# Study on the Modification Formula of Earth Material Compound with Cement and Gravel by Single Lattice Theory

Authors: Kun Zhang; Bai-Ru Lu; Yi-Hong Wang; Zhan Qu; Le Qu
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

Gravel and cement can effectively improve the low strength of traditional Earth materials. There have been few studies on test methods for raw soil-based composite admixtures. By introducing the theory of single lattice formula testing, the compressive strength of 10 formulas and 60 modified raw soil cubic specimens were tested. The failure process, failure mode, and compressive strength of specimens were studied. The effects of different formulations of raw soil on strength, peak displacement, and dispersion were analyzed. The results showed that cement content had a significant effect on the compressive strength and dispersion of earth material specimens. The optimal modification formula of cement, gravel, and raw soil was determined to be 0.1/0.08/0.82 (mass ratio). At the same time, the applicability of the test method was verified, which can be used as a reference for the experimental study on modification formulations of earth materials.


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## Body

## 1. Introduction

Earth materials are some of the oldest local building materials and are widely used in dwelling construction all over the world. In fact, one-third of the world’s population still lives in earthen dwellings [1, 2]. In the last twenty years, this original material has gained new attention from the construction industry because of its low energy consumption and excellent ability to regulate indoor temperature and humidity [3, 4]. As an important part of traditional architecture, it has the advantages of low energy consumption, good thermal performance, sound absorption, radiation, and environmental protection and can serve as a model for “green architecture” [5–7]. However, the weak strength of raw materials for native construction has always been considered a drawback by structural engineers. Therefore, there has been much research on the modification of raw earth materials all over the world. At present, the modification methods of raw soil materials can be divided into physical modification, mechanical modification, and chemical modification [8, 9].Physical modification is to select well graded soil material or add reinforced material into soil material [10]. Mechanical modification refers to the modern manufacturing process. Mechanical dynamic compaction is used to increase the compactness of raw soil foundation blocks [11]. Chemical modification is the incorporation of chemical reagents into materials [12]. Among the main curing agents in modifying earth materials are cement, lime, and fly ash. Spence and Cook of the University of Chichester have studied the modification of earth materials with a certain amount of cement and carried out experiments with Alfred et al. Their results showed that when the cement content is 5%, cement solidified. Also, adobe has good durability and water absorption performance [13, 14]. Stephane Hans of the University of Lyon and others have proposed the modification of raw soil materials with cement and lime and calculated the damage of existing native buildings in earthquakes [15]. Steve Burroughs of Australia has done stability testing to quantify the linear shrinkage of natural soils and the amount of cement and lime mixed into them [7]. Milani and Labaki studied the sustainability of the original soil materials and evaluated the modified soil materials mixed with cement and lime. The results showed that 7.5 wt% lime and 10 wt% cement have been mixed into the soil to obtain promising mixture materials [4]. Ziegler used fiber materials to modify the raw soil-based materials. This method formed a three-dimensional fiber network inside the materials, which significantly improved the compressive strength and shear strength of the raw soil-based materials [16].In a word, experimental results have been obtained in the earth modification field. However, the traditional orthogonal test cannot completely solve the problem of modifying material formulas. A single admixture and test methods are not unified from different studies and the existing research results scattered. Modified additives of materials often have complex chemical reactions in soil and modified earth materials are not green and environmentally friendly. In this study, based on the theory of single grid test design, the composite test of raw soil materials with multiple admixtures is carried out. Using physical modification methods according to research results from this research group, a modified formula test was carried out with gravel (gravel, 10–20 mm diameter), cement, and raw soil [17]. The modified formula of earth material was studied using the Voth single lattice model [18, 19]. Sixty specimens were produced for compressive strength testing and the strength, peak displacement, discrete reference parameters, parameter analysis, and elimination element regression were used to draw a triangular contour map. From these results, a reasonable modification test method and formula were presented. The research provided suggestions for similar studies in the future.


## 2. Voth Single Lattice Theory

In the design of the test, the test level is a factor of each factor accounted for in the proportion of the total, with the percentage of allP factors in the experiment being the percentage of the total. The total composition must be equal to 1 (100%). This kind of design is called formula design, also known as mix design [20].The level values ofP factors in formulation design were(1)x1,x2,x3,…,xp0≤xi≤1,0≤i≤p.Here,∑i=1pxi=1, xi is called the scaling factor. If using geometric terms, with p called the dimension, xi is a point on the P-dimensional plane in Hypothesis 3 factor formulation test. According to P component variables xi (0≤i≤p) in the formulation design. Here, y is the response variable and the mathematical model can be established:(2)y=fx1,x2,…,xp+ε,where ε is the random error and observation values y1,y2,…yn from reaction variables n times. The regression coefficient of the function was estimated and the regression equation obtained:(3)y^=fx1,x2,…,xp.The above is a polynomial of degreed and the formula test determined by the number of factors P and the number d of directional functions. A formula test was represented by Mp,d. Thus, it was also called a single lattice design. Here, p=n, d = m.A significance test of multivariate linear regression equation byF-test was used with the square sum (SST), the regression square sum (SSR), and the sum of squares of residuals (SSe), and F and R2 are calculated finally.After passing theF-test, the regression equation was solved and the isoline map drawn as a triangular coordinate map to determine the optimal mixing ratio.In the design of the single lattice formulation, there was no restriction onxi (0≤xi≤1 and 0≤i≤p). For the formulation design of restricted ingredients, restricted ingredients were expressed in terms of z. If there are lower bounds, upper bounds, or both upper and lower bounds for component zi, Li represented the lowest proportion of component i and the highest proportion of component Ui, that is, the upper and lower bounds of zi. The hypothesis was(4)z1+z2+⋯+zp=10≤Li≤zi≤10≤i≤p.Because the formula test area was simple, the original ingredients could be changed. A new set of components was defined in the executable formulation area and the change range of the new components still between 0 and 1. The appropriate formulation design structure model was obtained in the restricted area. The component was redefined as theL-coding component. The coding component xi of i was(5)xi=zi−Li1−L,L=∑i=1pLi≤1,where zi was the actual occupancy ratio after conversion.


## 3. Experimental

### 3.1. Experimental Design

According to the previous research results in this research group, adding cement materials to raw soil improves soil cohesion and adding gravel filled pores in the raw soil, thus improving the overall mechanical properties of the material. The cement content was not <3 wt% and the gravel content not <5 wt%. As raw soil was the main material, the content cannot be <75 wt%. Here,z1 was cement, z2 gravel, and z3 raw soil:(6)z1+z2+z3=1z1≥0.03z2≥0.05,z3≥0.75.Because there were three kinds of admixtures, the triangle single grid method was adopted, and ten points in the graph were selected as test-coding points. A single lattice test-coding diagram is shown in Figure1 and a limit zone diagram of cement, gravel, and raw soil content in Figure 2. The black spot in Figure 1 is the test mix ratio. Photocopy in Figure 2 is the limited area of material composition.Figure 1
(a) Triangular simplex design test points. (b) Limited areas of cement, gravel, and raw soil content.
(a)(b)Figure 2
Manufacturing equipment and test specimens. Raw material (a), mixture mixing, (b) schematic diagram of the device (c), specimen making (d), specimen (e), and specimen maintenance (f).
(a)(b)(c)(d)(e)(f)As can be seen from Section2 and formulas (4) and (5), z1nsi=0.03+0.17x1nsii=1–10, z2nsi=0.05+0.17x2nsii=1–10, and z3nsi=0.75+0.17x3nsii=1–10.The calculation results of the experimental grouping formula are shown in Table1.Table 1
Axis point extended simplex lattice design scheme for cement and gravel mixed with the modified raw soil test.
Test formationTest codingMix ratio of test materials (by wt)x1x2x3z1z2z3NS11000.200.050.75NS20100.030.220.75NS30010.030.050.92NS41/21/200.120.140.75NS501/21/20.030.140.84NS61/201/20.120.050.84NS72/31/61/60.140.080.78NS81/62/31/60.060.160.78NS91/61/62/30.060.080.86NS101/31/31/30.090.110.81


### 3.2. Materials

Soil samples used in this test were loess from the Chang’an District of Xi’an City in China. The optimal loess moisture content was 18.2% and maximum dry density at 2.04 g/cm3, according to “Standard of Geotechnical Experiment Methods” (Geotechnical test method standard, 1999) (GBT50123-1999), with the plastic limit at 15%, liquid limit at 26%, and plastic index at 11.3 [21]. The results are shown in Table 2. At the same time, the sieving test of loess is carried out, and the results are shown in Table 3 (the particle size distribution). The gravel was gravel 10–20 mm in diameter. The strength of the cement was 32.5 ordinary Portland cementSchool of Civil Engineering, Chang’an University, Xi’an 710061, China, Shaanxi Qinling Cement). Referring to previous research of this research group, the optimal moisture content of raw soil materials mixed with cement was determined [22]:(7)ω=ω0+0.27ϕ,where ω is the optimal moisture content of modified raw soil, ω0 at 18.2%, and φ the cement content. The moisture content of each group of specimens with a cement-modified formula is shown in Table 4.Table 2
The index of soil-based materials.
NameMaximum dry densityPd, (g/cm3)Optimum moisture contentw0, (%)Plastic limitwp, (%)Liquid limitwL, (%)Plastic indexLoess2.0418.2152611Table 3
Size distribution of loess screening result.
Particle size (mm)≥4.754.75～2.362.36～1.181.18～0.600.60～0.300.30～0.15≦0.15Particle content (%)0.0014.0720.6317.0014.008.4026.00Table 4
The moisture content of each group of specimens containing cement.
Test formationNS1NS2NS3NS4NS5NS6NS7NS8NS9NS10Moisture content0.230.190.190.210.190.210.220.200.200.20Before experimentation, the earth was passed through a 5 mm sieve and mixed to optimal moisture content; see Table4 for water mixing materials. Using a mold developed by this research group, 60 samples of 100 × 100 × 100 mm cubes were prepared from each earth using a method of jack compression. The preparation device and sample are shown in Figure 2 [23].


### 3.3. The Loading Device

Molded specimens were cured in a standard curing room and then tested for compressive strength. Here, the specimens were placed in a room at 25–30°C and 55–60% humidity to cure for 28 d. Using an MTS-500 universal uniaxial load tester (Manta Testing Systems Inc., Mississauga, Canada), test data regarding the relationship between displacement and load was automatically collected. A sample was centered and placed horizontally on the ball support, with the sample in close contact with the machine before test initiation. The loading rate was set to 1 mm/min and 30% of the peak load taken as the final condition of a test after peak loading, which ensured normal machine operation. The test procedure is shown in Figure3.Figure 3
Compressive strength tests of raw soil material cubes. Loading (a), cracking (b), damage (c), and completely destroyed (d).
(a)(b)(c)(d)At the initial stage of loading, cracks did not immediately appear on soil cube surfaces and no other visible cracks appeared. With increased load, specimen corners cracked, cracks developed along the specimen’s stress direction, and the specimen did not immediately break. As the load continued to increase to peak load, many vertical cracks appeared on the test piece surface, along with swelling cracks on the piece’s middle surfaces, and portions of the skin soil fell off. With further increased load, cracks continued to develop until the specimen was completely destroyed, accompanied by rapidly decreased load, bearing capacity loss, and failure mode appearing as a typical “hourglass.”


## 3.1. Experimental Design

According to the previous research results in this research group, adding cement materials to raw soil improves soil cohesion and adding gravel filled pores in the raw soil, thus improving the overall mechanical properties of the material. The cement content was not <3 wt% and the gravel content not <5 wt%. As raw soil was the main material, the content cannot be <75 wt%. Here,z1 was cement, z2 gravel, and z3 raw soil:(6)z1+z2+z3=1z1≥0.03z2≥0.05,z3≥0.75.Because there were three kinds of admixtures, the triangle single grid method was adopted, and ten points in the graph were selected as test-coding points. A single lattice test-coding diagram is shown in Figure1 and a limit zone diagram of cement, gravel, and raw soil content in Figure 2. The black spot in Figure 1 is the test mix ratio. Photocopy in Figure 2 is the limited area of material composition.Figure 1
(a) Triangular simplex design test points. (b) Limited areas of cement, gravel, and raw soil content.
(a)(b)Figure 2
Manufacturing equipment and test specimens. Raw material (a), mixture mixing, (b) schematic diagram of the device (c), specimen making (d), specimen (e), and specimen maintenance (f).
(a)(b)(c)(d)(e)(f)As can be seen from Section2 and formulas (4) and (5), z1nsi=0.03+0.17x1nsii=1–10, z2nsi=0.05+0.17x2nsii=1–10, and z3nsi=0.75+0.17x3nsii=1–10.The calculation results of the experimental grouping formula are shown in Table1.Table 1
Axis point extended simplex lattice design scheme for cement and gravel mixed with the modified raw soil test.
Test formationTest codingMix ratio of test materials (by wt)x1x2x3z1z2z3NS11000.200.050.75NS20100.030.220.75NS30010.030.050.92NS41/21/200.120.140.75NS501/21/20.030.140.84NS61/201/20.120.050.84NS72/31/61/60.140.080.78NS81/62/31/60.060.160.78NS91/61/62/30.060.080.86NS101/31/31/30.090.110.81


## 3.2. Materials

Soil samples used in this test were loess from the Chang’an District of Xi’an City in China. The optimal loess moisture content was 18.2% and maximum dry density at 2.04 g/cm3, according to “Standard of Geotechnical Experiment Methods” (Geotechnical test method standard, 1999) (GBT50123-1999), with the plastic limit at 15%, liquid limit at 26%, and plastic index at 11.3 [21]. The results are shown in Table 2. At the same time, the sieving test of loess is carried out, and the results are shown in Table 3 (the particle size distribution). The gravel was gravel 10–20 mm in diameter. The strength of the cement was 32.5 ordinary Portland cementSchool of Civil Engineering, Chang’an University, Xi’an 710061, China, Shaanxi Qinling Cement). Referring to previous research of this research group, the optimal moisture content of raw soil materials mixed with cement was determined [22]:(7)ω=ω0+0.27ϕ,where ω is the optimal moisture content of modified raw soil, ω0 at 18.2%, and φ the cement content. The moisture content of each group of specimens with a cement-modified formula is shown in Table 4.Table 2
The index of soil-based materials.
NameMaximum dry densityPd, (g/cm3)Optimum moisture contentw0, (%)Plastic limitwp, (%)Liquid limitwL, (%)Plastic indexLoess2.0418.2152611Table 3
Size distribution of loess screening result.
Particle size (mm)≥4.754.75～2.362.36～1.181.18～0.600.60～0.300.30～0.15≦0.15Particle content (%)0.0014.0720.6317.0014.008.4026.00Table 4
The moisture content of each group of specimens containing cement.
Test formationNS1NS2NS3NS4NS5NS6NS7NS8NS9NS10Moisture content0.230.190.190.210.190.210.220.200.200.20Before experimentation, the earth was passed through a 5 mm sieve and mixed to optimal moisture content; see Table4 for water mixing materials. Using a mold developed by this research group, 60 samples of 100 × 100 × 100 mm cubes were prepared from each earth using a method of jack compression. The preparation device and sample are shown in Figure 2 [23].


## 3.3. The Loading Device

Molded specimens were cured in a standard curing room and then tested for compressive strength. Here, the specimens were placed in a room at 25–30°C and 55–60% humidity to cure for 28 d. Using an MTS-500 universal uniaxial load tester (Manta Testing Systems Inc., Mississauga, Canada), test data regarding the relationship between displacement and load was automatically collected. A sample was centered and placed horizontally on the ball support, with the sample in close contact with the machine before test initiation. The loading rate was set to 1 mm/min and 30% of the peak load taken as the final condition of a test after peak loading, which ensured normal machine operation. The test procedure is shown in Figure3.Figure 3
Compressive strength tests of raw soil material cubes. Loading (a), cracking (b), damage (c), and completely destroyed (d).
(a)(b)(c)(d)At the initial stage of loading, cracks did not immediately appear on soil cube surfaces and no other visible cracks appeared. With increased load, specimen corners cracked, cracks developed along the specimen’s stress direction, and the specimen did not immediately break. As the load continued to increase to peak load, many vertical cracks appeared on the test piece surface, along with swelling cracks on the piece’s middle surfaces, and portions of the skin soil fell off. With further increased load, cracks continued to develop until the specimen was completely destroyed, accompanied by rapidly decreased load, bearing capacity loss, and failure mode appearing as a typical “hourglass.”


## 4. Results and Discussion

### 4.1. Compressive Strength Analysis

The average value of protest strength of each group of specimens is taken as the compressive strength of the test samples. The experimental results are shown in Table5.Table 5
Test results of compressive strength of specimens mixed with cement, gravel, and raw soil.
Test formationPeak load (kN)Peak displacement (mm)Compressive strengthAverage of strength (MPa)Standard deviationCoefficient of variationNS1115.123.5711.511.2900.112NS237.522.973.750.4470.119NS357.754.115.770.4920.085NS479.733.337.972.2020.276NS543.923.434.391.1080.252NS688.743.818.870.9710.109NS7111.023.5511.101.0670.096NS868.123.586.811.1780.173NS972.063.707.211.0050.139NS1069.153.706.910.7340.106Raw soil32.422.73.240.3930.121Referring to Table5, the compressive strength of the modified raw soil with gravel and cement is significantly higher than that of the raw soil. The compressive strength of the modified specimen is 1.16–3.55 times that of the raw soil specimen.Compared with NS3, NS5, and NS2 group, when the cement content is 3% and the gravel content is 5%, 13.5%, and 22%, the compressive strength of the specimen is 5.77 MPa, 4.39 MPa, and 3.75 MPa. The results show that when the cement content is fixed, the strength of the specimen decreases with the increase of the gravel content.Compared with NS3, NS6, and NS1 group, when the gravel content is 5% and the cement content is 3%, 11.5%, and 20%, the compressive strength of the specimen is 5.77 MPa, 8.87 MPa, and 11.51 MPa. The results show that when the gravel content is fixed, the strength of the specimen increases with the increase of the cement content.According to a large number of research results, the compressive strength of raw soil-based materials with gravel and cement is higher than that with gravel or cement alone [参考文献]. The strength of gravel is obviously higher than that of soil. However, when the gravel is mixed with the raw soil alone, the bond performance between the gravel and the soil is poor, which leads to the reduction of the time compression performance. As a cementitious material, cement can react with water molecules in the soil to consolidate soil particles and fill the pores in the soil, which improves the particle size distribution and increases the material density. The gravel is wrapped by cement soil and closely bonded, acting as coarse aggregate, playing a skeleton role, and resisting deformation when the material is under pressure, and the material strength is significantly improved.


### 4.2. Formula Discussion

The raw soil-modified cubic specimens consisted of sand (z1), gravel (z2), and raw soil (z3, Table 1). There was more than one reaction variable y in the formulation of specimens considering the test results. There were three parameters that reflected the advantages and disadvantages of the modified specimens. The first response variable was the cubic compressive strength (y1), the second response variable was the cubic peak displacement (y2), and the third response variable was the cubic Standard deviation (y3). Integrating Tables 1 and 3 yielded Table 6.Table 6
Axis point extended simplex lattice design and the result of lower limit component boundary in mixed cement and gravel modified with raw soil.
Test formationTest codingMix ratio of test materialsResponse variablesx1x2x3z1z2z3y1y2y3NS11000.200.050.7511.513.571.290NS20100.030.220.753.752.970.447NS30010.030.050.925.774.110.492NS41/21/200.120.140.757.973.332.202NS501/21/20.030.140.844.393.431.108NS61/201/20.120.050.848.873.810.971NS72/31/61/60.140.080.7811.103.551.067NS81/62/31/60.060.160.786.813.581.178NS91/61/62/30.060.080.867.213.701.005NS101/31/31/30.090.110.816.913.700.734(1) Regression analysis of response variabley1 was carried out with reference to Table 4. The calculation was simplified by examining the first-order polynomial for the regression model y^=fx=ax1+bx2+cx3 (Table 7), which showed that the coefficient of determination was R2 = 0.9249 by linear regression. According to the F-test, the linear regression equation was significant and the primary regression model concluded to be scientific and reasonable. The polynomial coefficients calculated by MATLAB are shown in Table 8. The corresponding regression equation was y1=12.23x1+4.13x2+5.92x3.Table 7
Regression analysis of compressive strength of raw soil mixed with cement and gravel.
Source parameterdfSSMSFp > FRegression3SSR54.31MSR18.10428.73RemarkableDeviation7SSe4.41MSe0.031Uncorrected summation10SST58.72R20.9249Table 8
Estimation of the partial regression coefficient ofy1 regression equation for compressive strength of mixed cement and gravel modified with raw soil.
Regression termPartial regression estimationt-testp>tRegression error of partial regression estimationx112.2321.340.00010.573x24.137.220.00010.571x35.9210.320.00010.571Combining regression equationy1, solving the equationymax  = 11.51, ymin  = 3.75，range Δ  = 7.76, y1¯  = 7.43, gradient ∇  = 1, using Mathcad software, the results of equation y1 are shown in Table 9.Table 9
Compressive strengthy1 regression equation calculations.
y1111098765x10.850.80.730.650.60.490.480.330.350.170.230.020.10x20.1500.2700.400.5200.6500.7700.90.51x300.200.3500.5100.6700.8300.9800.49The solution of the intersection point between the equation and the triangular coordinate axis is shown.According to the coordinates of the equation solution in Table9, a contour map corresponding to the equation was drawn in triangular coordinates using Origin 9.0 drawing software (Figure 4).Figure 4
Isogram of compressive strengthy1 regression equation.Parametersx1, x2, and x3 represented the admixtures of cement, gravel, and raw soil, respectively (Figure 4). The gradient of the isoline decreased from x1 point to the x2x3 coordinate axis step by step. The isoline basically deviated to the x1x2 coordinate axis. The average strength line fell on the x1x2 and x1x3 axes, which showed that the cement content of these three admixtures played a major role in compressive strength. The strength of gravel was higher than that of raw soil. Gravel particles here were mainly composed of quartz and the surfaces well encapsulated, adsorbed, and bonded with raw soil. However, when the gravel content increased, the gravel occupied a large amount of internal space, such that the soil particles did not bond well. With increased cement content, the hydration cementation of the cement was enhanced, which enhanced the cohesive force between soil particles and the material strength clearly increased. If only the compressive strength index was considered, the area of the enclosure area with y1 = 11 and x1 vertices was the optimal ratio area. From Figure 4, 0.8≤x1≤1, 0≤x2≤0.15, and 0≤x3≤0.2 were calculated from Section 3.1, with 0.17≤z1≤0.2, 0.05≤z2≤0.08, and 0.75≤z3≤0.78, and thus the optimal formula of cement, gravel, and raw soil was 0.18/0.06/0.76 (mass ratio, resp.).(2) The regression analysis of response variabley2 was carried out with reference to Table 6. The calculation was simplified by examining the first-order polynomial for the regression model y^=fx=ax1+bx2+cx3. The regression results showed the coefficient of determination at R2 = 0.9441 by linear regression (Table 10). According to the F-test, the linear regression equation was significant and the primary regression model concluded to be scientific and reasonable. The polynomial coefficients calculated using MATLAB are shown in Table 11. The corresponding regression equation was y2=3.6x1+3.08x2+4.04x3.Table 10
Regression analysis of peak displacement of raw soil mixed with cement and gravel.
Source parameterdfSSMSFp>FRegression3SSR0.94MSR0.313328.73RemarkableDeviation7SSe0.12MSe0.0170Uncorrected summation10SST1.06R20.9441Table 11
Estimation of the partial regression coefficient ofy2 regression equation for peak displacement of mixed cement and gravel modified with raw soil.
Regression termPartial regression estimationt-testp>tRegression error of partial regression estimationx13.636.850.00010.098x23.0831.570.00010.097x34.0441.380.00010.097Combining regression equationy2, solving the equation ymax  = 4.11, ymin  = 2.97, range Δ  = 1.14, y2¯  = 3.57, gradient ∇  = 0.1, using Mathcad software, the results of equation y2 are shown in Table 12.Table 12
Peak displacementy2 regression equation calculations.
y243.93.83.73.63.53.4x10.100.3200.5500.770100.800.620x200.0500.1500.250.0.3500.450.20.560.380.67x30.90.950.680.850.450.750.230.6500.5500.4400.33y23.33.23.1x10.4200.2300.040x20.580.770.770.870.960.97x300.2300.1400.03The solution of the intersection point between the equation and the triangular coordinate axis is shown.According to the coordinates of the equationy2 solution, a contour map corresponding to the equation was drawn in triangular coordinates using Origin 9.0 drawing software (Table 10, Figure 5).Figure 5
Isogram of peak displacementy2 regression equation.Parametersx1, x2, and x3 represented the admixtures of cement, gravel, and raw soil, respectively (Figure 5). The gradient of the isoline decreased from x3 point to the x1x2 coordinate axis step by step. The average line of peak displacement was located on the middle line of the coordinate triangle, which showed that the raw soil content of these three admixtures played a major role in peak displacement. With increased cement content, the hydration cementation of the cement was enhanced, which enhanced cohesive forces between soil particles and the peak displacement of the material decreased. The mean line was on the middle line of the coordinate triangle, which showed that the influences of gravel and cement on the peak displacement of specimens were basically the same. If only the peak displacement index was considered, the area of the siege with y2 = 4 and x3 vertices was the optimal ratio area. From Figure 5, 0≤x1≤0.1, 0≤x2≤0.05, and 0.9≤x3≤1were calculated from Section 3.1, with 0.03≤z1≤0.05,0.05≤z2≤0.06, and 0.9≤z3≤0.92, and thus the optimal formula for cement, gravel, and raw soil was 0.04/0.06/0.9 (mass ratio, resp.).(3) The regression analysis of response variabley3 was carried out with reference to Table 6. The calculation was simplified by examining the first-order polynomial for the regression model y^=fx=ax1+bx2+cx3. The regression results showed the coefficient of determination R2 = 0.8387 by linear regression (Table 13). According to the F-test, the linear regression equation was significant and the primary regression model concluded to be scientific and reasonable. The polynomial coefficients calculated using MATLAB are shown in Table 14. The corresponding regression equation was y3=1.51x1+1.04x2+0.6x3.Table 13
Regression analysis of the standard deviation of raw soil mixed with cement and gravel.
Source parameterdfSSMSFp>FRegression3SSR3.28MSR1.09312.14RemarkableDeviation7SSe0.63MSe0.09Uncorrected summation10SST3.91R20.8387Table 14
Estimation of the partial regression coefficient ofy3 regression equation for the standard deviation of mixed cement and gravel modified with raw soil.
Regression termPartial regression estimationt-testp>tRegression error of partial regression estimationx11.514.360.00030.347x21.042.980.0020.346x30.61.720.01290.346Combining regression equationy3, solving the equationymax  = 2.20, ymin  = 0.447, range Δ  = 1.755, y3¯  = 1.04, gradient ∇  = 0.1, using Mathcad software, the results of equation y3 are shown in Table 15.Table 15
Standard deviationy3 regression equation calculations.
y21.51.31.10.90.7x10.980.970.770.550.550.980.970.770.550.55x200.0300.45000.0300.450x30.0200.2300.450.0200.2300.45The solution of the intersection point between the equation and the triangular coordinate axis is shown.According to the coordinates of the equation solution in Table15, a contour map corresponding to the equation was drawn in triangular coordinates using Origin 9.0 drawing software (Figure 6).Figure 6
Isogram of standard deviationy3 regression equation.Parametersx1, x2, and x3 represented the admixtures of cement, gravel, and raw soil, respectively (Figure 6). The gradient of the isoline decreased from x1 point to the x2x3 coordinate axis step by step, which showed that the cement content of these three admixtures played a major role in the standard deviation. If only the standard deviation index was considered, the area of the siege with y3 = 7 and x3 vertices was the optimal ratio area. From Figure 6, 0≤x1≤0.11, 0≤x2≤0.23, and 0.77≤x3≤1 were calculated from Section 3.1, with 0.03≤z1≤0.05, 0.05≤z2≤0.09, and 0.88≤z3≤0.92, and thus the optimal formula of cement, gravel, and raw soil was 0.04/0.07/0.89 (mass ratio, resp.).(4) The optimal formulation should satisfy the three reaction variables at the same time. The strength of specimens also showed the material load resistance. The peak displacement of the specimen reflected the material deformation performance. The standard deviation of a test reflected the stability and discreteness of the material internal structure. Thus, a superposition of Figures4, 5, and 6 was studied. The optimal formulation area is shown in Figure 7. Here, y3 = 1.1, y2 = 3.6, and y1 = 8 in the shadow of the siege; solution of the equations 0.33≤x1≤0.55, 0≤x2≤0.28, and0.3≤x3≤0.67 were calculated from Section 3.1, with 0.08≤z1≤0.12 and 0.8≤z3≤0.86. Thus, the optimal formula for the cement, gravel, and raw soil was 0.1/0.08/0.82 (mass ratio, resp.).Figure 7
Optimal formulation region for synthesizing the three reaction variablesy1, y2, and y3.


## 4.1. Compressive Strength Analysis

The average value of protest strength of each group of specimens is taken as the compressive strength of the test samples. The experimental results are shown in Table5.Table 5
Test results of compressive strength of specimens mixed with cement, gravel, and raw soil.
Test formationPeak load (kN)Peak displacement (mm)Compressive strengthAverage of strength (MPa)Standard deviationCoefficient of variationNS1115.123.5711.511.2900.112NS237.522.973.750.4470.119NS357.754.115.770.4920.085NS479.733.337.972.2020.276NS543.923.434.391.1080.252NS688.743.818.870.9710.109NS7111.023.5511.101.0670.096NS868.123.586.811.1780.173NS972.063.707.211.0050.139NS1069.153.706.910.7340.106Raw soil32.422.73.240.3930.121Referring to Table5, the compressive strength of the modified raw soil with gravel and cement is significantly higher than that of the raw soil. The compressive strength of the modified specimen is 1.16–3.55 times that of the raw soil specimen.Compared with NS3, NS5, and NS2 group, when the cement content is 3% and the gravel content is 5%, 13.5%, and 22%, the compressive strength of the specimen is 5.77 MPa, 4.39 MPa, and 3.75 MPa. The results show that when the cement content is fixed, the strength of the specimen decreases with the increase of the gravel content.Compared with NS3, NS6, and NS1 group, when the gravel content is 5% and the cement content is 3%, 11.5%, and 20%, the compressive strength of the specimen is 5.77 MPa, 8.87 MPa, and 11.51 MPa. The results show that when the gravel content is fixed, the strength of the specimen increases with the increase of the cement content.According to a large number of research results, the compressive strength of raw soil-based materials with gravel and cement is higher than that with gravel or cement alone [参考文献]. The strength of gravel is obviously higher than that of soil. However, when the gravel is mixed with the raw soil alone, the bond performance between the gravel and the soil is poor, which leads to the reduction of the time compression performance. As a cementitious material, cement can react with water molecules in the soil to consolidate soil particles and fill the pores in the soil, which improves the particle size distribution and increases the material density. The gravel is wrapped by cement soil and closely bonded, acting as coarse aggregate, playing a skeleton role, and resisting deformation when the material is under pressure, and the material strength is significantly improved.


## 4.2. Formula Discussion

The raw soil-modified cubic specimens consisted of sand (z1), gravel (z2), and raw soil (z3, Table 1). There was more than one reaction variable y in the formulation of specimens considering the test results. There were three parameters that reflected the advantages and disadvantages of the modified specimens. The first response variable was the cubic compressive strength (y1), the second response variable was the cubic peak displacement (y2), and the third response variable was the cubic Standard deviation (y3). Integrating Tables 1 and 3 yielded Table 6.Table 6
Axis point extended simplex lattice design and the result of lower limit component boundary in mixed cement and gravel modified with raw soil.
Test formationTest codingMix ratio of test materialsResponse variablesx1x2x3z1z2z3y1y2y3NS11000.200.050.7511.513.571.290NS20100.030.220.753.752.970.447NS30010.030.050.925.774.110.492NS41/21/200.120.140.757.973.332.202NS501/21/20.030.140.844.393.431.108NS61/201/20.120.050.848.873.810.971NS72/31/61/60.140.080.7811.103.551.067NS81/62/31/60.060.160.786.813.581.178NS91/61/62/30.060.080.867.213.701.005NS101/31/31/30.090.110.816.913.700.734(1) Regression analysis of response variabley1 was carried out with reference to Table 4. The calculation was simplified by examining the first-order polynomial for the regression model y^=fx=ax1+bx2+cx3 (Table 7), which showed that the coefficient of determination was R2 = 0.9249 by linear regression. According to the F-test, the linear regression equation was significant and the primary regression model concluded to be scientific and reasonable. The polynomial coefficients calculated by MATLAB are shown in Table 8. The corresponding regression equation was y1=12.23x1+4.13x2+5.92x3.Table 7
Regression analysis of compressive strength of raw soil mixed with cement and gravel.
Source parameterdfSSMSFp > FRegression3SSR54.31MSR18.10428.73RemarkableDeviation7SSe4.41MSe0.031Uncorrected summation10SST58.72R20.9249Table 8
Estimation of the partial regression coefficient ofy1 regression equation for compressive strength of mixed cement and gravel modified with raw soil.
Regression termPartial regression estimationt-testp>tRegression error of partial regression estimationx112.2321.340.00010.573x24.137.220.00010.571x35.9210.320.00010.571Combining regression equationy1, solving the equationymax  = 11.51, ymin  = 3.75，range Δ  = 7.76, y1¯  = 7.43, gradient ∇  = 1, using Mathcad software, the results of equation y1 are shown in Table 9.Table 9
Compressive strengthy1 regression equation calculations.
y1111098765x10.850.80.730.650.60.490.480.330.350.170.230.020.10x20.1500.2700.400.5200.6500.7700.90.51x300.200.3500.5100.6700.8300.9800.49The solution of the intersection point between the equation and the triangular coordinate axis is shown.According to the coordinates of the equation solution in Table9, a contour map corresponding to the equation was drawn in triangular coordinates using Origin 9.0 drawing software (Figure 4).Figure 4
Isogram of compressive strengthy1 regression equation.Parametersx1, x2, and x3 represented the admixtures of cement, gravel, and raw soil, respectively (Figure 4). The gradient of the isoline decreased from x1 point to the x2x3 coordinate axis step by step. The isoline basically deviated to the x1x2 coordinate axis. The average strength line fell on the x1x2 and x1x3 axes, which showed that the cement content of these three admixtures played a major role in compressive strength. The strength of gravel was higher than that of raw soil. Gravel particles here were mainly composed of quartz and the surfaces well encapsulated, adsorbed, and bonded with raw soil. However, when the gravel content increased, the gravel occupied a large amount of internal space, such that the soil particles did not bond well. With increased cement content, the hydration cementation of the cement was enhanced, which enhanced the cohesive force between soil particles and the material strength clearly increased. If only the compressive strength index was considered, the area of the enclosure area with y1 = 11 and x1 vertices was the optimal ratio area. From Figure 4, 0.8≤x1≤1, 0≤x2≤0.15, and 0≤x3≤0.2 were calculated from Section 3.1, with 0.17≤z1≤0.2, 0.05≤z2≤0.08, and 0.75≤z3≤0.78, and thus the optimal formula of cement, gravel, and raw soil was 0.18/0.06/0.76 (mass ratio, resp.).(2) The regression analysis of response variabley2 was carried out with reference to Table 6. The calculation was simplified by examining the first-order polynomial for the regression model y^=fx=ax1+bx2+cx3. The regression results showed the coefficient of determination at R2 = 0.9441 by linear regression (Table 10). According to the F-test, the linear regression equation was significant and the primary regression model concluded to be scientific and reasonable. The polynomial coefficients calculated using MATLAB are shown in Table 11. The corresponding regression equation was y2=3.6x1+3.08x2+4.04x3.Table 10
Regression analysis of peak displacement of raw soil mixed with cement and gravel.
Source parameterdfSSMSFp>FRegression3SSR0.94MSR0.313328.73RemarkableDeviation7SSe0.12MSe0.0170Uncorrected summation10SST1.06R20.9441Table 11
Estimation of the partial regression coefficient ofy2 regression equation for peak displacement of mixed cement and gravel modified with raw soil.
Regression termPartial regression estimationt-testp>tRegression error of partial regression estimationx13.636.850.00010.098x23.0831.570.00010.097x34.0441.380.00010.097Combining regression equationy2, solving the equation ymax  = 4.11, ymin  = 2.97, range Δ  = 1.14, y2¯  = 3.57, gradient ∇  = 0.1, using Mathcad software, the results of equation y2 are shown in Table 12.Table 12
Peak displacementy2 regression equation calculations.
y243.93.83.73.63.53.4x10.100.3200.5500.770100.800.620x200.0500.1500.250.0.3500.450.20.560.380.67x30.90.950.680.850.450.750.230.6500.5500.4400.33y23.33.23.1x10.4200.2300.040x20.580.770.770.870.960.97x300.2300.1400.03The solution of the intersection point between the equation and the triangular coordinate axis is shown.According to the coordinates of the equationy2 solution, a contour map corresponding to the equation was drawn in triangular coordinates using Origin 9.0 drawing software (Table 10, Figure 5).Figure 5
Isogram of peak displacementy2 regression equation.Parametersx1, x2, and x3 represented the admixtures of cement, gravel, and raw soil, respectively (Figure 5). The gradient of the isoline decreased from x3 point to the x1x2 coordinate axis step by step. The average line of peak displacement was located on the middle line of the coordinate triangle, which showed that the raw soil content of these three admixtures played a major role in peak displacement. With increased cement content, the hydration cementation of the cement was enhanced, which enhanced cohesive forces between soil particles and the peak displacement of the material decreased. The mean line was on the middle line of the coordinate triangle, which showed that the influences of gravel and cement on the peak displacement of specimens were basically the same. If only the peak displacement index was considered, the area of the siege with y2 = 4 and x3 vertices was the optimal ratio area. From Figure 5, 0≤x1≤0.1, 0≤x2≤0.05, and 0.9≤x3≤1were calculated from Section 3.1, with 0.03≤z1≤0.05,0.05≤z2≤0.06, and 0.9≤z3≤0.92, and thus the optimal formula for cement, gravel, and raw soil was 0.04/0.06/0.9 (mass ratio, resp.).(3) The regression analysis of response variabley3 was carried out with reference to Table 6. The calculation was simplified by examining the first-order polynomial for the regression model y^=fx=ax1+bx2+cx3. The regression results showed the coefficient of determination R2 = 0.8387 by linear regression (Table 13). According to the F-test, the linear regression equation was significant and the primary regression model concluded to be scientific and reasonable. The polynomial coefficients calculated using MATLAB are shown in Table 14. The corresponding regression equation was y3=1.51x1+1.04x2+0.6x3.Table 13
Regression analysis of the standard deviation of raw soil mixed with cement and gravel.
Source parameterdfSSMSFp>FRegression3SSR3.28MSR1.09312.14RemarkableDeviation7SSe0.63MSe0.09Uncorrected summation10SST3.91R20.8387Table 14
Estimation of the partial regression coefficient ofy3 regression equation for the standard deviation of mixed cement and gravel modified with raw soil.
Regression termPartial regression estimationt-testp>tRegression error of partial regression estimationx11.514.360.00030.347x21.042.980.0020.346x30.61.720.01290.346Combining regression equationy3, solving the equationymax  = 2.20, ymin  = 0.447, range Δ  = 1.755, y3¯  = 1.04, gradient ∇  = 0.1, using Mathcad software, the results of equation y3 are shown in Table 15.Table 15
Standard deviationy3 regression equation calculations.
y21.51.31.10.90.7x10.980.970.770.550.550.980.970.770.550.55x200.0300.45000.0300.450x30.0200.2300.450.0200.2300.45The solution of the intersection point between the equation and the triangular coordinate axis is shown.According to the coordinates of the equation solution in Table15, a contour map corresponding to the equation was drawn in triangular coordinates using Origin 9.0 drawing software (Figure 6).Figure 6
Isogram of standard deviationy3 regression equation.Parametersx1, x2, and x3 represented the admixtures of cement, gravel, and raw soil, respectively (Figure 6). The gradient of the isoline decreased from x1 point to the x2x3 coordinate axis step by step, which showed that the cement content of these three admixtures played a major role in the standard deviation. If only the standard deviation index was considered, the area of the siege with y3 = 7 and x3 vertices was the optimal ratio area. From Figure 6, 0≤x1≤0.11, 0≤x2≤0.23, and 0.77≤x3≤1 were calculated from Section 3.1, with 0.03≤z1≤0.05, 0.05≤z2≤0.09, and 0.88≤z3≤0.92, and thus the optimal formula of cement, gravel, and raw soil was 0.04/0.07/0.89 (mass ratio, resp.).(4) The optimal formulation should satisfy the three reaction variables at the same time. The strength of specimens also showed the material load resistance. The peak displacement of the specimen reflected the material deformation performance. The standard deviation of a test reflected the stability and discreteness of the material internal structure. Thus, a superposition of Figures4, 5, and 6 was studied. The optimal formulation area is shown in Figure 7. Here, y3 = 1.1, y2 = 3.6, and y1 = 8 in the shadow of the siege; solution of the equations 0.33≤x1≤0.55, 0≤x2≤0.28, and0.3≤x3≤0.67 were calculated from Section 3.1, with 0.08≤z1≤0.12 and 0.8≤z3≤0.86. Thus, the optimal formula for the cement, gravel, and raw soil was 0.1/0.08/0.82 (mass ratio, resp.).Figure 7
Optimal formulation region for synthesizing the three reaction variablesy1, y2, and y3.


## 5. Conclusion

The compressive strength of raw soil modified by cement and gravel is significantly higher than that of plain soil. The compressive strength is 1.16–3.55 times that of raw soil. If the cement content is fixed, the compressive strength of the specimen decreases with the increase of the stone content. If the stone content is fixed, the compressive strength increases with the increase of cement content.In this study, the single lattice model formulation design method was introduced to effectively improve and control raw soil modification formulation tests affected by many factors. Through regression analysis, the isoline equation image solution was best for verifying this method, which showed that this method was feasible for the test study of raw soil modification. Using the combination of MATLAB modeling software, Mathcad calculation software, and Origin drawing software, programming calculations were performed to improve the efficiency of mathematical calculation.According to the equation solutions, a contour map of the triangular coordinate system was drawn, which effectively reflected the influence of the admixture on the results. The content of cement and gravel had a significant influence on the compressive strength and standard deviation of specimens. The raw soil content only affected peak displacement.Considering the compressive strength, peak displacement, and standard deviation, the optimal mass ratio of gravel, cement, and raw soil was 0.1/0.08/0.82 (mass ratio, resp.).The formulation model provided the method and basis for the standard method of raw soil mixed modification testing.



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*Source: 8136570-2020-12-21.xml*