Influence Analysis and Stepwise Regression of Coal Mechanical

energies

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

Peipeng Zhang 1, Jianpeng Wang 1,*, Lishuai Jiang 1 , Tao Zhou 2, Xianyang Yan 2, Long Yuan 1 and Wentao Chen 1

1 State Key Laboratory of Mining Disaster Prevention and Control, Shandong University of Science and Technology, Qingdao 266590, China; [email protected] (P.Z.); [email protected] (L.J.); [email protected] (L.Y.); [email protected] (W.C.) 2 Yanzhou Coal Mining Company Limited, Jining 272000, China; [email protected] (T.Z.); [email protected] (X.Y.) * Correspondence: [email protected]

Received: 13 April 2020; Accepted: 9 July 2020; Published: 15 July 2020

Abstract: Uniaxial compressive strength (UCS) and peak strain (PS) are essential indices for studying the mechanical properties of coal and rock masses, and they are closely related to mechanical parameters such as the elastic modulus (E), Poisson’s ratio (υ), cohesion (C) and internal friction angle (Φ) of coal and rock masses. This study took the No. 2-1 coal seam of Zhaogu No. 2 Mine, in Henan Province, China, as the research object. An RMT-150B servo testing machine was used to test all mechanical parameters, including the E, υ, C and Φ of coal and rock masses. Based on the principle of orthogonal testing, Three Dimensions Fast Lagrangian Analysis of Continua (FLAC3D) was used to select E, υ, C, Φ, tensile strength (Rm) and dilation angle (Ψ) as initial participation 6 factors. Using these six parameters and a five-level combination scheme (L25 (5 )), 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.

Keywords: uniaxial compressive strength; peak strain; orthogonal experiment; main controlling parameters; stepwise regression prediction

1. Introduction UCS and PS are significant indices in the study of the mechanical properties of coal and rock masses and have been used widely in the design of underground mining [1–5]. UCS and PS are closely related to other mechanical parameters of coal and rock masses, including E, υ, C, Φ,Rm 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

Energies 2020, 13, 3640; doi:10.3390/en13143640 www.mdpi.com/journal/energies Energies 2020, 13, 3640 2 of 26 rock mass surrounding roadways [6–12]. 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. While exploring the relationship between UCS, PS and mechanical parameters and investigating the prediction of UCS, predecessors have carried out extensive research based on accumulated engineering data [13–20]. For instance, He et al. [21] used the fuzzy positive correlation between UCS and E to establish a correlation between UCS and E for sedimentary rocks with 10 different types of structure. Yang et al. [22] used the SPSS 19.0 statistical software to establish six separate mathematical regression models, taking UCS as the dependent variable and E as the independent variable; they also obtained a variable quadratic nonlinear regression prediction equation. Mahdi et al. [23] indirectly estimated the reliability of predicting UCS, Brazilian tensile strength (BTS) and E of marlstone using a single compressive strength method. Considering the randomness and fuzziness inherent in the determination of rock UCS, Qi et al. [24] obtained 88 groups of test data through uniaxial compression experiments, taking UCS as the dependent variable and E and natural density as independent variables, and established a multiple linear regression model. Lin and Xu [25] used linear function, quadratic function and power function regression models to fit the relationship between UCS and E for coal measure strata based on differences in lithology, taking E as the independent variable and UCS as the dependent variable. Hu et al. [26] established a multiple linear regression prediction model for the UCS and E values of the Pengguan complex, adopting the measured UCS and E values as dependent variables and rock block density, Schmidt rebound hardness and the geological strength index as independent variables. Piotr et al. [27] measured three types of Young’s modulus according to International Society for Rock Mechanics (ISRM) methods in an indoor uniaxial compression experiment, obtained correlation expressions of the tangent modulus, secant modulus and average modulus with UCS and evaluated the consistency of three UCS–E mathematical models. Most previous studies have simply analyzed the correlation between single mechanical parameters and UCS, and little attention has been paid to the comprehensive correlation among multiple mechanical parameters, UCS and PS. Additionally, predictions of the UCS of coal seams have rarely been reported. Accordingly, this study took the No. 2-1 coal seam of Zhaogu No. 2 Mine in Henan Province as the case study. An RMT-150B servo testing machine was used for the uniaxial compression test, Brazilian splitting test and triaxial test of coal samples to determine the mechanical parameters. Based on orthogonal experiment, six mechanical parameters were selected: E, υ, C, Φ,Rm and Ψ. Using these 6 six factors and a five-level combination scheme (L25 (5 )), 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 Figure1. Energies 2020, 13, 3640 3 of 26 Energies 2020, 13, x FOR PEER REVIEW 3 of 28

FigureFigure 1. 1. TechnicalTechnical workflow. workflow. 2. Orthogonal Numerical Simulation Experiments 2. Orthogonal Numerical Simulation Experiments 2.1. Production of Coal Samples 2.1. Production of Coal Samples The main coal seam of Zhaogu No. 2 mine is the Permian Shanxi formation No. 2-1 coal seam. The main coal seam of Zhaogu No. 2 mine is the Permian Shanxi formation No. 2-1 coal seam. The The dip angle of the coal seam is between 2◦ and 6◦, and the average thickness is 6.16 m; this forms a dipthick, angle stable of the and coal near-horizontal seam is between coal 2° seam. and 6°, The and coal the is average high quality thickness anthracite is 6.16 with m; this medium forms ash,a thick, low stablesulfur, and high near-horizontal calorific value coal and seam. high The ash coal melting is high point. quality The anthracite coal seam with is simple medium in structure,ash, low sulfur, being highmainly calorific composed value of and lump high coal ash and me partiallylting point. filled The with coal calcite, seam is some simple of which in structure, contains being parting mainly and composedmudstone of interlayers. lump coal The and coal partially seam filled is characterized with calcite, by some the developmentof which contains of endogenous parting and fissures mudstone and interlayers.the high strength The coal of itsseam lump is coal.characterized The average by th naturale development apparent densityof endogenous of the coal fissures seam isand 1435 the kg high/m3 . 3 strengthTo ensureof its lump the reliabilitycoal. The average of the test natural data, ap fourparent sampling density boreholes of the coal were seam arranged is 1435 kg/m in the. track roadwayTo ensure of the the 1105 reliability working of face the intest the data, No. four 11 panel sampling area ofboreholes Zhaogu were No. 2arranged coal mine. in the The track coal roadwaysamples wereof the numbered 1105 working on site face and in wrapped the No. with11 panel cling area film of to Zhaogu prevent No. weathering. 2 coal mine. Coal The samples coal sampleswere carried were bynumbered specialized on site personnel. and wrapped According with tocling the film testing to prevent requirements weathering. of the Coal International samples wereSociety carried for Rock by Mechanicsspecialized (ISRM), personnel. the coalAccording samples to underwent the testing preliminary requirements processing of the International into fourteen Societystandard for specimens Rock Mechanics of diameter (ISRM), 50 mm the and coal height samples 100 mm, underwent of which threepreliminary specimens processing were further into fourteenprocessed standard into cylinders specimens of diameter of diameter 50 mm 50 andmm heightand height 30 mm. 100 Themm, prepared of which coal three samples specimens were were then furtherused for processed the uniaxial into compression cylinders of test,diameter Brazilian 50 mm splitting and height test and 30 triaxial mm. The compression prepared coal test tosamples obtain were then used for the uniaxial compression test, Brazilian splitting test and triaxial compression values for E, υ, C, Φ and Rm, as shown in Figure2. test to obtain values for E, υ, C, Φ and Rm, as shown in Figure 2.

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Figure 2. CoalCoal samples samples as as prepared prepared for experiment.

2.2. Mechanical Mechanical Testing Testing and Parameter Acquisition forfor CoalCoal SamplesSamples 2.2.1. Introduction of RMT-150B Servo Tester 2.2.1. Introduction of RMT-150B Servo Tester An RMT-150B rock mechanics multifunctional testing machine system was selected as the test An RMT-150B rock mechanics multifunctional testinging machinemachine systemsystem waswas selectedselected asas thethe testtest loading equipment, as shown in Figure3. The test machine was equipped with four types of sensors, loadingloading equipment,equipment, asas shownshown inin FigureFigure 3.3. TheThe testtest machinemachine waswas equippedequipped withwith fourfour typestypes ofof sensors,sensors, namely stroke, stress, displacement and pressure, with fourteen sensors installed in total. The stroke namely stroke, stress, displacement and pressure, with fourteenfourteen sensorssensors installedinstalled inin total.total. TheThe strokestroke sensor is used to monitor the stroke of the vertical hydraulic cylinder. The stress sensor is used to sensor is used to monitor the stroke of the vertical hydraulic cylinder. The stress sensor is used to monitor the stress exerted by the vertical and horizontal hydraulic cylinders on the sample being monitor the stress exerted by the vertical and horizontal hydraulic cylinders on the sample being tested. The displacement sensor is used to monitor the axial and lateral deformation of the specimen. tested.tested. TheThe displacementdisplacement sensorsensor isis usedused toto monitormonitor thethe axialaxial andand laterallateral deformationdeformation ofof thethe specimen.specimen. The pressure sensor is used to monitor the pressure in the triaxial compression. The above sensors have The pressure sensor is used to monitor the pressure in the triaxial compression. The above sensors the advantages of high precision and good stability, but human error or poor daily management and have the advantages of high precision and good stability, but human error or poor daily maintenance can lead to errors in the test results. Therefore, to avoid such errors, the testing machine management and maintenance can lead to errors in the test results. Therefore, to avoid such errors, is maintained by laboratory management personnel, and coal sample mechanical testing is performed thethe testingtesting machinemachine isis maintainedmaintained byby laboratorylaboratory managementmanagement personnel,personnel, andand coalcoal samplesample by an experienced operator to obtain accurate and reliable test data. mechanical testing is performed by an experienced operator to obtain accurate and reliable test data.

Figure 3. RMT-150BRMT-150B servo-control testing machine. 2.2.2. Mechanical Parameter Acquisition 2.2.2. Mechanical Parameter Acquisition Two test samples were damaged during processing and could not be used. Therefore, mechanical Two test samples were damaged during processing andand couldcould notnot bebe used.used. Therefore,Therefore, mechanicalmechanical tests were carried out on the remaining twelve specimens: four specimens underwent uniaxial teststests werewere carriedcarried outout onon thethe remainingremaining twelvetwelve specimens: four specimens underwent uniaxial compression testing, three underwent the Brazilian splitting test and ﬁve underwent triaxial compression testing, three underwent the Brazilian splitting test andand fivefive underwentunderwent triaxialtriaxial compression testing. The number of specimens tested is lower than the minimum suggested by ISRM compression testing. The number of specimens tested is lower than the minimum suggested by ISRM to to obtain reliable parameter estimates for the mechanical properties of rock. However, the mechanical obtain reliable parameter estimates for the mechanical properties of rock. However, the mechanical parameters provide a reference data for the orthogonal test and the variation of the mechanical parameters provide a reference data for the orthogonal test and the variation of the mechanical parameters has little inﬂuence on the results of orthogonal tests. Taking uniaxial compression testing as parameters has little influence on the results of orthogonal tests. Taking uniaxial compression testing as an example, a prepared specimen was placed on the pressure plate below the testing machine, and the an example, a prepared specimen was placed on the pressure plate below the testing machine, and axial and lateral displacement sensors were installed and adjusted, as shown in Figure4. thethe axialaxial andand laterallateral displacementdisplacement sensorssensors werewere installedinstalled andand adjusted,adjusted, asas shownshown inin FigureFigure 4.4.

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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 mountingFigure 4. Specimenplate; ⑧loading lower diagram: bearing ① upper bearing plate; ② axial displacement sensor Figure 4. Specimen loadingFigure diagram: 4.4. SpecimenSpecimen ① upper loadingloading bearing diagram:diagram: plate; ②O 1① axialupper upper displacement bearing bearing plate; plate;sensorO2 axial ② displacementaxial displacement sensor gripper;sensor FigureFigureFigure 4. 4. Specimen 4.Specimen Specimen loading loading loadingplate; diagram: diagram:Figure diagram:⑨ bearing ① 4.① ①Specimen upperupper upper seat bearingbearing bearingloading ⑩ plate;plate;locating plate; diagram: ②② ② axial axial pin;axial ① displacementdisplacement displacementupper⑪ adjustment bearing sensorsensor sensor plate; plate;gripper; ② ⑫axial ③foundation displacementtransverse displacementsupport; sensor ⑬ sensor; ④ transverse displacement sensor mounting gripper; ③③ ③transverse③ gripper;Odisplacement3 transverse ③ transversesensor; displacement④ ④④④ transversedisplacement sensor; displacementO4 sensor;transverse ④sensor displacementtransverse mounting displacement sensor⑤ mounting sensor seat; mountingO5 axial⑥ ⑦ ⑧ gripper;gripper;gripper; transversetransverse transverse displacementdisplacement displacementgripper; sensor;sensor;③ sensor; ⑭transverse transversetransverse transverse displacement displacementdisplacement displacement sensor; sensorsensor sensor ④ mounting mounting transversemounting displacementseat; axial displacement sensor mounting sensor; sample; sensor mounting plate; lower bearing seat; ⑤ ⑤axial⑤⑤ displacementbaseboard;displacement sensor; ⑥ ⑥and⑥ sensor;sample; located⑦⑦O 6⑦sample; sensor block. mountingO7 sensor mountingplate;⑧⑧⑧ ⑧ lower plate; bearingO8 lower bearing⑨ plate; O9⑩bearing seat ⑪ ⑫ ⑬ seat;seat;seat; axial axial axial displacement displacement displacementseat; sensor; seat;sensor;⑤ sensor; axial ⑤ axialsample; sample;displacement sample; displacement sensor sensor sensor mountingsensor; mounting sensor;mounting ⑥ plate; ⑥plate; plate;sample; sample; lower lower lower ⑦ bearing bearingsensorsensor bearing mounting mountingplate; plate; bearingplate; ⑧ ⑧seatlower lower bearinglocating bearing pin; adjustment plate; foundation support; plate;plate;plate; plate;⑨ ⑨ ⑨bearing ⑨ bearingbearing bearing seat seatseat seat ⑩ ⑩⑩ ⑩ locatinglocating locating pin;pin; pin; ⑪ ⑪ ⑪⑪ adjustment adjustment adjustmentadjustment plate;plate; plate;plate; ⑫⑫ ⑫ foundation⑫foundation foundation foundation support;support; support; support; ⑬⑬ ⑬ ⑬baseboard; and ⑭ located block. plate;locatingplate; ⑨ bearing pin;⑨ bearingadjustment seat seat ⑩ ⑩locating plate; locating pin;foundation pin; ⑪⑪ adjustmentadjustment support; plate;plate;baseboard; ⑫ ⑫ foundation foundation and located support; support; block. ⑬ ⑬ baseboard;baseboard;baseboard;baseboard; and and and and⑭ ⑭ ⑭ located ⑭ located located locatedDisplacement block.block. block. block. control was adopted for the uniaxial compression test, with a loading rate of 0.02 baseboard;baseboard; and and ⑭ located⑭ located block. block. mm/s,Displacement and loading controlwas carried was adoptedout under for computer the uniaxial control compression untilDisplacement the specimen test, control with was was a adopted loading damaged, for rate the uniaxialas of compression test, with a loading rate of 0.02 DisplacementDisplacementDisplacement control control control was was was adopted adopted adopted for for for the the the uniaxial uniaxial uniaxial compression compression compression test test test, ,with with, with a a loading aloading loading rate rate rate of of of0.02mm/s, 0.02 0.02 and loading was carried out under computer control until the specimen was damaged, as Displacement controlshown0.02 was mm inadopted/s, Figure and for loading 5. the The uniaxial Brazilian was carriedcompression splitting out undertest test, with was computer a controlledloading control rate byof until0.02the stroke, the specimen with a loading was damaged, rate of mm/s,mm/s,mm/s, and and and loading loading loading was was was carried carried Displacementcarried out Displacementout out under under under computer controlcomputer computer control wascontrol control controlwas adopted until adopteduntil until the the the forspecimen forspecimen specimen the the uniaxialuniaxial was was was damaged, damaged, damaged, compressioncompressionshown as as as in test,Figure test, with with5. Thea loading aBrazilian loading rate splitting rateof 0.02 of test 0.02 was controlled by the stroke, with a loading rate of mm/s, and loading was carried out under computer control until the specimen was damaged, as shownshownshown in in inFigure Figure Figure 5. 5. The5. The0.005as The Brazilian shownBrazilian Brazilian mm/s,m/min. insplitting splittingand splitting Figure A loadingconventional test test 5test. was Thewas waswas controlled controlled Brazilian controlled carried triaxial by byout by splittingthe thecompression the understroke, stroke, stroke, testcomputerwith with with was a a te loading aloading stloading controlled controlwas rate rate adoptedrate0.005 ofuntil of of bym/min. the the for specimen A stroke, triaxial conventional with wascompression atriaxialdamaged, loading compression testing, rateas of test was adopted for triaxial compression testing, shown in Figure 5. Themm/s, Brazilian and splitting loading test was was carried controlled out byunder the stroke, computer with controla loading until rate ofthe specimen was damaged, as 0.0050.0050.005 m/min. m/min. m/min. A A conventionalA conventional conventionalthat0.005 is,shownm triaxial triaxialσ/ min.triaxial1 > inσ compression 2compression A =Figurecompression σ conventional3, and 5. Thetest thetest test was Brazilianwasconfining was triaxialadopted adopted adopted splitting for compressionpressures for for triaxial triaxial triaxial test compression compression was werecompression test controlled 2, was 5, testing, testing,10, testing, adoptedthat 15by is, theand σ1 > stroke, for σ202 = triaxial σMPa.3, andwith Displacementthe compressiona loadingconfining rate pressures testing,controlof were 2, 5, 10, 15 and 20 MPa. Displacement control shown in Figure 5. The Brazilian splitting test was controlled by the stroke, with a loading rate of 0.005thatthat m/min.that is, is, is,σ σ1 >1σ > A1σ >σ 2 conventional=2σ = σ2 σ=3, 3σ ,and 3and, and the the the confining confining triaxial confining pressurescompression pressures pressures were were were test 2, 2, 5,2, 5,was 10,5, 10, 10, 15 adopted15 15and and and 20 20 20MPa.for MPa. MPa. triaxial Displacement Displacement Displacement compression control control controlwas testing,adopted for the triaxial testing. First, confining pressure was added in a static and horizontal wasthat is,adopted0.005σ1 >m/min.σ 2for= σAthe3 conventional, andtriaxial the confiningtesting. triaxial First, compression pressures confining were test pressure was 2, 5, adopted 10, 15was and for added triaxial 20 MPa. in compressiona Displacementstatic and testing, horizontal control thatwas wasis,was σadopted adopted1 >adopted σ2 = σfor for3 ,for andthe the the 0.005triaxial triaxialthe triaxial confining m/min.testing. testing. testing. First, First,Apressures First, conventional confin confin confin ingwereinging pressure pressure pressure2, triaxial5, 10, was was was15 added compression addedand added 20in in ina MPa.a static astatic static Displacement teand andst and washorizontal horizontal horizontal adoptedmanner, control with for a triaxial loading ratecompression of confining pressuretesting, of 0.1 MPa/s. Then, axial pressure was added until wasmanner, adoptedthat withis, σ1 fora> loadingσ2 the = σ3, triaxial and rate the of testing. confining confining First, pressures pressure confining were of 0.1 2, pressure 5,MPa/s. 10, 15 was Then,and added20 axial MPa. inpressure Displacement a static was and addedcontrol horizontal until wasmanner, manner,adoptedmanner, with with withfor a a theloading aloading loading triaxial rate rate rate oftesting. of confiningof confining1 confining First,2 pressure pressure3 pressure confin of ofing 0.1of 0.1 0.1 MPa/s.pressure MPa/s. MPa/s. Then, Then, Then,was axial axial addedaxial pressure pressure pressure in a was wasstatic was added added addedand until theuntilhorizontal until predetermined confining pressure value was reached, and the axial loading rate was 0.005 mm/s. that is,was σ adopted> σ = σ for, and the the triaxial confining testing. pressures First, confining were pressure2, 5, 10, was15 and added 20 inMPa. a static Displacement and horizontal control manner,thethethe predetermined predetermined withpredetermined a loading confining confining themanner,confining rate predetermined of pressure pressureconfining pressure with valuea value loading valuepressure was confiningwas was reac reac rate reacofhed,hed, 0.1hed, of and pressure MPa/s.and confiningand the the the axial Then,axial axial value loading pressureloading loadingaxial was rate pressurerate rate reached, ofwas was was 0.1 0.005 0.005 was0.005 MPa mm/s.and mm/s.added Themm/s./s. the mechanical Then, until axial axial loading parameters pressure rate and was the stress–strain0.005 added mm/s. until curv e of the coal sample were automatically recorded was adoptedmanner, withfor thea loading triaxial rate testing. of confining First, pressure confining of 0.1 pressure MPa/s. Then,was addedaxial pressure in a static was addedand horizontal until the ThepredeterminedTheThe mechanical mechanical mechanical parameters parameters confining parametersThethe predeterminedmechanical and andpressure and the the the stress–strain stress–strain stress–strain value parameters confiningwas curv curv curv reace eof eof andhed, theof the pressure the coal coaltheand coal sample samplestress–strain the sample valueaxial were were were loading wasautomatically automatically automatically curv reached, ratee of was recorded therecorded recorded and 0.005 bycoal the the mm/s.sample computer, axial loadingwere as shown automatically rate in Figure was 0.0056. recorded mm/s. manner,the predeterminedwith a loading confining rate of confining pressure value pressure was reached,of 0.1 MPa/s. and the Then, axial loadingaxial pressure rate was was 0.005 added mm/s. until Theby mechanicalby bythe the the computer, computer, computer, parameters as as shownas shownbyThe shown the and mechanicalin in Figureincomputer, Figure theFigure stress–strain 6. 6. 6. parameters as shown curv ein and of Figure the the coal stress–strain 6. sample were curve automatically of the coal recorded sample were automatically recorded the predeterminedThe mechanical confiningparameters pressureand the stress–strain value was curv reached,e of the and coal the sample axial were loading automatically rate was recorded 0.005 mm/s. by the computer, as shown in Figure 6. Theby the mechanicalby computer, the computer, parameters as shownas shown and in in Figure theFigure stress–strain6. 6. curve of the coal sample were automatically recorded by the computer, as shown in Figure 6.

Figure 5. Failure pattern of coal samples. FigureFigureFigure 5. 5. Failure 5.Failure Failure pattern pattern pattern of of coalof coal coal samples. samples. samples.

Figure 5. Failure pattern of coal samples. FigureFigure 5. 5. FailureFailure Failure patternpattern ofof coal coal samples. samples.

Figure 5. Failure pattern of coal samples.

Figure 6. Experimental curve of mechanics.

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The speciﬁc mechanical parameters are shown in Table1, in which the C and Φ were calculated by the ﬁve tests of the Mohr–Coulomb strength criterion. To ensure an adequate simulation of the numerical test and facilitate the orthogonal calculation, the mean values of some parameters were reduced.

Table 1. Mechanical parameters of coal samples.

Test E (GPa) υ UCS (MPa) C (MPa) Φ (◦) Ψ (◦)Rm (MPa) 3.20 0.35 20.6 //// 2.60 0.22 18.7 //// Uniaxial compression test 2.19 0.33 15.7 //// 3.22 0.30 25.4 //// ////// 1.23 Brazilian splitting ////// 0.12 test ////// 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

It is difficult to obtain an accurate Ψ through mechanical experiments, so Hoek and Brown [28] suggested that constant Ψ = Φ/4, Φ/8 and 0 should be used to describe the ideal post-peak behavior of rock masses, namely elasto-brittle, strain softening and ideal elastic–plastic deformation. Based on the mechanical test results of coal samples, this study determined that the Ψ is a constant value, Ψ = Φ/4. According to the classical plasticity theory, the Mohr–Coulomb strength criterion and the non-associated flow rule, it is assumed that the variation of material parameters is related to the softening parameters (plastic shear strain) [29–31]. Among them, the determination of FLAC3D numerical simulation softening parameters and material weakening method refer to the calculation formula in the literature [32] and the table command flow of FLAC3D numerical simulation is used to weaken the relevant parameters of strain softening (including C, Φ and Ψ).

2.3. Numerical Simulation Test

2.3.1. Establishment of Numerical Model The FLAC3D numerical simulation software, as an extension of a finite difference program, can analyze the mechanical properties and plastic flow failure of rock and soil masses and can characterize well the nonlinear mechanical response characteristics of rock and soil [33,34].Therefore, in this study, the cylinder model (diameter 50 mm height 100 mm) and the cuboid model (length 50 mm width × × 50 mm height 100 mm) were established by using FLAC3D finite difference software. Displacement × loads at a constant rate were applied to the top and bottom of the model, and no boundary conditions were applied around it, as shown in Figure7. A coal sample is an anisotropic inhomogeneous body with voids and developed joints; therefore, the numerical model was simplified as an isotropic homogeneous body, while the time units of load applications used were different. The mechanical experiment units were mm/s or mm/min, and the FLAC3D units were mm/step. The loading rate of mechanical testing was directly applied to the numerical model, and the results of the numerical simulation were quite different from those of mechanical testing. This is because the specimen for gap, joint development of anisotropic Energies 2020, 13, 3640 7 of 26

heterogeneousEnergies 2020, 13, x bodyFOR PEER and REVIEW the numerical simulation process model are simpliﬁed to ideal isotropic7 of 28 homogeneous body and there are no other deformations except for elastic deformation, such as crack closure and jointjoint development.development. Therefore, the loading rate in mechanical testing could not be directly used inin the the numerical numerical simulation. simulation. It is It necessary is necessary to determine to determine a reasonable a reasonable model shape, model a constitutive shape, a model,constitutive and amodel, loading rateand baseda loading on theoretical rate based analysis on andtheoretical many numerical analysis simulations.and many Therenumerical was asimulations. heavy workload There ofwas numerical a heavy simulations,workload of sonumerical in this paper simulations, we do notso in list this them paper in detail we do but not only list analyzethem in thedetail typical but only results analyze of each the condition. typical results of each condition.

(a) Cylinder model (b) Cuboid model

Figure 7. Mesh generation and boundary conditions of the computational model.

2.3.2. Constitutive Model Selection The constitutiveconstitutive relation relation is usedis used to represent to represen the mechanicalt the mechanical properties properties of materials of inmaterials numerical in simulationnumerical simulation testing, and testing, mechanical and mechanical response characteristicsresponse characteristics vary considerably vary considerably between dibetweenfferent constitutivedifferent constitutive models. Therefore, models. Theref it is importantore, it is important to select a constitutiveto select a constitutive model to study model the to mechanical study the characteristicsmechanical characteristics of coal accurately. of coal At present,accurately. strain-softening At present, models strain-softening and Mohr–Coulomb models and models Mohr– are typicallyCoulomb used models in mechanical are typically testing used ofin coalmechanical/rock masses testing with of coal/rock FLAC3D masses [35–38]. with FLAC3D [35–38]. The Mohr–CoulombMohr–Coulomb modelmodel isis thethe conventionalconventional modelmodel usedused toto representrepresent compression-shearcompression-shear failurefailure inin soilssoils andand rocks.rocks. The failure envelope for thisthis modelmodel correspondscorresponds toto aa Mohr–CoulombMohr–Coulomb criterioncriterion (shear(shear yieldyield function) function) with with tension tensio cuton cutoffff (tension (tension yield yield function) function) [39]. [39]. Once Once the stressthe stress reaches reaches the peak the strength,peak strength, the strength the strength of rock of rock will rapidlywill rapidly decrease decrease as the as deformationthe deformation continues continues to increase;to increase; this this is calledis called “strain “strain softening” softening” [40 [40].]. Therefore, Therefore, a a numerical numerical simulation simulation testtest cancan notnot onlyonly reflectreflect thethe elasticelastic stage and yield failure failure of of rock rock but but also also characterize characterize the the mechanical mechanical properties properties of rock of rock after after the peak. the peak. The Thestrain-softening strain-softening model model allows allows representation representation of ofnonlinear nonlinear materi materialal softening softening behavior behavior based based onon prescribedprescribed variations of thethe Mohr–CoulombMohr–Coulomb model propertiesproperties [[39].39]. In FLAC3D,FLAC3D, thethe strainstrain softeningsoftening model in in the the elastic elastic stage stage is exactly is exactly the same the same as the as Mohr–Coulomb the Mohr–Coulomb model [41]. model The [41 difference,]. The di however,fference, however, lies in the possibility that the C, Φ, Ψ and R may soften after the onset of plastic yield. In the lies in the possibility that the C, Φ, Ψ and Rm may softenm after the onset of plastic yield. In the Mohr– Mohr–CoulombCoulomb model, model,those properties those properties are assumed are assumed to rema toin remainconstant constant [39]. The [ 39yield]. The function, yield function, flow rule flow and rulestress and correction stress correction of the strain of the softening strain softening model ar modele consistent are consistent with those with of those the Mohr–Coulomb of the Mohr–Coulomb model model[42]. [42]. The shear yieldyield functionfunction ofof thethe strainstrain softeningsoftening modelmodel is:is: q s s = σ −σ + ff = σ 1 σ 3NNϕϕ+ 22cc NNϕϕ (1)(1) 1 − 3

Here, σ1 and σ3 represent the maximum and minimum principal stresses, respectively; C is Here, σ1 and σ3 represent the maximum and minimum principal stresses, respectively; C is cohesion; φ is the internal friction angle; and Nφ is defined as follows. cohesion; ϕ is the internal friction angle; and Nϕ is deﬁned as follows. = ()()+ ϕ − ϕ Nϕ 1 sin 1 sin (2) Nϕ = (1 + sin ϕ)/(1 sin ϕ) (2) − In the case of shear failure, according to the orthogonal flow law, the stress of shear failure can be modified as follows:

Energies 2020, 13, 3640 8 of 26

In the case of shear failure, according to the orthogonal flow law, the stress of shear failure can be modified as follows:   σN = σI λs α α N  1 1 1 2 ψ  − −  σN = σI α λs 1 N (3)  2 2 2 ψ  − −  σN = σI λs α α N 3 3 − 2 − 1 ψ Here, σ2 is the intermediate principal stress; σ1 and σ3 represent the maximum and minimum principal stresses, respectively; α1 and α2 are material constants defined in terms of the shear modulus and the bulk modulus, respectively; λt 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 Nψ is defined as follows: N = (1 + sin ψ)/(1 sin ψ) (4) ψ − It can be seen from the stress–strain curve (Figure6) obtained from the mechanical test of the coal sample that at the initial stage of loading, the axial strain of the coal sample increased rapidly due to the rapid closure of internal voids or micro cracks. With increasing applied load, the load was less than the yield strength, and the stress–strain curve essentially increased in a straight line, which showed that only reversible elastic deformation occurred in the specimen and that no plastic failure occurred. With a continuously increasing applied load, when the applied load approached the yield strength, the stress–strain curve began to show a concave shape, the specimen began to crack, and then it underwent irreversible plastic deformation. When the applied load reached the yield strength, the plastic failure range expanded rapidly, but the specimen retained a certain bearing capacity. However, the bearing capacity gradually decreased with increasing strain, showing a significant strain softening phenomenon. When the strain reached a certain value, the stress–strain curve rapidly decreased, which was mainly caused by the brittleness of the coal sample. Based on the above analysis, the strain softening model can better simulate the mechanical properties of coal samples. In the process of numerical simulation, to solve the problem of weakening parameters such as C and Φ in the strain softening stage, the formula in [32] can be used:

p + p = Nψε1 2ε3 0 (5) p p η = ε ε (6) 1 − 3  Kp K  − 1  Kp η η 0 η η1  − 1 ≤ ≤  K1 K2  K1 − (η η1) η1 η η2 K(η) = η2 η1 (7)  − − − ≤ ≤    ······ Kn 1 Kr  K − − (η η ) η η η n 1 η ηn 1 n 1 n 1 − − ∗− − − − − ≤ ≤ ∗ p p where ε1 is the axial plastic main strain; ε3 is the lateral plastic main strain; η is the softening coefficient; K represents C and Φ; p represents peak mechanical parameters; r represents residual mechanical parameters; η1, η2, ... , ηn 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. Combined with the results of the mechanical test shown in Table1, the softening coe fficients (η) of mechanical parameters at different stages can be obtained by Formulae (4)–(7), as shown in Table2.

Table 2. Variation of coal sample strength parameters at diﬀerent stages.

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 Energies 2020, 13, x FOR PEER REVIEW 9 of 27

Cumulative plastic strain 0.008 5.76 28.06 0.0133 Energies 2020, 13, 3640 9 of 26 Cumulative plastic strain 0.015 1.42 26.24 0.0250

2.3.3. Model shape Shape selection Selection In general, uniaxial compression testtest specimenspecimen shapesshapes includeinclude cylinderscylinders (50(50 mmmm × 100 mm) and × cuboids (50 mm × 50 mm × 100 mm). In In this expe experiment,riment, two models of cylinder and cuboid were × × established. The modelmodel adoptedadopted thethe weakenedweakened mechanical mechanical parameters parameters in in Table Table1. The1. The loading loading rates rates at theat the top top and and bottom bottom were were both both 1.9 1.9 ×10 105−5mm mm/step./step. TheThe stress–strainstress–strain curvecurve ofof thethe numerical model is × − shown in Figure8 8..

Figure 8. Stress–strain curves for different shape models. Figure 8. Stress–strain curves for different shape models. From the numerical simulation results, it can be seen that the cylinder model and the cuboid model From the numerical simulation results, it can be seen that the cylinder model and the cuboid underwent the same changes in the elastic stage, but their mechanical properties were quite different model underwent the same changes in the elastic stage, but their mechanical properties were quite after the peak. The weakening process of the cylinder model was slow after the peak. However, different after the peak. The weakening process of the cylinder model was slow after the peak. the strength of the cuboid model after the peak rapidly decreased after the initial slow fluctuations, However, the strength of the cuboid model after the peak rapidly decreased after the initial slow which accurately simulated the brittle failure of the coal body, and was consistent with the mechanical fluctuations, which accurately simulated the brittle failure of the coal body, and was consistent with testing results of the coal sample, as shown in Figure6. In addition, the yield strength of the cylinder the mechanical testing results of the coal sample, as shown in Figure 6. In addition, the yield strength model was 16.5 MPa and that of cuboid model was 24.4 MPa. The numerical model is homogeneous of the cylinder model was 16.5 MPa and that of cuboid model was 24.4 MPa. The numerical model is while coal samples are heterogeneous, with joints and voids; therefore, it is normal for the yield homogeneous while coal samples are heterogeneous, with joints and voids; therefore, it is normal for strength of the numerical simulation to be slightly higher than that of the mechanical experiment. the yield strength of the numerical simulation to be slightly higher than that of the mechanical Therefore, the cuboid model was selected for the orthogonal numerical simulation test. experiment. Therefore, the cuboid model was selected for the orthogonal numerical simulation test. 2.3.4. Mesh Number of Model Element Body Is Determined 2.3.4. Mesh Number of Model Element Body is Determined In FLAC3D software, the strain softening model is used to determine the degree of softening of the specimenIn FLAC3D parameters software, through the thestrain accumulated softening model plastic is strain used afterto determine the peak, the reflecting degree theof softening mechanical of propertiesthe specimen of the parameters specimen afterthrough the peak. the accumulated The size of the plastic model gridstrain has after an important the peak, influence reflecting on the developmentmechanical properties of plastic failureof the specimen in test samples. after the If the peak. grid The size size is too of large,the model it can grid indirectly has an enhance important the strengthinfluence of on the the test development sample, making of plastic it harder failure to damage in test it. samples. If the grid If sizethe isgrid too size small, is ittoo is toolarge, sensitive it can toindirectly loads and enhance the specimen the strength is weak of the and test easy sample, to damage. making In it this harder study, to damage three grid it. sizesIf the weregrid size designed: is too Schemesmall, it is A, too grid sensitive size 5 mm to loads long and5 mmthe specimen wide 2 is mm weak high, and with easy 5000 to damage. grids divided In this study, for test three samples; grid × × Schemesizes were B, designed: grid size Scheme 2.5 mm A, long grid size2.5 5 mm mm wide long * 52 mm mm wide high, * 2 with mm 20,000high, with grids 5000 divided grids divided for test × × samples;for test samples; and Scheme Scheme C, grid B, grid size size 1 mm 2.5 long mm long1 mm * wide2.5 mm2 wide mm high,* 2 mm with high, 125,000 with grids 20,000 divided grids × × fordivided test samples. for test samples; The stress–strain and Scheme curves C, grid of di ffsizeerent 1 mm mesh long sizes * 1 obtained mm wide by * numerical2 mm high, simulation with 125,000 are showngrids divided in Figure for9. test samples. The stress–strain curves of different mesh sizes obtained by numericalAccording simulation to the ar comparisone shown in of Figure Figures 9. 6 and9, the post-peak failure processes of the di fferent schemesAccording were quite to the di ffcomparisonerent. In Scheme of Figures A, the 6 post-peakand 9, the softeningpost-peak modulus failure processes of the sample of the was different small andschemes could were not accuratelyquite different. simulate In Scheme the brittleness A, the po ofst-peak the post-peak softening failure modulus of coal of samples. the sample The was post-peak small softeningand could modulus not accurately of Schemes simulate B and the C brittleness was relatively of the large, post-peak which couldfailure simulate of coal thesamples. brittleness The characteristicspost-peak softening of coal modulus samples. of However, Schemes when B and comparing C was relatively these two large, schemes, which the could stress–strain simulate curve the ofbrittleness Scheme Bcharacteristics rapidly decreased of coal following samples. aHowe shortver, fluctuation, when comparing while Scheme these Ctwo decreased schemes, directly the stress– and rapidlystrain curve after of the Scheme peak. SchemeB rapidly B wasdecreased more consistentfollowing witha short the fluctuation, mechanical while properties Scheme of coalC decreased samples

Energies 2020, 13, x FOR PEER REVIEW 10 of 27 Energies 2020, 13, x FOR PEER REVIEW 10 of 27 Energiesdirectly2020 and, 13 ,rapidly 3640 after the peak. Scheme B was more consistent with the mechanical properties10 of 26of directlycoal samples and rapidly(Figure after6). Therefore, the peak. Scheme Scheme B Bwas was sele moctedre consistent for this simulation: with the mechanicalthe grid size properties was 2.5 mm of (Figurecoallong samples * 2.56). mm Therefore, (Figure wide * 6).2 Scheme mm Therefore, high, B was and Scheme selected 20,000 B gridswas for sele thiswerected simulation: divided for this for simulation: thetest grid samples. size the wasgrid 2.5size mm was long 2.5 mm × 2.5long mm * 2.5 wide mm wide2 mm * 2 high, mm andhigh, 20,000 and 20,000 grids weregrids dividedwere divided for test for samples. test samples. ×

Figure 9. Comparison of numerical simulation results under different cell mesh sizes. Figure 9. Comparison of numerical simulation results under different cell mesh sizes. Figure 9. Comparison of numerical simulation results under different cell mesh sizes. 2.3.5.2.3.5. LoadingLoading RateRate DeterminationDetermination 2.3.5. Loading Rate Determination TheThe results results show show that that di differentfferent loading loading rates rates exerted exerted a a great great influence influence on on the the strength strength and and failure failure modesmodesThe of of rock resultsrock mass. mass. show ItIt hasthathas been beendifferent shownshown loading thatthat thethe rates peakpeak exer strengthstrengthted a great of of rockrock influence increases increases on with thewith strength increasing increasing and loading loading failure ratemodesrate [ 43[43]. ].of Through Throughrock mass. mechanicalmechanical It has been testing,testing, shown ititthat hashas the beenbeen peak shownsh strengthown thatthat of thethe rock peakpeak increases strengthstrength with ofof aincreasinga coal coal sample sample loading first first increasesrateincreases [43]. and Throughand then then decreases mechanicaldecreases with with testing, the the increase increaseit has been in in loading loading shown rate ratethat [ 44[44].the]. peak ThroughThrough strength ParticleParticle of a Flow Flowcoal CodesampleCode (PFC)(PFC) first numericalincreasesnumerical and simulationsimulation then decreases testing,testing, with itit waswas thefound increasefound thatthat in theloadingthe UCSUCS rate ofof coal [44].coal samples Throughsamplesincreases Particleincreases Flowwith with Code increased increased (PFC) loadingnumericalloading rate,rate, simulation andand thatthat thetesting,the failurefailure it was modemode found changeschanges that fromthefrom UCS X-shaped,X-shaped, of coal V-shaped V-shapedsamples increases oror Y-shapedY-shaped with toto increased aa singlesingle macroscopicloadingmacroscopic rate, fracture fractureand that surfacesurface the failure [[45].45]. InmodeIn addition,addition, changes thethe fr unitomunit X-shaped, usedused forfor the theV-shaped acceleration acceleration or Y-shaped raterate in in mechanical mechanicalto a single testingmacroscopictesting andand numerical numericalfracture surface simulationsimulation [45]. loadingInloading addition, isis didiffere thefferent. unitnt. used TheThe unitunitfor the ofof mechanicalaccelerationmechanical testingtestingrate in isismechanical mmmm/s/s oror mmtestingmm/min,/min, and andand numerical thethe unitunit ofof simulation FLAC3DFLAC3D isloading mm/step. mm/step. is Therefordiffere Therefore,nt.e, Thethe the loadingunit loading of ratemechanical rate of of mechanical mechanical testing test testis cannotmm/s cannot beor bemm/min,directly directly applied and applied the tounit to numerical numericalof FLAC3D simulation. simulation. is mm/step. To To Thereformake make the thee, thesimulation loading ratetest of resultsresults mechanical moremore accuratetestaccurate cannot andand be constantdirectlyconstant with appliedwith the the mechanicalto mechanical numerical properties properties simulation. of of coal coalTo samples, makesamples, the it it is simulationis necessary necessary test to to discuss discussresults the themore reasonable reasonable accurate value value and ofconstantof the the acceleration acceleration with the rate.mechanical rate. The The loading loading properties rates rates wereof werecoal 1.9 samples, 1.910 × 104,− 1.94it, 1.9is necessary 10× 105,−5 1.9, 1.9 to ×10 discuss 106−6and and the 1.9 1.9 reasonable ×10 107−7mm mm/step, /valuestep, × − × − × − × − andofand the the the acceleration remaining remaining numerical rate.numerical The loading simulation simulation rates conditions conditionswere 1.9 × remained remained10−4, 1.9 × consistent. consistent.10−5, 1.9 × The10The−6 and stress–strain stress–strain 1.9 × 10−7 curves mm/step,curves of of theandthe numerical numerical the remaining models models numerical with with di differentff simulationerent loading loading conditions rate rate conditions conditions remained were were consistent. recorded, recorded, The as as shownstress–strain shown in in Figure Figure curves 10 10.. of the numerical models with different loading rate conditions were recorded, as shown in Figure 10.

Figure 10. Stress–strain curves for diﬀerent loading rates. Figure 10. Stress–strain curves for different loading rates. It can be known fromFigure the 10. stress–strainStress–strain curves curves for different of diﬀerent loading loading rates. rates, from the yield It can be known from the stress–strain curves4 of different loading rates, from the yield strength, strength, when the loading rate V = 1.9 10− mm/step, the yield strength is 27.3 MPa. Compared −4 × withwhen theIt thecan mechanical loadingbe known rate testfrom V results= the 1.9 stress–strain × in10 Tablemm/step,1, curves the the deviation of yield different strength rate loading is 35.82%.is 27.3 rates, MPa. When from Compared thethe yield loading strength,with rate the whenmechanical the loading test5 results rate inV =Table 1.9 ×1, 10 the−4 mm/step, deviation therate yield is 35.82%. strength When is 27.3 the MPa.loading Compared rate V = 1.9with × 10the−5 V = 1.9 10− mm/step, the yield strength is 24.4 MPa and the deviation rate is 21.39%. When the mechanicalmm/step,× the test yield results strength in6 Table is 24. 1,4 MPathe deviation and the deviation rate is 35.82%. rate is 21.When39%. the When loading the loading rate V =rate 1.9 V × =10 1.9−5 loading rate V = 1.9 10− mm/step, the yield strength is 23.8 MPa and the deviation rate is 18.41%. mm/step,× 10−6 mm/step, the yield the strength yield× strength is 24.4 isMPa7 23.8 and MPa the and deviation the dev rateiation is 21.rate39%. is 18.41%. When theWhen loading the loading rate V = rate 1.9 When the loading rate V = 1.9 10− mm/step, the yield strength is 24.0 MPa and the deviation rate is −6 −7 × 19.4%.×V 10 = 1.9 mm/step, Therefore,× 10 mm/step, the when yield the the strength yield loading strength is 23.8 rate MPa wasis 24.0 low,and MPa the the and dev yield theiation strength deviation rate is tended 18.41%.rate isto 19.4%. When be stable. Therefore,the loading When when therate Vthe = loading1.9 × 10− rate7 mm/step, was low, the the yield yield strength strength5 is 24.0tended MPa to and be stable. the deviation When the rate loading is 19.4%. rate Therefore, was more when than loading rate was more than 1.9 10− mm/step, the yield strength of the numerical model rapidly the loading rate was low, the yield× strength tended to be stable. When the loading rate was more than

Energies 2020, 13, x FOR PEER REVIEW 11 of 28

It can be known from the stress–strain curves of different loading rates, from the yield strength, when the loading rate V = 1.9 × 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 VEnergies = 1.9 2020× 10, −137 ,mm/step, 3640 the yield strength is 24.0 MPa and the deviation rate is 19.4%. Therefore, 11when of 26 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. increased with the loading rate. Due to the low loading rate, the load of the model was similar to the Due to the low loading rate, the load of the model was similar to the static load, and the internal static load, and the internal damage and plastic damage of the model had sufficient time to develop, damage and plastic damage of the model had sufficient time to develop, thus the yield strength was thus the yield strength was low and the difference was small. With an increase in loading rate, the plastic low and the difference was small. With an increase in loading rate, the plastic failure time decreased, failure time decreased, the degree of damage in the model decreased and the strength continuously the degree of damage in the model decreased and the strength continuously increased [44].4 From the increased [44]. From the mechanical performance curve, when the loading rate V = 1.9 10− mm/step, mechanical performance curve, when the loading rate V = 1.9 × 10−4 mm/step, after the× model reaches after the model reaches the yield strength, the axial stress gradually decreases with the increase of axial the yield strength, the axial stress gradually decrea5 ses with the 7increase of axial strain. When the strain. When the loading speeds were V = 1.9 10− and 1.9 10− mm/step, when the model reached loading speeds were V = 1.9 × 10−5 and 1.9 × 10×−7 mm/step, when× the model reached the yield strength, the yield strength, as the strain increased, the stress decreased, slowly at first and then rapidly. When as the strain increased, the stress decreased,6 slowly at first and then rapidly. When the loading speed the loading speed was V = 1.9 10− mm/step, the axial stress rapidly decreased and the brittleness was V = 1.9 × 10−6 mm/step, the× axial stress rapidly decreased and the brittleness increased after the increased after the model reached the yield strength. Compared with the stress–strain curve of the model reached the yield strength. Compared with the stress–strain curve of the coal sample in Figure coal sample in Figure6, the stress–strain curve and the mechanical properties of the loading rate −5 −7 6, the stress–strain5 curve and7 the mechanical properties of the loading rate V = 1.9 × 10 and 1.9 × 10 V = 1.9 10− and 1.9 10− mm/step were more consistent with the mechanical test results of the mm/step× were more consistent× with the mechanical test results of the coal sample. However, when the coal sample. However, when the loading rate was V = 1.9 10 7 mm/step, the simulation time of a −7 − loading rate was V = 1.9 × 10 mm/step, the simulation time ×of a 5single model was significantly longer single model was significantly longer than for a rate of 1.9 10− mm/step. Based on the comparative than for a rate of 1.9 × 10−5 mm/step. Based on the comparative× analysis of the four loading rates, the analysis of the four loading rates, the selected loading rate of orthogonal numerical simulation test is selected loading rate of orthogonal numerical simulation test is 1.9 × 10−5 mm/step. 1.9 10 5 mm/step. × − 2.4. Orthogonal Orthogonal Experimental Experimental Method and Experimental Scheme

2.4.1. Orthogonal Orthogonal Experimental Method The orthogonalorthogonal experimentalexperimental method method involves involves an an overall overall design, design, comprehensive comprehensive comparison comparison and andstatistical statistical analysis analysis of a testof a carried test carried out in out an in orthogonal an orthogonal table. table. It also It includes also includes representative representative test points test pointsselected selected from a largefrom seta large of test set data; of test these data; test pointsthese test go throughpoints go testing, throug comprehensiveh testing, comprehensive comparison comparisonand summary and analysis summary via analysis the orthogonal via the orthogonal table. The specifictable. The design specific process design is shownprocessin is Figureshown 11in. FigureThe whole 11. The process whole of theprocess experiment of the ensuresexperiment that diensufferentres that levels different of each levels parameter of each are consideredparameter are the consideredsame number the of same times number in the scheme of times and in that the di schefferentme combinationsand that different of any combinations two parameters of occurany two the parameterssame number occur of times. the same Therefore, number the of uniformity times. Theref andore, rationality the uniformity of multiple and parametersrationality of at dimultiplefferent parameterslevels can be at guaranteed,different levels as cancan thebe guaranteed, reliability and as can representativeness the reliability and of testrepresentativeness results. Additionally, of test results.the number Additionally, of tests conducted the number can of be tests reduced conduc significantlyted can be without reduced aff significantlyecting the feasibility without of affecting the test theresults. feasibility Thus, ideal of testthe resultstest results. can be achievedThus, ideal based test on fewerresults representative can be achieved test points, based the sensitivityon fewer representativeof each parameter test topoints, the test the results sensitivity can be of determinedeach parameter and to optimal the test test results results can can be bedetermined obtained withand optimalthe minimum test results number can of be tests obtained [46]. with the minimum number of tests [46].

Figure 11. OrthogonalOrthogonal experimental experimental design design flow. ﬂow. 2.4.2. Orthogonal Experimental Research Level and Scheme Design 2.4.2. Orthogonal Experimental Research Level and Scheme Design Based on the mechanical properties and constitutive relations of the coal samples, the six mechanical Based on the mechanical properties and constitutive relations of the coal samples, the six parameters (E, υ, C, Φ,Rm and Ψ) measured in Table1 were taken as the initial participating factors of mechanical parameters (E, υ, C, Φ, Rm and Ψ) measured in Table 1 were taken as the initial the orthogonal experiment. Based on the results of previous studies considering the horizontal spacing participating factors of the orthogonal experiment. Based on the results of previous studies of diﬀerent parameters in orthogonal experiments [47,48], it was determined that each participating considering the horizontal spacing of different parameters in orthogonal experiments [47,48], it was 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,Rm and Ψ, respectively. The speciﬁc parameters adopted are presented in Table3. Energies 2020, 13, 3640 12 of 26

Table 3. Research level of orthogonal numerical simulation experimental parameters.

ABCDEF

E (GPa) υ C (MPa) Φ (◦)Rm (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

According to the orthogonal experiment design, a permutation combination scheme with six 6 parameters at five levels and twenty-five orthogonal numerical simulation schemes (L25 (5 )) 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 Table4.

Table 4. Orthogonal numerical simulation experimental schemes.

Mechanical Parameters Simulation Results Experimental 3 Scheme E (GPa) υ C (MPa) Φ (◦)Rm (MPa) Ψ (◦) UCS (MPa) PS (10− ) 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

3. Eﬀect of Mechanical Parameters on UCS and PS

3.1. Influence Analysis of Mechanical Parameters on UCS The range index was introduced to analyze the influence of mechanical parameters on UCS, based on orthogonality and the comprehensive comparability of the orthogonal experiment, both for additional analysis and to allow comprehensive comparability of the results with the orthogonal experiment. Range is used to express the variation in statistical data and to represent the difference between maximum and minimum values in the calculated mean UCS for a fixed value of one of the six parameters used in the orthogonal analysis. The size of the range reflects the degree of influence each parameter has under different levels of change; thus, it determines the degree of influence each parameter has on UCS and reveals how UCS varies with different parameters. The mean value and Energies 2020, 13, 3640 13 of 26 Energies 2020, 13, x FOR PEER REVIEW 13 of 27 range of of UCS UCS for for different different values values of ofvarious various parameters parameters were were calculated calculated fromfrom Table Table 4, as 4shown, as shown in Table in Table5. 5.

TableTable 5. AverageAverage and range for different diﬀerent values of each parameter.

A ABCDEFB C D E F UCSUCS (MPa) (MPa) E (GPa) υ C (MPa) Φ (°) Rm (MPa) Ψ(°) E (GPa) υ C (MPa) Φ (◦)Rm (MPa) Ψ (◦) AverageAverage 1 136.68 36.68 36.88 36.88 21.94 21.94 35.06 35.06 36.74 36.74 36.4236.42 AverageAverage 2 236.34 36.34 36.26 36.26 29.20 29.20 35.06 35.06 36.20 36.20 38.1238.12 AverageAverage 3 336.24 36.24 36.44 36.44 37.80 37.80 36.44 36.44 36.48 36.48 35.8835.88 AverageAverage 4 437.68 37.68 36.08 36.08 43.72 43.72 37.80 37.80 37.50 37.50 36.7436.74 AverageAverage 5 536.68 36.68 37.96 37.96 50.96 50.96 39.26 39.26 36.70 36.70 36.4636.46 Range 1.44 1.88 29.02 4.20 1.30 2.24 Range 1.44 1.88 29.02 4.20 1.30 2.24

As illustrated in in Table Table 44,, thethe rangesranges ofof E,E, υ,R, Rm,, ΨΨ and Φ are 1.44,1.44, 1.88, 1.30, 2.24 and 4.20, respectively. In In contrast, contrast, the the range range of of C C is is 29.02. 29.02. Thus, Thus, these these results results show show that that C C has the greatest influenceinfluence on UCS, followed by Φ, while E, υ,,R Rm andand ΨΨ have relatively little influence influence on UCS. To more intuitively demonstrate the influenceinfluence of each parameter on UCS, curve diagrams were produced to illustrate changes in horizontal compressive strength with changes in the average value (Table5 5)) of of the the corresponding corresponding compressivecompressive strengthstrength forfor eacheach parameterparameter inin TableTable4 .4. To To facilitate facilitate comparative analysisanalysis ofof thethe degree degree of of influence influence of of each each parameter parameter on on UCS, UCS, the th ordinatee ordinate range range of theof the six parameterssix parameters included included in the in curvethe curve diagrams diagrams was wa determineds determined to be to 20~52 be 20~52 MPa, MPa, and close-upand close-up plots plots were producedwere produced for the for parameters the parameters with relativelywith relatively small small amplitude amplitude changes, changes, as shown as shown in Figure in Figure 12. 12.

(a) E (b) υ

Figure 12. Cont.

Energies 2020, 13, 3640 14 of 26 Energies 2020, 13, x FOR PEER REVIEW 14 of 27

(e) Rm (f) Ψ

FigureFigure 12. 12. EffectEﬀect of of various various parameters parameters on on changes in rock compressive strength.

The following can be inferred based on the informationinformation presented in Table5 5 and and Figure Figure 12 12.. 1)(1) WhenWhen E E increases increases from from 2.81 2.81 to to 6.81 6.81 GPa, GPa, the the change change in in UCS UCS is minimal and the stress value fluctuatesfluctuates within within the range 36~3736~37 MPa.MPa. Therefore, thethe influenceinfluence of of the the elastic elastic modulus modulus on on UCS UCS is issmall. small. According According to theto the smaller smaller range range between between axis limits, axis limits, a nonlinear a nonlinear relationship relationship exists between exists betweenE and UCS. E and With UCS. increasing With increasing E, compressive E, compressi strengthve firststrength decreases first decreases before increasing before increasing and then anddecreasing then decreasing again; however, again; however, the fluctuation the fluctuatio range isn small,range indicatingis small, indicating that there that is a there relatively is a relativelystable critical stable range critical of UCSrange for of di UCSfferent for valuesdifferent of E,values as shown of E, as in shown Figure 12in a.Figure 12a. 2)(2) WithWith increasing increasing υυ,, UCS UCS exhibits exhibits a a fluctuating fluctuating curv curve.e. In In the the early early stage, stage, when when υ increases from 0.300.30 to 0.36,0.36, UCS UCS exhibits exhibits a decreasea decrease followed followed by an by increase an increase and a subsequentand a subsequent decrease, decrease, although althoughthe maximum the maximum change range change is only range 0.8 MPa.is only Conversely, 0.8 MPa. inConversely, the later stage, in the when laterυ stage,increases when from υ increases0.36 to 0.38, from the 0.36 compressive to 0.38, the strength compressive increases strength from 36.08 increases to 37.96 from MPa; 36.08 this increase to 37.96 of MPa; 1.88 MPathis increaseis relatively of 1.88 pronounced. MPa is relatively Thus, when pronounced.υ is small, Thus, changes when in UCS υ is aresmall, comparatively changes insmall UCS andare comparativelyUCS is not sensitive small and to changes UCS is not in Poisson’s sensitive ratio;to changes conversely, in Poisson’s when ratio;υ is comparatively conversely, when large, υ isthe comparatively sensitivity of large, UCS tothe changes sensitivity in υ ofincreases, UCS to changes as shown in in υ Figureincreases, 12b. as shown in Figure 12b. 3)(3) CC has has a a considerable considerable influence influence on on UCS UCS for for the the co coalal samples, with with UCS UCS ranging from 21.94 to 50.9650.96 MPa MPa for for different different C C levels. Based Based on on curve curve fo forr this this specific specific analysis, the fitting fitting equation betweenbetween C C and and UCS UCS is y = 3.628x3.628x + 0.8794,0.8794, with with R 22 == 0.9973.0.9973. There There is is a a clear clear linear linear relationship relationship betweenbetween C C and and UCS, UCS, with with a a range range of 29.02. Thus, Thus, UCS UCS is is sensitive sensitive to to changes in C, increasing linearly.linearly. The The results results also illustrate thatthat UCSUCS doesdoes not not converge converge to to a a critical critical range range with with variation variation in inC, C, as as shown shown in in Figure Figure 12 12c.c. 4) The influence of Φ on UCS can be divided into two stages: when Φ is less than 32°, UCS (4) The influence of Φ on UCS can be divided into two stages: when Φ is less than 32◦, UCS remains remains essentially unchanged with increasing Φ; conversely, when Φ is greater than 32°, UCS essentially unchanged with increasing Φ; conversely, when Φ is greater than 32◦, UCS increases increasesrapidly with rapidly increasing with Φincreasingand there Φ is aand significant there is linear a significant growth relationship linear growth between relationship the two betweenvariables. the Thus, two Φvariables.has a significant Thus, Φ impact has a onsignificant UCS and impact the degree on UCS of its and influence the degree is relatively of its influencelarge. Based is relatively on a comprehensive large. Based consideration on a comp ofrehensive the whole consideration curve, a nonlinear of the relationship whole curve, exists a nonlinear relationship exists between UCS and Φ: when Φ exceeds 32°, UCS increases linearly between UCS and Φ: when Φ exceeds 32◦, UCS increases linearly with increasing Φ and UCS withdoes increasing not have a Φ stable and UCS critical does range, not have as shown a stable inFigure critical 12 range,d. as shown in Figure 12d. 5) Based on the relationship between Rm and UCS, changes in the range of UCS with increasing (5) Based on the relationship between Rm and UCS, changes in the range of UCS with increasing Rm Rmare veryare very small, small, with with minimum minimum and maximumand maximum values values of 36.2 of and36.2 37.5 and MPa, 37.5 MPa, respectively respectively (a range (a rangeof only of 1.3 only MPa). 1.3 Therefore,MPa). Therefore, Rm has Rm little has influence little influence on UCS. on There UCS. is aThere significant is a significant nonlinear nonlinearrelationship relationship between the between two variables the two according variables to the according microscopic to analysisthe microscopic diagram, asanalysis shown diagram,in Figure as12 showne. in Figure 12e. 6) The relationship between Ψ and UCS indicates that, when Ψ is less than 10°, UCS first (6) The relationship between Ψ and UCS indicates that, when Ψ is less than 10◦, UCS first increases increases from 36.42 to 38.12 MPa and then decreases to 35.88 MPa with increasing Ψ (a range from 36.42 to 38.12 MPa and then decreases to 35.88 MPa with increasing Ψ (a range of 2.24 MPa). of 2.24 MPa). Conversely, when Ψ is greater than 10°, UCS increases from 35.88 to 36.74 MPa Conversely, when Ψ is greater than 10◦, UCS increases from 35.88 to 36.74 MPa and then decreases 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.

Energies 2020, 13, 3640 15 of 26

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. Energies 2020, 13, x FOR PEER REVIEW 16 of 28 3.2. Influence Analysis of Mechanical Parameters on PS 3.2. Influence Analysis of Mechanical Parameters on PS Along with strength properties, the deformation properties of coal can be considered critical mechanicalAlong properties. with strength The properties, deformation the anddeformation failure of properties coal masses of coal often can aff beect considered both the stability critical of rockmechanical surrounding properties. roadways The deformation and the dynamic and failure strength of coal of themasses rock often mass affect during both the the process stability of of coal mining.rock surrounding Therefore, roadways it is of great and importance the dynamic to strength be able of to the control rock mass rock surroundingduring the process roadways of coal and preventmining. dynamic Therefore, disasters it is of by great understanding importance to fully be deformationable to control and rock failure surrounding laws and roadways the mechanical and responseprevent characteristics dynamic disasters of coal. by understanding fully deformation and failure laws and the mechanical responseBased oncharacteristics stress–strain of curves,coal. the deformation of coal goes through three stages before reaching UCS: theBased compaction on stress–strain stage, curves, the elastic the deformation stage and theof coal plastic goes stage.through There three isstages a positive before correlationreaching UCS: the compaction stage, the elastic stage and the plastic stage. There is a positive correlation between between the stress and strain of coal masses during this process: the greater the deformation of the coal the stress and strain of coal masses during this process: the greater the deformation of the coal mass, the mass, the greater the required stress. When the coal mass reaches UCS, the bearing capacity of the coal greater the required stress. When the coal mass reaches UCS, the bearing capacity of the coal mass mass reaches its maximum; after exceeding UCS, the stress decreases with increasing strain, exhibiting reaches its maximum; after exceeding UCS, the stress decreases with increasing strain, exhibiting an an obvious strain softening phenomenon, as shown in Figure 13. Therefore, the strain corresponding to obvious strain softening phenomenon, as shown in Figure 13. Therefore, the strain corresponding to the thecoal coal mass mass peak peak is the is the PS, PS, and and its magnitude its magnitude has a has vital a vitalinfluence influence on the on mechanical the mechanical response response of the coal of the coalmass mass when when it is it damaged. is damaged. The The brittle brittle characteristic characteristicss of coal of coalfailure failure are dominant are dominant for low for PS, low which PS, which is is prone to inducing dynamic dynamic disaster; disaster; conversely, conversely, the theplastic plastic characteristics characteristics are dominant are dominant for higher for higher PS, PS,which which is isprone prone to toinducing inducing large-scale large-scale deformation deformation of the of rock the surrounding rock surrounding roadways roadways [49,50]. [This49,50 ]. Thisdemonstrates demonstrates that thatPS is PS one is oneof the of major the major indices indices able to able characterize to characterize the mechanical the mechanical response response of coal of coalmasses. masses.

FigureFigure 13.13. Whole stress–strain process process curve. curve.

BasedBased on on the the orthogonal orthogonal experimentexperiment results presen presentedted in in Table Table 4,4 ,the the mean mean and and range range of of PS PS at at diﬀdifferenterent levels levels of of each each parameter parameter considered considered werewere obtained,obtained, as shown shown in in Table Table 6.6.

Table 6. Statistics describing the influence of mechanical parameters on PS. Table 6. Statistics describing the inﬂuence of mechanical parameters on PS.

A ABCDEFB C D E F PS (10−3) 3 PS (10− ) E (GPa)E (GPa) υ υC (MPa)C (MPa) Φ(°)Φ ( ◦)RmR(MPa)m (MPa) Ψ (◦) Ψ(°) AverageAverage 1 1 6.24 6.24 3.85 3.85 2.53 2.53 3.87 3.87 3.97 3.97 3.91 3.91 AverageAverage 2 2 4.77 4.77 4.05 4.05 3.32 3.32 3.82 3.82 4.06 4.06 4.14 4.14 AverageAverage 3 3 3.78 3.78 4.24 4.24 4.24 4.24 3.93 3.93 3.94 3.94 4.15 4.15 Average 4 3.28 4.28 4.98 4.36 4.36 4.20 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 Average 5Range 2.74 3.50 4.33 0.48 5.68 3.15 4.78 0.96 0.49 4.43 0.46 4.37 Range 3.50 0.48 3.15 0.96 0.49 0.46

According to Table 6, the ranges in E and C are 3.50 and 3.15, respectively. It can be seen that E has the most pronounced influence on PS, followed by C; the ranges of υ, Φ, Rm and Ψ are relatively small, and their influences on PS are also small. To more intuitively characterize the influence of

Energies 2020, 13, 3640 16 of 26

According to Table6, the ranges in E and C are 3.50 and 3.15, respectively. It can be seen that E has the most pronounced influence on PS, followed by C; the ranges of υ, Φ,Rm and Ψ are relatively small,Energies and 2020 their, 13, influences x FOR PEER REVIEW on PS are also small. To more intuitively characterize the influence17 of 28 of various mechanical parameters on PS, curves were plotted to illustrate the relationships between these various mechanical parameters on PS, curves were plotted to illustrate the relationships between parameters and PS and the ordinates of all curves were unified with the same max and min values on these parameters and PS and the ordinates of all curves were unified with the same max and min the y-axis. Smaller ranges between axis limits were produced for the parameters with lower sensitivity, values on the y-axis. Smaller ranges between axis limits were produced for the parameters with as shownlower in sensitivity, Figure 14 as. shown in Figure 14.

(a) E (b) υ

(e) Rm (f) Ψ

FigureFigure 14. 14.Inﬂuence Influence of of variation variation inin each parameter parameter on on peak peak strain strain sensitivity. sensitivity.

The followingThe following inferences inferences can can be be made made based based onon Table 6 and Figure Figure 14. 14 . 1) Based on the curve in Figure 14a, the relationship between E and PS is approximately linear. (1) Based on the curve in Figure 14a, the relationship between E and PS is approximately linear. When When the other parameters remain constant, PS decreases with increasing E, which is closely the otherrelated parameters to the deformation remain constant,resistance PSof the decreases object as with characterized increasing by E, E. which For large is closely values relatedof E, to the deformationthe deformation resistance resistance of the is greater object asand characterized the deformation by E. is For less large pronounced; values of conversely, E, the deformation for resistancesmaller is values greater of and E, thethe deformationdeformation isresist lessance pronounced; is lesser conversely,and the deformation for smaller is values more of E, the deformationpronounced. 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

Energies 2020, 13, 3640 17 of 26

(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 indicatesEnergies 2020 that, 13, x PS FOR growth PEER REVIEW with increasing Poisson’s ratio occurs in two stages: in the first18 stage, of 28 υ increases from 0.3 to 0.34 and the PS growth rate is 10.13%; in the second stage, υ increases from limits indicates that PS growth with increasing Poisson’s ratio occurs in two stages: in the first 0.34 to 0.38 and the PS growth rate is only 2.12%. The growth rate of PS in the second stage is stage, υ increases from 0.3 to 0.34 and the PS growth rate is 10.13%; in the second stage, υ significantlyincreases lower from than0.34 to that 0.38 in and the firstthe PS stage, growth although rate is aonly relatively 2.12%. significantThe growth linearrate of relationship PS in the existssecond between stageυ andis significantly PS in each lowe stage,r than as shown that in in the Figure first stage,14b. although a relatively significant (3) Accordinglinear torelationship Figure 14 existsc, PS increasesbetween υlinearly and PS in from each 2.53 stage, to as 5.68 shown (range in Figure of 3.15) 14b. with increasing C. Other3) According than E, C to can Figure be considered 14c, PS increases to have linearly the largest from influence2.53 to 5.68 on (range PS. of 3.15) with increasing (4) The influenceC. Other than of Φ E,on C can PS be can considered also be consideredto have the largest to exhibit influence two stages.on PS. When Φ is less than 4) The influence of Φ on PS can also be considered to exhibit two stages. When Φ is less than 34°, 34◦, the fluctuation range of PS is very small with increasing Φ and Φ thus has no effect on PS. 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 influenceConversely, of Φ on when PS is Φ greater, is greater as than shown 34°, in PS Figure increases 14d. linearly with increasing Φ and the degree of influence of Φ on PS is greater, as shown in Figure 14d. (5) The influence of Rm on PS is variable, with no obvious regularity; the range is only 0.49 and Rm 5) The influence of Rm on PS is variable, with no obvious regularity; the range is only 0.49 and Rm has little influence on PS. However, PS tends broadly to increase with increasing R , as shown in has little influence on PS. However, PS tends broadly to increase with increasing mRm, as shown Figurein 14Figuree. 14e. (6) Based6) Based on the on curve the curve plotting plotting PS againstPS againstΨ, PSΨ, PS increases increases from from 3.91 3.91 to to 4.37 4.37 as asΨ Ψincreases increases from from 8 8°◦ to 12◦. Thisto 12°. fluctuation This fluctuation range is range small, is indicating small, indicating that the that influence the influence of Ψ on of PS Ψ is on small. PS is Although small. the mainAlthough plot indicates the main aplot lack indicates of any cleara lack linear of any relationship clear linear relationship between Ψ andbetween PS, theΨ and microscopic PS, the analysismicroscopic diagram analysis indicates diagram that PS indicates increases that linearly PS increases with increasing linearly withΨ before increasing leveling Ψ before off, and then exhibitingleveling off, a and further then linear exhibiting increase a further with Ψ linearat higher increase values with of ΨΨ ,at as higher shown values in Figure of Ψ 14, asf. shown in Figure 14f. The relationship between C and PS has been plotted for different values of E, as shown in Figure 15. The relationship between C and PS has been plotted for different values of E, as shown in Figure 15. The influence of C on PS is closely related to E. When E is 2.81 GPa, PS increases from 3.61 to 9.06 The influence of C on PS is closely related to E. When E is 2.81 GPa, PS increases from 3.61 to 9.06 (range (range of 5.45) as C increases from 5.88 to 13.88 MPa; conversely, when E is 6.81 GPa, PS increases of 5.45) as C increases from 5.88 to 13.88 MPa; conversely, when E is 6.81 GPa, PS increases from 1.85 to from 1.853.95 to(range 3.95 of (range 2.1) for of the 2.1) same for therange same of C. range Thus, ofthe C. influence Thus, the of C influence on PS decreases of C on with PS decreases increasing withE. increasingOverall, E. Overall,C and PS Cexhibit and PSa linear exhibit relati a linearonship relationshipregardless of regardlessthe value of ofE. the value of E.

FigureFigure 15. Inﬂuence 15. Influence degree degree of cohesion of cohesion on on peak peak strain strain under under di differentﬀerent elasticelastic modulus conditions. conditions.

3.3. Influence3.3. Influence Analysis Analysis of Mechanical of Mechanical Parameters Parameters on on Critical Critical Failure Failure Strength Strength ofof Coal Samples For theFor whole the whole stress–strain stress–strain curve, curve, UCS UCS represents represents the the maximum maximum load-bearingload-bearing capacity capacity of of the the coal coal mass,mass, and PS and can PS be can used be used to describe to describe the the deformation deformation characteristics characteristics ofof thethe coal mass mass under under uniaxial uniaxial compressioncompression failure. failure. Therefore, Therefore, both PSboth and PS UCS and canUCS be can used be to used characterize to characterize the critical the critical failure strengthfailure of a coalstrength mass. of a coal mass. BasedBased on analysis on analysis of theof the influence influence of of mechanical mechanical parameters parameters on on UCS UCS and and PS for PS coal for masses, coal masses, the the degreedegree of of influence influence ofof thethe mechanical parameters parameters on on UCS UCS decreases decreases in the in following the following order: order: C > Φ C>Ψ>> Φ υ > E > Rm. Conversely, the degree of influence of these parameters on PS decreases in the following >Ψ> υ > E > Rm. Conversely, the degree of influence of these parameters on PS decreases in the order: E > C > Φ > Rm > υ >Ψ. Thus, these mechanical parameters have different influences on UCS and following order: E > C > Φ > R > υ >Ψ. Thus, these mechanical parameters have different influences PS. To more fully reveal them influence of these parameters on UCS and PS, a normalization method on UCSwas and used PS. to To map more all fullyrange reveal data presented the influence in Tables of these 5 and parameters 6 onto the onrange UCS 0~1, and asPS, shown a normalization in Table 7. The normalization method is to scale the data that need to be processed to a small specific interval. To

Energies 2020, 13, 3640 18 of 26 method was used to map all range data presented in Tables5 and6 onto the range 0~1, as shown in Table7. The normalization method is to scale the data that need to be processed to a small speciﬁcEnergies interval. 2020, 13 To, x facilitate FOR PEER REVIEW data processing, the data were mapped to a range of 0~1 for processing.19 of 28 The normalization results are plotted as a histogram for processing and comparison, as shown in FigureEnergies 16facilitate. 2020 data, 13, x processing,FOR PEER REVIEW the data were mapped to a range of 0~1 for processing. The normalization19 of 28 results are plotted as a histogram for processing and comparison, as shown in Figure 16. facilitateTable data 7. Normalized processing, rangethe data of eachwereparameter mapped to corresponding a range of 0~1 tofor peak processing. stress and The peak normalization strain. results areTable plotted 7. Normalizedas a histogram range for of processi each parameterng and correspcomparison,onding as to shownpeak stress in Figure and peak 16. strain. E υ C Φ Rm Ψ E υ C Φ Rm Ψ Table 7. Normalized range of each parameter corresponding to peak stress and peak strain. Range (UCS)Range (UCS) 1.44 1.44 1.88 1.88 29.02 29.02 4.20 4.20 1.30 1.30 2.24 2.24 Range (PS)Range (PS) 3.50 3.50E 0.48 0.48υ 3.15 3.15C 0.96Φ 0.96 0.49R 0.49m Ψ 0.46 0.46 Range normalizationRangeRange normalization (UCS) (UCS) (UCS)0.036 1.44 0.036 0.047 1.88 0.047 0.724 29.02 0.724 0.105 4.20 0.105 0.032 1.30 0.032 2.24 0.056 0.056 Range normalizationRange normalizationRange (PS)(PS) (PS) 0.387 3.50 0.387 0.053 0.48 0.053 0.349 3.15 0.349 0.106 0.96 0.106 0.054 0.49 0.054 0.46 0.051 0.051 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

FigureFigure 16. Histogram 16. Histogram of the of the inﬂuence influence of of mechanical mechanical parameters on on peak peak stress stress and and strain. strain.

AccordingAccordingFigure to this 16. to comparativeHistogramthis comparative of the analysis, influence analysis, of only onlymechanical C C hashas anparameters important on peakinfluence influence stress onand onboth strain. both UCS UCS and PS, and PS, and E hasand aE significanthas a significant influence influence only only on on PS. PS. The The parameter parameter Φ hashas the the same same influence influence on UCS on UCS as on as on PS, andPS, all Accordingand other all other mechanical to thismechanical comparative parameters parameters analysis, considered considered only C has havehave an important a relatively influence small small influence on influence both UCSon both onand both UCSPS, UCS andand E PS. has The a significant results illustrate influence that only C has on PS.the Thegreatest parameter influence Φ has on the samecritical influence failure strength on UCS ofas coalon and PS. The results illustrate that C has the greatest influence on the critical failure strength of coal PS,masses and allunder other compression, mechanical andparameters that increased considered cohesion have cana relatively improve small the critical influence failure on strengthboth UCS of massesandcoal under PS. significantly. The compression, results Thus,illustrate and it can thatthat be increasedC inferred has the greatestthat cohesion the influencestability can improve ofon rock the thecriticalsurrounding critical failure failure roadwaysstrength strength of can coal be of coal significantly.massesenhanced under Thus, effectively compression, it can beby inferredgrouting and that and that increased an thechoring stability cohesion to improve of rock can Cimprove surrounding in the rock the criticalmass. roadways failure can strength be enhanced of effectivelycoal significantly. by grouting Thus, and anchoringit can be inferred to improve that the C instability the rock of rock mass. surrounding roadways can be enhanced4. Stepwise effectively Regression by grouting Prediction and of an UCSchoring to improve C in the rock mass. 4. Stepwise Regression Prediction of UCS According to the analysis described above, different mechanical parameters have different 4. Stepwise Regression Prediction of UCS Accordingdegrees of to influence the analysis on UCS: described C has the above, greatest di influencefferent mechanical on UCS, followed parameters by Φ, with have the di influencefferent degrees of of influenceotherAccording mechanical on UCS: to C parametersthe has analysis the greatest being descr relatively influenceibed above, small. on different UCS,Based followedon mechanic this, a regression byal Φparameters, with equation the have influence was different derived of other degrees of influence on UCS: C has the greatest influence on UCS, followed by Φ, with the influence of mechanicalby establishing parameters a predictive being relatively model small.of UCS Basedusing onstepwise this, aregression regression analysis, equation and was the derived main by othercontrolling mechanical parameters parameters were being used relativelyfor rapid predictionsmall. Based of onUCS, this, in a order regression to obtain equation relatively was derivedaccurate establishing a predictive model of UCS using stepwise regression analysis, and the main controlling byprediction establishing of values a predictive with fewer model variables. of UCS The usintechnicalg stepwise workflow regression for this isanalysis, shown inand Figure the 17.main parameterscontrolling were parameters used for rapid were used prediction for rapid of prediction UCS, in order of UCS, to obtainin order relatively to obtain relatively accurate accurate prediction of valuesprediction with fewer of values variables. with Thefewer technical variables. workflowThe technical for workflow this is shown for this in is Figure shown 17in .Figure 17.

Figure 17. Predictive regression analysis technical workflow.

4.1. Stepwise RegressionFigureFigure Equation 17. 17.Predictive Predictive Analysis regression ofregression Relevant analysis Explanatory technical technical Variables workflow. workﬂow. and Regression Model Establishment 4.1. Stepwise4.1. Stepwise Regression Regression Equation Equation Analysis Analysis of of Relevant Relevant Explanatory Variables Variables and and Regression Regression Model The stepwise regression method is introducing all the independent variables into the regression ModelEstablishment model and determining the significance level of each independent variable. Eliminating the Thenon-significant stepwiseThe stepwise regression independent regression method method variables is is introducing introducing and reducing all the thethe independent independentnumber of variables independent variables into intothe variables regression the regression are modelmodel and determiningand determining the the signiﬁcance significance levellevel of each independent independent vari variable.able. Eliminating Eliminating the the non-significant independent variables and reducing the number of independent variables are

Energies 2020, 13, 3640 19 of 26 non-significant independent variables and reducing the number of independent variables are advantageous to the fast predicting of the changing trend of dependent variables. At the same time, using the identified significant independent variables to establish the regression equation, the optimal interpretive parameter set can be obtained. Firstly, SPSS software was used to introduce all the participating factors into the regression model one by one. F test is generally used to compare the prediction model to the dataset for each factor and determine whether there is a significant difference in the correlation between the dependent variable and the independent variable. At the same time, the Student’s t-test was performed for the factors with obvious significance. When the significance level of other variables decreases due to the introduction of new variables, the model will automatically remove the variables with decreased significance, until no significant variables participate in the regression equation and no insignificant variables are removed by the equation, thus only the independent variables with significant effect are finally retained. In this paper, SPSS software was used to determine the independent variables of significance as C and Φ. Meanwhile, explanatory parameters of correlation regression test were recorded, including Variance Inflation Factor (VIF) value for multiple linear test, Durbin–Watson (D–W) value for autocorrelation in the residuals from the statistical regression analysis test, residual normality test and heteroscedasticity test. The specific parameters are shown in Table8.

Table 8. Related interpretation parameter sets of stepwise regression analysis.

Non-Standardized Standardization Coeﬃcient Coeﬃcient Adjusted t p VIF R2 F R2 B Standard Error Beta Constant 18.059 3.704 - 4.876 0.000 ** - − − 619.574 C 3.628 0.104 0.980 34.79 0.000 ** 1.000 0.983 0.981 (0.000 **) Φ 0.557 0.104 0.150 5.342 0.000 ** 1.000 Supplement: dependent variable—UCS. D–W value: 1.829. ** p < 0.01. Thank you for your advice.

In Table8, the value of the non-standardized coe fficient B is the value of the regression coefficient. B > 0 indicates that the influence of the independent variable on the dependent variable is positive and significant when p < 0.05, and the standard error indicates the degree of fluctuation of the regression coefficient. The standardized coefficient is the value of the regression coefficient for the related independent variable when the constant is 0; the t value is the intermediate variable for calculating the p value, and the p value is used to test the significance level of the coefficient. The smaller is the p value, the greater is the significance. R2 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 R2 is to 1, the more accurate is the model and the better is the fit of the regression equation. For this regression analysis, R2 = 0.983, which demonstrates that C and Φ can accurately represent changes in UCS. The adjusted R2 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. To ensure the reliability of regression model, further tests were undertaken to investigate the correlation of the stepwise regression model, as shown in Figure 18.

(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. Table8 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 Energies 2020, 13, x FOR PEER REVIEW 21 of 28

Energieswell2020 constructed., 13, 3640 20 of 26 2) Autocorrelation refers to correlation between the expected values of independent variables that Energieshave no 2020 significant, 13, x FOR PEER influence REVIEW on the dependent variables, which is determined by the D–W21 of 28 value. Ifmodel. the D–W Therefore, value is thenear mechanical 2 (specifically parameters 1.7–2.3), the of the model orthogonal is well construc experimentted. The were D–W found value to beof independentwell constructed. of each other, the model did not have multiple linear problems and the model was 2)this modelAutocorrelation is 1.829 andrefers the to correlationmodel construction between th wae expecteds demonstrated values of independentto be reasonable variables as there that was well constructed. no autocorrelationhave no significant among influence the on independent the dependent variab variables,les thatwhich had is determined no significant by the influence D–W value. on the (2) dependentAutocorrelationIf the D–W variables. value refers is near to correlation 2 (specifically between 1.7–2.3), the the expectedmodel is well values construc of independentted. The D–W variables value of that 3) Thehave thisresidual no model significant term is 1.829 represents influence and the model onthe the difference construction dependent between wa variables,s demonstrated the observed which istodetermined bevalue reasonable of each by as sample the there D–W was and value. the valueIf theno estimated D–W autocorrelation value by is the near among model. 2 (specifically the In independentgeneral, 1.7–2.3), the variabnormal theles modeldistribution that had is wellno issignificant constructed.used to testinfluence the The residual D–Won the value term: ifof the thisdependent test model residual isvariables. 1.829 data and conform the model to the construction normal distribution, was demonstrated the model to can be be reasonable considered as thereto be 3) The residual term represents the difference between the observed value of each sample and the wellwas noconstructed. autocorrelation Residual among normality the independent is used to variablestest the reliability that had noand significant periodicity influence of data onbased the value estimated by the model. In general, the normal distribution is used to test the residual term: ondependent experimental variables. sample data. The analysis of residual normality is only one of the reliability test if the test residual data conform to the normal distribution, the model can be considered to be (3) The residual term represents the difference between the observed value of each sample and the indiceswell to constructed. test the stepwise Residual regression normality is model. used to To te stjudge the reliability whether and the periodicityregression of model data based conforms value estimated by the model. In general, the normal distribution is used to test the residual term: to theon standard,experimental only sample the data.residual The analysisitems need of resi todual meet normality the normal is only distribution one of the reliability intuitively. test The residualif theindices test values residual to test of the dataregression stepwise conform regression analysis to the model.data normal in To distribution,this judge paper whether conform the the model regression to the can normalmodel be considered conforms distribution, to be indicatingwellto constructed. the standard,that the Residual stepwiseonly the normality residualregression items is mo used needdel to isto testreasonable, meet the the reliability normal as shown distribution and in periodicity Figure intuitively. 19. of data The based 4) Theon experimental residualheteroscedasticity values sample of regression was data. validated The analysis analysis bydata ofplo in residual ttingthis paper scatter normality conform diagram is to only the of onenormal the of independent thedistribution, reliability testand dependentindicesindicating to test variables that the stepwisethe stepwisewith regressionthe regression residual model. moterm.del To isThe reasonable, judge heteroscedasticity whether as shown the regression in Figurecan be 19. model compared conforms to the to 4)variance:the standard,The heteroscedasticity it is onlythe difference the residual was in itemsvalidated variance need bybetween to plo meettting the the scatter normal explaining diagram distribution variable of the intuitively. thatindependent omitted The fromand residual the dependent variables with the residual term. The heteroscedasticity can be compared to the modelvalues ofand regression the analysisunimportant data inexplained this paper conformvariable. toAccordingly, the normal distribution, to determine indicating the variance: it is the difference in variance between the explaining variable that omitted from the heteroscedasticity,that the stepwise regression the data modelcan be is reasonable,examined fo asr shownsigns of in Figureregularity 19. in the corresponding model and the unimportant explained variable. Accordingly, to determine the (4) The heteroscedasticity was validated by plotting scatter diagram of the independent and scatteredheteroscedasticity, points. If the the residualdata can isbe notably examined in creasedfor signs or of decreased regularity within the any corresponding increase in the independentdependentscattered variables points.variable, If with thethe residualregularity the residual is notablyis obvious; term. increased The then, heteroscedasticity orthe decreased model has with heteroscedasticity can any be increase compared in the and to the its constructionvariance:independent it is can the variable, dibeff erenceconsidered the regularity in variance poor. is obvious; betweenAll of then, thethe explainingtheindependent model has variable heteroscedasticity and thatdependent omitted and variables from its the consideredmodelconstruction and thehere unimportant werecan be plotted considered explained in a sca poor.tter variable. diagram,All ofAccordingly, the as independentshown toin determineFigure and 20.dependent the heteroscedasticity, variables the dataconsidered can be here examined were plotted for signsin a sca oftter regularity diagram, inas shown the corresponding in Figure 20. scattered points. If the The scatter diagrams in Figure 20a–g illustrate that the regression analysis data exhibit no residual is notably increased or decreased with any increase in the independent variable, the regularity.The When scatter the diagrams independent in Figure variables 20a–g illustrateare varied, that thethe residualregression items analysis do datanot exhibitexhibit noregular regularity is obvious; then, the model has heteroscedasticity and its construction can be considered increasesregularity. or decreases. When the Therefore,independent there variables is no are co varied,rrelation the betweenresidual itemsthe residual do not exhibititems regularand related increasespoor. All or of decreases. the independent Therefore, and there dependent is no correlation variables between considered the residual here were items plotted and related in a scatter variables and no heteroscedasticity, indicating that the model is well constructed. variablesdiagram, and as shownno heteroscedasticity, in Figure 20. indicating that the model is well constructed.

Figure 18. Correlation test of regression model. FigureFigure 18.18. Correlation test of regression model.model.

FigureFigure 19. 19.Normal Normal distributiondistribution of of residual residual data. data.

Figure 19. Normal distribution of residual data.

EnergiesEnergies2020 2020,, 1313,, 3640x FOR PEER REVIEW 2221 of of28 26

(a) Scatter diagram of elastic modulus

(b) Scatter diagram of Poisson’s ratio

(d) Scatter diagram of angle of internal friction

(e) Scatter diagram of tensile strength

Figure 20. Cont.

Energies 2020, 13, x FOR PEER REVIEW 23 of 28 Energies 2020, 13, 3640 22 of 26

(f) Scatter diagram of dilatancy angle

(g) Scatter diagram of uniaxial compressive strength

FigureFigure 20. 20. ScatterScatter diagram diagram of ofrelated related variables variables and and residuals residuals in the in the regression regression model. model.

InThe summary, scatter diagrams the indepe inndent Figure variables20a–g illustrate C and thatΦ exhibit the regression significant analysis correlations data exhibit with noUCS. regularity. As indicatedWhen the by independent the regression variables coefficients, are these varied, two the independent residual items variables do not have exhibit a positive regular relationship increases or withdecreases. UCS, which Therefore, is in line there with is nothe correlationresults of the between orthogonal the residualexperiment; items the and inte relatedrpretation variables parameters and no areheteroscedasticity, also considered to indicating be optimal. that In addition, the model test is wellparameters constructed. such as multicollinearity, autocorrelation, residualIn normality summary, and the heteroscedasticity independent variables are also in C line and withΦ theexhibit corresponding signiﬁcant standards. correlations Based with on the UCS. principleAs indicated of stepwise by the regressionregression coeanalysis,ﬃcients, a regression these two model independent for UCS, variables C and Φ have was a positiveestablished relationship and a functionalwith UCS, relationship which is in linebetween with thethe results main ofcontrolling the orthogonal parameters experiment; and theUCS interpretation was determined parameters as follows.are also considered to be optimal. In addition, test parameters such as multicollinearity, autocorrelation, residual normality and heteroscedasticity are also in line with the corresponding standards. Based on y = β0 + β1x1 + β2x2 (8) the principle of stepwise regression analysis, a regression model for UCS, C and Φ was established and wherea functional y is UCS relationship (MPa), x1 betweenis C (MPa), the x main2 is Φ controlling (in degrees) parameters and the coefficients and UCS was β1 determinedand β2 indicate as follows. the degree of influence of the main controlling parameters on UCS.

According to the principle of stepwisey = regressiβ0 + β1onx1 +analysis,β2x2 the standardized coefficient of the (8) regression model represents primarily the degree of influence of different independent variables on the dependentwhere y is variable. UCS (MPa), It comparesx1 is C the (MPa), relativex2 isimportanceΦ (in degrees) of all influencing and the coe parametersfficients β1 onand theβ 2dependentindicate the variabledegree after of influence nondimensionalization of the main controlling of the paramete parametersrs, and on the UCS. non-standardized coefficient should be used forAccording actual prediction to the principle using the of stepwiseregression regression model. The analysis, final stepwise the standardized regression coeequationfficient is ofas the follows.regression model represents primarily the degree of influence of different independent variables on the dependent variable. It compares the relative importance of all influencing parameters on the dependent y = 3.628x1 + 0.557x2 − 18.059 (9) variable after nondimensionalization of the parameters, and the non-standardized coefficient should be used for actual prediction using the regression model. The final stepwise regression equation is 4.2. Reliability Analysis of Stepwise Regression Equation of Coal Compressive Strength as follows. To further verify the accuracy andy = adaptability3.628x + 0.557 of thex stepwise18.059 regression in Equation (9) to the (9) 1 2 − prediction of the UCS of coal seams, the mechanical parameters of coal seams such as Zhaolou (ZL), Baodian4.2. Reliability (BD), Huaheng Analysis of (HH), Stepwise Xinhe Regression (XH) and Equation Dong oftan Coal (DT) Compressive were used. Strength The second regression equationTo further(σc = 11.31 verify + 4.19x the1 accuracy − 0.017x12 and, where adaptability x1 is the elastic of the modulus) stepwise regressionfor predicting in Equation the UCS (9)of the to the 11-2prediction coal roof of therock UCS in the of coalHuainan seams, mining the mechanical area was parameterscompared and of coal analyzed seams suchusing as the Zhaolou method (ZL), previouslyBaodian (BD), described Huaheng in [22]. (HH), The calculation Xinhe (XH) results and Dongtanare shown (DT) in Table were 9. used. The second regression

Energies 2020, 13, 3640 23 of 26

2 equation (σc = 11.31 + 4.19x 0.017x , where x is the elastic modulus) for predicting the UCS of 1 − 1 1 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 Table9.

Table 9. Prediction results of compressive strength of coal seams in diﬀerent coal mines.

Quadratic Stepwise Regression Regression E C Φ UCS Coal Sample Predicted Predicted (GPa) (MPa) (◦) (MPa) Error Error Value Value 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%

According to the compressive strength values of diﬀerent coal seams, the maximum error of the prediction results of a stepwise regression equation was seen for Zhaolou coal seam 3 (14.13%); the other prediction errors were less than 10%. The minimum error, of just 0.57%, was seen for 3down 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 > Rm. 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 > Rm > 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.

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. Energies 2020, 13, 3640 24 of 26

Author Contributions: Conceptualization, P.Z.and J.W.; methodology, P.Z.and L.J.; software, J.W.; validation, T.Z., L.Y.and W.C.; formal analysis, P.Z.;investigation, J.W.and X.Y.; resources, T.Z.; data curation, J.W.; writing—original draft preparation, P.Z.; writing—review and editing, P.Z.; visualization, J.W.; project administration, P.Z.; and funding acquisition, P.Z. All authors have read and agreed to the published version of the manuscript. Funding: The research for this study was sponsored by the National Natural Science Foundation of China (51704185), the Natural Science Foundation of Shandong Province (ZR2017BEE045), and A Project of Shandong Province Higher Educational Science and Technology Program (J17KA218). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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