# Influence Analysis and Stepwise Regression of Coal Mechanical Parameters on Uniaxial Compressive Strength Based on Orthogonal Testing Method

^{1}

^{2}

^{*}

## Abstract

**:**

_{m}) and dilation angle (Ψ) as initial participation factors. Using these six parameters and a five-level combination scheme (L

_{25}(5

^{6})), the influence of coal mechanical parameters on UCS and PS was investigated, using the software SPSS for stepwise regression analysis, and a uniaxial pressure-resistant regression prediction equation was established. The research showed that, under uniaxial compression conditions, the main parameters controlling UCS of coal masses are C and Φ; conversely, the main parameters controlling PS are E and C. UCS and PS exhibit significant linear relationships with these main controlling parameters. Here, a stepwise regression prediction equation was established through reliability verification analysis using the main controlling parameters. This prediction method produces very small errors and a good degree of fit, thus allowing the rapid prediction of UCS. The precision of the stepwise regression model depends on the number of test samples, which can be increased in the later stages of a design project to further improve the precision of the projection model.

## 1. Introduction

_{m}and Ψ. The mechanical parameters of coal masses exhibit considerable spatial variability owing to the effects of coal-forming geological conditions and structural geology. Diversity in these mechanical parameters leads to large variation in UCS and PS, which in turn has a considerable effect on the mechanical response characteristics of coal, including rock burst strength and the degree of deformation of the rock mass surrounding roadways [6,7,8,9,10,11,12]. Thus, the influence of mechanical parameters on UCS and PS has been researched and the main parameters controlling the characteristics of coal mass strength have been determined; accordingly, the rapid and accurate prediction of UCS of coal masses based on mechanical parameters has been made possible. Such knowledge is of critical importance for the prevention and control of rock failure.

_{m}and Ψ. Using these six factors and a five-level combination scheme (L

_{25}(5

^{6})), the influence of coal mechanical parameters on the uniaxial compression test and peak strain was investigated, with FLAC3D finite difference software used to simulate each scheme and determine the main control factors affecting the UCS and PS. SPSS software was used to establish a stepwise regression model for predicting UCS to obtain a regression equation for predicting UCS when adopting the main control factors and verify the reliability. To realize the above research contents, mechanical parameters such as UCS, C, Φ and υ of coal samples were obtained through mechanical tests, so as to facilitate the orthogonal numerical simulation test. Then, the FLAC3D simulation software was used to establish the numerical model. Based on the study of the mesh generation, constitutive model and loading rate, the numerical simulation conditions for the mechanical properties of coal samples were established. At the same time, according to the principle of orthogonal experiment, the horizontal spacing of six factors and five levels was determined, the horizontal parameters were listed, and the numerical simulation scheme of orthogonal experiment was determined. Finally, according to the numerical simulation results, the influence degree of each mechanical parameter on peak strength and peak strain was analyzed, in order to determine the main control factors. The stepwise regression prediction model of UCS was established by SPSS software, and the reliability of the model was verified. The specific technical roadmap is shown in Figure 1.

## 2. Orthogonal Numerical Simulation Experiments

#### 2.1. Production of Coal Samples

^{3}.

_{m}, as shown in Figure 2.

#### 2.2. Mechanical Testing and Parameter Acquisition for Coal Samples

#### 2.2.1. Introduction of RMT-150B Servo Tester

#### 2.2.2. Mechanical Parameter Acquisition

_{1}> σ

_{2}= σ

_{3}, and the confining pressures were 2, 5, 10, 15 and 20 MPa. Displacement control was adopted for the triaxial testing. First, confining pressure was added in a static and horizontal manner, with a loading rate of confining pressure of 0.1 MPa/s. Then, axial pressure was added until the predetermined confining pressure value was reached, and the axial loading rate was 0.005 mm/s. The mechanical parameters and the stress–strain curve of the coal sample were automatically recorded by the computer, as shown in Figure 6.

#### 2.3. Numerical Simulation Test

#### 2.3.1. Establishment of Numerical Model

#### 2.3.2. Constitutive Model Selection

_{m}may soften after the onset of plastic yield. In the Mohr–Coulomb model, those properties are assumed to remain constant [39]. The yield function, flow rule and stress correction of the strain softening model are consistent with those of the Mohr–Coulomb model [42].

_{1}and σ

_{3}represent the maximum and minimum principal stresses, respectively; C is cohesion; φ is the internal friction angle; and N

_{φ}is defined as follows.

_{2}is the intermediate principal stress; σ

_{1}and σ

_{3}represent the maximum and minimum principal stresses, respectively; α

_{1}and α

_{2}are material constants defined in terms of the shear modulus and the bulk modulus, respectively; λ

^{t}is the undetermined plasticity coefficient; superscripts N and I represent the new and old stress states of the element, respectively; Ψ is dilation angle; and N

_{ψ}is defined as follows:

_{1}, η

_{2}, …, η

_{n−1}are the softening parameter at the end of stages 1, 2, …, n − 1; and η* is the softening parameter when the discharge reaches the residual state.

#### 2.3.3. Model Shape Selection

^{−5}mm/step. The stress–strain curve of the numerical model is shown in Figure 8.

#### 2.3.4. Mesh Number of Model Element Body Is Determined

#### 2.3.5. Loading Rate Determination

^{−4}, 1.9 × 10

^{−5}, 1.9 × 10

^{−6}and 1.9 × 10

^{−7}mm/step, and the remaining numerical simulation conditions remained consistent. The stress–strain curves of the numerical models with different loading rate conditions were recorded, as shown in Figure 10.

^{−4}mm/step, the yield strength is 27.3 MPa. Compared with the mechanical test results in Table 1, the deviation rate is 35.82%. When the loading rate V = 1.9 × 10

^{−5}mm/step, the yield strength is 24.4 MPa and the deviation rate is 21.39%. When the loading rate V = 1.9 × 10

^{−6}mm/step, the yield strength is 23.8 MPa and the deviation rate is 18.41%. When the loading rate V = 1.9 × 10

^{−7}mm/step, the yield strength is 24.0 MPa and the deviation rate is 19.4%. Therefore, when the loading rate was low, the yield strength tended to be stable. When the loading rate was more than 1.9 × 10

^{−5}mm/step, the yield strength of the numerical model rapidly increased with the loading rate. Due to the low loading rate, the load of the model was similar to the static load, and the internal damage and plastic damage of the model had sufficient time to develop, thus the yield strength was low and the difference was small. With an increase in loading rate, the plastic failure time decreased, the degree of damage in the model decreased and the strength continuously increased [44]. From the mechanical performance curve, when the loading rate V = 1.9 × 10

^{−4}mm/step, after the model reaches the yield strength, the axial stress gradually decreases with the increase of axial strain. When the loading speeds were V = 1.9 × 10

^{−5}and 1.9 × 10

^{−7}mm/step, when the model reached the yield strength, as the strain increased, the stress decreased, slowly at first and then rapidly. When the loading speed was V = 1.9 × 10

^{−6}mm/step, the axial stress rapidly decreased and the brittleness increased after the model reached the yield strength. Compared with the stress–strain curve of the coal sample in Figure 6, the stress–strain curve and the mechanical properties of the loading rate V = 1.9 × 10

^{−5}and 1.9 × 10

^{−7}mm/step were more consistent with the mechanical test results of the coal sample. However, when the loading rate was V = 1.9 × 10

^{−7}mm/step, the simulation time of a single model was significantly longer than for a rate of 1.9 × 10

^{−5}mm/step. Based on the comparative analysis of the four loading rates, the selected loading rate of orthogonal numerical simulation test is 1.9 × 10

^{−5}mm/step.

#### 2.4. Orthogonal Experimental Method and Experimental Scheme

#### 2.4.1. Orthogonal Experimental Method

#### 2.4.2. Orthogonal Experimental Research Level and Scheme Design

_{m}and Ψ) measured in Table 1 were taken as the initial participating factors of the orthogonal experiment. Based on the results of previous studies considering the horizontal spacing of different parameters in orthogonal experiments [47,48], it was determined that each participating parameter in this experiment has 5 typical levels, with horizontal spacings of 1 GPa, 0.02, 2 MPa, 2°, 0.5 MPa and 1° for E, υ, C, Φ, R

_{m}and Ψ, respectively. The specific parameters adopted are presented in Table 3.

_{25}(5

^{6})) was selected. The primary purpose of this experiment was to study the influence of the selected mechanical parameters on UCS and PS. The main controlling parameters were determined; UCS was monitored and PS was recorded for different parameters and multiple levels in the numerical simulation test, as shown in Table 4.

## 3. Effect of Mechanical Parameters on UCS and PS

#### 3.1. Influence Analysis of Mechanical Parameters on UCS

_{m}, Ψ and Φ are 1.44, 1.88, 1.30, 2.24 and 4.20, respectively. In contrast, the range of C is 29.02. Thus, these results show that C has the greatest influence on UCS, followed by Φ, while E, υ, R

_{m}and Ψ have relatively little influence on UCS.

- (1)
- When E increases from 2.81 to 6.81 GPa, the change in UCS is minimal and the stress value fluctuates within the range 36~37 MPa. Therefore, the influence of the elastic modulus on UCS is small. According to the smaller range between axis limits, a nonlinear relationship exists between E and UCS. With increasing E, compressive strength first decreases before increasing and then decreasing again; however, the fluctuation range is small, indicating that there is a relatively stable critical range of UCS for different values of E, as shown in Figure 12a.
- (2)
- With increasing υ, UCS exhibits a fluctuating curve. In the early stage, when υ increases from 0.30 to 0.36, UCS exhibits a decrease followed by an increase and a subsequent decrease, although the maximum change range is only 0.8 MPa. Conversely, in the later stage, when υ increases from 0.36 to 0.38, the compressive strength increases from 36.08 to 37.96 MPa; this increase of 1.88 MPa is relatively pronounced. Thus, when υ is small, changes in UCS are comparatively small and UCS is not sensitive to changes in Poisson’s ratio; conversely, when υ is comparatively large, the sensitivity of UCS to changes in υ increases, as shown in Figure 12b.
- (3)
- C has a considerable influence on UCS for the coal samples, with UCS ranging from 21.94 to 50.96 MPa for different C levels. Based on curve for this specific analysis, the fitting equation between C and UCS is y = 3.628x + 0.8794, with R
^{2}= 0.9973. There is a clear linear relationship between C and UCS, with a range of 29.02. Thus, UCS is sensitive to changes in C, increasing linearly. The results also illustrate that UCS does not converge to a critical range with variation in C, as shown in Figure 12c. - (4)
- The influence of Φ on UCS can be divided into two stages: when Φ is less than 32°, UCS remains essentially unchanged with increasing Φ; conversely, when Φ is greater than 32°, UCS increases rapidly with increasing Φ and there is a significant linear growth relationship between the two variables. Thus, Φ has a significant impact on UCS and the degree of its influence is relatively large. Based on a comprehensive consideration of the whole curve, a nonlinear relationship exists between UCS and Φ: when Φ exceeds 32°, UCS increases linearly with increasing Φ and UCS does not have a stable critical range, as shown in Figure 12d.
- (5)
- Based on the relationship between R
_{m}and UCS, changes in the range of UCS with increasing Rm are very small, with minimum and maximum values of 36.2 and 37.5 MPa, respectively (a range of only 1.3 MPa). Therefore, Rm has little influence on UCS. There is a significant nonlinear relationship between the two variables according to the microscopic analysis diagram, as shown in Figure 12e. - (6)
- The relationship between Ψ and UCS indicates that, when Ψ is less than 10°, UCS first increases from 36.42 to 38.12 MPa and then decreases to 35.88 MPa with increasing Ψ (a range of 2.24 MPa). Conversely, when Ψ is greater than 10°, UCS increases from 35.88 to 36.74 MPa and then decreases to 36.46 MPa (a range of 0.86 MPa). This indicates that a nonlinear relationship exists between Ψ and UCS, with the influence of Ψ on UCS being more pronounced for small values of Ψ, as shown in Figure 12f.

#### 3.2. Influence Analysis of Mechanical Parameters on PS

_{m}and Ψ are relatively small, and their influences on PS are also small. To more intuitively characterize the influence of various mechanical parameters on PS, curves were plotted to illustrate the relationships between these parameters and PS and the ordinates of all curves were unified with the same max and min values on the y-axis. Smaller ranges between axis limits were produced for the parameters with lower sensitivity, as shown in Figure 14.

- (1)
- Based on the curve in Figure 14a, the relationship between E and PS is approximately linear. When the other parameters remain constant, PS decreases with increasing E, which is closely related to the deformation resistance of the object as characterized by E. For large values of E, the deformation resistance is greater and the deformation is less pronounced; conversely, for smaller values of E, the deformation resistance is lesser and the deformation is more pronounced.
- (2)
- PS increases slowly with increasing υ from 0.30 to 0.38; thus, the overall change in PS is small and the sensitivity of PS to υ can be considered relatively small. A smaller range between axis limits indicates that PS growth with increasing Poisson’s ratio occurs in two stages: in the first stage, υ increases from 0.3 to 0.34 and the PS growth rate is 10.13%; in the second stage, υ increases from 0.34 to 0.38 and the PS growth rate is only 2.12%. The growth rate of PS in the second stage is significantly lower than that in the first stage, although a relatively significant linear relationship exists between υ and PS in each stage, as shown in Figure 14b.
- (3)
- According to Figure 14c, PS increases linearly from 2.53 to 5.68 (range of 3.15) with increasing C. Other than E, C can be considered to have the largest influence on PS.
- (4)
- The influence of Φ on PS can also be considered to exhibit two stages. When Φ is less than 34°, the fluctuation range of PS is very small with increasing Φ and Φ thus has no effect on PS. Conversely, when Φ is greater than 34°, PS increases linearly with increasing Φ and the degree of influence of Φ on PS is greater, as shown in Figure 14d.
- (5)
- The influence of R
_{m}on PS is variable, with no obvious regularity; the range is only 0.49 and R_{m}has little influence on PS. However, PS tends broadly to increase with increasing R_{m}, as shown in Figure 14e. - (6)
- Based on the curve plotting PS against Ψ, PS increases from 3.91 to 4.37 as Ψ increases from 8° to 12°. This fluctuation range is small, indicating that the influence of Ψ on PS is small. Although the main plot indicates a lack of any clear linear relationship between Ψ and PS, the microscopic analysis diagram indicates that PS increases linearly with increasing Ψ before leveling off, and then exhibiting a further linear increase with Ψ at higher values of Ψ, as shown in Figure 14f.

#### 3.3. Influence Analysis of Mechanical Parameters on Critical Failure Strength of Coal Samples

_{m}. Conversely, the degree of influence of these parameters on PS decreases in the following order: E > C > Φ > R

_{m}> υ >Ψ. Thus, these mechanical parameters have different influences on UCS and PS. To more fully reveal the influence of these parameters on UCS and PS, a normalization method was used to map all range data presented in Table 5 and Table 6 onto the range 0~1, as shown in Table 7. The normalization method is to scale the data that need to be processed to a small specific interval. To facilitate data processing, the data were mapped to a range of 0~1 for processing. The normalization results are plotted as a histogram for processing and comparison, as shown in Figure 16.

## 4. Stepwise Regression Prediction of UCS

#### 4.1. Stepwise Regression Equation Analysis of Relevant Explanatory Variables and Regression Model Establishment

^{2}indicates the explanatory power of the independent variable with respect to the dependent variable; that is, it demonstrates the goodness of fit of the regression equation. The closer R

^{2}is to 1, the more accurate is the model and the better is the fit of the regression equation. For this regression analysis, R

^{2}= 0.983, which demonstrates that C and Φ can accurately represent changes in UCS. The adjusted R

^{2}excludes the influence of increases in the independent variable on the goodness of fit of the model. The model was checked using the F value test (F = 619.574, p < 0.05), which indicated that the regression model is effective: in the table, the * symbol against the F value indicates that at least two independent variables have an impact on the dependent variable. The basic parameters of the model thus meet the relevant standards, based on the analysis of related parameters considered in the stepwise regression.

- (1)
- Multicollinearity means that the correlation between explanatory variables (independent variables) in the regression model is too high to be estimated or predicted by the model. If there is a linear relationship between independent variables, the reliability of the regression parameters will be affected. The VIF value, also known as the variance expansion coefficient, can be used to measure the collinear severity of a regression model. Table 8 indicates that the VIF value was 1 for both C and Φ; this is far less than the standard value of 10 required to determine the collinearity of the model. Therefore, the mechanical parameters of the orthogonal experiment were found to be independent of each other, the model did not have multiple linear problems and the model was well constructed.
- (2)
- Autocorrelation refers to correlation between the expected values of independent variables that have no significant influence on the dependent variables, which is determined by the D–W value. If the D–W value is near 2 (specifically 1.7–2.3), the model is well constructed. The D–W value of this model is 1.829 and the model construction was demonstrated to be reasonable as there was no autocorrelation among the independent variables that had no significant influence on the dependent variables.
- (3)
- The residual term represents the difference between the observed value of each sample and the value estimated by the model. In general, the normal distribution is used to test the residual term: if the test residual data conform to the normal distribution, the model can be considered to be well constructed. Residual normality is used to test the reliability and periodicity of data based on experimental sample data. The analysis of residual normality is only one of the reliability test indices to test the stepwise regression model. To judge whether the regression model conforms to the standard, only the residual items need to meet the normal distribution intuitively. The residual values of regression analysis data in this paper conform to the normal distribution, indicating that the stepwise regression model is reasonable, as shown in Figure 19.
- (4)
- The heteroscedasticity was validated by plotting scatter diagram of the independent and dependent variables with the residual term. The heteroscedasticity can be compared to the variance: it is the difference in variance between the explaining variable that omitted from the model and the unimportant explained variable. Accordingly, to determine the heteroscedasticity, the data can be examined for signs of regularity in the corresponding scattered points. If the residual is notably increased or decreased with any increase in the independent variable, the regularity is obvious; then, the model has heteroscedasticity and its construction can be considered poor. All of the independent and dependent variables considered here were plotted in a scatter diagram, as shown in Figure 20.

_{0}+ β

_{1}x

_{1}+ β

_{2}x

_{2}

_{1}is C (MPa), x

_{2}is Φ (in degrees) and the coefficients β

_{1}and β

_{2}indicate the degree of influence of the main controlling parameters on UCS.

_{1}+ 0.557x

_{2}− 18.059

#### 4.2. Reliability Analysis of Stepwise Regression Equation of Coal Compressive Strength

_{c}= 11.31 + 4.19x

_{1}− 0.017x

_{1}

^{2}, where x

_{1}is the elastic modulus) for predicting the UCS of the 11-2 coal roof rock in the Huainan mining area was compared and analyzed using the method previously described in [22]. The calculation results are shown in Table 9.

_{down}coal seam of Dongtan coal mine. However, the error of the quadratic regression prediction results was large: the minimum error was 12.26%, and the maximum error was as high as 162.20%. It can be seen, therefore, that a secondary regression model of the compressive strength of roof strata is only suitable for predicting the UCS of coal seams with small elastic modulus, and it has large errors and poor universality. However, the stepwise regression prediction equation established in this study had the advantages of small errors, high accuracy and good universality.

## 5. Conclusions

- (1)
- The degree of influence of mechanical parameters on UCS decreases in the following order: C > Φ > Ψ > υ > E > R
_{m}. Thus, of these parameters, C has the greatest influence, followed by Φ. The other mechanical parameters considered have little influence on UCS for coal samples, and their relationships with UCS exhibit nonlinear characteristics. Thus, the main parameters controlling UCS are C and Φ. - (2)
- Different mechanical parameters have different degrees of influence on PS, with degree of influence decreasing in the following order: E > C > Φ > R
_{m}> υ > Ψ. Thus, E has the greatest influence on PS (negative linear relationship), followed by C (positive linear relationship). The other mechanical parameters considered have little influence on PS, and the main parameters controlling PS are E and C. - (3)
- The degree of influence of mechanical parameters on peak strength has been determined based on an orthogonal simulation experiment, with the mechanical parameters without obvious significance being eliminated by a stepwise regression analysis model. The stepwise regression equation is a mathematical model with C and Φ as independent variables and UCS as a dependent variable, and the reliability of the regression prediction equation was verified. The prediction results have small error, high fitting degree and good adaptability, indicating that the model can realize the prediction of UCS.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**Specimen loading diagram: ① upper bearing plate; ② axial displacement sensor gripper; ③ transverse displacement sensor; ④ transverse displacement sensor mounting seat; ⑤ axial displacement sensor; ⑥ sample; ⑦ sensor mounting plate; ⑧ lower bearing plate; ⑨ bearing seat ⑩ locating pin; ⑪ adjustment plate; ⑫ foundation support; ⑬ baseboard; and ⑭ located block.

Test | E (GPa) | υ | UCS (MPa) | C (MPa) | Φ (°) | Ψ (°) | R_{m} (MPa) |
---|---|---|---|---|---|---|---|

Uniaxial compression test | 3.20 | 0.35 | 20.6 | / | / | / | / |

2.60 | 0.22 | 18.7 | / | / | / | / | |

2.19 | 0.33 | 15.7 | / | / | / | / | |

3.22 | 0.30 | 25.4 | / | / | / | / | |

Brazilian splitting test | / | / | / | / | / | / | 1.23 |

/ | / | / | / | / | / | 0.12 | |

/ | / | / | / | / | / | 0.81 | |

Triaxial test | / | / | / | 10.88 | 40.5 | / | / |

/ | / | / | / | / | |||

/ | / | / | / | / | |||

/ | / | / | / | / | |||

/ | / | / | / | / | |||

Average | 2.81 | 0.30 | 20.1 | 10.88 | 40.5 | / | |

Adjusted parameter | 2.81 | 0.30 | 20.1 | 5.88 | 30 | 8 | 0.72 |

C/MPa | Φ/° | η | |
---|---|---|---|

Cumulative plastic strain 0 | 5.88 | 30 | 0 |

Cumulative plastic strain 0.008 | 5.76 | 28.06 | 0.0133 |

Cumulative plastic strain 0.015 | 1.42 | 26.24 | 0.0250 |

A | B | C | D | E | F | |
---|---|---|---|---|---|---|

E (GPa) | υ | C (MPa) | Φ (°) | R_{m} (MPa) | Ψ (°) | |

1 | 2.81 | 0.30 | 5.88 | 30 | 0.72 | 8 |

2 | 3.81 | 0.32 | 7.88 | 32 | 1.22 | 9 |

3 | 4.81 | 0.34 | 9.88 | 34 | 1.72 | 10 |

4 | 5.81 | 0.36 | 11.88 | 36 | 2.22 | 11 |

5 | 6.81 | 0.38 | 13.88 | 38 | 2.72 | 12 |

Experimental Scheme | Mechanical Parameters | Simulation Results | ||||||
---|---|---|---|---|---|---|---|---|

E (GPa) | υ | C (MPa) | Φ (°) | R_{m} (MPa) | Ψ (°) | UCS (MPa) | PS (10^{−3}) | |

1 | 2.81 | 0.30 | 5.88 | 30 | 0.72 | 8 | 20.1 | 3.61 |

2 | 2.81 | 0.32 | 7.88 | 32 | 1.22 | 9 | 27.9 | 4.88 |

3 | 2.81 | 0.34 | 9.88 | 34 | 1.72 | 10 | 36.1 | 5.99 |

4 | 2.81 | 0.36 | 11.88 | 36 | 2.22 | 11 | 44.9 | 7.65 |

5 | 2.81 | 0.38 | 13.88 | 38 | 2.72 | 12 | 54.4 | 9.06 |

6 | 3.81 | 0.30 | 7.88 | 34 | 2.22 | 12 | 29.2 | 3.78 |

7 | 3.81 | 0.32 | 9.88 | 36 | 2.72 | 8 | 37.7 | 4.95 |

8 | 3.81 | 0.34 | 11.88 | 38 | 0.72 | 9 | 47.0 | 6.06 |

9 | 3.81 | 0.36 | 13.88 | 30 | 1.22 | 10 | 46.9 | 6.01 |

10 | 3.81 | 0.38 | 5.88 | 32 | 1.72 | 11 | 20.9 | 2.78 |

11 | 4.81 | 0.30 | 9.88 | 38 | 1.22 | 11 | 39.5 | 4.14 |

12 | 4.81 | 0.32 | 11.88 | 30 | 1.72 | 12 | 40.6 | 4.23 |

13 | 4.81 | 0.34 | 13.88 | 32 | 2.22 | 8 | 49.0 | 5.03 |

14 | 4.81 | 0.36 | 5.88 | 34 | 2.72 | 9 | 21.9 | 2.34 |

15 | 4.81 | 0.38 | 7.88 | 36 | 0.72 | 10 | 30.2 | 3.15 |

16 | 5.81 | 0.30 | 11.88 | 32 | 2.72 | 10 | 42.3 | 3.75 |

17 | 5.81 | 0.32 | 13.88 | 34 | 0.72 | 11 | 51.2 | 4.36 |

18 | 5.81 | 0.34 | 5.88 | 36 | 1.22 | 12 | 22.9 | 2.09 |

19 | 5.81 | 0.36 | 7.88 | 38 | 1.72 | 8 | 31.5 | 2.75 |

20 | 5.81 | 0.38 | 9.88 | 30 | 2.22 | 9 | 40.5 | 3.47 |

21 | 6.81 | 0.30 | 13.88 | 36 | 1.72 | 9 | 53.3 | 3.95 |

22 | 6.81 | 0.32 | 5.88 | 38 | 2.22 | 10 | 23.9 | 1.85 |

23 | 6.81 | 0.34 | 7.88 | 30 | 2.72 | 11 | 27.2 | 2.05 |

24 | 6.81 | 0.36 | 9.88 | 32 | 0.72 | 12 | 35.2 | 2.67 |

25 | 6.81 | 0.38 | 11.88 | 34 | 1.22 | 8 | 43.8 | 3.20 |

UCS (MPa) | A | B | C | D | E | F |
---|---|---|---|---|---|---|

E (GPa) | υ | C (MPa) | Φ (°) | R_{m} (MPa) | Ψ (°) | |

Average 1 | 36.68 | 36.88 | 21.94 | 35.06 | 36.74 | 36.42 |

Average 2 | 36.34 | 36.26 | 29.20 | 35.06 | 36.20 | 38.12 |

Average 3 | 36.24 | 36.44 | 37.80 | 36.44 | 36.48 | 35.88 |

Average 4 | 37.68 | 36.08 | 43.72 | 37.80 | 37.50 | 36.74 |

Average 5 | 36.68 | 37.96 | 50.96 | 39.26 | 36.70 | 36.46 |

Range | 1.44 | 1.88 | 29.02 | 4.20 | 1.30 | 2.24 |

PS (10^{−3}) | A | B | C | D | E | F |
---|---|---|---|---|---|---|

E (GPa) | υ | C (MPa) | Φ (°) | R_{m} (MPa) | Ψ (°) | |

Average 1 | 6.24 | 3.85 | 2.53 | 3.87 | 3.97 | 3.91 |

Average 2 | 4.77 | 4.05 | 3.32 | 3.82 | 4.06 | 4.14 |

Average 3 | 3.78 | 4.24 | 4.24 | 3.93 | 3.94 | 4.15 |

Average 4 | 3.28 | 4.28 | 4.98 | 4.36 | 4.36 | 4.20 |

Average 5 | 2.74 | 4.33 | 5.68 | 4.78 | 4.43 | 4.37 |

Range | 3.50 | 0.48 | 3.15 | 0.96 | 0.49 | 0.46 |

E | υ | C | Φ | R_{m} | Ψ | |
---|---|---|---|---|---|---|

Range (UCS) | 1.44 | 1.88 | 29.02 | 4.20 | 1.30 | 2.24 |

Range (PS) | 3.50 | 0.48 | 3.15 | 0.96 | 0.49 | 0.46 |

Range normalization (UCS) | 0.036 | 0.047 | 0.724 | 0.105 | 0.032 | 0.056 |

Range normalization (PS) | 0.387 | 0.053 | 0.349 | 0.106 | 0.054 | 0.051 |

Non-Standardized Coefficient | Standardization Coefficient | t | p | VIF | R^{2} | Adjusted R^{2} | F | ||
---|---|---|---|---|---|---|---|---|---|

B | Standard Error | Beta | |||||||

Constant | −18.059 | 3.704 | - | −4.876 | 0.000 ** | - | 0.983 | 0.981 | 619.574 (0.000 **) |

C | 3.628 | 0.104 | 0.980 | 34.79 | 0.000 ** | 1.000 | |||

Φ | 0.557 | 0.104 | 0.150 | 5.342 | 0.000 ** | 1.000 |

Coal Sample | E (GPa) | C (MPa) | Φ (°) | UCS (MPa) | Stepwise Regression | Quadratic Regression | ||
---|---|---|---|---|---|---|---|---|

Predicted Value | Error | Predicted Value | Error | |||||

ZL3 coal seam | 4.32 | 6.4 | 25.2 | 22.36 | 19.2 | 14.13% | 29.09 | 30.1% |

BD3 _{down} coal seam | 13.93 | 8.2 | 30.24 | 26.63 | 28.53 | 7.15% | 66.38 | 149.36% |

BD 3 coal seam | 14.35 | 7.0 | 30 | 25.91 | 24.05 | 7.19% | 67.94 | 162.20% |

HH 3 coal seam | 7.87 | 5.4 | 32.3 | 18.22 | 19.67 | 7.15% | 43.23 | 137.28% |

XH 3 coal seam | 5.23 | 5.7 | 29.3 | 19.22 | 18.94 | 1.45% | 32.76 | 70.44% |

DT3 coal seam | 3.97 | 5.64 | 31.5 | 19.6 | 19.95 | 1.78% | 27.68 | 41.21% |

DT 3 _{up} coal seam | 3.43 | 5.52 | 33 | 20.82 | 20.35 | 2.26% | 25.48 | 22.39% |

DT 3 _{down} coal seam | 3.8 | 6.5 | 33 | 24.04 | 23.9 | 0.57% | 26.99 | 12.26% |

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## Share and Cite

**MDPI and ACS Style**

Zhang, P.; Wang, J.; Jiang, L.; Zhou, T.; Yan, X.; Yuan, L.; Chen, W.
Influence Analysis and Stepwise Regression of Coal Mechanical Parameters on Uniaxial Compressive Strength Based on Orthogonal Testing Method. *Energies* **2020**, *13*, 3640.
https://doi.org/10.3390/en13143640

**AMA Style**

Zhang P, Wang J, Jiang L, Zhou T, Yan X, Yuan L, Chen W.
Influence Analysis and Stepwise Regression of Coal Mechanical Parameters on Uniaxial Compressive Strength Based on Orthogonal Testing Method. *Energies*. 2020; 13(14):3640.
https://doi.org/10.3390/en13143640

**Chicago/Turabian Style**

Zhang, Peipeng, Jianpeng Wang, Lishuai Jiang, Tao Zhou, Xianyang Yan, Long Yuan, and Wentao Chen.
2020. "Influence Analysis and Stepwise Regression of Coal Mechanical Parameters on Uniaxial Compressive Strength Based on Orthogonal Testing Method" *Energies* 13, no. 14: 3640.
https://doi.org/10.3390/en13143640