Two dimensional stability evaluation of a single entry longwall mining system

The purpose of this study was to determine whether a single entry is as safe for the development of longwail panels than the present double entry system at the Sunnyside Coal Mine near Sunnyside, Utah. The work is a joint effort by the U.S. Bureau of Mines in cooperation with Kaiser Steel Corporation and the University of Utah. The study included field instrumentation and measurements as well as data processing and interpretation. In-situ and laboratory tests were conducted to estimate the physical and mechanical properties of rock types present at the Sunnyside Mine. Theoretical studies included prediction of stress concentrations and failure zones as well as comparisons between theoretical and field measurements of displacements about mine openings. A finite element computer program was used for theoretical study, comparison of field data, and for simulation of face advance.

TWO DIMENSIONAL STABILITY EVALUATION OF A SINGLE ENTRY LONGWALL MINING SYSTEM by Thummala Venkat Rao A thesis submitted to the faculty of the University of Utah in partial fulfillment of the requirements for the degree of Master of Science in Mining Engineering Department of Mining, Metallurgical, and Fuels Engineering University of Utah June 1974 UNIVERSITY OF UTAH GRADUATE SCI-TOOL SUPERVISORY COMMITTEE APPROVAL of a thesis submitted by Thumma1a Venkat Rao I have read this thesis and have found it to be of satisfactory quality for a master's degree. May 13, 1974 William Date G. Pariseau, Ph.D. Chainnan, Supervisory Committee I have read this thesis and have found it to be of satisfactory quality for a master's degree. May 138 1974 Date �rohn E. Willson, E.M. Member, Supervisory Committee T have read thj� thesis and have found it to be of satisfactory quality for a master's degree. May 13, Date 1974 UNIVERSITY OF UTAH GRADUATE SCHOOL FINAL READING APPROVAL To the Graduate Council of the University of Utah: Thummala Venkat Rao I have read the thesis of in its final fonn and have found that (1) its fonnat, citations, and bibliographic style are consistent and acceptable; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the Supervisory Committee and is ready for submission to the Graduate School. May 13, 1974 Date G. William Pariseau, Member, Supervisory Committee Approved for the Major Department Ferron A. Olson, Ph.D. Chainnan/Dean Approved for the Graduate Council A��',' Sterling '� . Dean of the Graduate School \ I Ph.D. ACKNOWLEDGMENTS The author wishes to express his deep gratitude and appreciation to Dr. William G. Pariseau for his guidance and encouragement which inspired free and objective thinking during this work. The author is indebted to Professor John E. Willson for his guidance and advice and also to Dr. Wolfgang R. Wawersik for his advice and cooperation in doing laboratory work. The financial support from U.S. Bureau of Mines and from the University of Utah is also kindly acknowledged. Finally, the author wishes to express his appreciation to his friends and his wife, Rajeswari, for their help and cooperation. TABLE OF CONTENTS ACKNOWLEDGMENTS Page iv LIST OF TABLES vii LIST OF ILLUSTRATIONS . vii i ABSTRACT I . II. III. xi INTRODUCTION 1 THEORETICAL BACKGROUND 6 Governing Equations. Yield Criteria Solution Technique 8 9 15 ANALYSIS OF MINE OPENINGS 23 Present and Proposed Mining Systems. . . . Applied Loads. . . . ~...... Material Properties . .•..... Finite Element Meshes for Mine Openings.. I V• RESULTS . . . . . . . . Double Entry Analysis. Single Entry Analysis. v. ANALYSIS AND DISCUSSION Comparison of Field and Computer Results Interpretation of Results . . . Stability Criteria . . . . . . . ... VI . VI I • 23 26 27 34 41 41 43 53 53 59 62 CONCLUSIONS . 65 REFERENCES 68 APPENDIX A. DETAILS OF MATERIAL CONSTANTS 72 APPENDIX B. DETAILS AND VERIFICATION OF FINITE ELEMENT PROGRAM . . . . . . . . . . . 74 Page APPENDIX C. ROCK PROPERTIES VARIABILITY STUDY 81 APPENDIX D. MODEL LOADING STUDY 92 APPENDIX E. MESH REFINEMENT STUDY APPENDIX F. WIDTH/HEIGHT RATIO STUDY . . VITA ... ... 98 . . 104 • 1 08 vi LIST OF TABLES Table Page 1. Mechanical Properties of Coal . • . 28 2• Mechanical Properties of Sandstone 29 3. Mechanical Properties of Coal Measure 4. Measured and Predicted Vertical Closures 58 5. Material Properties Used in Four ElasticPlastic Analyses of a Rectangular Opening 83 6. Random Rock Properties Defining 80 Subtypes 88 7. Displacements for Various Types of Model Load i ng .......... ... 96 Stresses for Various Types of Model Loading. 97 8. Rocks 31 LIST OF ILLUSTRATIONS Figure Page 1. Stress-strain curves for elastic-plastic and elastic-perfectly plastic materials. . 2. Stress-strain curve, elastic and plastic strains. . . . . . . . . . . . . . . . 10 Coulomb and Tresca Yield for an isotropic material . . . . . . . . . . . . . . . 12 Finite element mesh used for row of openings separated by pillars . . . . . . . . . . . 17 3. 4. 7 5 . . Typical two-dimensional triangular element 18 6. Present double entry longwa1l mining system. 24 7. Proposed single entry longwall mining system . . . ......... 25 Rock properties and geologic section for single and double entry analyses . . . . . 33 Mesh dimensions for single and double entry analyses . . • . . . . . . . . •. 35 Coal seam level dimension detail for single and double entry analyses. . . . . . . 36 Finite element mesh used for double entry analysis . . . . . . . . . . . . . . . 39 Finite element mes~ used for single entry analysis . . . . . . . . . . . . . . . 40 Predicted stress concentration near a double entry opening. . . . . . . . . . . . . . . 42 14. Stress contours around double entry opening. 44 15. Double entry opening displacements obtained using a coarse and a fine insert mesh. . . 45 8. 9. 10. 11. 12. 13. Figure 16. Page Predicted failure zones about a double entry opening 46 Predicted stress concentration about a single entry opening. . . . . . . . . . . . . . . 47 18. Stress contours around single entry opening. 49 19. Predicted displacements about a single entry opening 50 Predicted failure zones about a single entry opening 51 Proposed single entry mining system and location of test room 1 . . . . . . . . . . . . . . 54 22. Support system near test area ·1, mine 2 . 55 23. Different instruments and their location at tes t a rea 1 . . . . . . . . . . . . . . . . 56 Location of vertical closure stations in mine 2, test area 1 . . . . ........ 57 Comparison between single and double entry res u1 ts . . . . . . . . ........ 60 26. Flow diagram of finite element program 75 27. Elastic analysis of stress and displacement in a hollow cylinder under internal pressure. 77 Elastic analysis of stress in a transversely isotropic plate containing a circular hole under uniform pressure . . . . . . . . . . 79 Elastic plastic analysis of stress in a circular tunnel in rock . . . . . . . . . . . . 80 Stress concentration as a function of Poisson's ratio for a rectangular opening. . . 84 Element failures about a rectangular opening with different material properties . . . . 86 Element failures around single entry opening: a) conventional method with mean-values; b) with random material properties. . . . 90 Details of various types of model loading.. 93 17. 20. 21. 24. 25. 28. 29. 3~. 31. 32. 33. ix Figure 34. 35. 36. 37. 38. Page Dimensions of coarse and fine insert meshes for double entry analysis. . . . . • • . . . 100 Stress results for single and two pass from the same mesh and a fine insert mesh . . . . . . 101 Displacements from coarse and fine insert meshes . . . . . . . . . ... 103 Stress concentration as a function of width/ height ratio for rectangular openings. . . . 106 Displacement as a funation of width/height ratio for rectangular openings . . . . . 107 x ABSTRACT This thesis describes work performed under U.S. Bureau of Mines Contract #H022077 concerning an investigation of the relative merits of single and double entry longwall mining systems at the Sunnyside Mine of Kaiser Steel Corporation near Sunnyside, Utah. The Sunnyside Coal Mine has a present working depth of approximately 1800 feet with future expected depths to 3000 feet. is used. At present a two entry longwall system of mining However, lower production cost, improved ground control, and faster development of longwall panels provides incentive to develop longwall panels through a single entry system. Research consisted of experimental as well as theoretical investigations. Field study included instru- mentation, field measurements, and data processing. In-situ and laboratory tests were conducted to estimate the physical and mechanical properties of rock types present at the Sunnyside Mine. Theoretical study included prediction of stress concentrations, displacements, and failure zones around mine openings. A finite element technique was used for these purposes. All the finite element program branches used for the present study were verified by comparison of results to existing solutions of various problems. Predicted displacements around double entry opening compared favorably with the average vertical closure measured in the field. The magnitude of tensile stress concentration near the periphery of the single entry opening was slightly higher than in the double entry. Failure zones around single and double entry openings were approximately the same size. The results indicate, therefore, that the single entry system would be as safe as the existing double entry system. The views presented heretn are those of the author and do not necessarily reflect those of the U.S. Bureau of Mines. xii I. INTRODUCTION The purpose of this study was to determine whether a single entry is as safe for the development of longwall panels than the present double entry system at the Sunnyside Coal Mine near Sunnyside, Utah. The work is a joint effort by the U.S. Bureau of Mines in cooperation with Kaiser Steel Corporation and the University of Utah. The study included field instrumentation and measurements as well as data processing and interpretation. In-situ and laboratory tests were conducted to estimate the physical and mechanical properties of rock types present at the Sunnyside Mine. Theoretical studies included prediction of stress concentrations and failure zones as well as comparisons between theoretical and field measurements of displacements about mine openings. A finite element com- puter program was used for theoretical study, comparison of field data, and for simulation of face advance. In most of the United States coal and non-coal mines, longwall panels are developed by a three entry system (38): At the Sunnyside Mine longwall sections are developed by two entries and some advantages were realized in such a two entry system as compared to the three entries. In the proposed mining method longwall *Numbers in parentheses refer to corresponding items in the References. 2 panels are developed by a single entry on both sides of the panel. The proposed single entry system will be sup- ported by cribs along the center line and separated by fireproof brattice cloth, thus making it, in effect, a double entry without intervening coal pillars and crosscut intersections. During development, one-half of the single entry is used for fresh air and the other for return air. As the longwall face advances, half of the single entry will be caved and the remaining half will be used as return airway for the longwall section. The two entry system exposes less area of roof than the three entry system and decreases the magnitude of squeeze and severity of bounce conditions since the exposed area of excavation is reduced. The advantages of two entries over three can be extrapolated to the single entry system. There are advantages and risks involved in the single entry system as compared to the double entry. Some of the advantages are: a) Longwall panels can be developed at faster speed and development cost will be reduced. b) Less area of roof is exposed and absence of coal pillars would avoid pillar spalling and reduce stress concentrations. c) Asthe face advances, the roof would cave more easily behind the supports, since no pillar remnants are left in the gob. 3 d) Elimination of the second entry would reduce methane liberation because less coal surface area will be exposed. e) More desirable and efficient ventilation would be realized because the total amount of air will pass along only one exposed coal rib before reaching the face. f) Dust concentration would be reduced at the face since only the face is ventilated and dust would be carried directly into the return. Risks include those associated with geometry. The wider single entry would create higher stress concentration zones,greater convergence and increased failures than in the two entry. Since critical stresses and bounces occur during mining, stability of the opening and adequacy of cribs are important system questions. Failure of cribs could cause complete failure of the opening,and at the same time ventilation for the longwall section could be stopped. Information about stress concentration, convergence, and support requirements aid in evaluating the stability of mine openings. Input data for analyses include material properties, geological information, and mine opening geometry. Knowledge of material properties is one of the important factors for mine design problems. Some laboratory properties are available for the present study. Other material properties used for this analysis were selected by judgment after comparison between test values 4 and properties from the literature. A two dimensional finite element program {28}' based on an elastic-plastic material idealization serves as an analysis tool. Program capabilities include arbitrary geometry, in-situ stress, application of external forces and displacements, elastic and plastic properties for isotropic and anisotropic materials and time dependent material properties. Material properties may be linear or non- linear in the program. Finite element meshes with triangu- lar elements were used for the single and double entry study. • In the finite element model each triangular ele- ment can be assigned different material properties which need not be in numerical sequence. Openings in the meshes can be changed to the desirable geometry by changing coordinate factors. The program is also capable of handling premining stresses for an arbitrary geometry. Simulation of face advance can be included in the program. All the program branches used for these analyses were verified by comparing them with existing theoretical and experimental result~(30). Results obtained from the computer analyses are dependent on the refinement of mesh, mesh size, loading, and material model as well as more direct input such as material properties. Stress, displacement, and failure zone predictions were made for the single and double entrtes~ Predicted 5 stress concentrations around openings are comparable with existing experimental results. Applicability of the com- puter program for stability analysis of mine openings is demonstrated by comparison between closure predictions and mine convergence measurements from the Sunnyside Mine. Conclusions concerning stability of the proposed single entry system follow from these results. II. THEORETICAL BACKGROUND Most metals and rocks respond elastically to an initial application of load but tend to yield and deform permanently upon continued loading. The main character- istic of the mechanical behavior of an elastic material is adequately expressed by Hooke's law. No similar simple description of the mechanical behavior of plastic material seems possible. According to Drucker (7) an elastic body is reversible, nondissipative and time independent under isothermal conditions. All the work done on an elastic body is stored as strain energy and can be recovered on unloading. Plastic material denotes irreversibility and implies permanent or residual strain upon unloading. Stress-strain curves (Figure la) for work hardening material can be classified into elastic or perfectly elastic, ductile, and brittle according to their behavior and these are represented by OA, AB, and BC portions, respectively. An elastic-perfectly plastic material is supposed to deform elastically up to yield stress (Figure lb), but be able to sustain no stress greater than this so that it will flow indefinitely at this stress unless restricted by some outside agency or adjacent elastic region. Ductile behavior is not of great importance in rock mechanics, nevertheless rock can be assumed as an elastic and . perfectly a a Stress Stress B o £-strain (a ) a £-strain (b ) Fig. l.--Stress-strain curves for (a) elasticplastic and (b) elastic-perfectly plastic materials. 8 plastic material (4, 17, 31,33) that deforms permanently by some mechanism other than that responsible for .ductile behavior. The mechanical behavior of a perfectly plastic material is completely characterized by its yield function (8) . Governing Equations The governing equations describing the behavior of elastic plastic material are (12): (i) Stress dO' •• J1 ,d x j + X. 1 equilibrium equations = 0 ( 1) (ii) Equations of deformation geometry 1 s .. = -2(u . . + u· .) lJ 1,J J,l (2) (iii) Stress strain relations (constitutive equations) 0' •• lJ (3) and (iv) yield function y [ 0" •• , S lJ P .. ) = 0 lJ (4 ) Subscript notation and summation are in force. H are the elastic constants of the material, 0', s, u ijkl denote stress, strain, and displacement respectively; X refers to body force per unit volume, se and sP are the elastic and plastic strains, A is non-negative proportionality constant and Y is yield function. 9 The stress strain equations for elastic plastic material can be written in the incremental form. Stress strain curve OABC (Figure 2) in one dimensional uniaxial case rial. represents loading and unloading of the mate- Along the curve OA the material behaves completely elastic up to yield stress (a o ) and after point A it shows both elastic and plastic changes. In the elastic plastic region for each increment of stress (da) the total incremental strain (dE) will be sum of elastic and plastic strains, then dE .. = dE P .. + dEe .. lJ lJ lJ ( 5) , Total incremental strain is given by d£ .• lJ - H- l - • . k1 lJ d a k 1 + 1\'\ dVao . (6) lJ. Where dE P •. and d -E e .. are inc rem e ntal p1 a s tic and lJ lJ · . -1 e l astlc stralns; H ijkl are the inverse 'of elastic coefficients ' (linear case). Yield Criteria The elastic behavior of a material for tension or compression test is normally denoted by its tensile or compressive strengths, whereas for triaxial or polyaxial state of stres~ a single value of stress cannot be used to find the elastic limit. Therefore a functional relation is required among all the stress at the onset of yield which is known as the yield function or yield criterion (Y). The yield function is determined by experiment and testing. For any 10 B--.L o do Stress o ~~ ______________________ ~ Fig. 2.--Stress-strain curve, elastic and plastic strains are shown. 11 state of stress within the elastic range Y < 0; for such states of stress the material is called safe. Plastic flow may occur only under states of stress for which Y = o. The geometric representation of plastic states of stress is called the yield surface. The general types of yield conditions for isotropic material can be divided into two groups. The first group consists of those which are influenced by the intermediate principal stress, and the second consists of those that are not. The later group can be written in more general form (31 ) : I'Where m In = Ao m + B (7) ° m = 1/2 (0 1 + °3)' lm = 1/2 (°1 - °3),and A, B, and n (n ,2:1) are material constants; 01 and 03 are major and minor principal stresses, respectively. By substitution of different values for A, B, and n (31) in equation 7, various yield criteria can be formed which are in common use. For example, Tresca and Coulomb criteria are shown in Figure 3. For Tresca yield cri- terion: A = 0; B Co To = 2 - 2 and n = 1 (7 a ) and for Coulomb yield: A = Co - To Co + To ; B = Co To and n = (7b) Co+To Where Co and To are unaxial tensile and compressive strengths, compression is taken as positive. 12 T Tresca a Fig. 3.--Coulomb and Tresca yield for an isotropic material. 13 Since the process of yield must be independent of the choice of axes in an isotropic material, yield criterion should also be independent of choice of axes. This yield condition should be expressed in an invariant formulation which will include the effect of intermediate principal stress. This type of yield condition (31) is given in the form: IJ 2 I n / 2 = A, I, + B, (8 ) Where Al , B1 , and n (n ; 1) are material constants, 11 is the first stress invariant and J 2 is the second deviatoric stress invariant. These invariants are: 11 = Ox + 0 y + Oz J2 = 1 2 2 + 2 6" [(Oy oz) + (0 z - 0) (0 x - °y ) ] x 2 + 2 + 2 (8a) ° yz + ° zx ° xy or J 2 = "61 [(02 - 03) 2 + (03 - 01) 2 + (01 - 02) 2 ] In the case of plane strain analysis equation 8 is similar to equation 7. In the present work an extended Von Mises type of yield condition soils is used. appropriate to anisotropic rock and This was originally proposed by Pariseau, 1968 (31) , an dis g i ve n as: IF (Oy -oz)2 + G {oz - Ox)2 + H (ox - Oy)2 + L02yZ 2 (9 ) n/2 - (Uo + VO y + WO z ) - 1 = Y x XY Wh e re F, G, H, L, M, N U, V, W, and n (n ~ 1 ) are + M02zx + N0 14 material constants determined by experiment and x~ y, z refer to the principal axes of anisotropy. The value of n can be equal to one or two in the present work. The material constants of equation 9 will change if another reference coordinate system is adopted. = W = 0, the yield criterion equation For U = V 9 gives Hi11's (14) criterion which includes only the deviatoric stresses. The nine plastic moduli of equation 9 are expressab1e in terms of unconfined tensile, compressive and shear strengths (31). These are given in Appendix A. In the plane strain problem for isotropic material, for vanishing anisotropy the yield condition equation 9 represents a generalization In plane strain dE z = 0, where d Ez is the total strain increment and 0z is calculated from the of Coulomb yield. s t res sst r a i n r e 1 a t ion s ( 3 1). t'ft t h neg 1 e c t 0 f the e 1 a s tic component of the strain increment and after substituting the resulting value of 0z in equation 9 the yield criterion for isotropic material is: Io 2 xy + ro t x ; )2 0 YJ I n/2 - A2 - 2 (0 x + 0 y) + B2 ( 10 ) or (11 ) Where A2 , 8 2 , and n (n ~l) are material constants. Yield condition equation 11 is similar to the Coulomb condition equation 7, but the material constants in both the equations are not the same, therefore the stress analysis based on equations 11 and 7 will differ. 15 The yield condition equation 10 for anisotropic materials in plane strain is (31 ) given as: ( 0 x - 2 A3 = T 0 0 y r . A4 x + T + 0 2 xy ( 1 - c) n/2 ( 1 - c)-1/2 ( 12 ) 0 y + B3 Where c, A3 , A4 , 8 3 , and n ( n ~ 1 ) are material constants. In the present work, equation 9 is widely used. No assumption concerning the elastic component of the strain increment is made. However, all analyses are plane s t r a in, sot hat the tot a 1 s t r a i n inc rem e n t d E Z = o. The corresponrling stress increment dO z is calculated through an inverted form of the complete stress strain relations. Solution Technigue The finite element method of analysis is a powerful technique which can readily be applied to boundary value problems in solid mechanics. It is very well suited for handling rock mechanics problems because the usual assumptions that rock is an isotropic and homogeneous medium need not be made; any geometry and loading conditions are easily modeled. The method has been used for various problems in roc k me c han i c s (2, 2 3 , . 3 5, 3 9 ). I n fin i tee 1em e ntan a 1y- sis, different material properties can be assigned for different elements, which need not be in sequence. Be- cause of the above advantages and simplicity, the finite element technique is employed for the present work. are no suitable alternatives. There 16 f In the finite element method (6, 44) the continuum is subdivided by imaginary lines into a number of discrete pieces called finite elements. These elements are assumed to be interconnected at a number of nodal points situated on their boundaries. In the present study triangular ele- ments were used because two-dimensional irregular or c~r- vilinear shapes are most easily approximated by triangles. A finite element mesh used for an underground opening is shown in Figure 4. In the finite element method (6, 44), the displacements of the nodal points are usually the basic unknown parameters, u i and vi shown in Figure 5. The forces {F} acting at the nodal points and resulting nodal displacements{8} are related through a stiffness matrix ~ [K] char- acterizing the mechanical behavior of the body. The determination of the element stiffness matrix begins with an assumption defining the displacements within the element. The displacements within the element are defined (6,44) by two linear polynomials and given by: u (x, y) = al v (x, y) = + a x + a y 2 3 a + as x + a 6 y 4 ( 13 ) u(x, y) and v (x, y) define the variation of the x and y displacement components, respectively. The force displacement relation can be written as: ( 14 ) where: 17 --- - - - ____ --- __ ___ -- -- - - - --lI -1 I I I I I I I I I I I t I I I I I I I I I I I I ~----~~------r-----~~------~I I Fig. 4.--Finite element mesh used for row of openings separated by pillars. 18 o y x Fig. 5.--Typica1 two-dimensional triangular element. 19 {F} = . {~.} = = (14a) = (14b) and [K] = k11 k12 k13 k21 k22 k23 k31 k32 k33 (14c) where Ui , Vi' and ui' vi are the force and the corresponding displacement components in a common coordinate system and the subscri pt i represents node poi nt of each fi ni te element numbered in a counterclockwise direction. Symbols {F} and {oJ are column matrix and correspond to the listing of the nodal point forces and displacements in each element, respectively. The symbol [K] is a square matrix which represents the element stiffness matrix and in which 20 k1..J are submatrices which are again square matrices. The size of the submatrices will depend on the number of force components to be considered at nodes. The six constants ai in equation 13 can be evaluated by solving sets of simaltaneous equations 13, when the nodal point coordinates are inserted and the displacements equated to appropriate nodal displacements; for example: = u2 = u3 = ul a 1 + a 2 xl + a 3 Y1 ( 15 ) a l + a 2 x 2 + a 3 Y2 a 1 = a 2 x3 + a 3 Y3 The above equations can be written in matrix form: [N] · {u} = ( 16 ) {a} where {a} and {u} are column matrix, {u} is the listings of the displacement components within ~he element, and · {a} is the listing of displacements at nodal points shown in Fi gu re 5. Symbol [N] is a matrix linear in position. Matrix [N] is: 1 [N] = , xl Y, 0 0 0 x 2 Y2 0 0 0 1 x3 Y3 0 0 0 a 0 0 0 0 a , 0 a 0 1 1 (1 6a ) xl Y, x 2 Y2 x3 Y3 also: ·{u} = {u 1 u2 u3 v l v 2 3} (16b) {a} = {a, a 2 a 3 a 4 a 5 a 6 } ( 1 6c ) V an·d 21 Inverse of [N] matrix exists except for zero elements and the equation 16 can be written: = [N]-l {a} {u} ( 17 ) The geometry of strain equations are: £xx = au/ax £xy = 12 'ay [~ £yy = av/ay ( 18 ) +, lY, ax ] From the equations 17 and 18 , the strain-displacement relation can be written in the form: ' {e:} = '[8] or ' {e:} = {a} ( 19 ) [8] [N]-l {u} where [B] is the matrix of constant terms and {e:}isa listing of components of strain. Stresses can be computed through the constitutive equation 3, and given by: '{a} = [E] {e:} (20) '{a} is column matrix which represents the listings of stress and [E] is the material property matrix. The nodal point forces are made work equivalent to the surface tractions, body forces, and initial stresses acting on an element through an application of the virtual war kid en t i ty (44 ), g i ve n\ as: {F.} + {F.g} + {F. s }= [K] {u.} (21) 1 1 1 1 where ' {F i}, {Fig}, {Fi s } are the external applied nodal forces, nodal forces due to gravity, and nodal forces due to initial stress, respectively, and u.1, is listing of dis- placements adjacent to the ith node. the stiffness of the material. The matrix [K] is 22 The unknown displacements {u.} are computed from 1 equation 21. Once the displacements are known, strains and stresses can be calculated by the application of equations 19 and 20. Some of the details of the finite element program and verification of B. th~ program are presented in Appendix III. ANALYSIS OF MINE OPENINGS Present and Proposed Mining Systems At present a two-entry longwall system of mining is employed at the Sunnyside Mine, Utah. In this system, longwall panels are developed with two 24-foot wide entries that are separated by 26-foot wide coal pillars. Cross- cuts (18 foot wide) are driven between the two entries at approximately 115-foot centers. The average thickness of the working coal seam is about 7 feet. Dimensions of the longwall panel and other information is given in Figure 6. In the proposed mining method, panels are developed with one 26-foot wide entry, shown in Figure 7. The support system for the single entry includes roof bolts installed at 4-f90t centers and cribs along the center line of the entry, spaced 4 feet apart. Fire-resistant brattice cloth is installed along the crib line that separates the single entry into two portions. During development of the single entry, one side of the entry is used for intake air and the other for return air. Once the panel is developed, one side of the panel entry is used for fresh air and return air is sent through the other side. As the coal face advances, half of the single entry will be caved and the remaining portion 24 'I 11111/1/11/11/111 I; Ij / Ij //1/// I ;; / / 1/ 1/ / /1 / /: :~:==I I II I / / I / I / // / 1;/~~:(o7~:~!'~;;:'~<~~ U [II ~--------- 36oo 500 l 1 A ---"----+------f I II JI ]0 ---------PD o o ~-------I 0 t---I I P1 I I arll _--,tl'--_--,I to- --+-------f----B Doubl e Entry r::=J 0 l fJ / (a) Plan 24' Bleeder Entry , 26' Entry 24' Entry coal Pillar (b) Section On AB Fig. 6.--Present double entry longwall mining system. 3600~ Longwall face 500' (a) Plan Bleeder entry Single entry r-- 26 1 -----91,. . . .- - 500 -----e4oI·I_______-26 1 -- 1 (b) Section on AB Fig. 7.--Proposed single entry longwall mining system. N (J1 26 of the entry is supported by the cribs as shown in Figure 7. This remaining portion of the entry will be used as return airway for the longwall panel. Applied Loads Applied loads refer to the external forces capable of causing deformation of the mine openings. All under- ground rock is subjected to the weight of the overlying rock. Static loads include those of gravitational, tec- tonic and thermal origin; dynamic loads include those caused by blasting. Gravitational load is considered to be the major applied load in the present analysis. At the Sunnyside Mine, coal is being mined at a I depth of approximately 1750 feet from the surface and the coal seam is slightly dipping. Topography may affect the load due to gravity, but such effects are not discussed here. Development entries are driven by continuous miners, hence blasting is completely eliminated. thermal loads are not taken into account. Tectonic and The vertical premining stress component a v is computed as the specific weight multiplied by depth of the opening. The horizontal premining stress a H is calculated as M a v ' where M = v and v is Poisson's ratio. The applied loads are (l-v) achieved by replacing the material to be excavated by a set of stress boundary surface. conditionsacti~g at the excavation The magnitude of these stresses are exactly those 27 generated by the excavated material prior to its removal. The stresses resulting from the applied loads are t~en added to the premining stresses to obtain the final post-mining stresses. -Material Properties Material properties are an important component of input data. Properties of coal and sandstone were deter- mined by laboratory tests. These experiments were con- ducted by Bureau of Reclamation personnel for the Bureau of Mines. Coal and sandstone samples were cored from Sunnyside Mine No.2 near test area 1. Laboratoryexperi- ments include both static and dynamic testing. Dynamic rock properties have greater values (41) than statically determined properties. The difference between these values is from 10 to 30 percent. Statically determined properties were used in the present analysis. Laboratory properties of coal and sand- stone from 6 to 12 samples are summarized in Tables 1 and 2. · All the static measurements given in these tables were made on the first loading which was taken up to failure of the specimen. Shear moduli were computed from Young's mod- ult· and Poisson's ratios. Plastic properties include unconfined compressive, tensile, and shear strengths. Shear strengths for isotropic materials are computed from compressive and tensile strengths using the formulas in Appendix A. TABLE 1 MECHANICAL PROPERTIES OF COAL * Sample No. Young's Modulus 6 psi x 10 Poisson's Ra t i 0 Compressive T ENS I L E S T R E N G T H ~si Strength Indirect Direct (Co) psi Parallel Perpendicu1ar to to Bedding Bedding 1 0.24 O. 31 1 ,360 210 240 55 2 0.42 O. 31 3,920 110 280 32 3 0.40 0.39 5,600 185 170 32 4 0.30 O. 15 2,950 165 160 34 5 O. 19 6 0.20 0.23 2,900 150 200 7 0.37 0.31 4,200 175 125 8 0.50 0.45 2,400 9 10 0.42 0.24 5, 140 11 0.31 0.17 2,980 12 O. 31 o. 1 6 * These data were supplied by the N ex> u. S. Bureau of Mines, Spokane. TABLE 2 .MECHAN I CAL PROPERTIES OF SANDSTONE * Young's Modulus 6 psi x 10 Poisson's Ratio Compressive Strength psi Specific Gravity 2.29 0.05 1 7 ,550 2.41 1 7 ,900 2.37 0.04 17,350 2.39 1 .23 0.03 1 5,850 2.37 II 1 .69 0.05 1 Massive sandstone 2.87 2 II 2.71 0.06 3 II 4.64 0.12 4 II 3.32 0.08 5 II 2.66 0.06 Sample No. Rock Type 1 Interbedded sandstone 2 II 3 II 1 .67 0.07 4 II 1 . 21 5 II 6 2.40 2.36 2.37 19,700 2.37 2.37 20,000 2.41 N \0 * These data were supplies by the U.S. Bureau of Mines, Spokane. 30 A typical geological section of the Sunnyside Mine is shown in Figure 8. Rock types may be geologically different but have similar material properties. Layers Qutside the section are assigned average properties; the stresses in this region are not affected by the excavation at the coal seam level. Laboratory test values were not available for all layers. Consequently some of the properties were estimated from an abbreviated survey of the technical literature. These are given in Table 3. The material properties used for the final analyses were selected by judgment from laboratory and literature. These are presented in Figure 8. In the present work the material is assumed as isotropic and homogeneous within each layer. More complicated material behavior can be handled by the computer program. Rock properties used in the present study are about midrange of those values obtained from laboratory tests. But the in-situ properties can vary from the laboratory values. Rock properties' within each layer itself may not be constant due to the presence of fractures and inhomogeneity in material. In general, a standard deviation of 30 percent of the mean or greater is observed in rock properties data. Rock properties could exert a considerable influence on stress prediction and factor of safety. When the factors of safety are high, rock properties variability may not greatly influence stability of the opening, but TABLE 3: MECHANICAL PROPERTIES OF COAL MEASURE Rock Type Referenee Siltstone (Utah) Coal (Utah) Sandstone (Utah) Coal (Utah) Sandstone (Virginia) Siltstone (Virginia) Sandstone (W . Virginia) Sandstone (U.S.) #15 #15 #15 #15 #15 #15 #15 #34 Shale Sandstone (Ohio) Siltstone (New Jersey) Shale #34 #5 #5 #10 Shale #3 Sandstone (Utah) Sandstone (Utah) #43 #43 *Perpendieu1ar to bedding Unit Weight pef Young's Modulus 6 psi x 10 Poisson's Ratio 0.38 0.22 0.09 to O. 13 0.01 0.07 27.0 to 49.0 1 5. 500 27.700 0.654 2.980 1 . 709 3.670 7 . 711 136.2 1 62.0 10.63 to 12.28 0.09 to 0.13 137.28 1 .900 143.50 2.770 Compressive Strength ~si x 1000 ** * 3.869 4.541 3.600 8.987 6.861 3.642 4.030 17.860 16~250 3.483 2.073 15.550 18.0 to 22.0 11.000 10.700 17.800 0.487 5.494 2.760 3.870 ROCKS ** Parallel to bedding Tensile Strength ~si x 100 ** * 1.70 4.30 w TABLE 3--Continued Rock Type Reference Unit Weight pcf Young's Modulus 6 psi x 10 Poisson'·s Ratio Compressive Strength psi x 1000 ** Shale (Utah) Shale (Michigan) Sandstone (Utah) # 43 #1 #1 174.72 173.47 143.52 8.44 7.50 3. 15 0.09 O. 1 5 0.03 31.300 28.600 13.100 Tensile Strength psi x 100 * ** 1. 6 * Perpendicular to bedding ** Parallel to bedding W N 33 0 Overburden Massive sandstone Coal Layered sandstone M.S. SandySha 1e Coal M.S. Layered sandstone Unit Weight pcf 10 6 psi 144.0 1 .700 0.25 1.000 0.500 148~2 3.000 0.20 19.000 8.500 75~0 0.350 0.30 3.500 1.300 149.5 1 . 500 0.10 1 7 . 500 7.000 148.2 3.000 0.20 19.000 8.500 170.0 4.500 0.10 15.000 0.000 75.0 0.350 0.30 3.500 1 .300 148.2 3.500 0.20 19.500 10.000 149.5 1 .500 0.10 17.500 7.000 E Co 3 10 psi To 10 2 si Dimensions in feet Fig. 8.--Rock properties and geologic section for single and double entry analyses. 34 where margins of safety are low, a stability analysis as based on the mean values is questionable. In the elas- tic-plastic analysis, variability of plastic moduli discloses more failed elements which are safe of rock properties. a~ mean values More detailed study of rock proper- ties uncertainty is discussed in Appendix C. Finite Element Meshes for Mine Openings Mesh information and material properties are the basic components of the input for finite element analysis. Meshes are prepared by drawing the model to scale. Each element in the mesh is identified by its nodal points and nodes are located by their coordinates. are taken from the drawing. Nodal coordinates Mesh plots were generated by the computer for verification of the meshes. Accuracy of stress and displacements from an analysis will depend on mesh size, boundary influence, number and size of elements around the openings, and model loading. As a ru1e-of-thumb, a stress concentration caused by two dimensional excavation decays inversely with the square of distance from the opening. The magnitude of stress at a distance of ten times the dimen~on of th~ outer edge of the opening may not be affected by the excavations. satisfies All the meshes used in the present work this distance rule. The dimensions of,the meshes used for single and double entry are shown in Figures 9 and 10. 35 Surface . Double entry Single entry l750~ 500' r -,-Insert mesh ----L- ~2 30'-1 120' ~ ~ 120'· • --. C 1 1-- t-- 260 , ... 7' Coal seam 500' ! Fig. 9.--Mesh dimensions for single and double entry analyses. 36 l __ ~37' ~ __________________ 463 ' Mesh side v!/lwF;77;;7:~77S7:::777Z777ZZZZZZZZll7Zm WIll/III ~~ PM 7" X 24' entry (one of two) 7 ' x 26 ' pillar (half pillar shown) (a) doubl e entry Mesh side 13 247 t . ~ I-- 7' X 1 -------~ 26' entry ' (h'alf shown) (b) single entry Fig. lO.--Coal seam level dimension detail for single and double entry analyses. 37 Stresses near the side and bottom boundaries should not be affected by the excavations. is determined by a simple test. Adequacy of mesh sizes Pre~ and post-mining stress adjacent to the mesh boundaries should be mately the same. approxi~ Though the mesh size is limited by the number of elements and nodal points, all the meshes used for the analysis are of adequate size. Single and double entry meshes extend to the face. sur~ For the meshes that are not extended to the surface, the effect of the overburden can be replaced by equivalent nodal forces, but by this system of loading the stiffness effect of the overburden cannot be replaced without influencing displacement predictions. Displacement predictions mainly depend on the type of loading. When the overburden effect is replaced by its equivalent nodal point forces, the value of displacements are a little higher. Some of the results for different types of model loading are discussed in Appendix D. In general, stress gradients are very high periphery of the underground openings. near the Though coarse mesh- es are satisfactory for prediction of displacements, the same meshes may not be sufficient for estimation of stress. Detailed information about stress history near the periphery of the opening requires a more refined mesh. One of the difficulties in preparing a refined mesh is that the element size is controlled by very thin rock layers and numbers of elements in a mesh are limited by 38 computer core storage. To overcome this problem p two- pass solution technique is used. This point is discussed in greater detail and illustrated in Appendix E. The computer program (28) is sensitive to changes in the geometry of the opening and has th~ ability to dis- criminate between rectangular openings of even slightly different dimensions. The double and single entry systems have width to height ratios of 3.4 and 3.7 to 1, respectively. In the present analysis it is important to the cal- culation of stress concentrations near the periphery of the single and double entries. Slight changes in the width to height ratios are in fact reflected in stress and displacement predictions by the computer program. More information on this study is discussed in Appendix F. Portions of the finite element meshes used for double and single entry analyses are shown in Figures 11 and 12. The elements around the openings are made very small for accurate stress ·predictions near periphery of the openings. Near the opening, the width of each element is 12 inches and the height is 6 11 • 500' --- -- - -- ~ ---- ------I I I I I I / I / Fig. ll.--Finite element mesh used for double entry analysis. 2250' 40 60' T 2250' 1 I I I I I I I I I I Fig. l2.--Finite element mesh used for single entry analysis. IV. RESULTS The output results from finite element analyses consist of listings of stresses, strains, displacements, and results of the elastic-plastic analysis. In the double and single entry analyses, the effect of layering is included. Material properties in both cases are the same as shown in Figure 8. Average nodal point stresses were com- puted from element stresses; these stresses were taken from the elastic part of the solution. Stress concentrations, displacements, stress contours, and failure zone predic- ' tions for the double and single entry systems are presented here. Double Entry Analysis A two-pass technique was , used in the double entry analysis. Displacements from coarse and fine meshes are in close agreement, but there are differences in stress predictions. The difference in stress is due to high stress gradient around the periphery of opening and mesh refinement. Figure 13 shows the boundary stress concentration about a double entry opening, Maximum tensile stress con- centration is 1.63 and it occurs along the centerline of the opening. Compressive stress concentratiori is slightly 42 coarse mesh 6 o fine insert mesh 4 3 2 °t v -S- 1 + compression o tension -3 , I , I ~ I I 7/:'7~~' pllla. ///'//// I I I I I I I , I I I I i I I I - --- -1-- - - --1-+-+- * = , I I ~I"-\t-"<""'\\ pillar / """"" ,,, 3.43 Fig. 13.--Predicted stress concentration near a double entry opening. 43 higher in the pillar than in solid coal. Maximum predicted compressive stress concentration is 2.02. stress Sv is approximately 1750 psi. The vertical Stress concentrations from both fine and coarse mesh are shown in Figure 13. Stress contours near the opening are shown in Figure 14. These contour lines were drawn by joining the lines along centroids of elements having the same magriitude of stress. Tensile stress is predominant both in the roof and floor. Tensile stress in the roof drops down to 400 psi in massive sandstone due to low Young's modulus as compared to shale modulus. Peak compressive stress concentra- tion is localized near the corners of the opening and is higher than in the coal ribs. Figure 15 shows displacements about the opening. Results from coarse and a fine mesh insert mesh are in close agreement. Displacements from elastic and elastic- plastic analyses are within 3 percent. For practical pur- poses they can be taken as equal. Figure 16 shows the failed elements about a double entry opening. and floor. Tensile stresses are present in the roof The failure zone in the roof extends through the shale and decreases in massive sandstone. Failure zone extends 4.5 feet into the roof and 1.S feet in the floor. No rib failures are indicated. Single Entry Analysis Figure 17 shows the boundary stress concentrations 50 1 I , -2.3 0.4~ --- ~O.~ I ~O.~~ (ge 2.5 1.8 I 5 I I I; ca 1e -I opening x -----)GA -3.0 ~l -2. 2.3 2.7 -:::::0: 0 . 6 O. 4 I 1.0 _~ 0.8 ----- I 1• 1 --- -2.3 T 2250· --------- -- --- --------- --- -- I I ~ - Compression: Vertical stress + Tension: Horizontal stress Fig. 14.--Stress contours around double entry opening (contour values are multiples of 1000 psi). 45 ~ 24' T 1 Opening 7' 1 W H = 3.4 0 Fine insert mesh 6 Coarse mesh 1" r--1 Displacement scale Fig. l5.--Double entry opening displacements obtained using a coarse and a fine insert mesh. 46 Opening A 6 I- 5' - compression 0y - tension { ' °x °x ' 0y - both tensi on 1scale Fig. l6.--Predicted failure zones about a double entry opening. 47 3 2 0 1 at -sv + Compression 0 - Tension -1 -2 Opening I I -r- -- - - - - - - - - - - - - - t -F_u_;----]- w_ H - 3.7 Fig. 17.--Predicted stress concentration about a single entry opening. 48 about a single entry opening. are ~hown The stress concentrations on one side of the centerline of the opening. Maximum elastic tensile stress concentration is 1,73 and compressive stress concentration near the corner is 1.84. Tensile stress concentration in this case is about 6 percent higher than in the double entry, whereas maximum compressive rib stress concentration in the single entry system is 6 percent less than in the double entry system. Ab- sence of coal pillars in the single entry system reduces the critical compressive stresses in the rib. Stress contours around the single entry opening are shown in Figure 18. roof and floor. Tensile stresses are present in the There is about a 200 psi increase in ten- sile stress in the floor and roof over the double entry. Higher compressive stresses are present near centers of the opening. Figure 19 shows displacements about the single entry opening. Displacements from both elastic and elastic-plastic analyses are presented. Though results from the elastic-plastic analysis are somewhat higher, for practical purposes they are equal. Maximum vertical closure is represented along the centerline of the opening. Figure 20 shows the zone of failed elements about the single entry opening. Failure zone in the roof extends into shale and sandstone. There are more failed elements in shale than in the sandstone. Maximum height of the 49 T ----- ------- ,-- ,-- 2250' --- -- I ,-- ,-- I I I I I I I 2.9 r 2. 1 2.5 I y I scale I x I I I I I - Compression: + Tension: Vertical stress Horizontal stress Fig. 1B.--Stress contours around single entry opening (contour values are multiples of 1000 psi). 1 50 13' ·1 I I I I Opening I 7' II I I I I I I 0.5 H o 0 8 ~ 11 Scal e Elastic Elastic-plastic Fig. 19.--Predicted displacements about a single entry opening. 51 I I I I tl Opening I I I x I .. { cr x A cr r 2 1'1 cry x' - compression tension cr y - both tension Scale Fig. 20.--Predicted failure zones about a single entry opening. 52 failure zone in the roof is about 4.5 feet and in the floor 1.5 feet. V. ANALYSIS AND DISCUS$ION Compafis6h "6f "Field and " Comput~~Results Field observations of vertical closuie from Mine 2, Test Room 1 of the Sunnyside Mine are available for com- parison with computer results. room is shown in Figure 21. The location of the test The support system and the instrumentation are shown in Figure 22. The support system consists of 3/4-inch diameter, 6-foot long roof bolts spaced approximately on 4-foot centers sUpporting landing mats. Timber posts and cribs are installed wherever addi- tional support is required. Figure 23 shows various types of instruments used in the test room and their location. The locations of vertical closure points used for comparison with computer results are shown in Figure 24. Table 4 is a tabulation of computer predictions and field measurements of vertical closure at the Sunnyside Mine. Field measurements show maximum vertical closure near the rib of the pillar but predicted maximum vertical closure occurs at the center of the opening. ments from elastic and elastic~plastic Displace- solutions are close. The average computer predictions for the double entry system are approximately 10 percent higher than the field measurements. The effect of roof bolting is not included Area instrumented ~350 nu nn n~ O~ I --I_ I / I 3000' 1~~~~c::::2~c::::::::Jt:::Jc::::J I Test area 1 "T r 1 500 ' Proposed flU flll un Single entry Panels j on O~ un on Dil 1 Ma in haulage Fig. 2l.--Proposed single entry mining system and location of test room 1. 55 ."....-- ---- ----... ...... . ..... " \ \, \ I Pilla r • , I , g .a-fl o C I 9 _.... ~ __ - - - - - • Adva nce I e, : -g,' B ............ - ..... 13-8 /~ 8 B B B 9 I 8 0 iii .-.. Pilla r 0"/ Q 0 0: ,,-- ---:lff---------.-:-- .. ·f.] ~t i.. _____ ~__1_ ~~--~--~~---~-~-~~~~----~-------- • Timb er post o Rock bol ts - Land ing mat loca tions ~ Cribs Fig. 22.-- Supp ort syste m near test area 1, mine 2~ 56 l!F ~,~ __R_B~~4 ---l;- - Ib __ 19 ----- -{c:l~ J:O-Rs--l5 - -~:- CL-20 CL-l 0 ~ RB-14 CL-9 ~ - - ... ' ..... ~ -" CL-~ CL-19~ oRB - 1 3 • 0 RB-1 2 ~ 0 mt l ,.~;" ~--- RB- 22 CL-1B-.. - - 0 1"7 '; 0 RB-23 0 r 14 ,15 _12 .?~'~ __ ~ BE=-9 Advance BE-8 BE-7 Pillar : RB-ll; CL_h ~RB-1Q ~E-6 {I BE-5 14 _) ORB-9{ _--.f. ~~ __4 _______ --C-L~3-~--:CL-6 • RB 8'-''t]' --CL-lt 00 0 RB-27 o 5 8 oRB-21 1.2 RB-5 A f) CL-1 3 (I) RB - 1 7 • 0 RB-26 -4 "CL-S ~ ~ CL-16 oRB-20 C[-7 ~ CL - 2 RB-3 e RB- 7 :l§ - --------- • RB _ '1 6 ~'3 CL -1 ·C L- 4. ~-RJ3,:~_·EiE ~l---------- -.. + RB-19 4r~ 11 rnI 0 RB- 25 BE-3 BE-2 BE-l @@- Left entry Right entry o Roof bqlt (hydraulic) RB o Roof bolt (strain gauge) • Vertical closure 0' Crib load cell (strain gauge) CL [ill Cri b load ce 11 (hydra u1 i c) ~Horizontal ~ closure Brass rod extensometer Fig. 23.--Different instruments and their location at test area 1. 57 ------- .... --'---------- --------- --- -- .-. - -- .... _ ...... .".,... _ - - - - - - , , ~ ~ @ .... -... .,.",..--- .... ,,, " ,.""........ \ , I ,• I \ ---------------- l \ \ _ /• /' .",..--..",.. -- Left entry @- Ri ght entry 6 CL-19 6 CL-18 .,,- ,I ,I ........ ----.,- .... - 6 I --_.-."..----- Vertical closure CL t \.----- ------6 --CL-=-17--- - -- ----6 @- CL . . 20 ____ ,-, - - - - - - - . . . . , . , U A CL-16 Fig. 24.--Location of vertical closure stations in mine 2, test area 1. 58 TABLE 4 MEASURED AND PREDICTED VERTICAL CLOSURES * M E A S U RED CL Station Vertical Closure 15 0.237 20 0.323 PRE D I C T E 0 Elastic Solution Double Single Elastic-Plastic Solution Single Double 0.367 0.359 0.367 0.364 0.642 0.660 0.664 0.359 0.413 0.364 0.453 0.480 0.464 (0.384) ** Average: 0.280 16 0.426 0.651 19 0.292 (0.641) Average: 0.359 17 0.877 0.413 18 0.475 (0.401 ) Average: 0.676 Average 0.438 0.477 (0.475) * All measurements and predictions were i n inches ** Coarse mesh results 59 in the finite element analysis which would reduce the predicted values and bring the averages into better agreement. Though the roof span is greater in the single entry system than in the double entry,displacements are higher in the latter system due to pillar squeeze. also appear to reflect pillar squeeze. Field data Predicted vertical displacement in the pillar is less than field measurements which indicates that coal modulus from laboratory tests is higher than the actual field value. No attempt was made to change material properties for matching computer displacement results and field measurements in greater detail. Interpretation of Results A comparison of results for the double and single entry analyses are shown in Figure 25. Opening geometry, vertical closure, stress and failure zone in both the systems are compared. Width to height ratios in single and double entry systems are 3.7 and 3.4 respectively. opening in both cases is the same. opening in a si~gle The height of the Since the width of the entry system is more than in the double entry system, one can expect higher values ' of displacements or vertical closure, higher stress concentration, and a more extensive failure zone in the single entry. In both cases, layering effect is included and each layer represents its own mechanical behavior. But the differences in the results for the two systems are generally very small. W_ 1) Geometry Tension 60 Double entry Single entry H- w = 3.7 IT Compression 3.4 Tension Compression ~ompression I I T 7~ Openi ng Opening 26' ~- 24' 2) Predicted stress concentrations ----I Tension -1.62 Tension -1 .73 Compression In pillar -2.02 In solid coal -1.8 Compression -1 .84 3) Maximum vertical closure 0.660" 0.664" 4) Predicted failure zone Massive S~le ~ndstone ~~ I 2.0' ~.5' I Opening ...,Jr ~ Massive sandstone ..:::::::: 1.5' Massive phale sandstone : ~ ~ t::] 1/ 2.0' ---::--- 2.5' Opening E: -..--.. \Massive sandstone t 0.5 Strengths Massive sandstone Co = 19000 psi To = 850 psi Shale Co = 15000 psi To = 1000 psi Coal Co = 3500 psi 130 psi To = Massive sandstone Co = 19000 psi 850 psi To = Shale Co = 15000 psi To = 1000 psi Coal Co = 3500 psi 130 psi To = Fig. 25.--Comparison between single and double entry results. 61 Vertical closure in the single and double entries are very close and for practical purposes they are equal. Tensile stress concentration in the single entry is approximately 6 percent higher than in the double entry system, whereas critical compressive stress concentration near the opening is higher in the double entry. Tensile stress con- centration in the single entry system is higher because the length of the roof span is greater, The results show that absence of pillars in the single entry reduces the value of critical compressive stress in the sides of the opening. The predicted failure zone from elastic-plastic analysis for the single and double entries extends in the roof and floor. Failure zone extends into the immediate roof (shale and massive sandstone) layers. No failures are indicated in the remote roof that consists of layered sandstone. The floor is a massive sandstone and the failure zone is less in the floor than in the roof. In single and double entry systems the failed zone in the roof is almost the same size. But in double entry,failed elements in the floor are less than in the other case. Rib failures were not indicated in both the cases due to higher strength of coal and triaxial state of stress in the rib elements. The failure zone in the roof extends into two layers and the height of this zone is approximately 4.5 f~et~ ..~ti­ ficial roof supports can control the failure zone and'growth of this zone around the opening. Six foot long roof bolts 62 are used for controlling fractured zones in the rnof. How- ever, the results from the single and double entry analyses show that the difference in ~esults for both systems are v e r y . s rna 1 1. ' Stability Criteria Stability (25,32,36 )evaluations of surface and underground openings in rock are based mainly on considerat ions of strength and stress. The stability of the open- ing will depend on the ability of the rock and artificial support system to resist the applied loads. In general, stability of an opening is determined on the basis of safety factors. The safety factor FS (11, 37) is defined as the ratio of the relevant values of strength to load, that is: Safety factor IFS I = strength load A structure is stable when the load acting on it is smaller than the strength of the structure. Failure occurs when load acting on it is greater than the strength of the structure or when load exceeds the strength. These statements in terms of the safety factor are as follows: FS >1 stable structure FS <1 failed or unstable structure As stated above, factors of safety greater than one are considered indicative of stability and factors of safety less than one indicate that the opening or structure 63 would be unstable. Failure or instability are used to evaluate any kind of operational hazard, such as collapse or failure of an opening, caused by application of load leading to unserviceability of the structure or opening over its intended lifetime. The object of a stability analysis is to determine whether or not the opening will remain useful over its intended life. Complete stability of an opening can be determined when the safety factors are known. But the safety factors in rock mechanics are not well defined and direct methods are not available for computing safety fac~ tors. In the present study stability of the single entry is determined from the results of the elastic and plastic analyses. elastic~ Elastic analysis includes calculation of elastic stress concentrations and displacements around openings. Elastic~plastic analysis is utilized to deter- mine the failure or collapse conditions of the underground openings. Elastic-plastic stability analysis involves the usual considerations of applied loads, material properties, and opening geometry. Results of the elastic analysis were presented in the previous sections. Elastic stress concentrations and displacements around single and double entries are comparable and close. The size of the failed or plastic zone is indicative of the risk of collapse. The failure zone . 64 around the opening indicates the regions whose safety factors are one. Local failures may not lead to the unstable or collapse conditions. The predicted localized failure zones can be controlled by using artificial supports (16, 18, 20, 22, 27). VI. CONCLUSIONS An existing finite element computer program has been used for double and single entry stability analyses. All program branches used in the present work are fully verified by comparison with existing analytic and numerical solutions of various problems. Applicability of the solu- tion technique to stability analysis is assessed by comparison of closure predictions and mine measurements obtained in an instrumented double entry section of the Sunnyside Mine. Predicted vertical closure values are, on average, in good agreement with vertical closure measurements in the mine. The effect of layering is included in both the dou- ble and single entry analyses, although the divided layers are considered to be isotropic. Field measurements indi- cate that maximum vertical closure occurs near the rib of the coal pillar. But the maximum predicted vertical clo- sure is along the centerline of the opening. The coal Young's modulus and strength determined by experimental procedure are therefore probably higher than the actual in-situ properties. Predicted vertical closure is approximately 10 percent higher than the measured values. The effect of roof bolting is not 66 included in the analysis, which would reduce the predicted closure values. Stability evaluation of the single entry system requires the knowledge of stress concentrations, ments and failure zones around the opening. displace~ Stress field, displacement and failure zone predictions have been made for the single and double entry systems. The vertical clo- sure of the single entry is higher than in the double entry, but the closure values in both the cases are equal for practical purposes. Tensile stress concentration in the single entry system is approximately 6 percent higher than' in the double entry design. However the results indicate that absence of pillars in the single entry reduces the value of the compressive rib stresses around the opening. Permanent deformation and failure are important features of the single and double entry systems. Pre- dicted failure zones in the single and double entries are of the same size. tends in two roof Failure zone in the single entry exlayers~ The height of the failure zone in the roof is approximately four feet; this can be easily controlled by using roof bolts. In the single and double entry systems the differences in stress concentration, displacements, and failure zones are small. Dangerous pillar stress concentrations and wide intersections are eliminated in the single entry system. The results of the present investigation indicate 67 that the single entry is as stable and safe as the present double entry on development, but the effect of supports and influence of mining operations has not been considered in the present work. The resul ts of the present investigation offer en,couragement, and further research should be continued for a more complete assessment of stability and safety of the single entry longwall mining system. Suggestions for fur- ther research include: 1. Reinforcement effects of bolts and cribs should be included in future research. The effect of bolting can be introduced by applying two point loads for each bolt. Nonlinear behavior of wooden cribs can also be included. 2. Adequacy of cribs and caving patterns are important factors in the single entry stability analysis. 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Inf., 1966 .. 6. Desai, C. S., and Abel, J. F., "Introduction to the Fi ni te El ement Method," VNR Company, New York, 1972 . 7. Drucker, D. C. IIA Definition of Stable Inelastic Materia1,11 J. Applied Mechanics, Vol. 24, March 1959, pp. 101-106. 8. Drucker, D. C., Prager, W., and Greenberg, H. J., "Extended Limit Design Theorems for Continuous Media,1I Q. Applied Math., Vol. 9, No.4, 1952, pp. 381-389. 9. Duncan, J. M., and Goodman, R. E., "Finite Element Analysis of Slopes in Jointed Rock," Contract Report S-68-3, University of California, Berkeley, 1968. 10. 0f Min e Roc k , ~I 1. . Emery, C. L., IIIn-situ Measurements Applied to Mine Design,1I 6th Sym. on Rock Mechanics at Rolla, October 1964. 69 11. Freudenthal, A.M., Garrelts, J.M., and Shinozuka~ M., "T h e An a 1y sis 0 f S t r u c t u r a 1 Sa f e ty Pro c. AS CE Volume 92, N. ST1, pp. 267-325, Feb. 1966. ,II 12. Fung, Y.C., "Foundations of Solid Mechanics,1I Prentice-Hall, Inc. (1965) 13 . Ham mer s 1 e y, J. M., and Han d s com b, D. C" II M0 n t e Car 1 0 Met hod s ," J 0 h n Wi 1 ey and Son s, Inc., N. Y., 1965 14. Hill, R., "The Mathematical Theory of Plasticity," Oxford. The Clarendon Press, 1950. 15. Holland, C.T., "Cause and Occurrence of Coal Mine Bumps," Mining Engineering, Transactions AIME, pp. 996, September 1958. 16. Jacobi, 0., "The Origin of Roof Fall in Starting Faces with the Caving System," Int. J. Rock Mech. Min . Sc i ., Vol. 1, p p. 31 3 - 3 1 8, 1 9 6 5 • 17. Jaeger, J.C. and Cook, N.G. W., "Fundamentals of Rock Mechanics," Chapman and Hall Ltd. 1969. 18. Kenny, P., "The Cavi ng of the Waste on Longwa 11 Faces," Int. J. Rock Mech. Min.Sci., Vol. 6, pp. 541-555, 1969. 19. Lekhuitskii, S.G., "Theory of Elasticity of an Anisotropic Elastic Body," Holden-Day, Inc., San Franc i sco, 1963. 20. Lewis, A.T., and Wllson, J.W., "The Introduction of Composite Pack Support at Western Holdings, Ltd.," Papers and Discussions 1968-69, Association of Mine Managers of South Africa, The Chamber of Mines of South Africa, pp. 561-583. 21. Love, A.E.H., "A Treatise on the Mathematical Theory of Elasticity," Dover Publications, N.Y., 1944. 22. Margo, E., and Bradley, R.K.O., "An Analysis of the Load Compression Characteristics of Conventional Chock Packs," Journal of the South African Institute of Mining and Metallurgy, pp. 364-401, April 1966. 70 23. Nair, K., Sandhu., R.S., and Wilson, E. L., IITimedependent Analysis of Underground Cavities Under an Arbitrary Initial Stress Field," Tenth Symposium on Rock Mechanics, Austin Texas, AIME Publications, 1968. 24. Naude, T. R., liThe Pioneering of Full Mechanized Longwa11 Coal Mining in South Africa," Journal of the South African Institute of Mining and Metallurgy pp. 322-350, February 1967. 25. Obert, L., and Duvall, W., IIDesign and Stability of Excavations in Rock-Subsurface,1I SME Mining Engineering Handbook, Edited by Given, I.A., Vol. 1, pp. (710) - (7-47), SocietyofMining Engineers, AIME, N.Y., 1973. 26. Obert, L. Duvall, W. I., and Merrill, R. H., UDesign of Underground Openings in Competent Rock,u Bulletin 587, U.S. Bureau of Mines, pp. 8-17,1960. 27. Panek, L.A., IIDesign for Bolting Stratified Rock," Transactions, Society of Mining Engineers of AIME, Vol. 229, pp. 113-119, June. 1964. 28. Pariseau, W. G., IIA Finite Element Program Based on the Elastic-Plastic Material Idealization," University of Utah, 1972. 29. Pariseau, W.G., IIInfluence of Rock Properties Variability on Mine Opening Stability Analysis," Ninth Canadian Symposium on Rock Mechanics, Montreal, Quebec, 1973. 30. Pariseau, W.G. IIInterpretation of Rock Mechanics Oat a : Sin 9 1 e En try Sy s t em, Sun ny sid e Min e, Uta h , First Annual Report, U.S.Bureau of Mines Contract #H022077, June 1973. \ 1.1 31 . Pariseau, W.G., "Plasticity Theory for Anisotropic Rocks and Soils," lOth Symposium on Rock Mechanics, Au s tin, Te x as, 1 968 . 32. Pariseau, W. G. IIRock Mechanics and Risk in"Open Pit Mining,1I Proc. Eleventh International Symposium on Computer Applications in the Mineral Industry, University of Arizona, Tucson, April 16-20, 1973, pp. A106-Al24. 33. Pariseau, W.G. and Fairhurst, C., "The Force-Penetration Characteristic for Wedge Penetration into Rock," Int. J. Rock Mech. Min. Sci. Vol. 4, pp. 165-180, 1967. 71 34. Phillips, D.W., "Investigation of the Physical Properties of Coal Measured Rock," Transactions AIME, Vol. 82, pp. 432-450, 1931. 35. Reyes, S.F., IIElastic-Plastic Analysis of Underground Openings by the Finite Element Method,1I Ph.D. Thesis University of Illinois, Urbana, 1966. 36. Salamon, M.D.G., "Stability, Instability and Design of Pillar Workings,1I Int. J. Rock. Mech. Min.Sci., Vol. 7, pp. 613-631,1970. 37. Salamon, M.D.G., and Muro, A.H., IIA Study of the Strength of Coal Pillars,1I Journal of the South African Institute of Mining and Metallurgy, pp. 55-67, September 1967. 38. Shields, J.J., IILongwa11 Mining in Bituminous Coal Mines with Planers, Shearer Loaders and Self Advancing Hydraulic Roof Supports,1I Information Circular-8321, U.S. Bureau of Mines, 1967. 39. Stacey, T. R., IIThree-dimentional Finite Element Stress Analysis Applied to Two Problems in Rock Mechanics,1I Journal of the South African Institute of Mining and Metallurgy, Vol. 71, May 1972. 40. Su, Y.L., Wayng, Y.J., and Stefanko, R., IIFinite Element Analysis of Underground Stresses Utilizing Stochastically Simulated Material Properties," Rock Mechanics Theory and Practice (Ed. Sowerton, W.H.), AIME/SME, N. Y., pp. 253-266, 1970. 41. Sutherland, R.B., IIS ome Dynamic and Static Properties of Rock,1I Rock Mechanics, Edited by Fairhurst, C., pp. 473-491, Pergamon Press, New York 1963. 42.· Wagner, H.M., IIPrincip1es of Operations Research,1I Prentice-Hall, Inc., N.J. 1969. 43. Winders, S. L., IIPhysical Properties of Mine Rock ll Part II, U.S.B.M., RI 4727, 1950. 44. Zienkiewicz, O.C., liThe Finite Element Method in Engi neeri ng Sci ence, II McGraw-Hi 11, London, 1971. , APPENDIX A DETAILS OF MATERIAL CONSTANTS For isotropic material yield conditions dependent on the intermediate principal stress, equation 8 can be rewritten as(4, 31): IJ I n / 2 = A (A-1 ) 1+ B Where (A-2) and I = 11 _ 2- Taking compression as positive, values for A and B ··c an be f 0 un d by sub s tit uti ng 0 x = C, u nco nfin e d com pre s s i ve strength, and T" uncon fi ned tens i 1 e strength, and n = 0y = 0 xy = 0 1. C I? T 72 = CA 2 + B = -TA -2- + B (A-3) Solving for A and B from (A-3): - C A = 12 T T + C TC B = 12 T+C TC and R = 2 T+C 13 (A-4) 73 Where R is unconfined shear strength; A, B, and R define the yield parameters for isotropic extended Von Mises yield in terms of unconfined compressive and tensile strengths. A generalization of equation 9 for anisotropic material is given by Pariseau, 1968 (31): F (a y -a z )2+G(a z -a x )2+H(a x -a y )2+La 2yz +Ma 2 zx +N a 2xy n/2 I I (A-5) - (Ua x + Va y + Wa z ) = 1 The material constants (F, G, H, L, M, N, U, V, and W are defined in terms of unconfined compressive, tensile and shear strengths in the principal axes of anisotropy. ing the same procedure used for equations A-4 and n 2F = 1 [( 1 +1 ) 2 + 4" 1 Ty Cy (1T z + 1 )2 Cz (tx+ tx )2] Repeat= 1 (31): [( tz+tz )2 + (1Tx + 1Cx )2 (1 + 1 )2] Ty Cy + 1 )2] 1 [(1 + 1 )2 + (1 + 1 )2 - (1 2H = 4" Tz Cz Ty Cy Tx ex 2G = 4" U = 1/2 (l/Tx -l/C x ) V (1 IT y 1 ICy) W = 112 (l/T z L = 1 I R2 l/C z ) = .1 12 (A-6) 2 M = 1/5 N = l/T2 Wher e Cx' Cy ' Cz , Tx , Ty , Tz , R, 5, Tar e un con fin e d compressive, tensile and shear strengths referred to the principal axes of anisotropY (31). APPENDIX B DETAILS AND VERIFICATION OF THE FINITE ELEMENT PROGRAM The finite element program is capable of handling a wide variety of problems including arbitrary sequences of cuts and fills, face advances and a number of material complexities. The two dimensional (plane strain, plane stress and axial symmetry) problems involving anisotropic, time dependent, nonhardening, gravitating elastic-plastic materials may be initially stressed. Nonlinearities in mate- rial properties are approximated. Linear and nonlinear extended VonMises type of yield conditions for anisotropic geologic media are used in the program (28, 30). Ihe program consists of " a main l"ine and followed by eight subroutines. The mainline acts as an executive rou- tine and reads most of the input data and other information of the problem. A brief flow diagram of the program is shown in Figure 26. Element stiffness matrices are formed by Elstif and Assem adds and subtracts element stiffness from the master stiffness matrix. Writer prints the re- sults of the elastic and elastic-plastic analyses. The Elyeld subroutine locates the intersection of load paths 75 Read Input Oa ta Element Nodal Points Nodal Point Coordinates Material Properties Boundary Conditions l Calculate nodal' point forces due to gravity and initial stresses j Introduce excavation surface, prescribed forces, and displacements ~ Generate master stiffness matrix for thermal and time dependent problems, call T-moduli and T-force 1 Calculate strains and stresses caused by displacements and add to initial stress ~ Print results of elastic solution l· 1 Only elastic solution required Elastic-plastic solution is required 1 l Stop l I (continue) I l Elastic-plastic analysis; solve for incremental displacements 1 Calculate s tra ins· and check for failed elements 1 Update s ti ffness for failed elements and compute stress ! Print results of failed elements in each load · increment ! ~nt elastic-plastic solution after final . lncrement I l I Stop I Fig. 26.--Flow diagram for finite element program. 76 with the present yield surface. The Axes routine trans- forms forces and displacements to local coordinates or global coordinates as required. T-module reads the time dependent material properties and T-force computes time dependent loads. The assembled system of equations has the form F = [K] {u}, where {F} and {u} are the nodal point force and displacement respectively and [K] is the master stiffness matrix. For solving the system of equations Gauss-Seidel line iteration method is used. Where the node displacement component is prescribed the corresponding row of equations is skipped during the iteration. In the case of nonlinear stress strain analysis the final load is applied in a prescribed number of load increments. The system of equations is solved by iteration for each load increment. Program Verification All branches of the finite element program used for the present work are verified and results are compared with available analytic and numerical solutions (9, 19, 21, 35). Each example problem verifies gram. ~ifferent branches of the pro~ Three example problems are shown here: Example Problem 1: Elastic analysis of hollow cylinder under internal pressure. For this problem the analytic solution is known and shown in Figure 27. Results from the finite element method agree within the plotting accuracy of the analytic solution (21). This example problem verifies the 77 y o a P 0 0 Exact solution (Ref. #21) Finite element solution o x a b 2.0 4.0 1. 5 3.0 at U p 1.0 2.0 -a p 0.5 o 1 .0 r a r 1 .0 2.0 Fig. 27 .--Elastic analysis of stress and displacement in a hollow cylinder under internal pressure. 78 linear, homogeneous isotropic stress, strain, displacement, and boundary condition branches of the program. Average nodal point stresses are given in the example problems. Example Problem 2: Elastic analysis of circular hole under internal pressure. This example verifies the anisotropic elastic branches of the program. This example was run under plane stress conditions to compare with analytic and finite element results. The results are presented in Figure 28. In Figure 28 the results are in good agreement with the analytic results (19) and present results predict more maximum stress than those mentioned in reference #9. Better results can be obtained by decreasing the size of the elements around periphery of the hole. Example Problem 3: Elastic-plastic analysis of circular tunnel in rock, loaded by external forces. This problem has been solved numerical~y (35) and provides a test of the linear yield, isotropic elastiGplastic stress and displacement branches of the program. The yield condition, one appropriate to rock, is used in this analysis. The circular tunnel in rock is under nonuniform external loading. Results are shown in Figure 29 which are in good agreement with those reported in reference #35. The verification of the program is discussed in greater detail in reference #30. 79 2 3.0 p 2.0 e 1. 0 0 -6A-~~~ Ex(act solut.ion Ref. #19) Finite element Reference: E =1.2 x 10 o ~~~ o __ ~ __ ~ ____ 1~0 ~ __ ~-i~ 2.0 (9 ) 5 psi l 5 E =0.6 x 10 psi 2 G = 7000 psi __ ~~ 3.0 Fig. 28 .--E1astic analysis of stress in a transversely isotropic plate containing a circular hole under uniform pressure. 80 1.0 ksi l 11 11 ~0.4 ksi ~ --~--------~~ 2.0 o I~fi- 8 R ....... ~ A Reference: o 1.5 - o (35) Present program .,.. I Vertical stress on OA 1. 0 V) V) OJ ~ ...., (/) 0.5 Horizontal stress on OA 2R 3R Fig. 29.--Elastic plastic analysis of stress in a circular tunnel in rock. APPENDIX C ROCK PROPERTIES VARIABITY STUDY Uniform material properties are used in the conventional finite element stress analysis. These values are usually midrange or average values determined from laboratory tests. Because the rock is not perfectly homogeneous and isotropic, the laboratory material properties in the same rock may change from sample to sample. Moreover in the usual laboratory testing the effect of moisture, temperature, and structural defects are not taken into account. These factors may further reduce the values deter- mined by laboratory tests. A standard deviation of 30 per- cent of the average values is not uncommon in rock properties testing procedures (41). Variability of material properties would influence the stability of mine openings" land the stability analyses of openings determined by using average values are therefore questionable. Two different methods were used to introduce the effect of material properties variability into the stress analysis (30).· These are: 1) A conventional stress analysis procedure where the material properties vary for different runs; and 82 2) A modified conventional method in which the effect of variability is directly introduced into stress analysis (29, 32, 40). The finite element mesh used for the first case represents a typical rectangular opening and pillar in a long row of such entries at a depth of 1750 feet. The width to height ratio of the opening is 3-4 which is the same ratio as in the double entry system. The material properties used for four elastic and elastic-plastic analyses are given in Table 5. Figure 30 shows stress concentrations around the periphery of the opening for four sets of rock properties. These stresses are taken from the elastic part of the anlysis. Uniform rock proper- ties were used for this case and the material assumed as homogeneous and isotropic. Results from , Figure 30 show that with increasing Poisson's ratio, the tensile stress concentration in the roof tend to vanish. Critical compressive stress is not affected much by changes in Poisson's ratio. The average pillar stress can be calculated from formula (26) a p = 0v (f-R) where 0p and 0v are pillar and vertical stresses and R is the extraction ratio, which is defined as the ratio of area mined to original area. According to this formula for a given extraction ratio, the pillar stress in Figure 30 should have stress concentration of 2.08. Results from the finite element analysis are close to this value. But 83 TAB LE 5 MATERIAL PROPERTIES USED IN FOUR ELASTIC-PLASTIC ANALYSES OF A RECTANGULAR OPENING P R0 P E RT Y E (10 6 psi) v Co (10 3 psi) To (10 2 psi) 1 3.0 o. 1 5 5.0 2.5 2 3.0 0.25 10.0 4.5 3 3.0 0.38 15.0 7.0 4 3.0 0.45 10 .0 4.5 Run 84 6 0 0 5 0 4 °t ~ • No. l-E=3xl0 6 psi No. 2-E=3xl0 6 psi v= 0.25 No. 3-E=3xl0 6 psi No . 4-E=3xl0 6 psi v= 0.38 v= 0.45 o. 1 5 v= 3 2 1 2 o i=~~~~~~~~~~~~+-=C~o~mLP~r~e~s~s~i~on~___ - Tension -1 I I I .I I I I 1 J J , / I / w = 3.4 "IT Fig. 30 .--Stress concentration as a function of Poisson's ratio for a rectangular opening. 85 the critical stress concentration given in Figure 30 is approximately 3.1 and near skin of the pillar is about 2.2. Higher peak compressive stress concentrations in the pillar are due to sharp corners. The horizontal pre- mining stress 0H is computed from the relation 0H ( 1v_ v -) and vis Poi s son's rat; 0 . = cr v This i nd i cat est hat the variation of Poisson's ratio changes the magnitude of tangential stress around the opening. Plastic properties used for elastic-plastic analysis are given in Table 5. Results of the elastic analysis show that the rock properties influence the growth of the tensile region in the roof. Figure 31 shows the failure zone in the roof and floor for four elastic-plastic cases. Young's modulus in all the four cases is constant. Pois- son's ratio, tensile strength, and compressive strength were varied. Tensile failure zone in the roof and floor for the first two cases is of appreciable size. In these two runs Poisson's ratio and tensile strength are relatively low and the converse is true for runs 3 and 4. None of the elements in the pillar failed. The material properties were changed in the conventional stress analysis but in each run the rock is assumed to be homogeneous. It is more realistic when the are changed from point to point. properti'es ~ In the modified conven- tional method each element in the mesh is assigned different material properties (the material within the 86 I Opening I I Opening II, I Run 1 Run 2 51 ~ ~ , I t~I I I Opening t'I Opening I I I I Run 3 Run 4 Fig. 31 .--Element failures about a rectangular opening with different material properties. 87 element is homogeneous). For this purpose, the finite element method is well suited for handling complex rock properties, and the usual assumption that rock ii made of uniform material need not be made. The variation of material properties in the stress analysis is directly introduced by using the Monte Carlo simulation technique (13,42). This type of approach has been applied to elas- tic finite element analysis (40) and elastic-plastic analy sis (29, 3"2) 0 f min e 0 pen i n g s . In the modified conventional analysis Monte Carlo simulation randomly assigns each element properties fng to a specified distribution. accord~ The distribution speci- fied is determined from the test data. The distribution of test data can be obtained from a histogram. A single entry mesh with eight rock layers has been used for this study. Special computer coding in the pro- gram divides each layer into ten subtypes and computes material properties for each subtype with standard deviations of 10 percent mean values. These values are given in Table 6. Figure 32 shows predicted failure zones in the roof and floor of a single entry. Figure 32a is from conven- tional analysis with eight layers and based on mean values of test data;,Figure 32b : is based on Monte Carlo simulation of properties variability within the main layer. Failed ele- ments represent the tensile stress concentration in the 88 TABLE 6 RANDOM ROCK PROPERTIES DEFINING 80 SUBTYPES Type E psi 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 .9350+06 .1105+07 · 1275+07 .1448+07 · 1615+07 .1785+07 · 1955+07 .2125+07 .2295+07 .2465+07 · 1650+07 · 1950+07 .2250+07 .2550+07 .2850+07 .3150+07 .3450+07 .3750+07 .4050+07 .4350+07 · 1925+06 .2275+06 .2625+06 .2975+06 .3325+06 .3675+06 .4025+06 .4375+06 .4725+06 .5075+06 .8250+06 .9750+06 .1125+07 .1275+07 .1425+07 · 1575+07 · 1725+07 · 1875+07 .2025+07 .2175+07 G psi v .4110+06 .4753+06 .5368+06 .5959+06 .6525+06 .7069+06 .7592+06 .8095+06 .8579+06 .9046+06 .7432+06 .8628+06 .9783+06 .1090+07 · 1197+07 · 1302+07 .1402+07 .1500+07 · 1594+07 · 1686+07 .8262+05 .9519+05 · 1071 +06 · 1185+06 .1298+06 .1397+06 · 1496+06 .1591+06 · 1681 +06 .1768+06 .3910+06 .4577+06 .5233+06 .5876+06 .6507+06 .7127+06 .7735+06 .8333+06 .8921+06 .9498+06 .1375+00 · 1625+00 .1875+00 .2125+00 .2375+00 .2625+00 .2875+00 .3125+00 .3375+00 .3625+00 · 1100+00 · 1300+00 · 1500+00 .1700+00 .1900+00 .2100+00 .2300+00 .2500+00 .2700+00 .2900+00 · 1650+00 · 1950+00 .2250+00 .2550+00 .2850+00 .3150+00 .3450+00 .3750+00 .4050+00 .4350+00 .5500-01 .6500-01 .7500-01 .8500-01 .9500-01 .1050+00 .1150+00 · 1250+00 · 1350+00 .1450+00 C psi To psi Ro psi .5500+03 .6500+03 .7590+03 .8500+03 .9500+03 .1050+04 .1150+04 · 1250+04 .1350+04 · 1450+04 .1045+05 .1236+05 · 1425+05 .1615+05 · 1805'+0.5 · 1995+05 .2185+05 .2375+05 .2565+05 .2755+05 .1926+04 .2275+04 .2625+04 .2975+04 .3325+04 .3675+04 .4025+04 .4375+04 .4725+04 .5075+04 .9625+04 .1137+05 · 1312+05 .1487+05 .'1662+05 .1837+05 .2012+05 .2188+05 .2362+05 .2537+05 .2750+02 .3250+02 .3750+02 .4250+02 .4750+02 .5250+02 .5750+02 .6250+02 .6750+02 .7250+02 .4675+03 .5525+03 .6375+03 .7225+03 .8075+03 .8925+03 .-9775+03 .1062+04 · 1147+04 .1232+04 .9900+02 .1170+03 · 1350+03 · 1530+03 .1710+03 .1890+03 .2070+03 .2250+03 .2430+04 .2610+03 .3850+03 .4550+03 .5250+03 .5950+03 .6650+03 .7350+03 .8050+03 .8750+03 .9450+03 · 1015+ 0 4 .3024+02 .3574+02 .4124+02 .4674+02 .5224+02 .5774+02 .6323+02 .6373+02 .7423+02 .7943+02 .5167+03 .6107+03 .7046+03 .7985+03 .8925+03 .9864+03 .1080+04 .1174+04 · 1268+04 .1362+04 .1087+03 · 1285+03 · 1463+03 .1650+03 .1878+03 .2076+03 .2273+03 .2471+03 .2669+03 .2866+03 .4275+03 .5052+03 .5829+03 .6606+03 .7383+03 .8161+03 .8938+03 .9715+03 .1049+04 .1127+04 0 TABLE 6.--Continued Type 41 42 43 44 45 46 47 48 . 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 E 89 G psi psi \) · 1650+07 · 1950+07 .2250+07 .2550+07 .2850+07 .3150+07 .3450+07 .3750+07 .4050+07 .4350+07 .2475+07 .2925+07 .3375+07 .3825+07 .4275+07 .4725+07 .5175+07 .5625+07 .6075+07 .6525+07 · 1925+06 .2275+06 .2625+06 .2975+06 .3325+06 .3675+06 .4025+06 .4375+06 .4725+06 .5075+06 · 1925+07 .2275+07 .2625+07 .2975+07 .3325+07 .3675+07 .4025+07 .4375+07 .4725+07 .5075+07 .7432+06 .8628+06 .9763+06 .1090+07 .1197+07 · 1302+07 · 1402+07 .1500+07 · 1594+07 .1686+07 .1173+07 · 1373+07 .1570+07 · 1763+07 · 1952+07 .2138+07 .2321+07 .2500+07 .2876+07 .2849+07 .8262+05 .9519+05 · 1071 +06 .1185+06 .1294+06 · 1397+06 · 1496+06 .1591+06 .1681+06 · 1768+06 .8671+06 .1007+07 .1141+07 .1271+07 · 1397+07 .1519+07 .1636+07 · 1750+07 .1860+07 · 1967+07 · 11 00+00 .1300+00 · 1500+00 .1700+00 .1900+00 .2100+00 .2300+00 .2500+00 .2700+00 .2900+00 .5500-01 .6500-01 .7500-01 .8500-01 .9550-01 .1050+00 .1150+00 .1250+00 .1350+00 .1450+00 .1650+00 .1950+00 .2250+00 .2550+00 .2850+00 .3150+00 .3450+00 .3750+00 .4050+00 .4350+00 · 1100+00 · 1300+00 · 1500+00 .1700+00 .1900+00 .2100+00 .2300+00 .2500+00 .2700+00 .2900+00 Co To R psi psi psi .1045+05 · 1235+05 · 1425+05 .1615+05 · 1805+05 · 1995+05 .2185+05 .2375+05 .2565+05 .2755+05 .8250+04 .9750+04 .1126+05 · 1275+05 · 1425+05 · 1575+05 .1725+05 .1875+05 .2025+05 .2175+05 · 1925+04 .2275+04 .2625+04 .2975+04 .3325+04 .3675+04 .4025+04 .4375+04 .4725+04 .5075+04 .1072+05 · 1267+05 .1462+05 · 1657+05 · 1852+05 .2047+05 .2242+05 .2437+05 .2632+05 .2827+05 .4675+03 .5525+03 .6375+03 .7225+03 .8075+03 .8925+03 .9775+03 .1062+04 .1147+04 · 1232+04 .5500+03 .6500+03 .7500+03 .8500+03 .9500+03 .1050+04 · 1150+04 · 1250+04 · 1350+04 · 1450+04 .9900+02 · 1170+03 · 1350+03 · 1530+03 .1710+03 · 1890+03 .2070+03 .2250+03 .2430+03 .2610+03 .5500+03 .6500+03 .7500+03 .8500+03 .9500+03 .1050+04 .1150+04 · 1250+04 · 1350+04 · 1450+04 .5167+03 .6107+03 .7046+03 .7985+03 .8925+03 .9864+03 .1080+04 .1174+04 .1268+04 · 1362+04 .5954+03 .7036+03 .8119+03 .9202+03 .1028+04 · 1137+04 · 1245+04 · 1353+04 · 1461 +04 · 1570+04 .1087+03 .1285+03 · 1483+03 · 1680+03 .1878+03 .2076+03 .2273+03 .2471+03 .2669+03 .2866+03 .6041+03 .7139+03 .8238+03 .9336+03 .1043+04 · 1153+04 · 1263+04 · 1373+04 · 1483+04 · 1593+04 0 90 I I I I t: Opening I I (a ) 2' ~e , I Opening tl I I ( b) Fig. 32 .--Element failures around single entry opening: a) conventional method with mean-values; b) with random material properties. 91 roof and floor. When random properties are assigned, the failed elements extend farther toward the rib of the pillar than in the conventional method. A conventional anal- ysis using low strength properties may indicate less failure and more stability, but simulated rock properties variability may result in a larger failure zone and collapsible conditions. APPENDIX D MODEL LOADING STUDY Displacements and stresses from the finite element analysis woul d depend on the type of model 1oadi ng. These values may be higher or lower depending on different types of loads. results, Meshes that extend to surface would give best but small meshes are more economical because they contai n 1 ess elements and requi re 1 ess computati onal effort~ Six examples are taken to compare the results between full and small meshes. These examples are illus- trated schematically in Figure 33,and details are given below. a) The full mesh is to the surface. Initial 2250~ gr~vity to the weight of the overburden. x 500' iri size and extends stresses are computed due Nodal forces around the opening are calculated to the excavated surface by a set of stress boundary conditions and these stresses are added to the initial gravity loads. This type of model loading gives more accurate stresses and displacements. b) The mesh used in this case is small and 240' x 230' in size, which is also a part of full mesh. Small mesh nodes on the external boundaries will exactly match the full mesh. Displacements in X and Y directions are 93 Surface C:J (a ) (b) Surface Surface 7777177777 (c ) /7 1630' 1630' (e ) (d ) Pass-l ( f) Fig. 33 .--Details of various types of model loading. 94 taken from the output of full coarse mesh for the small mesh. ~nd used as input In this way, the effect of the removed material is introduced by prescribing the displacements on the external boundary nodal points of small mesh. c) This model loading is the same as in (b) except the stresses generated by the excavated material are not added to the gravity loads. Because the effect of excava- tion is not included in this model and stresses obtained in this case are equal to the unit weight of material times depth d) In this case, the effect of overburden is introduced by burying the small mesh at the same depth as in the full mesh. This can be obtained by simply adding the difference in depth to all nodal point coordinates. Grav- ity loads are computed as unit weight times depth and the stresses generated by the excavated material are calculated by a set of stress boundary conditions on the excavation surface. These stresses due to excavation'are added to the gravity loads. The results from this model are within 4 to 9 percent as compared to the results from full mesh and require less computer time because of smaller mesh. e) In this case the problem is solved in two passes. The same small mesh as in model (d) is used for this, two pass solution. In the first pass the model is buried at its original depth and the gravity stresses are computed from material unit weight and depth. Stresses due to 95 excavation are calculated in the second pass by a. set of stress Doundary conditions acting at the excavation boundary. These stresses generated by the excavated material are added to the gravity loads in the second pass. f) This type of model loading is quite similar to the previous one. In the first pass the mesh is not buried at its original depth like in the previous case, whereas the effect of the overburden is introduced by application of overburden equivalent forces on the top nodal points of the mesh. Stresses caused by excavated material are com- puted by a set of boundary conditions on the excavation boundary and these stresses are added to the initial gravity stresses in the second pass. Results for all these models are summarized in Tables 7 and 8. Vertical displacements for the nodes in the roof are given in the tables. For models (a), (b) the displacements and stresses are approximately equal. Very low values of stresses and displacements were predicted in model (c) as compared to the other cases. Dis- placements and stresses from model (d) are close (4 to 9 pe~cent) to the values obtained from the full mesh. The last two models predict low stresses and displacements. These are approximately 22 percent less than in the first case. Though the smaller mesh in (b) predicts very close values as in the full mesh, input information had to be taken from the full mesh run. Therefore, in model (b) 96 TABLE 7 DISPLACEMENTS FOR VARIOUS TYPES OF MODEL LOADING Model r Node No. 11 6 11 7 118 119 120 121 143 146 147 153 155 156 157 158 159 (c} (a } (b } Vertical dis~lacement .34047 .41397 .45291 .45236 .42086 .32524 .22145 .13412 . 1 2506 -.06090 -.10442 -.13428 -.15789 -.12488 -.05852 .34034 .41346 .45216 .45154 .42004 .32464 .22159 .13527 .12629 -.06233 -.11455 -.13526 -.13210 -.11273 -.05497 ( d) {e) {f) in inches: .10123 .10402 .01527 . 10497 .10337 .09919 .09199 .08615 .08451 .07106 .06866 .06786 .06785 .06862 .07110 .36963 .44380 .48232 .48056 .44668 .34693 .23839 . 15466 .13745 -.15365 -.10817 -.13025 -.12788 -.10916 -.05119 .25381 .32047 .35546 .35355 .32558 .24097 .16081 .07913 .07227 -.09082 -.14278 -.16544 -.15874 -.14006 -.08206 .26116 .32842 .36369 .36182 .33345 .24776 .16562 .08355 .07723 -.08922 -.14144 -.16407 -.15766 -.13890 -.08077 97 TABLE 8 STRESSES FOR VARIOUS TYPES OF MODEL LOADING Model r Node No. 118 (c } (a } (b } (d) (f} (e} Nodal Qoint stresses near o~ening (~si}: 1484.2 1470.9 - 1 36 . 1 1564.0 1147.3 1176.3 119 1635.9 1621.4 - 1 34. 3 1709.2 1261.4 1293.6 120 1112.57 1099.3 - 163.2 1157.1 713. 1 745.5 121 -1863.9 -1854.2 -1769.8 -1923.3 -2053.1 -2044.3 143 -2509.5 -2470.5 -1044.9 -2590.7 -3005.2 -2980.3 147 -2618.3 -2598.8 -1048.2 -2688.8 -3101.9 -3077.7 the input data have to be taken from the full mesh and this procedure is long and takes more computer time which may not be economical. When the smaller meshes are used for stress analysis, model (d) gives reasonable results which are only 4 to 9 percent higher than the values from full mesh. Therefore, the model (d) is preferred to the others and at the same time saves computer time and extra work. APPENDIX E MESH REFINEMENT STUDY Stress concentrations around underground openings are necessary for design. Because stress gradients are high near the periphery of the openings, a relatively fine mesh ;s required to detect dangerous stress concentrations. Failure zones which are masked by large elements are revealed by a more refined mesh (30). In the refined meshes, 35 to 45 percent of the total elements are placed near the periphery of the opening. , Element sizes are controlled by thin rock layers and maximum number of elements in a mesh are limited by computer core storage capacity. More refined meshes re- quired for accurate stress analysis may easily exceed computer core storage capacity. To overcome this diffi- culty, a two-pass problem-solving approach is used. The two-pass solving system needs more computer time than the single pass runs, but remains within computer core storage limits. The two-pass solution technique consists of making an initial pass with full coarse mesh and then making a second pass with a small refined insert mesh. The insert mesh includes the mine opening but matches the first 99 (coarse) mesh at nodal points away from the opening. Dis- placements from the first coarse mesh output are used as input to the fine insert mesh. These displacements are prescribed on the external boundaries of the fine insert mesh. In this way the effect of the removed material on the insert mesh is introduced. The dimensions of the coarse mesh and fine insert mesh used for double entry study are shown in Figure 34. The fine mesh has 62 nodal points around the opening and the coarse mesh has only 14 nodal points. Coarse insert and fine insert meshes contain the same number of nodal points on their external boundaries. The dimensions of the openings in both cases are the same. The width of refined mesh for the single entry is less than the double entry mesh and fits within the limitations of core storage. Therefore, a two-pass solution procedure is not required for the single entry analysis. Reliability of the two-pass solution is verified by using a mesh that fits entirely in core. Conventional run results were compared to the results obtained by the two-pass procedure using a part of the original mesh. In both these cases uniform material properties were used. Results were in good agreement as shown in Figure 35. Results from the two-pass solution, those obtained by using a fine mesh, are also given in Figure 35. Coarse mesh results show 0.12 maximum tensile stress concentration in the roof, whereas fine mesh predicts maximum 100 Surface 1630' 2250 t 13' 71 I I ___~CJ Insert or fi ne 240 ~ coa~ me~~i 380 '. I I I I LI~ -"'" 230...1.-- 270 I 500' Coarse mesh Fig. 34.--0imensions of coarse and fine insert meshes for double entry analysis. 101 • 5 0 4 0 Full mesh Course insert mesh Fine insert mesh D r---' I ~ I I IL ___ .JI 3 at SV I I 2 + compression - tension w_ H - 3.43 Stress concentration around opening Fig. 35 .--Stress results for single and two pass from the same mesh and a fine insert mesh. 102 a critical tensile stress concentration of 0.36 The more reliable fine mesh estimate is three times greater than the coarse mesh estimate of peak tensile stress concentration. Estimates of compressive stress concentration are much closer. Coarse mesh predicts 2.8 and fine mesh 3.2 compressive stress concentration. Figure 36 shows a comparison of displacements obtained from coarse and fine meshes for a double entry system. Displacements from coarse and fine meshes are within 2 percent. 103 Opening w o If = 3.4 Fine insert mesh o Coarse insert mesh ~DisPlacement scale Fig. 36 .--Displacements from coarse and fine insert meshes. APPENDIX F WIDTH/HEIGHT RATIO STUDY In the double entry system at the Sunnyside Mine, the opening has a width to height ratio of 3.42. The proposed single entry system has a width to height ratio approximately 9 percent greater than in the double entry. Since the differences in width/height ratios are small, a question arises concerning the ability of the computer program to discriminate between the two width/height ratios. A study was made to see whether or not the computer program would discriminate between small changes in the opening geometry (30). Maximum displacements and stress concentrations around rectangular openings are functions of the width/ height ratio. These stresses and displacements near the periphery of the opening will increase as width/height rat i 0 inc rea s e s . A ty.p:i cal open i n g and a pi 11 a r ina 1 on g row of entries was used for this analysis. Uniform mate- erial properties (no layering effect) were assumed for this study. The coal seam depth was 1750 feet which is the same as at the Sunnyside Mine. The results of this study are shown in Figures 37 and 38. 105 Figure 37 sho~s the boundary stress concentration about rectangular openings of width/height ratios of 3.0, 3 . 2, 3. 4, 3. 6, 3. 8 and 4. 0 '. The d iff ere nc e bet wee nth e width/height ratios is approximately 6.5 percent and this is less than the difference between the single and double entries. There is a little difference in the critical ten- sile stress concentration near rectangular opening which is approximately constant for width/height ratios from 3.0 to 4.0. There is a small but definite increase in the critical compressive stress concentration near the corner of the opening. Compressive stress concentration for width/height ratio of 3~0 is about 3.3 and for width/height ratio of 4.0 the concentration is 3.6. Predicted differences in compressive stress concentration between width/height ratios of 3.0 and 4~0 is 0.3 and the difference from the analytical method gives 0.9. The extraction ratio in all the cases is constant, so the average stress in the pillar is very close. The displacements near the openings of different width/height ratios are shown in Figure 38. Maximum vertical closure increases from 2.5 inches to 3.4 inches as the width/height ratio increases from 3.0 to 4.0. This study indicates that the computer program is sensitive to . small changes in the opening geometry. 106 • [xperiment, W = 4.0 (Ref. #26) H Finite element: o *= 3. a 6W H =3.2 6 0 5 o!i = 3.6 H • W = 3.8 <:> W H = 4.0 W H = 3.4 \) = H 0.25 4 3 °t Sv 2 1 + Compression - Tension -1 3.0 I I I .--------- - - - 4.0 - - --- Fig.37 .--Stress concentration as a function of width/height ratio for rectangular opening. 107 • w • H = 4.0 • • H= W 3.8 0--0 .. H = W H = 3.2 W 3.6 6----l:l W H = 3.0 • I ~ W 3.0 H = 1 T 2.1 II If i 1 lO.95" 11 .6 11 T I ~ I I I Opening I I I I I 1 ~ I I 1 . 31\ I I 0.9 II TI: ra t i 0 w H = 3.0 ·1 w 4.0 H= ~I Fig. 38.-- Dis placement as a function of width/height for rectangular openings.