveness of the method in predicting the RM values of the experimental mixtures prepared with both soil types.

4.4. Mixtures Design Optimization for Pavement Construction

Aiming to apply the soil–LFS–FA mixtures in structural layers of asphalt pavements, response surfaces were obtained with a design optimization process of Soil C and V mixtures, as presented in Figure 12, Figure 13 and Figure 14. The CBR-S criterion was not considered during the optimization process of Soil V mixtures since the mathematical model developed for this property presented a low R² value.

In mixtures prepared with Soil C, the greatest desirability values were observed for mixtures located at the left bottom corner of the diagram, regardless of the criteria considered in the analysis. For application in base layers of flexible pavements [Figure 12a], the M3 mixture presented a desirability value close to 0.66. Moreover, the M3 mixture presented desirability values of 0.795 and 0.960, when applied in the subbase [Figure 13a] and subgrade reinforcement layers [Figure 14a], respectively. Since the M3 mixture presented the greatest desirability value, this mixture can be considered the most adequate construction material for the base, subbase and subgrade reinforcement layers of flexible pavements.

For Soil V mixtures used in base and subbase applications [Figure 11b and Figure 12b], the highest desirability values were verified for mixtures located at the left bottom corner of the diagram. With desirability values equal to 0.33 and 0.34 when considering criteria for application in base and subbase layers, respectively, the M3 mixture best met the criteria. Thus, it is possible to infer that this mixture presents the best performance when used in the construction of base and subbase layers of asphalt pavements. Regarding the application of the mixtures as subgrade reinforcement [Figure 14b], the desirability value increased towards the lower central edge of the diagram, so that the M4 mixture presented the highest desirability value (0.603). Therefore, the M4 mixture can be considered the most suitable material for the construction of subgrade reinforcement layers of asphalt pavements among the mixtures analyzed in the present work.

Since the M3 and M4 mixtures presented the highest desirability values among the mixtures analyzed in this research, they were able to concomitantly achieve the UCS-, CBR-, CBR-S- and RM-defined criteria and, therefore, can be considered mixtures with optimized performance. Although these mixtures presented the best performance, it is important to highlight that their desirability values were lower than 1.0, which indicates that they did not simultaneously meet all the technical criteria considered in this analysis.

5. Conclusions

This study presents original research on the concomitant use of LFS and FA as sustainable stabilizers for distinct soils, providing novel insights into their synergistic and antagonistic effects on the mechanical properties of pavement layers. The simplex-centroid methodology, desirability function, response surfaces and mathematical models revealed the mixture proportions of soil, LFS and FA that comply with pavement design regulations. The main conclusions obtained from this study are listed below:

• The addition of LFS and FA to the natural soils provided UCS, CBR index and RM increases up to 130.5%, 324.4% and 132.6%, respectively;
• LFS and FA generally provided higher mechanical properties improvements in sandy soils than in clayey soils;
• The use of industrial by-products caused small changes in the CBR-S of the natural soils, indicating that their application does not result in a significant increase in the expansion of the soil samples;
• The UCS, CBR and RM descriptive mathematical models showed R² values ranging from 0.760 to 0.998, which evidences the simplex-centroid accuracy in the description of these properties;
• The DoE methodology and the desirability function were useful for optimizing the mechanical properties, offering mixture proportions that meet the requirements established in the literature. The best proportions were 80% soil, 20% LFS and 0% FA for both soil samples;
• The comparison between estimated and actual values showed that the DoE was an effective tool for predicting and optimizing the properties of stabilized soils.

Future research should explore the interactions between other types of soils and sustainable stabilizer agents, using the DoE of mixtures in a simplex-centroid network. Examining the mechanical performance of sustainable materials for road construction, along with conducting comprehensive life cycle assessments (LCA), could unveil new strategies to enhance sustainability in the construction field. Future research should also explore the potential of incorporating LFS and FA into asphalt mixtures, offering additional insights into the development of sustainable and high-performance materials for pavement construction.