Particle Packing Optimization for CCR-GGBS-FA Binder Stone Waste Pavement Base Material

Abstract

Stone waste refers to the waste stone particles generated from mining and stone processing to finished products that are not utilized in a resourceful manner. In this study, a CGF solid waste-based binder (abbreviated as CGF), with calcium carbide residue (CCR), ground granulated blast-furnace slag (GGBS), and fly ash (FA) as components, was developed to solidify the stone waste. Through “treating waste with waste”, the resource utilization of solid waste was realized. In order to improve the performance of the new material, this paper proposes the MAA-SW model for stone waste pavement base material based on the MAA model, establishes the relationship with the target gradation, and obtains the ideal gradation composition of stone waste through the calculation of the response surface analysis so as to obtain an energy-saving stone waste pavement base material with excellent performance.

1. Introduction

Stone waste is generated from the process of mining and stone processing to the finished products that are not used resourcefully. Due to the poor engineering nature of stone waste, it cannot be directly used resourcefully, and the disposal method is mainly centralized stockpiling [1]. In addition to consuming a large amount of land resources and raising the operating costs of enterprises, the stacking of stone waste has a high content of fine particles, which can cause dust pollution, and there is also the safety hazard of landslides, which poses a threat to human safety, after it is stacked into a mountain. calcium carbide residue, ground granulated blast-furnace slag, fly ash, and other industrial solid waste piling processes also exist in the ecological environment.

In the field of resource utilization, concrete or other materials for geotechnical filling can be prepared by mixing cement and other cementitious materials with rock waste, which is one of the main ways of rock waste resource utilization at present [2,3,4,5]. Stone waste can be mixed with other geotechnical materials or cementitious materials to be used as geotechnical filling materials, and its application and research have mainly focused on pavement base engineering, while the current research mainly focuses on the laying of pavement base and pavement subgrade [6].

In pavement base layer materials, the strength of the skeleton plays an important role in their mechanical properties, and the strength of the skeleton is, in turn, determined by the grading of coarse and fine aggregates. Due to the wide distribution of the particle-size gradation of primary stone waste, which leads to its poor engineering properties, the grading of primary stone waste will need to be optimized and designed to improve the mechanical and durability properties of the stone waste pavement base material so as to meet the engineering requirements of the pavement base layer.

In 1892, the French scholar Feret confirmed that the strength of concrete is mainly related to the volume contents of cement, water, and air, and based on this, he proposed a theoretical equation for calculating the strength of concrete, as shown in Equation (1) [7]:

$$f_c=k{({\displaystyle \frac{c}{c+v+a}})}^2.$$

(1)

where f_c represents the compressive strength of concrete; c, v, and a represent the volumes of cement, water, and air in the wet mix, respectively; and k is a constant.

In order to optimize the gradation and improve its close-packed pile, Fuller published the Fuller gradation curve in 1907 [8], as shown in Equation (2).

$$Y=100{(\frac{D_0}{D_{\max }})}^n.$$

(2)

where Y denotes the number of particles passing through the D₀ aperture; D_max denotes the maximum particle size in the system; and n is a coefficient related to the aggregate type.

The Fuller gradation curve is regarded as a semi-empirical and semi-theoretical model that has been successful in improving the close packing and compressive strength of concrete by optimizing the aggregate gradation. However, there is a large error in the prediction of this curve when large flowability or other types of concrete need to be formulated.

Andreasen and Andersen proposed the classical model of continuous particle packing in 1930 [9], called the A-A model, as shown in Equation (3). Among the many ultra-high-performance concrete proportion design methods based on the theory of closest packing, the design method based on the MAA model is more accurate due to its advantage of considering the particle-size distribution [10].

$$P(D)\hspace{0.17em}=\hspace{0.17em}(\frac{D}{D_{\max }}{)}^q$$

(3)

where P(D) is the percentage of total solids (including concrete aggregate and cementitious materials) less than a certain particle size D, %; D is the particle size, μm; D_max is the maximum particle size, μm; and q is the distribution modulus.

Under the condition that the total solids are determined, the A-A model can be used to calculate the gradation of the total solids corresponding to the tightest packing, i.e., the target gradation. However, the disadvantage of the A-A model is that it fails to reflect the effect of the smallest-size particles in the total solids on the target gradation. Therefore, Brouwers et al. [11] and Yu et al. [12] modified the A-A model by introducing the minimum particle size D_min of finite small particles and a new distribution modulus q, and proposed the modified A-A model, i.e., the MAA model. The modified A-A model (Modified Andreasen and Andersen, MAA) is shown in Equation (4).

$$P(D)\hspace{0.17em}=\hspace{0.17em}\frac{D^q-D_{\min }^q}{D_{\max }^q-D_{\min }^q}.$$

(4)

where P(D), D, and D_max are the same as in Equation (3); D_min is the smallest particle size in the total solids; q is the distribution modulus, and the larger the value of q, the greater the curvature of the curve of the target gradation; the q value of ordinary self-compacting concrete is taken to be 0.21~0.24 [6].

Based on the A-A model, the MAA-SW model is proposed in this paper, i.e., by adjusting the content of each component in the stone waste to form a fitted gradation, and a relationship is established with the target gradation. By sieving the as-built stone wastes, a number of test groups with different gradations were initially designed, and the specimen group with the highest density was selected. However, due to the differences between different cementitious material systems, the selection of the specimen group with the highest density is not necessarily the most reasonable grading interval range theoretically, which may result in missing the optimum grading performance. Subsequently, the response surface design optimization was carried out based on stone waste particles of different particle sizes, and the second-order regression equations between stone waste particles of different particle sizes and solidified stone waste samples were established so as to fit and optimize the experimental group with the optimal mechanical properties, and to derive the ideal composition of stone waste gradation, thus obtaining energy-saving road subgrade materials with excellent performance.

In the theory of closest packing, the packing of materials is classified into a structure of closest packing with equal particles and a structure of closest packing with unequal particles; the stone waste–CGF belongs to the latter because of the large span of the stone waste gradation, as shown in Figure 1.

The MAA model was used to calculate the target gradation considering the smallest particle size in the total solids, and the total solids’ gradation design was carried out with this as the target. The maximum particle size of the virgin stone waste is 4.75 mm, and the minimum particle size is 0.0001 mm, in which the content of stone powder is high, while the particle size of each component of CGF is less than 0.03 mm; thus, the D_max and D_min in the virgin stone waste–CGF system are 4.75 and 0.0001 mm, respectively.

In summary, the particle grading of the test stone waste was redesigned, the maximum dry density test and unconfined compressive test were conducted after obtaining the target grading of the stone waste through sieving, and the MAA-SW model was used to improve the engineering performance of the CGF stone waste materials. The comprehensive utilization of stone waste and tailings resources not only alleviates the contradiction between the supply and demand of sand and gravel, but also saves land space and alleviates the problem of carbon emissions generated by the cement manufacturing industry. Through the comprehensive utilization of stone waste and tailings resources, the high-value utilization of solid waste can be achieved and the value of solid waste utilization can be enhanced. The design of stone waste gradation can further enhance the economic and environmental benefits of stone waste materials.

2. Materials and Methods

2.1. Test Materials

The materials used in this test were stone waste and CGF all-solid-waste binder. This study investigated stone waste material extracted from the stone waste mountain in Wulian County. Particle-size grading was performed on the original, as-received soil samples, as shown in Figure 2A. Stone waste with particle sizes less than 0.075 mm was categorized as stone powder, and waste with particle sizes between 0.075 and 4.75 mm was categorized as stone chips. After sampling, particle-size grading was performed on the as-received stone waste soil samples, and different particle sizes were identified via sieving, as shown in Figure 2A. The samples with different particle sizes were categorized as stone chips, fine stone dust, and ultrafine stone dust, as shown in Figure 3. The content of stone chips in the as-received stone waste accounted for 26.2% of the composition, fine stone dust accounted for 54.9%, and ultrafine stone dust accounted for 18.2%. Fine-grained particles smaller than 0.075 mm exceeded 70 percent of the total weight. The stone waste contains fine-grained soil type, with an inhomogeneity coefficient of 85, a curvature coefficient of 0.42, and poor gradation. The chemical and mineral composition of the stone waste is shown in Figure 4.

The specific gravity of the stone waste is 2.68, the natural water content is 15.6%, the stone chips and fine stone powder have no plasticity, and the liquid and plastic limits of the ultrafine stone powder are 39.5% and 18.9%, respectively. The plasticity index of the stone waste is 20.6, which indicates the presence of clay with a low liquid limit. In summary, the Wulian County stone waste consists of poorly graded soil, which is characterized as fine-grained soil without plasticity. The SiO₂ content of the stone waste measured using the XRF test reached 68.19%, and the Al₂O₃ content reached 15.23%. The diffraction pattern produced using X-ray analysis of the stone waste crystals was examined to analyze the atomic distribution pattern inside the crystals, and then the composition of the physical phase of the stone waste minerals was qualitatively and quantitatively analyzed. The raw data of the XRD pattern (Figure 4) of the stone waste sample were imported into Jade 6.0 software, and after searching/matching, it was determined that the main crystalline phase of the sample is quartz (SiO₂, PDF card no. 85-0796). Moreover, it was found that the sample contains sodium feldspar (Na(AlSi₃O₈), PDF#83-1658), calcium feldspar (Ca(Al₂Si₂O₈), PDF#73-1435), and microplagioclase feldspar (KAlSi₃O₈, PDF#19-0932). These results establish that the mineral composition of the waste from the stone waste mountain in Wulian County is mainly quartz and feldspar.

The CGF material consists of an alkali activator and pozzolanic material. The alkali activator consists primarily of calcium carbide slag (CCR), and the pozzolanic material consists of blast-furnace slag (GGBS) and fly ash (FA) (Figure 2B). CCR is a by-product of the hydrolysis of calcium carbide in the production of ethylene, with a pH value greater than 12.0, which provides an alkaline environment [7]. GGBS is the waste slag produced in the process of ironmaking, which is a highly reactive pozzolanic material. FA is the fly ash collected by the baghouse of a coal-fired power plant, which is a low-reactivity pozzolanic materials [7]. The strength values of the CGF cementitious material’s net slurry and cured pure clay-grained soil, pure powder-grained soil, and pure sand-grained soil were determined to be no less than that of cement (P.O 42.5) under the same conditions based on the results of the strength tests conducted in previous studies [13,14,15,16,17,18,19]. The particle-size distribution of the components in CGF is shown in Figure 3, and the mineral composition is shown in Figure 5.

2.2. Test Methodology and Programme

The maximum dry density test was carried out using the vibration hammering method to determine the grading of the stone waste in the tightest state, and then the unconfined compressive strength test was carried out to determine the strength characteristics of the solidified stone waste in the tightest state. The solidified stone waste scheme is shown in

Table 1. Test scheme of solidified stone waste.

Stone Waste	Type of Binder	Gradation Group	Curing Time (Days)	Tests	Number of Test Groups
Raw stone waste	CGF	1	7, 28	Maximum density test, UCST	3
Stone waste		17

.

2.2.1. Maximum Dry Density Test

Under the condition that D_max and D_min are unchanged, by adjusting the distribution ratio of each component of stone waste, a fitting curve with good correlation with the target gradation can be formulated. Therefore, based on the MAA model, this paper proposes to adjust the content of each component in the stone waste to form the fitted gradation without considering the method of cementitious materials, and establish the relationship with the target gradation by calculation; this is named the MAA-SW model (Modified Andreasen and Andersen-Stone Waste, MAA-SW), as shown in Equation (5), which has a good correlation with the target gradation.

$$P(D)\hspace{0.17em}=\hspace{0.17em}\frac{D^{\lambda }-D_{\min }^{\lambda }}{D_{\max }^{\lambda }-D_{\min }^{\lambda }}.$$

(5)

where P(D) is the percentage of stone waste smaller than a certain particle size D, in %; D is the particle size of stone waste, in mm; D_max is the maximum particle size of stone waste, in mm; D_min is the minimum particle size of stone waste; λ is the distribution modulus, and the larger the value of λ, the larger the curvature of the curve of the target gradation. Based on the MAA-SW model, it can be determined that when the fitted gradation is closer to the target gradation, the better the correlation is proved, i.e., the closer the particles’ stacking state.

In soil mechanics, close packed refers to the degree of close-packed arrangement of solid particles in soil with a single-grain structure and is an important indicator of the state of coarse-grained soils. In this study, the physical index of density γ_d(max) (corresponding to the minimum pore ratio e_min) in the tightest packing state with relatively stable measured values was used as an evaluation index for the advantages and disadvantages of the particle gradation of similar stone waste, i.e., the maximum density of the stone waste particles was measured by compacting only the stone waste particles without the addition of water and cementing materials. The greater the density γ_d(max) in the densest state, the more the grading particles are closely arranged, and, thus, the more the stone waste is structurally stable, with high strength, and not easily compressed. On the contrary, if the particles are loosely arranged and the structure is in an unstable state, the grading of the stone waste is poor.

This test refers to the “standard of geotechnical test methods (GB/T50123-2019)” in the “vibration hammering method” to determine the density γ_d(max) of the stone waste in the most compact state. The specific steps are as follows:

(1) Process the raw stone waste for drying using two sieves of 0.075 and 0.005 mm. Sieve out the stone chips (particle size 0.075–4.75 mm), fine stone powder (particle size 0.005–0.075 mm), and ultrafine stone powder (particle size ≤ 0.005 mm) to obtain the raw material for the test, as shown in Figure 6.

(2) Prepare a vibrating hammermill with an inner diameter of 5 cm and a height of 12.7 cm, as shown in Figure 6, and weigh its mass.

(3) After mixing, pour the stone waste into a container for vibration processing. This process should be repeated three times according to the different stone waste particle sizes. The amount of material poured into the container for each treatment should produce a final volume that is slightly larger than one-third of the container after vibration. The vibration process is conducted 240 times with a vibrating fork on both sides of the container for 2 min. At the same time, a hammer is used to carry out abrasive surface hammering 120 times. The third sample should be poured directly into the mouth of the container ring. The vibration and compaction process is shown in Figure 6.

(4) Remove the ring, scrape off excess stone waste from the top of the container with a soil repair knife, and weigh the mass of the container and stone waste.

(5) Calculate the density of the stone waste in the densest state. For two parallel measurements, the maximum allowable parallel difference between the two measurements is ±0.03 g/cm³. Take the arithmetic average of the two measurements as the density state (γ_d(max)). The maximum density formula is shown in Equation (6).

$${\gamma }_{d(\max )}={\displaystyle \frac{m_s}{V}}.$$

(6)

where γ_d(max) is the density in the densest state; m_s is the mass of the stone waste; and V is the volume of the container, which is 250 cm³.

2.2.2. Compressive Strength Test without Lateral Limit

Place the blended graded stone waste in an oven at 105 °C for 24 h, and then mix it with the cementitious materials proportionally to form a mixture of stone waste and cementitious materials. Next, weigh the required amount of water according to the water/cement ratio of 1.0, add it to the mixture, and then mix it evenly for 5 min. Then, pour the mixed slurry into the test molds in three layers and chisel the surface layer of the slurry after tamping each layer. Place the specimens into a standard maintenance box (95% relative humidity and 20 °C) for 24 h, demold after 24 h of maintenance, and continue until the set curing age is reached. The cured specimens were subjected to the unconfined compressive test, and the stress–strain curve of each specimen was recorded.

Through the process of sieving the original stone waste, three different groups were obtained for this study: stone chips, fine stone powder, and ultrafine stone powder. Then, using Equation (5), the three groups were fitted to the target gradation and blended to be consistent with the mixed stone waste.

2.3. Stone Waste Maximum Density Test and Results

The preliminary results identified seven groups (λ were 0.14, 0.17, 0.20, 0.23, 0.26, 0.29, and 0.32) among the stone waste of different particle sizes. The density test under the densest state was used to examine each group of stone waste. The cumulative curves of the particle-size gradation of each group of stone waste after blending are shown in Figure 7.

The density test results are based on the most compact state of different grades of stone waste at maximum density, as shown in

Table 2. Test results of stone waste density.

λ	Mass of Sample/g	γd(max)/(g/cm³)
0.14	377.64	1.515
0.17	381.53	1.531
0.20	391.34	1.570
0.23	398.32	1.598
0.26	397.26	1.594
0.29	387.93	1.556
0.32	384.35	1.542

. The test data can be seen with the increase in the value of λ. The maximum density values of the specimens show a trend of increasing and then decreasing when λ = 0.23, and γ_d is the highest; however, the results show the density of the 7 groups based on the best value, not the theoretical existence of the maximum value. Therefore, it was determined that the stone waste gradation method requires further optimization.

3. Design and Optimization of a Stone Waste Gradation Method Based on Response Surface Methodology

3.1. Response Surface Methodology

Compared with the orthogonal test method commonly used at present, the response surface method can obtain the function expression between the influencing factors and the response value in order to obtain the optimal combination between the influencing factors and the optimal solution of the response value. Moreover, when the test factor level values are higher, if an orthogonal test is used, a large number of tests need to be carried out, which is cumbersome. Response surface methodology has been widely used in many fields because of the small number of tests required and the high precision of prediction; this method was first used for the fitting of physical tests. In recent years, the response surface method has become an optimization theory method that is widely used in China and abroad in the fields of medicine, chemistry, environment, materials, and others.

Since CGF stone waste is a brand-new gelling system, the influences of CGF gelling materials, water, etc., were not considered in this study, and the value of λ was not continuous. Therefore, the results obtained based on the theory of closest stacking contain errors that must be further calculated and analyzed.

The results indicate that λ = 0.23, P(2) = 80%, P(0.075) = 33%, and P(0.005) = 13%, which means that the 2 mm, 0.075 mm, and 0.005 mm sieve pass rates of the stone waste material are 80%, 33%, and 13%, respectively. Moreover, the stone debris accounted for 67% of the waste material (of which 20% consisted of 2~4.75 mm and 0.075~2 mm particles, and 13% consisted of the stone waste material), stone chips of 0.075~2 mm accounted for 47%, fine stone powder accounted for 20%, and ultrafine stone powder accounted for 13%.

CGF was added to the optimized stone waste materials to investigate the mechanical property changes in the optimized CGF stone waste-cured specimens. Based on the considerations of engineering economy and applicability, the dosage of CGF was set to be 10%. During an indoor test conducted in a previous study, it was determined that the specimens were better formed when the water/cement ratio was 1.0; therefore, the test was set to have a water/cement ratio of 1.0.

Response surface design was used to optimize the stone waste gradation method by including a wider range of values and fitting the second-order equation. In this test, the Box–Behnken design response surface method was selected for optimization of the stone waste gradation design, and subsequently, a three-factor, three-level experimental design was adopted, with the three influencing factors set to be 2 mm, 0.075 mm, and 0.005 mm sieve pass rates, represented by X₁, X₂, and X₃, respectively, and with the specimens’ unconfined 7 and 28 d compressive strengths as the two response values, denoted by Y₇ and Y₂₈, respectively. Since the relationship between the three influencing factors and the response values in this test is not linear, a higher-order polynomial, i.e., a second-order equation, is required.

P(2) = 80%, P(0.075) = 33%, and P(0.005) = 13% were used as the centroids of X₁, X₂, and X₃, respectively, and the ranges of variation, X₁, X₂, and X₃, were set at 70–90%, 23–43%, and 8–18%, respectively. The three levels of the influencing factors were coded as −1 (low), 0 (medium), and +1 (high). The Box–Behnken design scheme is shown in

Table 3. Three kinds of influencing factors and their coding level.

Influencing Factor	Codes	Coding Level
		−1	0	+1
2 mm sieve pass rates	X1	70%	80%	90%
0.075 mm sieve pass rates	X2	23%	33%	43%
0.005 mm sieve pass rates	X3	8%	13%	18%

. The number of trials was calculated using Equation (7) as follows:

$$\mathrm{C}=2b(b-1)+a$$

(7)

where C is the number of tests to be conducted, a represents the number of test center points, and b represents the number of influencing factors. Set the number of test center points a to take the value of 5, and b is the number of influencing factors that takes the value of 3, so this test has a total of 17 groups.

After determining the level scheme of each factor, the experimental design of the stone waste gradation method was carried out. A multiple regression technique was used to fit a quadratic multinomial model of the Box–Behnken design method, and the corresponding test scheme is shown in Table 3.

Based on the 17 groups of the proportioning scheme in

Table 4. Experimental design of response surface method.

Test Group	Influencing Factor
	2 mm Sieve Pass Rates (%)	0.075 mm Sieve Pass Rates (%)	0.005 mm Sieve Pass Rates (%)
1	80	33	13
2	80	33	13
3	80	43	18
4	90	43	13
5	80	23	18
6	80	43	8
7	80	23	8
8	70	43	13
9	80	33	13
10	90	23	13
11	80	33	13
12	70	23	13
13	80	33	13
14	90	33	18
15	90	33	8
16	70	33	8
17	70	33	18

, specimen preparation was carried out first. The materials were prepared according to the required stone waste of different grain sizes, the samples were prepared according to Section 3, and then the unconfined compressive strength test was carried out after the specified curing age was achieved. In the 17 groups of the proportioning scheme, three parallel specimens were made for each group of different ages, and the compressive strength was taken as the average of the three. The results of the test are shown in Figure 8.

3.2. Variance Analysis

The interaction between the three influencing factors and the two response values was fitted by establishing a second-order regression equation, as shown in Equation (8):

$$y={\beta }_0+{\displaystyle \sum _{i=1}^k{\beta }_i}{x}_i+{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^k{\beta }_{ij}{x}_i{x}_j}}+{\displaystyle \sum _{i=1}^k{\beta }_{ii}}{x}_i^2.$$

(8)

where y is the value of the unconfined compressive strength; x_i and x_j are the actual values of each influencing factor; β₀ is the intercept value of the equation; β_i is the coefficient of a single factor; β_ij is the coefficient of the interaction of two factors; β_ii is the coefficient of squaring; and k is the number of influencing factors.

After deriving the unconfined compressive strength of each group of specimens, the second-order variance was fitted to the data, followed by the analysis of variance (ANOVA) for the 7 and 28 d unconfined compressive strengths, as shown in

Table 5. ANOVA results.

Variance Model	7 d			28 d
	F-Value	p-Value	Coefficients of the Equation	F-Value	p-Value	Coefficients of the Equation
Model	14.24	0.0010	−44,833.79	9.73	0.0033	−7060.53
X1	25.36	0.0015	+934.93	24.72	0.0016	+351.52
X2	9.14	0.0193	+601.30	12.61	0.0093	+21.59
X3	18.12	0.0038	+144.23	27.40	0.0012	−317.71
X1X2	7.14	0.0319	−3.39	0.9380	0.3651	+1.24
X1X3	1.31	0.2901	+2.90	6.47	0.0384	+6.52
X2X3	6.01	0.0440	+6.22	2.74	0.1417	+4.24
X12	17.04	0.0044	−5.10	4.68	0.0672	−2.70
X22	22.22	0.0022	−5.82	3.05	0.1244	−2.18
X32	15.46	0.0057	−19.43	3.67	0.0969	−9.56

.

The F-value test is an alias joint hypothesis test, also known as the variance ratio test. The variance is a measure of dispersion, and the larger the value, the greater the dispersion. The p-value test can determine the degree of reliability of the fitted equations; the smaller the p-value, the better the effect of the model fit and, thus, the closer the model to real-world situations. When p ≤ 0.05, the response surface model is judged to be more significant, indicating that the model is statistically comparable. From the ANOVA, it can be seen that the p-value of 7 and 28 d unconfined compressive strength of this model is less than 0.05, which indicates that the model is universal and the fitting is better. In the single-factor effect, the p-values of X₁, X₂, and X₃ are all less than 0.05, indicating that all three play a significant role in the formation of specimen strength.

According to the equation coefficients, the 7 and 28 d unconfined compressive strength prediction formulas are shown in Equations (9) and (10), respectively:

$$\begin{array}{l}{Y}_7=-44833.79+934.93X_1+601.30X_2+144.23X_3-3.39X_1{X}_2\\ \hspace{1em} +2.90X_1{X}_3+6.22X_2{X}_3-5.10{X_1}^2-5.82{X_2}^2-19.43{X_3}^2\end{array}$$

(9)

$$\begin{array}{l}{Y}_{28}=-7060.53+351.52X_1+21.59X_2-317.71X_3+1.24X_1{X}_2\\ \hspace{1em} +6.52X_1{X}_3+4.24X_2{X}_3-2.70{X_1}^2-2.18{X_2}^2-9.56{X_3}^2\end{array}.$$

(10)

where Y₇ and Y₂₈ are the 7 and 28 d unconfined compressive strength of the specimen, respectively; X₁, X₂, and X₃ are the 2 mm, 0.075 mm, and 0.005 mm sieve pass rates.

3.3. Analysis of the Fit of Second-Order Equations

(1) Residual Distribution Test

In mathematical statistics, the residual is the difference between the actual observed value and the estimated value (fitted value). This difference reflects the error between the predicted value of the model and the actual observed value and is an important indicator for assessing the performance of a regression model.

When the regression model is correct, the residuals can be regarded as the observed values of the error. By observing the normal probability distribution plot of the residuals of the 7 and 28 d unconfined compressive strength test data, as shown in Figure 9, it can be seen that the residuals are mainly distributed in and immediately around a straight line, which indicates a high degree of model fit. Since there are only three variables in this experiment, deleting points with VIF < 10 may result in less data, which may lead to failure to obtain a clear pattern.

(2) Residuals and prediction

By looking at Figure 9, it can be seen that the distribution of the points in the residual vs. prediction plots for the 7 and 28 d unconfined compressive strengths is very discrete, confirming that the fit is good for this test.

(3) Residuals and test

Although this test was set for only three influencing factors, there might be other influencing conditions that cause greater interference. The residuals and a test plot were used to determine whether additional influencing factors exist that could cause greater interference to the test results other than the three influencing factors. By looking at Figure 9, it can be seen that the residuals of the 7 and 28 d unconfined compressive strengths and the test plot inside the point distribution are discrete, with few fluctuations, confirming that, aside from the three influencing factors set up in this test, no other influencing factors affect the test results.

(4) Residuals and influencing factors

The plot of residuals versus influencing factors analyzes the relationship between the three influencing factors and the residuals. If the points are dispersed and random, the model is better fitted as a whole. The plots of residuals versus influencing factors for the 7 and 28 d unconfined compressive strengths in Figure 10 are in line with the characteristics of the points being dispersed and random, which proves that the second-order equations are better fitted.

3.4. Interaction Analysis

(1) Seven-day interaction

The results of the fit analysis of the test data showed that no other influences aside from the three influencing factors (X₁, X₂, and X₃) were present, so the influencing factors were analyzed with 2-by-2 correspondence. Under the interaction between two factors, the p-values of X₁ and X₂ in the 7 d ANOVA results were the smallest among the two-factor interactions, which showed that X₁ and X₂ significantly affected the unconfined compressive strength of the solidified stone waste specimens at 7 d. The results of the 7 d ANOVA results showed that X₁ and X₂ were the smallest among the two-factor interactions. Figure 11 shows the contour plots and response surface plots of the 7 d mechanical strength model.

The effect of the interaction between X₁ and X₂ on the mechanical properties of the solidified stone waste specimens can be seen more intuitively through the contour plots and response surface plots. When the amount of X₃ is controlled at the midpoint of the dosage range, i.e., the 0 level, the unconfined compressive strength of the specimens shows a tendency to increase and then decrease with the dosages of X₁ and X₂.

(2) Twenty-eight-day interaction

Under the interaction between two factors, the p-values of X₁ and X₃ in the 28 d ANOVA results were the smallest among the interactions between two factors, which indicates that X₁ and X₃ significantly affect the unconfined compressive strength of the solidified stone waste specimens at 28 d. The contour plots and the response surface plots of the 28 d mechanical strength model are shown in Figure 12.

When the amount of X₂ is controlled at the midpoint of the dosage range, i.e., the 0 level, the unconfined compressive strength of the specimens shows a tendency to increase and then decrease with the dosages of X₁ and X₃.

3.5. Strength Results of Solidified Stone Waste after Optimized Grading

The unconfined compressive strength of the solidified stone waste specimens showed a tendency to increase and then decrease under the interaction between two factors, and the single-factor effect was significant. Therefore, theoretically, there exists an optimal grading design of X₁, X₂, and X₃ that can make the compressive strength of the solidified stone waste specimens reach the maximum.

Figure 8 shows the actual 28 d strength (mixing ratio: 20%) and Figure 11 and Figure 12 show the predicted 28 d strength (mixing ratio: 20%); the conclusion can be drawn after comparing the two groups of strengths. Through fitting optimization, the predicted 7 and 28 d unconfined compressive strengths of the solidified stone waste specimens were 6383 and 8907 kPa when X₁, X₂, and X₃ were 85.4, 36.6, and 16.5, respectively.

To verify the error between the predicted and actual strengths of the solidified stone waste specimens, the specimens were prepared according to the optimized stone waste gradation method, and strength tests were carried out after curing to the specified age. The measured unconfined compressive strengths of the specimens at 7 and 28 d were 6538 and 9103 kPa, respectively, and the actual values were closer to the predicted values with smaller errors, indicating that this second-order equation is more accurate and scientific. In the case of the same amount of binder and the same curing age, compared with the CGF virgin stone waste material, the 28 d unconfined compressive strength of the optimized graded solidified stone waste specimen was improved by about 2.1 MPa, which was about 29.8% higher, and it was 2.6 MPa higher than that of the cemented virgin stone waste material, with an increase in strength of approximately 39.3%.

4. Discussion and Conclusions

In this study, the applicability of the most-compact stacking theory for stone wastes was determined, and a grading design method, i.e., the MAA-SW model, applicable to stone wastes was constructed. Initially, seven groups of granular graded stone wastes were designed, and the test group with the highest density was selected to control the CGF dosing of 10% and the water/cement ratio of 1.0. Subsequently, based on the response surface design methodology, 2 mm sieve pass rate (X₁), 0.075 mm sieve pass rate (X₂), and 0.005 mm sieve pass rate (X₃) were identified as the three influencing factors, and the unconfined compressive strengths of the specimens at 7 and 28 d (Y₇, Y₂₈) were selected as the two response values. A second-order equation was fitted, and the test results were analyzed using ANOVA. The following conclusions are obtained:

(1) According to the ANOVA results, X₁, X₂, and X₃ all play an important role in the 7 and 28 d unconfined compressive strengths of the solidified rock waste specimens; X₁ and X₂ significantly affect the unconfined compressive strength of the solidified rock waste specimens at 7 d, whereas it is X₁ and X₃ that have a significant effect at 28 d.

(2) Second-order equations were fitted to the response surface test data to derive the strength prediction equations between 2 mm sieve pass (X₁), 0.075 mm sieve pass (X₂), and 0.005 mm sieve pass (X₃) rates and the unconfined compressive strengths of the specimens at 7 and 28 d (Y₇, Y₂₈); the strength prediction formulas were derived when X₁, X₂, and X₃ were 85.4, 36.6, and 16.5, respectively. The solidified rock waste specimen has the greatest strength, and the predicted 7 and 28 d unconfined compressive strengths are 6383 and 8907 kPa, respectively, while the error between the predicted and actual values is small. Compared with the CGF solidified rock waste material, the 28 d unconfined compressive strength of the optimized graded specimen was improved by about 2.1 MPa, which was about 29.8% higher, and it was about 2.6 MPa higher than the strength of the cemented solidified rock waste material, with the strength being improved by about 39.3%.